Subgroups — C₂×#20288492 #28292411

Order 13568.

12982 conjugacy classes of subgroups.

The subgroup lattice has 596 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	2	yes	maximal, complemented
2	#20288492	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811292	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×D₂₁₂	C₂	1	3	yes	maximal, complemented
6	C₂×#16811258	C₂	1	3	yes	maximal, complemented
7	C₂×#16811257	C₂	1	6	yes	maximal, complemented
8	C₂×#16811189	C₂	1	6	yes	maximal, complemented
9	C₂×#16811170	C₂	1	3	yes	maximal, complemented
10	C₂×#127×D₅₃	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×#11719949	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133263	—	2	32	no	—
14	C₂×#13133263	C₂²	1	32	yes	complemented
15	C₂²×#11719949	C₂²	1	1	yes	characteristic
16	C₂²×#11719949	C₂²	1	2	yes	complemented
17	C₂²×#11719945	C₂²	1	6	yes	complemented
18	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
19	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
20	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
21	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
22	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
23	#16811295	C₂²	1	16	yes	complemented
24	#16811292	C₂²	1	32	yes	complemented
25	D₄×D₂₁₂	C₂²	1	48	yes	complemented
26	#16811258	C₂²	1	48	yes	complemented
27	#16811257	C₂²	1	96	yes	complemented
28	#16811189	C₂²	1	96	yes	complemented
29	#16811170	C₂²	1	48	yes	complemented
30	C₂×#13133216	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13133214	C₂²	1	3	yes	complemented
32	C₂×#13133213	C₂²	1	6	yes	complemented
33	C₂×#13133213	C₂²	1	6	yes	complemented
34	C₂×#13133213	C₂²	1	3	yes	complemented
35	#127×D₅₃	C₂²	1	48	yes	complemented
36	C₂×#13133203	C₂²	1	6	yes	complemented
37	C₂×#13133192	C₂²	1	6	yes	complemented
38	C₂×#13133192	C₂²	1	3	yes	complemented
39	C₂×#13133191	C₂²	1	6	yes	complemented
40	C₂×#13133190	C₂²	1	6	yes	complemented
41	C₂×#13133188	C₂²	1	6	yes	complemented
42	C₂×#13133188	C₂²	1	3	yes	complemented
43	C₂×#13133187	C₂²	1	12	yes	complemented
44	C₂×#13133185	C₂²	1	12	yes	complemented
45	C₄×#11719949	C₂²	1	16	yes	complemented
46	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
48	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
49	C₂×#13133179	C₂²	1	3	yes	complemented
50	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
51	C₂×#13133175	C₂²	1	3	yes	complemented
52	C₂×C₄²×D₅₃	C₂²	1	3	yes	complemented
53	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
54	C₅₃×#550	C₂²	1	16	yes	complemented
55	C₂×#127×C₅₃	C₂²	1	3	yes	complemented
56	C₂×#121×C₅₃	C₂²	1	6	yes	complemented
57	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	complemented
59	C₂⁴×D₅₃	C₂³	1	12	yes	—
60	#13133263	—	2	256	no	—
61	C₂×#11719949	—	2	16	no	—
62	C₂×#11719949	D₄	1	16	yes	complemented
63	C₂×#11719949	C₂³	1	8	yes	—
64	C₂×#11719949	C₂³	1	16	yes	complemented
65	C₂×#11719945	—	2	48	no	—
66	C₂×#11719945	C₂³	1	48	yes	complemented
67	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
