Subgroups — C₂×#20288384 #28292405

Order 13568.

12366 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	1	yes	maximal, characteristic, complemented
2	#20288384	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811292	C₂	1	2	yes	maximal, complemented
4	C₂×#16811284	C₂	1	2	yes	maximal, complemented
5	C₂×#16811261	C₂	1	6	yes	maximal, complemented
6	C₂×D₄×C₂²⋊D₅₃	C₂	1	6	yes	maximal, complemented
7	C₂×#16811201	C₂	1	3	yes	maximal, complemented
8	C₂×#16811187	C₂	1	3	yes	maximal, complemented
9	C₂×#16811134	C₂	1	3	yes	maximal, complemented
10	C₂²×#13133191	C₂	1	3	yes	maximal, complemented
11	C₂×#16811106	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133263	—	2	16	no	—
14	C₂×#13133263	C₂²	1	16	yes	complemented
15	C₂²×#11719949	C₂²	1	2	yes	complemented
16	C₂²×#11719945	C₂²	1	3	yes	complemented
17	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
18	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
19	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
20	#16811292	C₂²	1	32	yes	complemented
21	#16811284	C₂²	1	32	yes	complemented
22	#16811261	C₂²	1	96	yes	complemented
23	D₄×C₂²⋊D₅₃	C₂²	1	96	yes	complemented
24	#16811201	C₂²	1	48	yes	complemented
25	#16811187	C₂²	1	48	yes	complemented
26	C₂×#13133213	C₂²	1	6	yes	complemented
27	C₂×#13133206	C₂²	1	6	yes	complemented
28	#16811134	C₂²	1	48	yes	complemented
29	C₂×#13133200	C₂²	1	6	yes	complemented
30	C₂×#13133196	C₂²	1	3	yes	complemented
31	C₂×#13133196	C₂²	1	1	yes	characteristic, complemented
32	C₂×#13133193	C₂²	1	6	yes	complemented
33	C₂×#13133191	C₂²	1	3	yes	complemented
34	C₂×#13133191	C₂²	1	6	yes	complemented
35	C₂×#13133191	C₂²	1	6	yes	complemented
36	C₂×#13133191	C₂²	1	6	yes	complemented
37	C₂×#13133191	C₂²	1	3	yes	complemented
38	C₂×#13133191	C₂²	1	48	yes	complemented
39	C₂×#13133189	C₂²	1	12	yes	complemented
40	C₂×#13133188	C₂²	1	6	yes	complemented
41	C₂×#13133186	C₂²	1	6	yes	complemented
42	C₂×#13133185	C₂²	1	12	yes	complemented
43	#16811106	C₂²	1	16	yes	complemented
44	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	complemented
45	C₂×D₄×Dic₅₃	C₂²	1	6	yes	complemented
46	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
47	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
48	C₂×#13133176	C₂²	1	3	yes	complemented
49	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	complemented
50	C₂²×#11719921	C₂²	1	6	yes	complemented
51	C₂²×#41×C₅₃	—	2	16	no	—
52	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
53	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
54	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
55	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
56	C₂²×#41×C₅₃	C₂²	1	16	yes	complemented
57	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
59	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	complemented
60	C₂⁴×D₅₃	C₂³	1	6	yes	—
61	#13133263	—	2	128	no	—
62	C₂×#11719949	—	2	16	no	—
63	C₂×#11719949	C₂³	1	16	yes	complemented
64	C₂×#11719945	—	2	24	no	—
65	C₂×#11719945	C₂³	1	24	yes	complemented
66	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
67	C₂²×D₂₁₂	—	2	24	no	—
68	C₂²×D₂₁₂	C₂³	1	3	yes	—
69	C₂²×D₂₁₂	C₂³	1	6	yes	—
70	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
71	C₂²×C₂²⋊D₅₃	—	4	24	no	—
72	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
73	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
