Subgroups — C₂²×#16811136 #28290751

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719949	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811136	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133202	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133193	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133192	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13133191	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133179	C₂	1	2	yes	maximal, complemented
10	C₂²×#13133177	C₂	1	2	yes	maximal, complemented
11	C₂²×#13133175	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#11719949	—	2	16	no	—
15	C₂²×#11719949	C₂²	1	24	yes	complemented
16	C₂³×D₂₁₂	C₂²	1	2	yes	—
17	C₂³×D₂₁₂	C₂²	1	2	yes	—
18	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
19	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
20	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₄×D₅₃	—	2	16	no	—
22	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
25	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₄×D₅₃	C₂²	1	24	yes	complemented
27	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
28	#16811136	C₂²	1	256	yes	direct product factor
29	C₂×#13133202	C₂²	1	48	yes	complemented
30	C₂×#13133193	C₂²	1	24	yes	complemented
31	C₂×#13133192	C₂²	1	24	yes	complemented
32	C₂×#13133191	C₂²	1	24	yes	complemented
33	C₂×#13133187	C₂²	1	48	yes	complemented
34	C₂×#13133179	C₂²	1	48	yes	complemented
35	C₂×#13133177	C₂²	1	48	yes	complemented
36	C₂×#13133175	C₂²	1	24	yes	complemented
37	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂²×#11719919	C₂²	1	2	yes	complemented
39	C₂²×#11719921	C₂²	1	2	yes	—
40	C₂²×#11719921	C₂²	1	2	yes	—
41	C₂²×#11719921	C₂²	1	2	yes	complemented
42	C₂²×#11719921	C₂²	1	2	yes	complemented
43	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
44	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
47	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
50	C₂⁴×D₅₃	—	2	8	no	—
51	C₂⁴×D₅₃	—	2	8	no	—
52	C₂⁴×D₅₃	—	2	8	no	—
53	C₂⁴×D₅₃	C₂³	1	2	yes	—
54	C₂⁴×D₅₃	C₂³	1	12	yes	—
55	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
56	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
57	C₂×#11719949	—	2	192	no	—
58	C₂×#11719949	C₂³	1	64	yes	complemented
59	C₂²×D₂₁₂	—	2	16	no	—
60	C₂²×D₂₁₂	—	2	16	no	—
61	C₂²×D₂₁₂	—	2	8	no	—
62	C₂²×D₂₁₂	C₂³	1	24	yes	—
63	C₂²×D₂₁₂	C₂³	1	24	yes	—
64	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
65	C₂²×C₂²⋊D₅₃	—	2	16	no	—
66	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
67	C₂³×Dic₅₃	—	2	8	no	—
68	C₂³×Dic₅₃	C₂³	1	2	yes	—
69	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂³×Dic₅₃	C₂³	1	12	yes	—
71	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×D₅₃	—	2	8	no	—
73	C₂²×C₄×D₅₃	—	2	8	no	—
74	C₂²×C₄×D₅₃	—	2	8	no	—
75	C₂²×C₄×D₅₃	—	2	16	no	—
76	C₂²×C₄×D₅₃	—	2	192	no	—
77	C₂²×C₄×D₅₃	D₄	1	16	yes	complemented
78	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