68	C₂²×D₂₁₂	—	2	48	no	—
69	C₂²×D₂₁₂	C₂³	1	3	yes	—
70	C₂²×D₂₁₂	C₂³	1	6	yes	—
71	C₂²×D₂₁₂	C₂³	1	6	yes	—
72	C₂²×D₂₁₂	C₂³	1	6	yes	—
73	C₂²×D₂₁₂	C₂³	1	3	yes	—
74	C₂²×D₂₁₂	C₂³	1	48	yes	complemented
75	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
76	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
77	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
78	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
79	C₂×D₄×D₅₃	—	2	48	no	—
80	C₂×D₄×D₅₃	—	2	48	no	—
81	C₂×D₄×D₅₃	—	4	24	no	—
82	C₂×D₄×D₅₃	—	4	24	no	—
83	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
84	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
85	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
86	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
87	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
88	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
89	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
90	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
91	#13133216	C₂³	1	8	yes	complemented
92	#13133214	C₂³	1	24	yes	complemented
93	#13133213	C₂³	1	48	yes	complemented
94	#13133213	C₂³	1	24	yes	complemented
95	#13133213	C₂³	1	48	yes	complemented
96	#13133203	C₂³	1	48	yes	complemented
97	#13133192	C₂³	1	48	yes	complemented
98	#13133192	C₂³	1	24	yes	complemented
99	#13133191	C₂³	1	48	yes	complemented
100	#13133190	C₂³	1	48	yes	complemented
101	#13133188	C₂³	1	48	yes	complemented
102	#13133188	C₂³	1	24	yes	complemented
103	#13133187	C₂³	1	96	yes	complemented
104	#13133185	C₂³	1	96	yes	complemented
105	Q₈×Dic₅₃	C₂³	1	8	yes	complemented
106	C₄×D₂₁₂	C₂³	1	24	yes	complemented
107	C₄×D₂₁₂	C₂³	1	24	yes	complemented
108	#13133179	C₂³	1	24	yes	complemented
109	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
110	#13133175	C₂³	1	24	yes	complemented
111	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
112	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
113	C₂×#11719919	C₂³	1	2	yes	—
114	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
115	C₂×#11719921	C₂³	1	6	yes	—
116	C₂×#11719921	C₂³	1	6	yes	—
117	C₂×#11719921	C₂³	1	6	yes	—
118	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
120	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
121	C₄²×D₅₃	C₂³	1	24	yes	complemented
122	C₂×#41×C₅₃	—	2	16	no	—
123	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
124	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
126	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
127	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
128	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
129	#127×C₅₃	C₂³	1	24	yes	complemented
130	#121×C₅₃	C₂³	1	48	yes	complemented
131	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
132	C₄×D₄×C₅₃	C₂³	1	24	yes	complemented
133	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
134	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
135	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
136	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