74	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
75	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
76	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
77	C₂×D₄×D₅₃	—	2	48	no	—
78	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
79	C₂³×Dic₅₃	C₂³	1	6	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
81	#13133213	C₂³	1	48	yes	complemented
82	#13133206	C₂³	1	48	yes	complemented
83	#13133200	C₂³	1	48	yes	complemented
84	#13133196	C₂³	1	8	yes	complemented
85	#13133196	C₂³	1	24	yes	complemented
86	#13133193	C₂³	1	48	yes	complemented
87	#13133191	C₂³	1	48	yes	complemented
88	#13133191	C₂³	1	48	yes	complemented
89	#13133191	C₂³	1	24	yes	complemented
90	#13133191	C₂³	1	48	yes	complemented
91	#13133191	C₂³	1	24	yes	complemented
92	#13133189	C₂³	1	96	yes	complemented
93	#13133188	C₂³	1	48	yes	complemented
94	#13133186	C₂³	1	48	yes	complemented
95	#13133185	C₂³	1	96	yes	complemented
96	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
97	D₄×Dic₅₃	C₂³	1	48	yes	complemented
98	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
99	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
100	#13133176	C₂³	1	24	yes	complemented
101	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
102	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
103	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
105	C₂×#11719919	C₂³	1	2	yes	—
106	C₂×#11719919	C₂³	1	6	yes	—
107	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
108	C₂×C₂²⋊Dic₅₃	C₂³	1	2	yes	—
109	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
110	C₂×#11719921	C₂³	1	6	yes	—
111	C₂×#11719921	C₂³	1	12	yes	—
112	C₂×#11719921	C₂³	1	6	yes	—
113	C₂×#11719921	C₂³	1	48	yes	complemented
114	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
115	C₂⁵×C₅₃	C₂³	1	3	yes	—
116	C₂×#41×C₅₃	—	2	128	no	—
117	C₂×#41×C₅₃	—	2	8	no	—
118	C₂×#41×C₅₃	—	2	24	no	—
119	C₂×#41×C₅₃	—	2	24	no	—
120	C₂×#41×C₅₃	D₄	1	16	yes	complemented
121	C₂×#41×C₅₃	C₂³	1	8	yes	—
122	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
123	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
124	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
125	C₂²×Q₈×C₅₃	—	2	8	no	—
126	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
127	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×Q₈×C₅₃	C₂³	1	8	yes	complemented
129	C₂²×D₄×C₅₃	—	2	24	no	—
130	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
131	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
132	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
133	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
134	C₂³×C₄×C₅₃	—	2	24	no	—
135	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
136	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
137	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
138	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
139	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
140	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
141	C₂³×C₄×C₅₃	C₂³	1	24	yes	complemented
142	C₂⁴×C₅₃	—	2	12	no	—
143	C₂⁴×C₅₃	—	2	12	no	—
144	C₂⁴×C₅₃	—	4	6	no	—
145	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
146	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
147	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
148	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂³×D₅₃	—	2	24	no	—
