Subgroups — C₂×#20288489 #28292412

Order 13568.

12982 conjugacy classes of subgroups.

The subgroup lattice has 496 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	#20288489	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811292	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×C₂²⋊D₅₃	C₂	1	12	yes	maximal, complemented
5	C₂×#16811190	C₂	1	3	yes	maximal, complemented
6	C₂×#16811187	C₂	1	3	yes	maximal, complemented
7	C₂×#16811169	C₂	1	3	yes	maximal, complemented
8	C₂²×#13133213	C₂	1	3	yes	maximal, complemented
9	C₂×#41×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133263	—	2	16	no	—
12	C₂×#13133263	C₂²	1	16	yes	complemented
13	C₂²×#11719949	C₂²	1	2	yes	complemented
14	C₂²×#11719945	C₂²	1	3	yes	complemented
15	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	12	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
18	#16811292	C₂²	1	64	yes	complemented
19	D₄×C₂²⋊D₅₃	C₂²	1	192	yes	complemented
20	#16811190	C₂²	1	48	yes	complemented
21	#16811187	C₂²	1	48	yes	complemented
22	#16811169	C₂²	1	48	yes	complemented
23	C₂×#13133215	C₂²	1	1	yes	characteristic, complemented
24	C₂×#13133213	C₂²	1	6	yes	complemented
25	C₂×#13133213	C₂²	1	12	yes	complemented
26	C₂×#13133213	C₂²	1	3	yes	complemented
27	C₂×#13133213	C₂²	1	48	yes	complemented
28	C₂×#13133206	C₂²	1	3	yes	complemented
29	C₂×#13133204	C₂²	1	3	yes	complemented
30	C₂×#13133191	C₂²	1	12	yes	complemented
31	C₂×#13133189	C₂²	1	12	yes	complemented
32	C₂×#13133188	C₂²	1	12	yes	complemented
33	C₂×#13133186	C₂²	1	12	yes	complemented
34	C₂×#13133185	C₂²	1	12	yes	complemented
35	#41×Dic₅₃	C₂²	1	16	yes	complemented
36	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	complemented
37	C₂×D₄×Dic₅₃	C₂²	1	6	yes	complemented
38	C₂×C₄×C₂²⋊D₅₃	C₂²	1	12	yes	complemented
39	C₂×#13133176	C₂²	1	3	yes	complemented
40	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	complemented
41	C₂×H(5,𝔽₂)×C₅₃	—	2	16	no	—
42	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	16	yes	complemented
43	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
44	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
45	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
46	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
47	C₂³×D₄×C₅₃	C₂²	1	6	yes	complemented
48	C₂⁴×D₅₃	C₂³	1	6	yes	—
49	#13133263	—	2	128	no	—
50	C₂×#11719949	—	2	16	no	—
51	C₂×#11719949	C₂³	1	16	yes	complemented
52	C₂×#11719945	—	2	24	no	—
53	C₂×#11719945	C₂³	1	24	yes	complemented
54	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
55	C₂²×D₂₁₂	—	2	24	no	—
56	C₂²×D₂₁₂	C₂³	1	6	yes	—
57	C₂²×D₂₁₂	C₂³	1	3	yes	—
58	C₂²×D₂₁₂	C₂³	1	24	yes	complemented
59	C₂²×C₂²⋊D₅₃	—	4	48	no	—
60	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
61	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	96	yes	complemented
64	C₂×D₄×D₅₃	—	2	48	no	—
65	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
66	C₂³×Dic₅₃	C₂³	1	6	yes	—