137	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
138	C₂³×D₅₃	—	2	48	no	—
139	C₂³×D₅₃	—	4	24	no	—
140	C₂³×D₅₃	—	4	24	no	—
141	C₂³×D₅₃	C₂⁴	1	6	yes	—
142	C₂³×D₅₃	C₂⁴	1	3	yes	—
143	C₂³×D₅₃	C₂⁴	1	48	yes	—
144	#41×C₅₃	—	2	64	no	—
145	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
146	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
147	C₂×D₄×C₅₃	—	2	24	no	—
148	C₂×D₄×C₅₃	—	2	12	no	—
149	C₂×D₄×C₅₃	—	4	12	no	—
150	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
153	C₂²×C₄×C₅₃	—	2	24	no	—
154	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
159	C₂×C₂²⋊D₅₃	—	2	24	no	—
160	C₂×C₂²⋊D₅₃	—	2	24	no	—
161	C₂×C₂²⋊D₅₃	—	2	24	no	—
162	C₂×C₂²⋊D₅₃	—	4	24	no	—
163	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
164	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
165	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
166	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
167	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
168	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
169	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
170	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
171	#11719949	—	2	64	no	—
172	#11719949	C₂×D₄	1	64	yes	complemented
173	D₄×D₅₃	—	2	192	no	—
174	D₄×D₅₃	—	2	192	no	—
175	D₄×D₅₃	—	4	96	no	—
176	D₄×D₅₃	—	4	96	no	—
177	#11719945	—	2	192	no	—
178	C₂×D₂₁₂	—	2	24	no	—
179	C₂×D₂₁₂	—	2	24	no	—
180	C₂×D₂₁₂	—	2	12	no	—
181	C₂×D₂₁₂	—	2	192	no	—
182	C₂×D₂₁₂	—	4	12	no	—
183	C₂×D₂₁₂	C₂×D₄	1	24	yes	complemented
184	C₂×D₂₁₂	C₂⁴	1	12	yes	—
185	C₂×D₂₁₂	C₂⁴	1	24	yes	—
186	C₂×D₂₁₂	C₂⁴	1	24	yes	—
187	C₂×D₂₁₂	C₂⁴	1	24	yes	—
188	C₂×D₂₁₂	C₂⁴	1	12	yes	—
189	C₂×C₄×D₅₃	—	2	24	no	—
190	C₂×C₄×D₅₃	—	2	12	no	—
191	C₂×C₄×D₅₃	—	2	12	no	—
192	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	complemented
193	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
194	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
195	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
196	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
197	C₂×Dic₁₀₆	—	2	8	no	—
198	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
199	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
201	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
202	C₄²×C₅₃	C₂⁴	1	12	yes	—
203	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
204	#11719921	C₂⁴	1	24	yes	—
205	#11719921	C₂⁴	1	24	yes	—
206	#11719921	C₂⁴	1	24	yes	—
207	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
208	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
209	#11719919	C₂⁴	1	8	yes	—
210	C₄×Dic₅₃	C₂⁴	1	4	yes	—
211	C₄×Dic₅₃	C₂⁴	1	12	yes	—
212	C₄×Dic₅₃	C₂⁴	1	12	yes	—
213	C₂³×C₅₃	—	2	12	no	—
214	C₂³×C₅₃	—	2	12	no	—
215	C₂³×C₅₃	—	2	12	no	—
216	C₂³×C₅₃	C₂⁵	1	12	yes	—
217	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
218	C₂²×D₅₃	—	2	6	no	—
219	C₂²×D₅₃	—	2	6	no	—
220	C₂²×D₅₃	—	2	12	no	—
221	C₂²×D₅₃	—	2	12	no	—
222	C₂²×D₅₃	—	2	12	no	—
223	C₂²×D₅₃	—	2	96	no	—
224	C₂²×D₅₃	—	4	48	no	—
225	C₂²×D₅₃	—	4	48	no	—
226	C₂²×D₅₃	—	8	96	no	—