150	C₂³×D₅₃	—	4	12	no	—
151	C₂³×D₅₃	—	4	12	no	—
152	C₂³×D₅₃	C₂⁴	1	6	yes	—
153	C₂³×D₅₃	C₂⁴	1	24	yes	—
154	#41×C₅₃	—	2	32	no	—
155	#41×C₅₃	—	2	96	no	—
156	#41×C₅₃	—	2	96	no	—
157	#41×C₅₃	C₂×D₄	1	64	yes	complemented
158	C₂×Q₈×C₅₃	—	2	32	no	—
159	C₂×Q₈×C₅₃	—	2	12	no	—
160	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
161	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
162	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
163	C₂×D₄×C₅₃	—	2	24	no	—
164	C₂×D₄×C₅₃	—	2	96	no	—
165	C₂×D₄×C₅₃	—	2	12	no	—
166	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
167	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
168	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
169	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
170	C₂²×C₄×C₅₃	—	2	12	no	—
171	C₂²×C₄×C₅₃	—	2	4	no	—
172	C₂²×C₄×C₅₃	—	2	12	no	—
173	C₂²×C₄×C₅₃	—	2	12	no	—
174	C₂²×C₄×C₅₃	—	2	12	no	—
175	C₂²×C₄×C₅₃	—	2	96	no	—
176	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
177	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
179	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
182	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
183	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
184	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
186	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
187	C₂×C₂²⋊D₅₃	—	2	24	no	—
188	C₂×C₂²⋊D₅₃	—	2	24	no	—
189	C₂×C₂²⋊D₅₃	—	4	24	no	—
190	C₂×C₂²⋊D₅₃	—	4	12	no	—
191	C₂×C₂²⋊D₅₃	—	4	96	no	—
192	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
193	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
194	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
195	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
196	C₂²×Dic₅₃	—	4	12	no	—
197	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
198	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
199	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
200	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
201	#11719949	—	2	64	no	—
202	D₄×D₅₃	—	2	192	no	—
203	#11719945	—	2	96	no	—
204	C₂×D₂₁₂	—	2	24	no	—
205	C₂×D₂₁₂	—	2	12	no	—
206	C₂×D₂₁₂	—	2	96	no	—
207	C₂×D₂₁₂	C₂⁴	1	12	yes	—
208	C₂×D₂₁₂	C₂⁴	1	24	yes	—
209	C₂×C₄×D₅₃	—	2	24	no	—
210	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
211	C₂×Dic₁₀₆	—	2	4	no	—
212	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
213	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
214	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
215	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
216	#11719921	C₂⁴	1	24	yes	—
217	#11719921	C₂⁴	1	48	yes	—
218	#11719921	C₂⁴	1	24	yes	—
219	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
220	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
221	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
222	#11719919	C₂⁴	1	8	yes	—
223	#11719919	C₂⁴	1	24	yes	—
224	C₄×Dic₅₃	C₂⁴	1	24	yes	—
225	C₂³×C₅₃	—	2	2	no	—
226	C₂³×C₅₃	—	2	6	no	—
227	C₂³×C₅₃	—	2	6	no	—
228	C₂³×C₅₃	—	2	2	no	—
229	C₂³×C₅₃	—	2	24	no	—
230	C₂³×C₅₃	—	2	6	no	—
231	C₂³×C₅₃	—	2	6	no	—
232	C₂³×C₅₃	—	2	24	no	—
233	C₂³×C₅₃	—	2	6	no	—
234	C₂³×C₅₃	—	4	12	no	—
235	C₂³×C₅₃	—	4	48	no	—
236	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
237	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	complemented
238	C₂³×C₅₃	C₂⁵	1	6	yes	—
239	C₂³×C₅₃	C₂⁵	1	6	yes	—
240	C₂³×C₅₃	C₂⁵	1	2	yes	—