67	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
68	#13133215	C₂³	1	8	yes	complemented
69	#13133213	C₂³	1	96	yes	complemented
70	#13133213	C₂³	1	24	yes	complemented
71	#13133213	C₂³	1	48	yes	complemented
72	#13133206	C₂³	1	24	yes	complemented
73	#13133204	C₂³	1	24	yes	complemented
74	#13133191	C₂³	1	96	yes	complemented
75	#13133189	C₂³	1	96	yes	complemented
76	#13133188	C₂³	1	96	yes	complemented
77	#13133186	C₂³	1	96	yes	complemented
78	#13133185	C₂³	1	96	yes	complemented
79	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
80	D₄×Dic₅₃	C₂³	1	48	yes	complemented
81	C₄×C₂²⋊D₅₃	C₂³	1	96	yes	complemented
82	#13133176	C₂³	1	24	yes	complemented
83	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
84	C₂×#11719919	C₂³	1	4	yes	—
85	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
86	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
87	C₂×#11719921	C₂³	1	12	yes	—
88	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
90	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
91	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
92	C₂⁵×C₅₃	C₂³	1	6	yes	—
93	H(5,𝔽₂)×C₅₃	—	2	128	no	—
94	C₂×#41×C₅₃	—	2	16	no	—
95	C₂×#41×C₅₃	—	2	24	no	—
96	C₂×#41×C₅₃	D₄	1	16	yes	complemented
97	C₂×#41×C₅₃	C₂³	1	8	yes	—
98	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
99	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
100	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×D₄×C₅₃	—	2	48	no	—
103	C₂²×D₄×C₅₃	—	2	24	no	—
104	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
105	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
106	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
107	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
108	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
109	C₂²×D₄×C₅₃	C₂³	1	48	yes	complemented
110	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
111	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
112	C₂⁴×C₅₃	—	2	24	no	—
113	C₂⁴×C₅₃	—	4	24	no	—
114	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
117	C₂³×D₅₃	—	2	24	no	—
118	C₂³×D₅₃	—	4	24	no	—
119	C₂³×D₅₃	C₂⁴	1	6	yes	—
120	C₂³×D₅₃	C₂⁴	1	24	yes	—
121	#41×C₅₃	—	2	64	no	—
122	#41×C₅₃	—	2	96	no	—
123	#41×C₅₃	C₂×D₄	1	64	yes	complemented
124	C₂×Q₈×C₅₃	—	2	4	no	—
125	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
126	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
127	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
128	C₂×D₄×C₅₃	—	2	24	no	—
129	C₂×D₄×C₅₃	—	2	192	no	—
130	C₂×D₄×C₅₃	—	2	12	no	—
131	C₂×D₄×C₅₃	—	2	24	no	—
132	C₂×D₄×C₅₃	—	2	96	no	—
133	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
134	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
135	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
136	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
137	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
138	C₂²×C₄×C₅₃	—	2	24	no	—
139	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