227	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
228	C₂²×D₅₃	C₂⁵	1	12	yes	—
229	C₂²×D₅₃	C₂⁵	1	6	yes	—
230	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
231	D₄×C₅₃	—	2	48	no	—
232	D₄×C₅₃	—	2	24	no	—
233	D₄×C₅₃	—	4	24	no	—
234	C₂×C₄×C₅₃	—	2	4	no	—
235	C₂×C₄×C₅₃	—	2	6	no	—
236	C₂×C₄×C₅₃	—	2	48	no	—
237	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
238	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	complemented
239	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
240	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
241	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
243	C₂²⋊D₅₃	—	2	48	no	—
244	C₂²⋊D₅₃	—	2	48	no	—
245	C₂²⋊D₅₃	—	2	48	no	—
246	C₂²⋊D₅₃	—	4	48	no	—
247	C₂×Dic₅₃	—	2	4	no	—
248	C₂×Dic₅₃	—	2	6	no	—
249	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
250	C₂×Dic₅₃	H(5,𝔽₂)	1	4	yes	complemented
251	C₂×Dic₅₃	C₂⁵	1	4	yes	—
252	C₂×Dic₅₃	C₂⁵	1	6	yes	—
253	C₂×Dic₅₃	C₂⁵	1	2	yes	—
254	C₂×Dic₅₃	C₂⁵	1	2	yes	—
255	D₂₁₂	—	2	48	no	—
256	D₂₁₂	—	2	48	no	—
257	D₂₁₂	—	2	24	no	—
258	D₂₁₂	—	4	24	no	—
259	D₂₁₂	C₂²×D₄	1	48	yes	complemented
260	C₄×D₅₃	—	2	48	no	—
261	C₄×D₅₃	—	2	24	no	—
262	C₄×D₅₃	—	2	24	no	—
263	C₄×D₅₃	C₂²×D₄	1	48	yes	complemented
264	Dic₁₀₆	—	2	16	no	—
265	C₂×#3498	—	53	1	no	maximal, Sylow
266	C₂²×C₅₃	—	2	12	no	—
267	C₂²×C₅₃	—	2	12	no	—
268	C₂²×C₅₃	—	2	12	no	—
269	C₂²×C₅₃	—	4	24	no	—
270	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
271	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
272	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
273	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
274	C₂×D₅₃	—	2	6	no	—
275	C₂×D₅₃	—	2	6	no	—
276	C₂×D₅₃	—	2	24	no	—
277	C₂×D₅₃	—	2	12	no	—
278	C₂×D₅₃	—	2	12	no	—
279	C₂×D₅₃	—	2	12	no	—
280	C₂×D₅₃	—	4	24	no	—
281	C₂×D₅₃	—	8	96	no	—
282	C₂×D₅₃	C₂³×D₄	1	12	yes	—
283	C₄×C₅₃	—	2	4	no	—
284	C₄×C₅₃	—	2	6	no	—
285	C₄×C₅₃	C₂³×D₄	1	12	yes	—
286	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	complemented
287	Dic₅₃	—	2	4	no	—
288	Dic₅₃	—	2	6	no	—
289	Dic₅₃	C₂³×D₄	1	4	yes	—
290	Dic₅₃	C₂×H(5,𝔽₂)	1	4	yes	complemented
291	C₂²×H(5,𝔽₂)	—	53	2	no	—
292	#3498	—	53	32	no	—
293	C₂×#550	—	53	2	no	—
294	C₂×#550	—	53	1	no	—
295	C₂×#550	D₅₃	1	1	yes	characteristic, complemented
296	C₂×D₄²	—	53	3	no	—
297	C₂×D₄²	—	53	6	no	—
298	C₂×D₄²	—	53	3	no	—
299	C₂×#535	—	53	6	no	—
300	C₂×#532	—	53	3	no	—
301	C₂²×#127	—	53	3	no	—
302	C₂×C₄×#41	—	53	1	no	—
303	C₂×C₅₃	—	4	12	no	—
304	C₂×C₅₃	#3498	1	4	yes	complemented
305	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
306	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
307	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
308	D₅₃	—	2	12	no	—
309	D₅₃	—	4	12	no	—
310	C₂×H(5,𝔽₂)	—	53	32	no	—
311	C₂×H(5,𝔽₂)	—	106	32	no	—
312	C₂²×#41	—	53	1	no	—
313	C₂²×#41	—	53	2	no	—
314	C₂²×#41	—	53	6	no	—
315	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
316	C₂³×D₄	—	53	6	no	—
317	C₂³×D₄	—	53	6	no	—
318	C₂³×D₄	—	53	6	no	—
319	C₂³×D₄	—	53	6	no	—
320	C₂³×D₄	—	53	6	no	—
321	#550	—	53	32	no	—