241	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
242	C₂²×D₅₃	—	2	12	no	—
243	C₂²×D₅₃	—	2	12	no	—
244	C₂²×D₅₃	—	2	12	no	—
245	C₂²×D₅₃	—	2	48	no	—
246	C₂²×D₅₃	—	4	24	no	—
247	C₂²×D₅₃	—	4	24	no	—
248	C₂²×D₅₃	—	8	48	no	—
249	C₂²×D₅₃	C₂⁵	1	12	yes	—
250	Q₈×C₅₃	—	2	24	no	—
251	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
252	D₄×C₅₃	—	2	48	no	—
253	D₄×C₅₃	—	2	24	no	—
254	D₄×C₅₃	C₂²×D₄	1	48	yes	complemented
255	C₂×C₄×C₅₃	—	2	2	no	—
256	C₂×C₄×C₅₃	—	2	6	no	—
257	C₂×C₄×C₅₃	—	2	24	no	—
258	C₂×C₄×C₅₃	—	2	8	no	—
259	C₂×C₄×C₅₃	—	2	24	no	—
260	C₂×C₄×C₅₃	—	2	24	no	—
261	C₂×C₄×C₅₃	—	2	24	no	—
262	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
264	C₂×C₄×C₅₃	C₂²×D₄	1	48	yes	complemented
265	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
266	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
267	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
268	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
269	C₂²⋊D₅₃	—	2	48	no	—
270	C₂²⋊D₅₃	—	2	48	no	—
271	C₂²⋊D₅₃	—	4	48	no	—
272	C₂²⋊D₅₃	—	4	24	no	—
273	C₂×Dic₅₃	—	2	4	no	—
274	C₂×Dic₅₃	—	2	12	no	—
275	C₂×Dic₅₃	—	2	4	no	—
276	C₂×Dic₅₃	—	4	24	no	—
277	C₂×Dic₅₃	C₂⁵	1	4	yes	—
278	C₂×Dic₅₃	C₂⁵	1	12	yes	—
279	C₂×Dic₅₃	C₂⁵	1	4	yes	—
280	D₂₁₂	—	2	24	no	—
281	D₂₁₂	—	2	48	no	—
282	C₄×D₅₃	—	2	48	no	—
283	Dic₁₀₆	—	2	8	no	—
284	C₂×#3488	—	53	1	no	maximal, Sylow
285	C₂²×C₅₃	—	2	2	no	—
286	C₂²×C₅₃	—	2	6	no	—
287	C₂²×C₅₃	—	2	6	no	—
288	C₂²×C₅₃	—	2	2	no	—
289	C₂²×C₅₃	—	2	24	no	—
290	C₂²×C₅₃	—	2	8	no	—
291	C₂²×C₅₃	—	2	6	no	—
292	C₂²×C₅₃	—	2	6	no	—
293	C₂²×C₅₃	—	2	6	no	—
294	C₂²×C₅₃	—	4	48	no	—
295	C₂²×C₅₃	—	4	12	no	—
296	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
297	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
298	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
299	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	complemented
300	C₂²×C₅₃	#585	1	2	yes	—
301	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
302	C₂×D₅₃	—	2	12	no	—
303	C₂×D₅₃	—	2	12	no	—
304	C₂×D₅₃	—	2	12	no	—
305	C₂×D₅₃	—	4	24	no	—
306	C₂×D₅₃	—	8	48	no	—
307	C₄×C₅₃	—	2	2	no	—
308	C₄×C₅₃	—	2	6	no	—
309	C₄×C₅₃	C₂³×D₄	1	4	yes	—
310	C₄×C₅₃	C₂³×D₄	1	12	yes	—
311	Dic₅₃	—	2	4	no	—
312	Dic₅₃	—	2	12	no	—
313	Dic₅₃	—	2	4	no	—
314	C₂²×H(5,𝔽₂)	—	53	1	no	—
315	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
316	#3488	—	53	32	no	—
317	C₂×#550	—	53	2	no	—
318	C₂×#548	—	53	2	no	—
319	C₂×#546	—	53	6	no	—
320	C₂×D₄²	—	53	6	no	—
321	C₂×#537	—	53	3	no	—
322	C₂×#535	—	53	3	no	—
323	C₂×#525	—	53	3	no	—
324	C₂²×#121	—	53	3	no	—
325	C₂×#520	—	53	1	no	—
326	C₂×C₅₃	—	2	12	no	—
327	C₂×C₅₃	—	2	4	no	—
328	C₂×C₅₃	—	4	6	no	—
329	C₂×C₅₃	#3488	1	4	yes	complemented
330	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
331	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
333	D₅₃	—	4	12	no	—
334	C₂×H(5,𝔽₂)	—	53	16	no	—
335	C₂×H(5,𝔽₂)	—	106	16	no	—
336	C₂²×#41	—	2	16	no	—
337	C₂²×#41	—	53	2	no	—
338	C₂²×#41	—	53	3	no	—
339	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
340	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
341	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
342	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic, complemented
343	C₂²×#41	C₂×D₅₃	1	16	yes	complemented
344	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
345	C₂³×D₄	—	53	6	no	—
346	C₂³×D₄	—	53	6	no	—