82	C₂²×C₄×D₅₃	C₂³	1	64	yes	complemented
83	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
84	#13133202	C₂³	1	128	yes	complemented
85	#13133193	C₂³	1	64	yes	complemented
86	#13133192	C₂³	1	64	yes	complemented
87	#13133191	C₂³	1	64	yes	complemented
88	#13133187	C₂³	1	128	yes	complemented
89	#13133179	C₂³	1	128	yes	complemented
90	#13133177	C₂³	1	128	yes	complemented
91	#13133175	C₂³	1	64	yes	complemented
92	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
93	C₂×#11719919	C₂³	1	24	yes	complemented
94	C₂×#11719921	C₂³	1	24	yes	—
95	C₂×#11719921	C₂³	1	24	yes	—
96	C₂×#11719921	C₂³	1	24	yes	complemented
97	C₂×#11719921	C₂³	1	24	yes	complemented
98	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
99	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×Q₈×C₅₃	—	2	8	no	—
101	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
102	C₂³×C₄×C₅₃	—	2	8	no	—
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
105	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
107	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	2	4	no	—
114	C₂⁴×C₅₃	#41	1	8	yes	complemented
115	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂³×D₅₃	—	2	128	no	—
118	C₂³×D₅₃	—	2	4	no	—
119	C₂³×D₅₃	—	2	4	no	—
120	C₂³×D₅₃	—	2	4	no	—
121	C₂³×D₅₃	—	2	4	no	—
122	C₂³×D₅₃	—	2	48	no	—
123	C₂³×D₅₃	—	2	48	no	—
124	C₂³×D₅₃	—	2	48	no	—
125	C₂³×D₅₃	—	2	8	no	—
126	C₂³×D₅₃	—	2	8	no	—
127	C₂³×D₅₃	—	2	8	no	—
128	C₂³×D₅₃	C₂×D₄	1	8	yes	—
129	C₂³×D₅₃	#41	1	8	yes	complemented
130	C₂³×D₅₃	C₂⁴	1	12	yes	—
131	C₂³×D₅₃	C₂⁴	1	16	yes	—
132	C₂³×D₅₃	C₂⁴	1	6	yes	—
133	C₂³×D₅₃	C₂⁴	1	6	yes	—
134	C₂×Q₈×C₅₃	—	2	48	no	—
135	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	2	8	no	—
138	C₂²×C₄×C₅₃	—	2	8	no	—
139	C₂²×C₄×C₅₃	—	2	48	no	—
140	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
146	C₂×C₂²⋊D₅₃	—	2	96	no	—
147	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
148	C₂²×Dic₅₃	—	2	4	no	—
149	C₂²×Dic₅₃	—	2	8	no	—
150	C₂²×Dic₅₃	—	2	48	no	—
151	C₂²×Dic₅₃	C₂×D₄	1	8	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
153	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
154	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
155	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
156	#11719949	—	2	256	no	—
157	C₂×D₂₁₂	—	2	96	no	—
158	C₂×D₂₁₂	—	2	96	no	—
159	C₂×D₂₁₂	—	2	48	no	—
160	C₂×D₂₁₂	C₂⁴	1	32	yes	—
161	C₂×D₂₁₂	C₂⁴	1	32	yes	—
162	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
163	C₂×C₄×D₅₃	—	2	48	no	—
164	C₂×C₄×D₅₃	—	2	48	no	—
165	C₂×C₄×D₅₃	—	2	48	no	—
166	C₂×C₄×D₅₃	—	2	96	no	—
167	C₂×C₄×D₅₃	—	2	256	no	—
168	C₂×C₄×D₅₃	C₂×D₄	1	96	yes	complemented
169	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
170	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
171	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
172	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
173	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