322	#550	—	53	16	no	—
323	#550	C₂×D₅₃	1	16	yes	complemented
324	D₄²	—	53	48	no	—
325	D₄²	—	53	96	no	—
326	D₄²	—	53	48	no	—
327	#535	—	53	96	no	—
328	#532	—	53	48	no	—
329	C₂×C₄⋊Q₈	—	53	1	no	—
330	C₂×#127	—	53	3	no	—
331	C₂×#127	—	53	6	no	—
332	C₂×#127	—	53	6	no	—
333	C₂×#127	—	53	48	no	—
334	C₂×#127	—	53	3	no	—
335	C₂×#127	C₂×D₅₃	1	3	yes	complemented
336	C₂×#124	—	53	6	no	—
337	C₂×#121	—	53	3	no	—
338	C₂×#121	—	53	6	no	—
339	C₂×#121	—	53	6	no	—
340	C₂×#121	—	53	3	no	—
341	C₂×#121	—	53	6	no	—
342	C₂×#121	—	53	6	no	—
343	C₂×#121	C₂×D₅₃	1	6	yes	complemented
344	C₂×#120	—	53	12	no	—
345	C₂×#120	—	53	12	no	—
346	C₄×#41	—	53	16	no	—
347	C₂×C₄×Q₈	—	53	1	no	—
348	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
349	C₂×C₄×D₄	—	53	6	no	—
350	C₂×C₄×D₄	—	53	3	no	—
351	C₂×C₄×D₄	—	53	3	no	—
352	C₂×C₄×D₄	—	53	3	no	—
353	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	complemented
354	C₂×#117	—	53	3	no	—
355	C₂²×C₄²	—	53	3	no	—
356	C₅₃	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
357	C₂⁵	—	53	12	no	—
358	H(5,𝔽₂)	—	106	256	no	—
359	C₂×#41	—	2	16	no	—
360	C₂×#41	—	53	16	no	—
361	C₂×#41	—	53	8	no	—
362	C₂×#41	—	53	16	no	—
363	C₂×#41	—	53	48	no	—
364	C₂×#41	—	106	48	no	—
365	C₂×#41	—	106	16	no	—
366	C₂×#41	C₂²×D₅₃	1	16	yes	complemented
367	C₂²×Q₈	—	53	2	no	—
368	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
369	C₂²×D₄	—	53	6	no	—
370	C₂²×D₄	—	53	6	no	—
371	C₂²×D₄	—	53	6	no	—
372	C₂²×D₄	—	53	12	no	—
373	C₂²×D₄	—	53	48	no	—
374	C₂²×D₄	—	53	48	no	—
375	C₂²×D₄	—	53	48	no	—
376	C₂²×D₄	—	53	48	no	—
377	C₂²×D₄	—	53	3	no	—
378	C₂²×D₄	—	53	3	no	—
379	C₂²×D₄	—	53	6	no	—
380	C₂²×D₄	—	53	6	no	—
381	C₂²×D₄	—	53	48	no	—
382	C₂²×D₄	—	53	6	no	—
383	C₂²×D₄	—	106	48	no	—
384	C₂²×D₄	—	106	48	no	—
385	C₂²×D₄	—	106	48	no	—
386	C₂²×D₄	—	212	24	no	—
387	C₂²×D₄	—	212	24	no	—
388	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
389	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
390	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
391	C₂³×C₄	—	53	3	no	—
392	C₂³×C₄	—	53	3	no	—
393	C₂³×C₄	—	53	3	no	—
394	C₂³×C₄	—	53	6	no	—
395	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
396	C₄⋊Q₈	—	53	8	no	—
397	#127	—	53	24	no	—
398	#127	—	53	48	no	—
399	#127	—	53	48	no	—
400	#127	—	53	24	no	—
401	#127	C₂²×D₅₃	1	24	yes	complemented
402	#124	—	53	48	no	—
403	#121	—	53	24	no	—
404	#121	—	53	48	no	—
405	#121	—	53	48	no	—
406	#121	—	53	24	no	—
407	#121	—	53	48	no	—
408	#121	—	53	48	no	—
409	#121	C₂²×D₅₃	1	48	yes	complemented
410	#120	—	53	96	no	—
411	#120	—	53	96	no	—
412	C₄×Q₈	—	53	8	no	—
413	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
414	C₄×D₄	—	53	48	no	—
415	C₄×D₄	—	53	24	no	—
416	C₄×D₄	—	53	24	no	—
417	C₄×D₄	—	53	24	no	—
418	C₄×D₄	C₂²×D₅₃	1	24	yes	complemented
419	#117	—	53	24	no	—
420	C₂×C₄⋊C₄	—	53	3	no	—
421	C₂×C₄⋊C₄	—	53	2	no	—
422	C₂×C₄⋊C₄	—	53	1	no	—
423	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
424	C₂×C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	—	53	6	no	—
426	C₂×C₂²⋊C₄	—	53	6	no	—