347	C₂³×D₄	—	53	3	no	—
348	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
349	C₂⁴×C₄	C₂×D₅₃	1	3	yes	complemented
350	#550	—	53	32	no	—
351	#548	—	53	32	no	—
352	#546	—	53	96	no	—
353	D₄²	—	53	96	no	—
354	#537	—	53	48	no	—
355	#535	—	53	48	no	—
356	C₂×#127	—	53	6	no	—
357	C₂×#124	—	53	6	no	—
358	#525	—	53	48	no	—
359	C₂×#123	—	53	6	no	—
360	C₂×C₂²⋊Q₈	—	53	6	no	—
361	C₂×C₂²⋊Q₈	—	53	1	no	—
362	C₂×C₂²⋊Q₈	—	53	3	no	—
363	C₂×#121	—	53	6	no	—
364	C₂×#121	—	53	12	no	—
365	C₂×#121	—	53	3	no	—
366	C₂×#121	—	53	6	no	—
367	C₂×#121	—	53	6	no	—
368	C₂×#121	—	53	6	no	—
369	C₂×#121	—	53	3	no	—
370	C₂×#121	—	53	48	no	—
371	C₂×#120	—	53	6	no	—
372	C₂×#120	—	53	12	no	—
373	#520	—	53	16	no	—
374	C₂×C₄×Q₈	—	53	2	no	—
375	C₂×C₄×D₄	—	53	6	no	—
376	C₂×C₄×D₄	—	53	6	no	—
377	C₂×C₄×D₄	—	53	6	no	—
378	C₂×#117	—	53	3	no	—
379	C₂²×C₄⋊C₄	—	53	3	no	—
380	C₂²×C₂²⋊C₄	—	53	6	no	—
381	C₅₃	C₂×#3488	1	1	yes	characteristic, complemented, Sylow
382	C₂⁵	—	53	6	no	—
383	C₂⁵	C₂²×D₅₃	1	3	yes	—
384	H(5,𝔽₂)	—	106	128	no	—
385	C₂×#41	—	2	128	no	—
386	C₂×#41	—	2	8	no	—
387	C₂×#41	—	2	24	no	—
388	C₂×#41	—	2	24	no	—
389	C₂×#41	—	53	16	no	—
390	C₂×#41	—	53	24	no	—
391	C₂×#41	—	106	16	no	—
392	C₂×#41	—	106	24	no	—
393	C₂×#41	C₂²⋊D₅₃	1	16	yes	complemented
394	C₂×#41	C₂²×D₅₃	1	8	yes	—
395	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
396	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
397	C₂×#41	C₂²×D₅₃	1	8	yes	complemented
398	C₂²×Q₈	—	2	8	no	—
399	C₂²×Q₈	—	53	1	no	—
400	C₂²×Q₈	C₂²×D₅₃	1	3	yes	—
401	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
402	C₂²×Q₈	C₂²×D₅₃	1	8	yes	complemented
403	C₂²×D₄	—	2	24	no	—
404	C₂²×D₄	—	53	6	no	—
405	C₂²×D₄	—	53	6	no	—
406	C₂²×D₄	—	53	6	no	—
407	C₂²×D₄	—	53	12	no	—
408	C₂²×D₄	—	53	48	no	—
409	C₂²×D₄	—	53	48	no	—
410	C₂²×D₄	—	53	3	no	—
411	C₂²×D₄	—	53	6	no	—
412	C₂²×D₄	—	53	24	no	—
413	C₂²×D₄	—	106	48	no	—
414	C₂²×D₄	—	106	24	no	—
415	C₂²×D₄	—	212	24	no	—
416	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
417	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
418	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
419	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
420	C₂³×C₄	—	2	24	no	—
421	C₂³×C₄	—	53	6	no	—
422	C₂³×C₄	—	53	6	no	—
423	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
424	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
425	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
426	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
427	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
428	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
429	C₂³×C₄	C₂²×D₅₃	1	24	yes	complemented
430	#127	—	53	48	no	—
431	#124	—	53	48	no	—
432	#123	—	53	48	no	—
433	C₂²⋊Q₈	—	53	48	no	—
434	C₂²⋊Q₈	—	53	8	no	—
435	C₂²⋊Q₈	—	53	24	no	—
436	#121	—	53	96	no	—
437	#121	—	53	48	no	—
438	#121	—	53	24	no	—
439	#121	—	53	48	no	—
440	#121	—	53	48	no	—
441	#121	—	53	48	no	—
442	#121	—	53	24	no	—
443	#120	—	53	48	no	—
444	#120	—	53	96	no	—
445	C₄×Q₈	—	53	16	no	—
446	C₄×D₄	—	53	48	no	—
447	C₄×D₄	—	53	48	no	—
448	C₄×D₄	—	53	48	no	—
449	#117	—	53	24	no	—
450	C₂×C₄⋊C₄	—	53	3	no	—
451	C₂×C₄⋊C₄	—	53	6	no	—
452	C₂×C₄⋊C₄	—	53	24	no	—
453	C₂×C₄⋊C₄	—	53	1	no	—
454	C₂×C₄⋊C₄	—	53	2	no	—
455	C₂×C₄⋊C₄	—	53	6	no	—
456	C₂×C₂²⋊C₄	—	53	6	no	—
457	C₂×C₂²⋊C₄	—	53	2	no	—
458	C₂×C₂²⋊C₄	—	53	6	no	—
459	C₂×C₂²⋊C₄	—	53	6	no	—