140	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
145	C₂×C₂²⋊D₅₃	—	2	24	no	—
146	C₂×C₂²⋊D₅₃	—	2	48	no	—
147	C₂×C₂²⋊D₅₃	—	4	192	no	—
148	C₂×C₂²⋊D₅₃	—	4	48	no	—
149	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
150	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
151	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
152	C₂²×Dic₅₃	—	2	24	no	—
153	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
154	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
155	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
156	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
157	#11719949	—	2	64	no	—
158	D₄×D₅₃	—	2	192	no	—
159	#11719945	—	2	96	no	—
160	C₂×D₂₁₂	—	2	24	no	—
161	C₂×D₂₁₂	—	2	12	no	—
162	C₂×D₂₁₂	—	2	96	no	—
163	C₂×D₂₁₂	C₂⁴	1	24	yes	—
164	C₂×D₂₁₂	C₂⁴	1	12	yes	—
165	C₂×C₄×D₅₃	—	2	24	no	—
166	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
167	C₂×Dic₁₀₆	—	2	4	no	—
168	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
169	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
170	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
171	#11719921	C₂⁴	1	48	yes	—
172	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
173	#11719919	C₂⁴	1	16	yes	—
174	C₄×Dic₅₃	C₂⁴	1	4	yes	—
175	C₄×Dic₅₃	C₂⁴	1	12	yes	—
176	C₄×Dic₅₃	C₂⁴	1	24	yes	—
177	C₂³×C₅₃	—	2	12	no	—
178	C₂³×C₅₃	—	2	12	no	—
179	C₂³×C₅₃	—	2	24	no	—
180	C₂³×C₅₃	—	2	48	no	—
181	C₂³×C₅₃	—	4	48	no	—
182	C₂³×C₅₃	—	8	48	no	—
183	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
184	C₂³×C₅₃	C₂⁵	1	12	yes	—
185	C₂³×C₅₃	C₂⁵	1	6	yes	—
186	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
187	C₂²×D₅₃	—	2	12	no	—
188	C₂²×D₅₃	—	2	24	no	—
189	C₂²×D₅₃	—	2	48	no	—
190	C₂²×D₅₃	—	4	48	no	—
191	C₂²×D₅₃	—	8	48	no	—
192	C₂²×D₅₃	C₂⁵	1	12	yes	—
193	Q₈×C₅₃	—	2	8	no	—
194	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
195	D₄×C₅₃	—	2	48	no	—
196	D₄×C₅₃	—	2	24	no	—
197	D₄×C₅₃	—	2	48	no	—
198	D₄×C₅₃	C₂²×D₄	1	48	yes	complemented
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	48	no	—
201	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	48	yes	complemented
204	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
205	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
206	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
207	C₂²⋊D₅₃	—	2	48	no	—
208	C₂²⋊D₅₃	—	2	96	no	—
209	C₂²⋊D₅₃	—	4	96	no	—
210	C₂×Dic₅₃	—	2	4	no	—
211	C₂×Dic₅₃	—	2	12	no	—
212	C₂×Dic₅₃	—	2	48	no	—
213	C₂×Dic₅₃	H(5,𝔽₂)	1	8	yes	complemented
214	C₂×Dic₅₃	C₂⁵	1	4	yes	—
215	C₂×Dic₅₃	C₂⁵	1	12	yes	—
216	C₂×Dic₅₃	C₂⁵	1	4	yes	—
217	D₂₁₂	—	2	24	no	—
218	D₂₁₂	—	2	48	no	—
219	C₄×D₅₃	—	2	48	no	—
220	Dic₁₀₆	—	2	8	no	—
221	C₂×#3498	—	53	1	no	maximal, Sylow
222	C₂²×C₅₃	—	2	12	no	—
223	C₂²×C₅₃	—	2	24	no	—
224	C₂²×C₅₃	—	2	12	no	—
225	C₂²×C₅₃	—	2	24	no	—
226	C₂²×C₅₃	—	4	24	no	—
227	C₂²×C₅₃	—	8	48	no	—
228	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
229	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