427	C₂×C₂²⋊C₄	—	53	6	no	—
428	C₂×C₂²⋊C₄	—	53	6	no	—
429	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
430	C₂×C₄²	—	53	3	no	—
431	C₂×C₄²	—	53	3	no	—
432	C₂×C₄²	—	53	1	no	—
433	C₂×C₄²	—	53	24	no	—
434	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
435	C₂⁴	—	53	3	no	—
436	C₂⁴	—	53	6	no	—
437	C₂⁴	—	53	48	no	—
438	C₂⁴	—	106	48	no	—
439	C₂⁴	—	212	24	no	—
440	C₂⁴	—	212	24	no	—
441	C₂⁴	C₂³×D₅₃	1	6	yes	—
442	#41	—	2	64	no	—
443	#41	—	53	64	no	—
444	#41	—	106	192	no	—
445	#41	—	106	64	no	—
446	C₂×Q₈	—	53	8	no	—
447	C₂×Q₈	—	106	8	no	—
448	C₂×Q₈	D₄×D₅₃	1	8	yes	complemented
449	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
450	C₂×D₄	—	2	24	no	—
451	C₂×D₄	—	2	12	no	—
452	C₂×D₄	—	4	12	no	—
453	C₂×D₄	—	53	24	no	—
454	C₂×D₄	—	53	24	no	—
455	C₂×D₄	—	53	24	no	—
456	C₂×D₄	—	53	48	no	—
457	C₂×D₄	—	53	24	no	—
458	C₂×D₄	—	53	12	no	—
459	C₂×D₄	—	53	12	no	—
460	C₂×D₄	—	53	24	no	—
461	C₂×D₄	—	53	24	no	—
462	C₂×D₄	—	53	24	no	—
463	C₂×D₄	—	106	192	no	—
464	C₂×D₄	—	106	24	no	—
465	C₂×D₄	—	106	192	no	—
466	C₂×D₄	—	106	24	no	—
467	C₂×D₄	—	106	24	no	—
468	C₂×D₄	—	106	192	no	—
469	C₂×D₄	—	106	24	no	—
470	C₂×D₄	—	106	24	no	—
471	C₂×D₄	—	106	12	no	—
472	C₂×D₄	—	212	12	no	—
473	C₂×D₄	—	212	96	no	—
474	C₂×D₄	—	212	96	no	—
475	C₂×D₄	—	212	24	no	—
476	C₂×D₄	C₂³×D₅₃	1	12	yes	—
477	C₂×D₄	C₂³×D₅₃	1	24	yes	—
478	C₂×D₄	C₂³×D₅₃	1	24	yes	—
479	C₂²×C₄	—	2	24	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	1	no	—
482	C₂²×C₄	—	53	24	no	—
483	C₂²×C₄	—	53	12	no	—
484	C₂²×C₄	—	53	3	no	—
485	C₂²×C₄	—	53	2	no	—
486	C₂²×C₄	—	53	12	no	—
487	C₂²×C₄	—	53	12	no	—
488	C₂²×C₄	—	53	24	no	—
489	C₂²×C₄	—	106	12	no	—
490	C₂²×C₄	—	106	24	no	—
491	C₂²×C₄	—	106	12	no	—
492	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
493	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
494	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
495	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
496	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
497	C₄⋊C₄	—	53	12	no	—
498	C₄⋊C₄	—	53	8	no	—
499	C₄⋊C₄	—	53	4	no	—
500	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
501	C₄⋊C₄	C₂³×D₅₃	1	4	yes	—
502	C₂²⋊C₄	—	53	24	no	—
503	C₂²⋊C₄	—	53	24	no	—
504	C₂²⋊C₄	—	53	24	no	—
505	C₂²⋊C₄	—	53	24	no	—
506	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
507	C₄²	—	53	4	no	—
508	C₄²	—	53	12	no	—
509	C₄²	—	53	12	no	—
510	C₄²	C₂³×D₅₃	1	12	yes	—
511	C₂³	—	2	12	no	—
512	C₂³	—	2	12	no	—
513	C₂³	—	2	12	no	—
514	C₂³	—	53	12	no	—
515	C₂³	—	53	6	no	—
516	C₂³	—	53	12	no	—
517	C₂³	—	106	96	no	—
518	C₂³	—	106	6	no	—
519	C₂³	—	106	6	no	—
520	C₂³	—	106	12	no	—
521	C₂³	—	106	12	no	—
522	C₂³	—	106	12	no	—
523	C₂³	—	424	96	no	—
524	C₂³	—	212	48	no	—
525	C₂³	—	212	48	no	—
526	C₂³	C₂⁴×D₅₃	1	12	yes	—
527	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
528	Q₈	—	106	16	no	—
529	Q₈	C₂×D₄×D₅₃	1	16	yes	complemented
530	D₄	—	2	48	no	—
531	D₄	—	2	24	no	—
532	D₄	—	4	24	no	—
533	D₄	—	53	48	no	—
534	D₄	—	106	48	no	—
535	D₄	—	106	48	no	—
536	D₄	—	106	48	no	—
537	D₄	—	106	48	no	—