174	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
175	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
176	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
177	#11719921	C₂⁴	1	32	yes	—
178	#11719921	C₂⁴	1	32	yes	—
179	#11719921	C₂⁴	1	32	yes	complemented
180	#11719921	C₂⁴	1	32	yes	complemented
181	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
182	#11719919	C₂⁴	1	32	yes	complemented
183	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
184	C₂³×C₅₃	—	2	32	no	—
185	C₂³×C₅₃	—	2	12	no	—
186	C₂³×C₅₃	—	2	12	no	—
187	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
188	C₂³×C₅₃	C₂×#41	1	4	yes	—
189	C₂³×C₅₃	C₂×#41	1	4	yes	—
190	C₂³×C₅₃	C₂×#41	1	24	yes	complemented
191	C₂³×C₅₃	C₂⁵	1	4	yes	—
192	C₂³×C₅₃	C₂⁵	1	3	yes	—
193	C₂²×D₅₃	—	2	384	no	—
194	C₂²×D₅₃	—	2	32	no	—
195	C₂²×D₅₃	—	2	32	no	—
196	C₂²×D₅₃	—	2	12	no	—
197	C₂²×D₅₃	—	2	12	no	—
198	C₂²×D₅₃	—	2	12	no	—
199	C₂²×D₅₃	—	2	12	no	—
200	C₂²×D₅₃	—	2	32	no	—
201	C₂²×D₅₃	—	2	32	no	—
202	C₂²×D₅₃	—	2	32	no	—
203	C₂²×D₅₃	—	2	24	no	—
204	C₂²×D₅₃	—	2	24	no	—
205	C₂²×D₅₃	—	2	24	no	—
206	C₂²×D₅₃	—	4	32	no	—
207	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
208	C₂²×D₅₃	C₂×#41	1	24	yes	complemented
209	C₂²×D₅₃	C₂⁵	1	8	yes	—
210	C₂²×D₅₃	C₂⁵	1	4	yes	—
211	C₂²×D₅₃	C₂⁵	1	4	yes	—
212	Q₈×C₅₃	—	2	32	no	—
213	C₂×C₄×C₅₃	—	2	12	no	—
214	C₂×C₄×C₅₃	—	2	24	no	—
215	C₂×C₄×C₅₃	—	2	24	no	—
216	C₂×C₄×C₅₃	—	2	32	no	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
218	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
219	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
220	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
221	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
222	C₂²⋊D₅₃	—	2	64	no	—
223	C₂×Dic₅₃	—	2	12	no	—
224	C₂×Dic₅₃	—	2	24	no	—
225	C₂×Dic₅₃	—	2	32	no	—
226	C₂×Dic₅₃	C₂²×D₄	1	24	yes	—
227	C₂×Dic₅₃	C₂⁵	1	4	yes	—
228	C₂×Dic₅₃	C₂⁵	1	8	yes	—
229	C₂×Dic₅₃	C₂⁵	1	4	yes	—
230	D₂₁₂	—	2	64	no	—
231	D₂₁₂	—	2	64	no	—
232	D₂₁₂	—	2	32	no	—
233	C₄×D₅₃	—	2	32	no	—
234	C₄×D₅₃	—	2	32	no	—
235	C₄×D₅₃	—	2	32	no	—
236	C₄×D₅₃	—	2	64	no	—
237	C₄×D₅₃	C₂²×D₄	1	64	yes	complemented
238	C₂²×#525	—	53	1	no	maximal, Sylow
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	48	no	—
242	C₂²×C₅₃	#525	1	16	yes	complemented
243	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
244	C₂²×C₅₃	C₂²×#41	1	6	yes	—
245	C₂²×C₅₃	C₂²×#41	1	6	yes	—
246	C₂²×C₅₃	C₂²×#41	1	8	yes	complemented
247	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
248	C₂×D₅₃	—	2	128	no	—
249	C₂×D₅₃	—	2	4	no	—
250	C₂×D₅₃	—	2	4	no	—
251	C₂×D₅₃	—	2	4	no	—
252	C₂×D₅₃	—	2	4	no	—
253	C₂×D₅₃	—	2	48	no	—
254	C₂×D₅₃	—	2	48	no	—
255	C₂×D₅₃	—	2	8	no	—
256	C₂×D₅₃	—	2	8	no	—
257	C₂×D₅₃	—	2	8	no	—
258	C₂×D₅₃	—	4	48	no	—
259	C₂×D₅₃	C₂³×D₄	1	8	yes	—
260	C₂×D₅₃	C₂²×#41	1	8	yes	complemented
261	C₄×C₅₃	—	2	4	no	—
262	C₄×C₅₃	—	2	8	no	—
263	C₄×C₅₃	—	2	8	no	—
264	C₄×C₅₃	C₂³×D₄	1	8	yes	—
265	Dic₅₃	—	2	4	no	—
266	Dic₅₃	—	2	8	no	—