230	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
231	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
232	C₂×D₅₃	—	2	12	no	—
233	C₂×D₅₃	—	2	24	no	—
234	C₂×D₅₃	—	4	24	no	—
235	C₂×D₅₃	—	8	48	no	—
236	C₄×C₅₃	—	2	4	no	—
237	C₄×C₅₃	C₂³×D₄	1	4	yes	—
238	C₄×C₅₃	C₂³×D₄	1	12	yes	—
239	Dic₅₃	—	2	4	no	—
240	Dic₅₃	—	2	12	no	—
241	Dic₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
242	C₂²×H(5,𝔽₂)	—	53	1	no	—
243	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
244	#3498	—	53	32	no	—
245	C₂×#550	—	53	4	no	—
246	C₂×D₄²	—	53	12	no	—
247	C₂×#535	—	53	3	no	—
248	C₂×#535	—	53	3	no	—
249	C₂×#532	—	53	3	no	—
250	C₂²×#127	—	53	3	no	—
251	C₂×C₄×#41	—	53	1	no	—
252	C₂×C₅₃	—	2	12	no	—
253	C₂×C₅₃	—	4	12	no	—
254	C₂×C₅₃	#3498	1	4	yes	complemented
255	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
256	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
257	D₅₃	—	4	12	no	—
258	C₂×H(5,𝔽₂)	—	2	16	no	—
259	C₂×H(5,𝔽₂)	—	53	16	no	—
260	C₂×H(5,𝔽₂)	—	106	16	no	—
261	C₂×H(5,𝔽₂)	C₂×D₅₃	1	16	yes	complemented
262	C₂²×#41	—	53	2	no	—
263	C₂²×#41	—	53	3	no	—
264	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
265	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
266	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
267	C₂³×D₄	—	53	12	no	—
268	C₂³×D₄	—	53	6	no	—
269	C₂³×D₄	—	53	3	no	—
270	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
271	C₂³×D₄	C₂×D₅₃	1	6	yes	complemented
272	#550	—	53	64	no	—
273	D₄²	—	53	192	no	—
274	#535	—	53	48	no	—
275	#535	—	53	48	no	—
276	#532	—	53	48	no	—
277	C₂×C₄⋊Q₈	—	53	1	no	—
278	C₂×#127	—	53	3	no	—
279	C₂×#127	—	53	6	no	—
280	C₂×#127	—	53	12	no	—
281	C₂×#127	—	53	48	no	—
282	C₂×#124	—	53	3	no	—
283	C₂×#124	—	53	3	no	—
284	C₂×#121	—	53	12	no	—
285	C₂×#121	—	53	12	no	—
286	C₂×#121	—	53	12	no	—
287	C₂×#120	—	53	12	no	—
288	C₂×#120	—	53	12	no	—
289	C₄×#41	—	53	16	no	—
290	C₂×C₄×Q₈	—	53	2	no	—
291	C₂×C₄×D₄	—	53	6	no	—
292	C₂×C₄×D₄	—	53	12	no	—
293	C₂×#117	—	53	3	no	—
294	C₂²×C₄²	—	53	3	no	—
295	C₅₃	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
296	C₂⁵	—	53	6	no	—
297	C₂⁵	C₂²×D₅₃	1	6	yes	—
298	H(5,𝔽₂)	—	2	128	no	—
299	H(5,𝔽₂)	—	106	128	no	—
300	C₂×#41	—	2	16	no	—
301	C₂×#41	—	2	24	no	—
302	C₂×#41	—	53	16	no	—
303	C₂×#41	—	53	24	no	—
304	C₂×#41	—	106	16	no	—
305	C₂×#41	—	106	24	no	—
306	C₂×#41	C₂²⋊D₅₃	1	16	yes	complemented
307	C₂×#41	C₂²×D₅₃	1	8	yes	—
308	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
309	C₂×#41	C₂²×D₅₃	1	16	yes	complemented
310	C₂²×Q₈	—	53	1	no	—
311	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
312	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
313	C₂²×D₄	—	2	48	no	—
314	C₂²×D₄	—	2	24	no	—
315	C₂²×D₄	—	53	12	no	—
316	C₂²×D₄	—	53	6	no	—
317	C₂²×D₄	—	53	24	no	—
318	C₂²×D₄	—	53	96	no	—
319	C₂²×D₄	—	53	48	no	—
320	C₂²×D₄	—	53	6	no	—
321	C₂²×D₄	—	53	3	no	—
322	C₂²×D₄	—	53	24	no	—
323	C₂²×D₄	—	106	48	no	—
324	C₂²×D₄	—	106	24	no	—