538	D₄	—	106	48	no	—
539	D₄	—	106	24	no	—
540	D₄	—	212	24	no	—
541	D₄	—	212	48	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	6	no	—
544	C₂×C₄	—	2	48	no	—
545	C₂×C₄	—	53	4	no	—
546	C₂×C₄	—	53	4	no	—
547	C₂×C₄	—	53	2	no	—
548	C₂×C₄	—	53	2	no	—
549	C₂×C₄	—	53	48	no	—
550	C₂×C₄	—	53	6	no	—
551	C₂×C₄	—	53	4	no	—
552	C₂×C₄	—	106	24	no	—
553	C₂×C₄	—	106	4	no	—
554	C₂×C₄	—	106	6	no	—
555	C₂×C₄	—	106	48	no	—
556	C₂×C₄	—	106	24	no	—
557	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
558	C₂×C₄	#13133263	1	4	yes	complemented
559	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
560	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
561	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
562	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
563	C₂²	—	2	12	no	—
564	C₂²	—	2	12	no	—
565	C₂²	—	2	12	no	—
566	C₂²	—	4	24	no	—
567	C₂²	—	53	12	no	—
568	C₂²	—	106	6	no	—
569	C₂²	—	106	24	no	—
570	C₂²	—	106	6	no	—
571	C₂²	—	106	12	no	—
572	C₂²	—	106	12	no	—
573	C₂²	—	106	12	no	—
574	C₂²	—	424	96	no	—
575	C₂²	—	212	24	no	—
576	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
577	C₂²	H(5,𝔽₂)×D₅₃	1	2	yes	—
578	C₂²	C₂×#13133263	1	2	yes	—
579	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
580	C₄	—	2	4	no	—
581	C₄	—	2	6	no	—
582	C₄	—	53	4	no	—
583	C₄	—	53	4	no	—
584	C₄	—	106	4	no	—
585	C₄	—	106	6	no	—
586	C₄	C₂²×D₄×D₅₃	1	12	yes	—
587	C₄	C₂×#13133263	1	4	yes	complemented
588	C₂	—	4	12	no	—
589	C₂	—	106	12	no	—
590	C₂	—	212	12	no	—
591	C₂	#20288492	1	4	yes	direct product factor
592	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
593	C₂	C₂×H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
594	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#550	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#550	D₅₃
hypocenter	C₅₃	C₂×#3498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#550

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. #20288492	C₂	direct (N × H)
3. C₂×#16811295	C₂	semidirect
4. C₂×#16811292	C₂	semidirect
5. C₂×D₄×D₂₁₂	C₂	semidirect
6. C₂×#16811258	C₂	semidirect
7. C₂×#16811257	C₂	semidirect
8. C₂×#16811189	C₂	semidirect
9. C₂×#16811170	C₂	semidirect
10. C₂×#127×D₅₃	C₂	semidirect
11. C₂×C₄×#11719949	C₂	semidirect
12. C₂×C₅₃×#550	C₂	semidirect
14. C₂×#13133263	C₂²	semidirect
16. C₂²×#11719949	C₂²	semidirect
17. C₂²×#11719945	C₂²	semidirect
18. C₂³×D₂₁₂	C₂²	semidirect
19. C₂²×D₄×D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
21. C₂²×D₄×D₅₃	C₂²	semidirect
22. C₂²×D₄×D₅₃	C₂²	semidirect
23. #16811295	C₂²	semidirect
24. #16811292	C₂²	semidirect
25. D₄×D₂₁₂	C₂²	semidirect
26. #16811258	C₂²	semidirect
27. #16811257	C₂²	semidirect
28. #16811189	C₂²	semidirect
29. #16811170	C₂²	semidirect
30. C₂×#13133216	C₂²	semidirect
31. C₂×#13133214	C₂²	semidirect
32. C₂×#13133213	C₂²	semidirect
33. C₂×#13133213	C₂²	semidirect
34. C₂×#13133213	C₂²	semidirect
35. #127×D₅₃	C₂²	semidirect
36. C₂×#13133203	C₂²	semidirect
37. C₂×#13133192	C₂²	semidirect
38. C₂×#13133192	C₂²	semidirect