267	Dic₅₃	C₂³×D₄	1	8	yes	—
268	C₂³×#41	—	53	1	no	—
269	C₂⁵×C₄	—	53	1	no	—
270	C₂×#525	—	53	48	no	—
271	C₂²×#123	—	53	2	no	—
272	C₂²×C₂²⋊Q₈	—	53	1	no	—
273	C₂²×C₂²⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
274	C₂²×#121	—	53	1	no	—
275	C₂²×#121	—	53	1	no	—
276	C₂²×#120	—	53	2	no	—
277	C₂²×C₄×D₄	—	53	2	no	—
278	C₂²×C₄×D₄	—	53	2	no	—
279	C₂²×#117	—	53	1	no	—
280	C₂×C₅₃	—	2	8	no	—
281	C₂×C₅₃	C₂×#525	1	12	yes	complemented
282	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
283	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
284	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
285	D₅₃	—	2	8	no	—
286	D₅₃	—	2	8	no	—
287	D₅₃	—	4	8	no	—
288	C₂⁶	—	53	1	no	—
289	C₂²×#41	—	53	24	no	—
290	C₂²×#41	—	106	16	no	—
291	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
292	C₂³×D₄	—	53	2	no	—
293	C₂³×D₄	—	53	2	no	—
294	C₂³×D₄	—	53	1	no	—
295	C₂³×D₄	—	53	2	no	—
296	C₂⁴×C₄	—	53	1	no	—
297	C₂⁴×C₄	—	53	1	no	—
298	C₂⁴×C₄	—	53	1	no	—
299	C₂⁴×C₄	—	53	1	no	—
300	C₂⁴×C₄	—	53	1	no	—
301	C₂⁴×C₄	—	53	24	no	—
302	C₂⁴×C₄	—	53	2	no	—
303	C₂⁴×C₄	—	106	16	no	—
304	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
305	#525	—	53	256	no	—
306	C₂×#123	—	53	48	no	—
307	C₂×C₂²⋊Q₈	—	53	24	no	—
308	C₂×C₂²⋊Q₈	C₂×D₅₃	1	24	yes	complemented
309	C₂×#121	—	53	24	no	—
310	C₂×#121	—	53	24	no	—
311	C₂×#120	—	53	48	no	—
312	C₂×C₄×D₄	—	53	48	no	—
313	C₂×C₄×D₄	—	53	48	no	—
314	C₂×#117	—	53	24	no	—
315	C₂²×C₄⋊C₄	—	53	2	no	—
316	C₂²×C₄⋊C₄	—	53	1	no	—
317	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
318	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
319	C₂²×C₂²⋊C₄	—	53	2	no	—
320	C₂²×C₂²⋊C₄	—	53	2	no	—
321	C₂²×C₂²⋊C₄	—	53	2	no	—
322	C₂²×C₂²⋊C₄	—	53	2	no	—
323	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
324	C₂²×C₄²	—	53	2	no	—
325	C₅₃	C₂²×#525	1	1	yes	characteristic, complemented, Sylow
326	C₂⁵	—	53	1	no	—
327	C₂⁵	—	53	1	no	—
328	C₂⁵	—	53	12	no	—
329	C₂⁵	—	53	2	no	—
330	C₂⁵	—	106	8	no	—
331	C₂⁵	—	106	8	no	—
332	C₂⁵	—	106	8	no	—
333	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
334	C₂×#41	—	53	64	no	—
335	C₂×#41	—	106	192	no	—
336	C₂²×Q₈	—	2	8	no	—
337	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
338	C₂²×D₄	—	53	24	no	—
339	C₂²×D₄	—	53	24	no	—
340	C₂²×D₄	—	53	24	no	—
341	C₂²×D₄	—	53	12	no	—
342	C₂²×D₄	—	106	16	no	—
343	C₂²×D₄	—	106	16	no	—
344	C₂²×D₄	—	106	16	no	—
345	C₂²×D₄	—	106	8	no	—
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	—	53	1	no	—
348	C₂³×C₄	—	53	16	no	—
349	C₂³×C₄	—	53	12	no	—
350	C₂³×C₄	—	53	12	no	—
351	C₂³×C₄	—	53	2	no	—
352	C₂³×C₄	—	53	1	no	—
353	C₂³×C₄	—	53	12	no	—
354	C₂³×C₄	—	53	24	no	—
355	C₂³×C₄	—	53	12	no	—
356	C₂³×C₄	—	53	12	no	—
357	C₂³×C₄	—	53	64	no	—
358	C₂³×C₄	—	106	8	no	—
359	C₂³×C₄	—	106	192	no	—
360	C₂³×C₄	—	106	16	no	—
361	C₂³×C₄	—	106	8	no	—
362	C₂³×C₄	—	106	8	no	—
363	C₂³×C₄	—	106	8	no	—
364	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