325	C₂²×D₄	—	212	48	no	—
326	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
327	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
328	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
329	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
330	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
331	C₂²×D₄	C₂²×D₅₃	1	48	yes	complemented
332	C₂³×C₄	—	53	6	no	—
333	C₂³×C₄	—	53	6	no	—
334	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
335	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
336	C₄⋊Q₈	—	53	8	no	—
337	#127	—	53	24	no	—
338	#127	—	53	48	no	—
339	#127	—	53	96	no	—
340	#124	—	53	24	no	—
341	#124	—	53	24	no	—
342	#121	—	53	96	no	—
343	#121	—	53	96	no	—
344	#121	—	53	96	no	—
345	#120	—	53	96	no	—
346	#120	—	53	96	no	—
347	C₄×Q₈	—	53	16	no	—
348	C₄×D₄	—	53	48	no	—
349	C₄×D₄	—	53	96	no	—
350	#117	—	53	24	no	—
351	C₂×C₄⋊C₄	—	53	6	no	—
352	C₂×C₄⋊C₄	—	53	4	no	—
353	C₂×C₂²⋊C₄	—	53	6	no	—
354	C₂×C₂²⋊C₄	—	53	12	no	—
355	C₂×C₂²⋊C₄	—	53	12	no	—
356	C₂×C₄²	—	53	3	no	—
357	C₂×C₄²	—	53	6	no	—
358	C₂×C₄²	—	53	24	no	—
359	C₂×C₄²	—	53	1	no	—
360	C₂⁴	—	2	24	no	—
361	C₂⁴	—	4	24	no	—
362	C₂⁴	—	53	6	no	—
363	C₂⁴	—	53	24	no	—
364	C₂⁴	—	106	24	no	—
365	C₂⁴	—	212	24	no	—
366	C₂⁴	C₂³×D₅₃	1	6	yes	—
367	C₂⁴	C₂³×D₅₃	1	3	yes	—
368	C₂⁴	C₂³×D₅₃	1	24	yes	—
369	#41	—	2	64	no	—
370	#41	—	2	96	no	—
371	#41	—	106	64	no	—
372	#41	—	106	96	no	—
373	#41	C₂×C₂²⋊D₅₃	1	64	yes	complemented
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	53	4	no	—
376	C₂×Q₈	—	106	4	no	—
377	C₂×Q₈	C₂×C₂²⋊D₅₃	1	8	yes	complemented
378	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
379	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
380	C₂×D₄	—	2	24	no	—
381	C₂×D₄	—	2	192	no	—
382	C₂×D₄	—	2	24	no	—
383	C₂×D₄	—	2	12	no	—
384	C₂×D₄	—	2	96	no	—
385	C₂×D₄	—	53	48	no	—
386	C₂×D₄	—	53	24	no	—
387	C₂×D₄	—	53	96	no	—
388	C₂×D₄	—	53	24	no	—
389	C₂×D₄	—	53	12	no	—
390	C₂×D₄	—	106	48	no	—
391	C₂×D₄	—	106	24	no	—
392	C₂×D₄	—	106	192	no	—
393	C₂×D₄	—	106	24	no	—
394	C₂×D₄	—	106	12	no	—
395	C₂×D₄	—	106	96	no	—
396	C₂×D₄	—	212	192	no	—
397	C₂×D₄	—	212	48	no	—
398	C₂×D₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
399	C₂×D₄	C₂³×D₅₃	1	24	yes	—
400	C₂×D₄	C₂³×D₅₃	1	24	yes	—
401	C₂×D₄	C₂³×D₅₃	1	12	yes	—
402	C₂×D₄	C₂³×D₅₃	1	12	yes	—
403	C₂²×C₄	—	2	24	no	—
404	C₂²×C₄	—	53	2	no	—
405	C₂²×C₄	—	53	24	no	—
406	C₂²×C₄	—	53	6	no	—
407	C₂²×C₄	—	53	2	no	—
408	C₂²×C₄	—	53	24	no	—
409	C₂²×C₄	—	106	24	no	—
410	C₂²×C₄	—	106	24	no	—
411	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
412	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
413	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
414	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
415	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
416	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
417	C₄⋊C₄	—	53	24	no	—
418	C₄⋊C₄	—	53	16	no	—
419	C₂²⋊C₄	—	53	24	no	—
420	C₂²⋊C₄	—	53	48	no	—
421	C₂²⋊C₄	—	53	48	no	—