460	C₂×C₂²⋊C₄	—	53	12	no	—
461	C₂×C₂²⋊C₄	—	53	48	no	—
462	C₂×C₂²⋊C₄	—	53	6	no	—
463	C₂×C₄²	—	53	6	no	—
464	C₂⁴	—	2	12	no	—
465	C₂⁴	—	2	12	no	—
466	C₂⁴	—	4	6	no	—
467	C₂⁴	—	53	6	no	—
468	C₂⁴	—	53	24	no	—
469	C₂⁴	—	106	24	no	—
470	C₂⁴	—	212	12	no	—
471	C₂⁴	—	212	12	no	—
472	C₂⁴	C₂³×D₅₃	1	3	yes	—
473	C₂⁴	C₂³×D₅₃	1	12	yes	—
474	C₂⁴	C₂³×D₅₃	1	3	yes	—
475	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
476	#41	—	2	32	no	—
477	#41	—	2	96	no	—
478	#41	—	2	96	no	—
479	#41	—	106	64	no	—
480	#41	—	106	96	no	—
481	#41	C₂×C₂²⋊D₅₃	1	64	yes	complemented
482	C₂×Q₈	—	2	32	no	—
483	C₂×Q₈	—	2	12	no	—
484	C₂×Q₈	—	53	4	no	—
485	C₂×Q₈	—	106	4	no	—
486	C₂×Q₈	C₂×C₂²⋊D₅₃	1	8	yes	complemented
487	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
488	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
489	C₂×D₄	—	2	24	no	—
490	C₂×D₄	—	2	96	no	—
491	C₂×D₄	—	2	12	no	—
492	C₂×D₄	—	53	24	no	—
493	C₂×D₄	—	53	24	no	—
494	C₂×D₄	—	53	24	no	—
495	C₂×D₄	—	53	48	no	—
496	C₂×D₄	—	53	12	no	—
497	C₂×D₄	—	53	24	no	—
498	C₂×D₄	—	106	24	no	—
499	C₂×D₄	—	106	24	no	—
500	C₂×D₄	—	106	192	no	—
501	C₂×D₄	—	106	12	no	—
502	C₂×D₄	—	106	24	no	—
503	C₂×D₄	—	106	96	no	—
504	C₂×D₄	—	212	12	no	—
505	C₂×D₄	—	212	24	no	—
506	C₂×D₄	—	212	96	no	—
507	C₂×D₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
508	C₂×D₄	C₂³×D₅₃	1	24	yes	—
509	C₂×D₄	C₂³×D₅₃	1	12	yes	—
510	C₂×D₄	C₂³×D₅₃	1	12	yes	—
511	C₂²×C₄	—	2	12	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	12	no	—
514	C₂²×C₄	—	2	12	no	—
515	C₂²×C₄	—	2	12	no	—
516	C₂²×C₄	—	2	96	no	—
517	C₂²×C₄	—	53	2	no	—
518	C₂²×C₄	—	53	6	no	—
519	C₂²×C₄	—	53	2	no	—
520	C₂²×C₄	—	53	24	no	—
521	C₂²×C₄	—	53	24	no	—
522	C₂²×C₄	—	106	24	no	—
523	C₂²×C₄	—	212	12	no	—
524	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
525	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
526	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
527	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
528	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
529	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
530	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
531	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
532	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
533	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
534	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
535	C₄⋊C₄	—	53	12	no	—
536	C₄⋊C₄	—	53	24	no	—
537	C₄⋊C₄	—	53	4	no	—
538	C₄⋊C₄	—	53	8	no	—
539	C₄⋊C₄	—	53	24	no	—
540	C₂²⋊C₄	—	53	24	no	—
541	C₂²⋊C₄	—	53	8	no	—
542	C₂²⋊C₄	—	53	24	no	—
543	C₂²⋊C₄	—	53	24	no	—
544	C₂²⋊C₄	—	53	48	no	—
545	C₂²⋊C₄	—	53	24	no	—
546	C₄²	—	53	24	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	6	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	6	no	—
551	C₂³	—	2	24	no	—
552	C₂³	—	2	6	no	—
553	C₂³	—	2	6	no	—
554	C₂³	—	2	24	no	—
555	C₂³	—	2	6	no	—
556	C₂³	—	4	12	no	—
557	C₂³	—	4	48	no	—
558	C₂³	—	53	12	no	—
559	C₂³	—	106	48	no	—
560	C₂³	—	106	12	no	—
561	C₂³	—	106	12	no	—
562	C₂³	—	106	12	no	—
563	C₂³	—	424	48	no	—
564	C₂³	—	212	24	no	—
565	C₂³	—	212	24	no	—
566	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
567	C₂³	#13133263	1	4	yes	complemented
568	C₂³	C₂⁴×D₅₃	1	6	yes	—
569	C₂³	C₂⁴×D₅₃	1	6	yes	—
570	C₂³	C₂⁴×D₅₃	1	2	yes	—
571	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
572	Q₈	—	2	24	no	—