365	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
366	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
367	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
368	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
369	#123	—	53	128	no	—
370	C₂²⋊Q₈	—	53	64	no	—
371	C₂²⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
372	#121	—	53	64	no	—
373	#121	—	53	64	no	—
374	#120	—	53	128	no	—
375	C₄×D₄	—	53	128	no	—
376	C₄×D₄	—	53	128	no	—
377	#117	—	53	64	no	—
378	C₂×C₄⋊C₄	—	53	12	no	—
379	C₂×C₄⋊C₄	—	53	24	no	—
380	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
381	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
382	C₂×C₂²⋊C₄	—	53	24	no	—
383	C₂×C₂²⋊C₄	—	53	24	no	—
384	C₂×C₂²⋊C₄	—	53	24	no	—
385	C₂×C₂²⋊C₄	—	53	24	no	—
386	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
387	C₂×C₄²	—	53	24	no	—
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	2	4	no	—
390	C₂⁴	—	53	8	no	—
391	C₂⁴	—	53	8	no	—
392	C₂⁴	—	53	6	no	—
393	C₂⁴	—	53	6	no	—
394	C₂⁴	—	53	12	no	—
395	C₂⁴	—	53	16	no	—
396	C₂⁴	—	106	48	no	—
397	C₂⁴	—	106	48	no	—
398	C₂⁴	—	106	128	no	—
399	C₂⁴	—	106	4	no	—
400	C₂⁴	—	106	4	no	—
401	C₂⁴	—	106	4	no	—
402	C₂⁴	—	106	4	no	—
403	C₂⁴	—	106	48	no	—
404	C₂⁴	—	106	8	no	—
405	C₂⁴	—	106	8	no	—
406	C₂⁴	—	106	8	no	—
407	C₂⁴	#11719949	1	8	yes	complemented
408	C₂⁴	C₂³×D₅₃	1	6	yes	—
409	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
410	#41	—	106	256	no	—
411	C₂×Q₈	—	2	48	no	—
412	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
413	C₂×D₄	—	53	32	no	—
414	C₂×D₄	—	53	32	no	—
415	C₂×D₄	—	53	32	no	—
416	C₂×D₄	—	53	16	no	—
417	C₂×D₄	—	106	96	no	—
418	C₂×D₄	—	106	96	no	—
419	C₂×D₄	—	106	96	no	—
420	C₂×D₄	—	106	48	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	2	48	no	—
425	C₂²×C₄	—	53	8	no	—
426	C₂²×C₄	—	53	6	no	—
427	C₂²×C₄	—	53	96	no	—
428	C₂²×C₄	—	53	16	no	—
429	C₂²×C₄	—	53	12	no	—
430	C₂²×C₄	—	53	6	no	—
431	C₂²×C₄	—	53	16	no	—
432	C₂²×C₄	—	53	16	no	—
433	C₂²×C₄	—	53	16	no	—
434	C₂²×C₄	—	53	16	no	—
435	C₂²×C₄	—	53	32	no	—
436	C₂²×C₄	—	106	48	no	—
437	C₂²×C₄	—	106	256	no	—
438	C₂²×C₄	—	106	96	no	—
439	C₂²×C₄	—	106	48	no	—
440	C₂²×C₄	—	106	48	no	—
441	C₂²×C₄	—	106	4	no	—
442	C₂²×C₄	—	106	8	no	—
443	C₂²×C₄	—	106	48	no	—
444	C₂²×C₄	D₄×D₅₃	1	8	yes	—
445	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
446	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
447	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
448	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
449	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
450	C₄⋊C₄	—	53	16	no	—
451	C₄⋊C₄	—	53	32	no	—
452	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
453	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
454	C₂²⋊C₄	—	53	32	no	—
455	C₂²⋊C₄	—	53	32	no	—
456	C₂²⋊C₄	—	53	32	no	—
457	C₂²⋊C₄	—	53	32	no	—
458	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
459	C₄²	—	53	32	no	—
460	C₂³	—	2	32	no	—
461	C₂³	—	2	12	no	—