422	C₄²	—	53	4	no	—
423	C₄²	—	53	12	no	—
424	C₄²	—	53	24	no	—
425	C₂³	—	2	12	no	—
426	C₂³	—	2	12	no	—
427	C₂³	—	2	24	no	—
428	C₂³	—	2	48	no	—
429	C₂³	—	4	48	no	—
430	C₂³	—	8	48	no	—
431	C₂³	—	53	12	no	—
432	C₂³	—	106	48	no	—
433	C₂³	—	106	12	no	—
434	C₂³	—	106	24	no	—
435	C₂³	—	424	48	no	—
436	C₂³	—	212	48	no	—
437	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
438	C₂³	C₂⁴×D₅₃	1	12	yes	—
439	C₂³	C₂⁴×D₅₃	1	6	yes	—
440	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	106	8	no	—
443	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	complemented
444	D₄	—	2	48	no	—
445	D₄	—	2	48	no	—
446	D₄	—	2	24	no	—
447	D₄	—	106	96	no	—
448	D₄	—	106	48	no	—
449	D₄	—	106	48	no	—
450	D₄	—	106	24	no	—
451	D₄	—	212	96	no	—
452	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	complemented
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	48	no	—
455	C₂×C₄	—	53	8	no	—
456	C₂×C₄	—	53	4	no	—
457	C₂×C₄	—	53	12	no	—
458	C₂×C₄	—	53	4	no	—
459	C₂×C₄	—	106	48	no	—
460	C₂×C₄	—	106	4	no	—
461	C₂×C₄	—	106	12	no	—
462	C₂×C₄	—	106	48	no	—
463	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
464	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
465	C₂×C₄	C₂²×C₂²⋊D₅₃	1	48	yes	complemented
466	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
467	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
468	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
469	C₂²	—	2	24	no	—
470	C₂²	—	2	12	no	—
471	C₂²	—	2	12	no	—
472	C₂²	—	2	24	no	—
473	C₂²	—	4	24	no	—
474	C₂²	—	8	48	no	—
475	C₂²	—	106	12	no	—
476	C₂²	—	106	24	no	—
477	C₂²	—	424	48	no	—
478	C₂²	—	212	24	no	—
479	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
480	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
481	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
482	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
483	C₄	—	2	4	no	—
484	C₄	—	53	8	no	—
485	C₄	—	106	4	no	—
486	C₄	—	106	12	no	—
487	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
488	C₄	C₂³×C₂²⋊D₅₃	1	4	yes	—
489	C₂	—	2	12	no	—
490	C₂	—	4	12	no	—
491	C₂	—	212	12	no	—
492	C₂	#20288489	1	4	yes	direct product factor
493	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
494	C₂	C₂×H(5,𝔽₂)×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×#3498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. #20288489	C₂	direct (N × H)
3. C₂×#16811292	C₂	semidirect
4. C₂×D₄×C₂²⋊D₅₃	C₂	semidirect
5. C₂×#16811190	C₂	semidirect
6. C₂×#16811187	C₂	semidirect
7. C₂×#16811169	C₂	semidirect
8. C₂²×#13133213	C₂	semidirect
9. C₂×#41×Dic₅₃	C₂	semidirect
10. C₂²×H(5,𝔽₂)×C₅₃	C₂	semidirect
12. C₂×#13133263	C₂²	semidirect
13. C₂²×#11719949	C₂²	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂³×D₂₁₂	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. #16811292	C₂²	semidirect
19. D₄×C₂²⋊D₅₃	C₂²	semidirect
20. #16811190	C₂²	semidirect
21. #16811187	C₂²	semidirect
22. #16811169	C₂²	semidirect
23. C₂×#13133215	C₂²	semidirect