39. C₂×#13133191	C₂²	semidirect
40. C₂×#13133190	C₂²	semidirect
41. C₂×#13133188	C₂²	semidirect
42. C₂×#13133188	C₂²	semidirect
43. C₂×#13133187	C₂²	semidirect
44. C₂×#13133185	C₂²	semidirect
45. C₄×#11719949	C₂²	semidirect
46. C₂×Q₈×Dic₅₃	C₂²	semidirect
47. C₂×C₄×D₂₁₂	C₂²	semidirect
48. C₂×C₄×D₂₁₂	C₂²	semidirect
49. C₂×#13133179	C₂²	semidirect
50. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
51. C₂×#13133175	C₂²	semidirect
52. C₂×C₄²×D₅₃	C₂²	semidirect
53. C₂²×#41×C₅₃	C₂²	semidirect
54. C₅₃×#550	C₂²	semidirect
55. C₂×#127×C₅₃	C₂²	semidirect
56. C₂×#121×C₅₃	C₂²	semidirect
57. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
58. C₂×C₄×D₄×C₅₃	C₂²	semidirect
62. C₂×#11719949	D₄	semidirect
64. C₂×#11719949	C₂³	semidirect
66. C₂×#11719945	C₂³	semidirect
74. C₂²×D₂₁₂	C₂³	semidirect
83. C₂×D₄×D₅₃	C₂³	semidirect
84. C₂×D₄×D₅₃	C₂³	semidirect
85. C₂×D₄×D₅₃	C₂³	semidirect
86. C₂×D₄×D₅₃	C₂³	semidirect
91. #13133216	C₂³	semidirect
92. #13133214	C₂³	semidirect
93. #13133213	C₂³	semidirect
94. #13133213	C₂³	semidirect
95. #13133213	C₂³	semidirect
96. #13133203	C₂³	semidirect
97. #13133192	C₂³	semidirect
98. #13133192	C₂³	semidirect
99. #13133191	C₂³	semidirect
100. #13133190	C₂³	semidirect
101. #13133188	C₂³	semidirect
102. #13133188	C₂³	semidirect
103. #13133187	C₂³	semidirect
104. #13133185	C₂³	semidirect
105. Q₈×Dic₅₃	C₂³	semidirect
106. C₄×D₂₁₂	C₂³	semidirect
107. C₄×D₂₁₂	C₂³	semidirect
108. #13133179	C₂³	semidirect
109. C₄×C₂²⋊D₅₃	C₂³	semidirect
110. #13133175	C₂³	semidirect
121. C₄²×D₅₃	C₂³	semidirect
123. C₂×#41×C₅₃	C₂³	semidirect
129. #127×C₅₃	C₂³	semidirect
130. #121×C₅₃	C₂³	semidirect
131. C₄×Q₈×C₅₃	C₂³	semidirect
132. C₄×D₄×C₅₃	C₂³	semidirect
145. C₂×Q₈×C₅₃	C₂×D₄	semidirect
172. #11719949	C₂×D₄	semidirect
183. C₂×D₂₁₂	C₂×D₄	semidirect
192. C₂×C₄×D₅₃	C₂×D₄	semidirect
230. Q₈×C₅₃	C₂²×D₄	semidirect
238. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
250. C₂×Dic₅₃	H(5,𝔽₂)	semidirect
259. D₂₁₂	C₂²×D₄	semidirect
263. C₄×D₅₃	C₂²×D₄	semidirect
286. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
290. Dic₅₃	C₂×H(5,𝔽₂)	semidirect
295. C₂×#550	D₅₃	semidirect
304. C₂×C₅₃	#3498	semidirect
315. C₂²×#41	C₂×D₅₃	semidirect
323. #550	C₂×D₅₃	semidirect
335. C₂×#127	C₂×D₅₃	semidirect
343. C₂×#121	C₂×D₅₃	semidirect
348. C₂×C₄×Q₈	C₂×D₅₃	semidirect
353. C₂×C₄×D₄	C₂×D₅₃	semidirect
356. C₅₃	C₂×#3498	semidirect
366. C₂×#41	C₂²×D₅₃	semidirect
401. #127	C₂²×D₅₃	semidirect
409. #121	C₂²×D₅₃	semidirect
413. C₄×Q₈	C₂²×D₅₃	semidirect
418. C₄×D₄	C₂²×D₅₃	semidirect
448. C₂×Q₈	D₄×D₅₃	semidirect
529. Q₈	C₂×D₄×D₅₃	semidirect
558. C₂×C₄	#13133263	semidirect
587. C₄	C₂×#13133263	semidirect
591. C₂	#20288492	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
#20288492	2
C₂×#16811295	2
C₂×#16811292	2
C₂×D₄×D₂₁₂	2
C₂×#16811258	2
C₂×#16811257	2
C₂×#16811189	2
C₂×#16811170	2
C₂×#127×D₅₃	2
C₂×C₄×#11719949	2
C₂×C₅₃×#550	2
C₂×#3498	53

Subgroups — C2×#20288492 #28292411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288492 #28292411