573	Q₈	—	106	8	no	—
574	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	complemented
575	D₄	—	2	48	no	—
576	D₄	—	2	24	no	—
577	D₄	—	106	48	no	—
578	D₄	—	106	48	no	—
579	D₄	—	106	24	no	—
580	D₄	—	106	48	no	—
581	D₄	—	212	24	no	—
582	D₄	—	212	48	no	—
583	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	complemented
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	6	no	—
586	C₂×C₄	—	2	24	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	24	no	—
589	C₂×C₄	—	2	24	no	—
590	C₂×C₄	—	2	24	no	—
591	C₂×C₄	—	53	4	no	—
592	C₂×C₄	—	53	12	no	—
593	C₂×C₄	—	53	4	no	—
594	C₂×C₄	—	106	4	no	—
595	C₂×C₄	—	106	12	no	—
596	C₂×C₄	—	106	4	no	—
597	C₂×C₄	—	106	48	no	—
598	C₂×C₄	—	212	24	no	—
599	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
600	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
601	C₂×C₄	C₂²×C₂²⋊D₅₃	1	48	yes	complemented
602	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
603	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
604	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
605	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
606	C₂²	—	2	8	no	—
607	C₂²	—	2	24	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	6	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	6	no	—
612	C₂²	—	2	6	no	—
613	C₂²	—	2	6	no	—
614	C₂²	—	2	6	no	—
615	C₂²	—	4	48	no	—
616	C₂²	—	4	12	no	—
617	C₂²	—	106	12	no	—
618	C₂²	—	106	12	no	—
619	C₂²	—	106	12	no	—
620	C₂²	—	424	48	no	—
621	C₂²	—	212	24	no	—
622	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
623	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
624	C₂²	C₂×#13133263	1	2	yes	—
625	C₂²	C₂×#13133263	1	4	yes	complemented
626	C₂²	#16811459	1	2	yes	—
627	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	2	6	no	—
630	C₄	—	106	4	no	—
631	C₄	—	106	12	no	—
632	C₄	—	106	4	no	—
633	C₄	C₂³×C₂²⋊D₅₃	1	4	yes	—
634	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
635	C₂	—	2	4	no	—
636	C₂	—	2	12	no	—
637	C₂	—	4	6	no	—
638	C₂	—	212	12	no	—
639	C₂	#20288384	1	4	yes	direct product factor
640	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
641	C₂	C₂²×#13133263	1	1	yes	characteristic
642	C₂	C₂×#16811459	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂×#3488
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. #20288384	C₂	direct (N × H)
3. C₂×#16811292	C₂	semidirect
4. C₂×#16811284	C₂	semidirect
5. C₂×#16811261	C₂	semidirect
6. C₂×D₄×C₂²⋊D₅₃	C₂	semidirect
7. C₂×#16811201	C₂	semidirect
8. C₂×#16811187	C₂	semidirect
9. C₂×#16811134	C₂	semidirect
10. C₂²×#13133191	C₂	semidirect
11. C₂×#16811106	C₂	semidirect
12. C₂³×#41×C₅₃	C₂	semidirect
14. C₂×#13133263	C₂²	semidirect
15. C₂²×#11719949	C₂²	semidirect
16. C₂²×#11719945	C₂²	semidirect
17. C₂³×D₂₁₂	C₂²	semidirect
18. C₂³×C₂²⋊D₅₃	C₂²	semidirect
19. C₂²×D₄×D₅₃	C₂²	semidirect
20. #16811292	C₂²	semidirect
21. #16811284	C₂²	semidirect
22. #16811261	C₂²	semidirect
23. D₄×C₂²⋊D₅₃	C₂²	semidirect
24. #16811201	C₂²	semidirect
25. #16811187	C₂²	semidirect
26. C₂×#13133213	C₂²	semidirect
27. C₂×#13133206	C₂²	semidirect
28. #16811134	C₂²	semidirect
29. C₂×#13133200	C₂²	semidirect
30. C₂×#13133196	C₂²	semidirect
31. C₂×#13133196	C₂²	semidirect
32. C₂×#13133193	C₂²	semidirect
33. C₂×#13133191	C₂²	semidirect
34. C₂×#13133191	C₂²	semidirect
35. C₂×#13133191	C₂²	semidirect
36. C₂×#13133191	C₂²	semidirect