24. C₂×#13133213	C₂²	semidirect
25. C₂×#13133213	C₂²	semidirect
26. C₂×#13133213	C₂²	semidirect
27. C₂×#13133213	C₂²	semidirect
28. C₂×#13133206	C₂²	semidirect
29. C₂×#13133204	C₂²	semidirect
30. C₂×#13133191	C₂²	semidirect
31. C₂×#13133189	C₂²	semidirect
32. C₂×#13133188	C₂²	semidirect
33. C₂×#13133186	C₂²	semidirect
34. C₂×#13133185	C₂²	semidirect
35. #41×Dic₅₃	C₂²	semidirect
36. C₂×Q₈×Dic₅₃	C₂²	semidirect
37. C₂×D₄×Dic₅₃	C₂²	semidirect
38. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
39. C₂×#13133176	C₂²	semidirect
40. C₂²×C₄×Dic₅₃	C₂²	semidirect
42. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
44. C₂²×#41×C₅₃	C₂²	semidirect
45. C₂²×#41×C₅₃	C₂²	semidirect
46. C₂³×D₄×C₅₃	C₂²	semidirect
47. C₂³×D₄×C₅₃	C₂²	semidirect
51. C₂×#11719949	C₂³	semidirect
53. C₂×#11719945	C₂³	semidirect
58. C₂²×D₂₁₂	C₂³	semidirect
63. C₂²×C₂²⋊D₅₃	C₂³	semidirect
65. C₂×D₄×D₅₃	C₂³	semidirect
68. #13133215	C₂³	semidirect
69. #13133213	C₂³	semidirect
70. #13133213	C₂³	semidirect
71. #13133213	C₂³	semidirect
72. #13133206	C₂³	semidirect
73. #13133204	C₂³	semidirect
74. #13133191	C₂³	semidirect
75. #13133189	C₂³	semidirect
76. #13133188	C₂³	semidirect
77. #13133186	C₂³	semidirect
78. #13133185	C₂³	semidirect
79. Q₈×Dic₅₃	C₂³	semidirect
80. D₄×Dic₅₃	C₂³	semidirect
81. C₄×C₂²⋊D₅₃	C₂³	semidirect
82. #13133176	C₂³	semidirect
91. C₂×C₄×Dic₅₃	C₂³	semidirect
96. C₂×#41×C₅₃	D₄	semidirect
98. C₂×#41×C₅₃	C₂³	semidirect
99. C₂×#41×C₅₃	C₂³	semidirect
108. C₂²×D₄×C₅₃	C₂³	semidirect
109. C₂²×D₄×C₅₃	C₂³	semidirect
123. #41×C₅₃	C₂×D₄	semidirect
125. C₂×Q₈×C₅₃	C₂×D₄	semidirect
133. C₂×D₄×C₅₃	C₂×D₄	semidirect
139. C₂²×C₄×C₅₃	C₂×D₄	semidirect
194. Q₈×C₅₃	C₂²×D₄	semidirect
198. D₄×C₅₃	C₂²×D₄	semidirect
203. C₂×C₄×C₅₃	C₂²×D₄	semidirect
213. C₂×Dic₅₃	H(5,𝔽₂)	semidirect
241. Dic₅₃	C₂×H(5,𝔽₂)	semidirect
243. C₂²×H(5,𝔽₂)	D₅₃	semidirect
254. C₂×C₅₃	#3498	semidirect
261. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
265. C₂²×#41	C₂×D₅₃	semidirect
266. C₂²×#41	C₂×D₅₃	semidirect
270. C₂³×D₄	C₂×D₅₃	semidirect
271. C₂³×D₄	C₂×D₅₃	semidirect
295. C₅₃	C₂×#3498	semidirect
306. C₂×#41	C₂²⋊D₅₃	semidirect
308. C₂×#41	C₂²×D₅₃	semidirect
309. C₂×#41	C₂²×D₅₃	semidirect
330. C₂²×D₄	C₂²×D₅₃	semidirect
331. C₂²×D₄	C₂²×D₅₃	semidirect
373. #41	C₂×C₂²⋊D₅₃	semidirect
377. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
398. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
411. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
443. Q₈	C₂²×C₂²⋊D₅₃	semidirect
452. D₄	C₂²×C₂²⋊D₅₃	semidirect
465. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
492. C₂	#20288489	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
#20288489	2
C₂×#16811292	2
C₂×D₄×C₂²⋊D₅₃	2
C₂×#16811190	2
C₂×#16811187	2
C₂×#16811169	2
C₂²×#13133213	2
C₂×#41×Dic₅₃	2
C₂²×H(5,𝔽₂)×C₅₃	2
C₂×#3498	53

Subgroups — C2×#20288489 #28292412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288489 #28292412