37. C₂×#13133191	C₂²	semidirect
38. C₂×#13133191	C₂²	semidirect
39. C₂×#13133189	C₂²	semidirect
40. C₂×#13133188	C₂²	semidirect
41. C₂×#13133186	C₂²	semidirect
42. C₂×#13133185	C₂²	semidirect
43. #16811106	C₂²	semidirect
44. C₂×Q₈×Dic₅₃	C₂²	semidirect
45. C₂×D₄×Dic₅₃	C₂²	semidirect
46. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
47. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
48. C₂×#13133176	C₂²	semidirect
49. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
50. C₂²×#11719921	C₂²	semidirect
53. C₂²×#41×C₅₃	C₂²	semidirect
54. C₂²×#41×C₅₃	C₂²	semidirect
55. C₂²×#41×C₅₃	C₂²	semidirect
56. C₂²×#41×C₅₃	C₂²	semidirect
57. C₂³×Q₈×C₅₃	C₂²	semidirect
58. C₂³×D₄×C₅₃	C₂²	semidirect
59. C₂⁴×C₄×C₅₃	C₂²	semidirect
63. C₂×#11719949	C₂³	semidirect
65. C₂×#11719945	C₂³	semidirect
70. C₂²×D₂₁₂	C₂³	semidirect
76. C₂²×C₂²⋊D₅₃	C₂³	semidirect
78. C₂×D₄×D₅₃	C₂³	semidirect
81. #13133213	C₂³	semidirect
82. #13133206	C₂³	semidirect
83. #13133200	C₂³	semidirect
84. #13133196	C₂³	semidirect
85. #13133196	C₂³	semidirect
86. #13133193	C₂³	semidirect
87. #13133191	C₂³	semidirect
88. #13133191	C₂³	semidirect
89. #13133191	C₂³	semidirect
90. #13133191	C₂³	semidirect
91. #13133191	C₂³	semidirect
92. #13133189	C₂³	semidirect
93. #13133188	C₂³	semidirect
94. #13133186	C₂³	semidirect
95. #13133185	C₂³	semidirect
96. Q₈×Dic₅₃	C₂³	semidirect
97. D₄×Dic₅₃	C₂³	semidirect
98. C₄×C₂²⋊D₅₃	C₂³	semidirect
99. C₄×C₂²⋊D₅₃	C₂³	semidirect
100. #13133176	C₂³	semidirect
104. C₂×C₄⋊Dic₅₃	C₂³	semidirect
113. C₂×#11719921	C₂³	semidirect
120. C₂×#41×C₅₃	D₄	semidirect
122. C₂×#41×C₅₃	C₂³	semidirect
123. C₂×#41×C₅₃	C₂³	semidirect
124. C₂×#41×C₅₃	C₂³	semidirect
128. C₂²×Q₈×C₅₃	C₂³	semidirect
133. C₂²×D₄×C₅₃	C₂³	semidirect
141. C₂³×C₄×C₅₃	C₂³	semidirect
157. #41×C₅₃	C₂×D₄	semidirect
160. C₂×Q₈×C₅₃	C₂×D₄	semidirect
166. C₂×D₄×C₅₃	C₂×D₄	semidirect
176. C₂²×C₄×C₅₃	C₂×D₄	semidirect
237. C₂³×C₅₃	H(5,𝔽₂)	semidirect
251. Q₈×C₅₃	C₂²×D₄	semidirect
254. D₄×C₅₃	C₂²×D₄	semidirect
264. C₂×C₄×C₅₃	C₂²×D₄	semidirect
299. C₂²×C₅₃	C₂×H(5,𝔽₂)	semidirect
315. C₂³×#41	D₅₃	semidirect
329. C₂×C₅₃	#3488	semidirect
340. C₂²×#41	C₂×D₅₃	semidirect
341. C₂²×#41	C₂×D₅₃	semidirect
342. C₂²×#41	C₂×D₅₃	semidirect
343. C₂²×#41	C₂×D₅₃	semidirect
344. C₂³×Q₈	C₂×D₅₃	semidirect
348. C₂³×D₄	C₂×D₅₃	semidirect
349. C₂⁴×C₄	C₂×D₅₃	semidirect
381. C₅₃	C₂×#3488	semidirect
393. C₂×#41	C₂²⋊D₅₃	semidirect
395. C₂×#41	C₂²×D₅₃	semidirect
396. C₂×#41	C₂²×D₅₃	semidirect
397. C₂×#41	C₂²×D₅₃	semidirect
402. C₂²×Q₈	C₂²×D₅₃	semidirect
419. C₂²×D₄	C₂²×D₅₃	semidirect
429. C₂³×C₄	C₂²×D₅₃	semidirect
481. #41	C₂×C₂²⋊D₅₃	semidirect
486. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
507. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
524. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
567. C₂³	#13133263	semidirect
574. Q₈	C₂²×C₂²⋊D₅₃	semidirect
583. D₄	C₂²×C₂²⋊D₅₃	semidirect
601. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
625. C₂²	C₂×#13133263	semidirect
639. C₂	#20288384	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133263	2
#20288384	2
C₂×#16811292	2
C₂×#16811284	2
C₂×#16811261	2
C₂×D₄×C₂²⋊D₅₃	2
C₂×#16811201	2
C₂×#16811187	2
C₂×#16811134	2
C₂²×#13133191	2
C₂×#16811106	2
C₂³×#41×C₅₃	2
C₂×#3488	53

Subgroups — C2×#20288384 #28292405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288384 #28292405