462	C₂³	—	2	12	no	—
463	C₂³	—	53	24	no	—
464	C₂³	—	53	24	no	—
465	C₂³	—	53	4	no	—
466	C₂³	—	53	4	no	—
467	C₂³	—	53	8	no	—
468	C₂³	—	106	32	no	—
469	C₂³	—	106	32	no	—
470	C₂³	—	106	384	no	—
471	C₂³	—	106	32	no	—
472	C₂³	—	106	32	no	—
473	C₂³	—	106	12	no	—
474	C₂³	—	106	12	no	—
475	C₂³	—	106	12	no	—
476	C₂³	—	106	12	no	—
477	C₂³	—	106	32	no	—
478	C₂³	—	106	24	no	—
479	C₂³	—	106	24	no	—
480	C₂³	—	106	24	no	—
481	C₂³	—	212	32	no	—
482	C₂³	C₂×D₄×D₅₃	1	4	yes	—
483	C₂³	#41×D₅₃	1	4	yes	—
484	C₂³	C₂×#11719949	1	4	yes	—
485	C₂³	C₂×#11719949	1	24	yes	complemented
486	C₂³	C₂⁴×D₅₃	1	4	yes	—
487	C₂³	C₂⁴×D₅₃	1	3	yes	—
488	Q₈	—	2	32	no	—
489	D₄	—	106	64	no	—
490	D₄	—	106	64	no	—
491	D₄	—	106	64	no	—
492	D₄	—	106	32	no	—
493	C₂×C₄	—	2	12	no	—
494	C₂×C₄	—	2	24	no	—
495	C₂×C₄	—	2	24	no	—
496	C₂×C₄	—	2	32	no	—
497	C₂×C₄	—	53	24	no	—
498	C₂×C₄	—	53	4	no	—
499	C₂×C₄	—	53	64	no	—
500	C₂×C₄	—	53	8	no	—
501	C₂×C₄	—	53	4	no	—
502	C₂×C₄	—	106	32	no	—
503	C₂×C₄	—	106	64	no	—
504	C₂×C₄	—	106	32	no	—
505	C₂×C₄	—	106	32	no	—
506	C₂×C₄	—	106	12	no	—
507	C₂×C₄	—	106	24	no	—
508	C₂×C₄	—	106	32	no	—
509	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
510	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
511	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
512	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
513	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
514	C₂²	—	2	48	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	53	8	no	—
518	C₂²	—	53	8	no	—
519	C₂²	—	106	128	no	—
520	C₂²	—	106	48	no	—
521	C₂²	—	106	48	no	—
522	C₂²	—	106	4	no	—
523	C₂²	—	106	4	no	—
524	C₂²	—	106	4	no	—
525	C₂²	—	106	4	no	—
526	C₂²	—	106	8	no	—
527	C₂²	—	106	8	no	—
528	C₂²	—	106	8	no	—
529	C₂²	—	212	48	no	—
530	C₂²	#16811136	1	16	yes	direct product factor
531	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
532	C₂²	C₂×#41×D₅₃	1	6	yes	—
533	C₂²	C₂²×#11719949	1	6	yes	—
534	C₂²	C₂²×#11719949	1	8	yes	complemented
535	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
536	C₄	—	2	4	no	—
537	C₄	—	2	8	no	—
538	C₄	—	2	8	no	—
539	C₄	—	53	8	no	—
540	C₄	—	106	4	no	—
541	C₄	—	106	8	no	—
542	C₄	C₂²×D₄×D₅₃	1	8	yes	—
543	C₂	—	2	8	no	—
544	C₂	—	106	8	no	—
545	C₂	—	106	8	no	—
546	C₂	—	212	8	no	—
547	C₂	C₂×#16811136	1	12	yes	direct product factor
548	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
549	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
550	C₂	C₂³×#11719949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₂²⋊Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	C₂²×#525
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719949	C₂	semidirect
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂×#16811136	C₂	direct (N × H)
4. C₂²×#13133202	C₂	semidirect
5. C₂²×#13133193	C₂	semidirect
6. C₂²×#13133192	C₂	semidirect
7. C₂²×#13133191	C₂	semidirect
8. C₂²×#13133187	C₂	semidirect
9. C₂²×#13133179	C₂	semidirect
10. C₂²×#13133177	C₂	semidirect
11. C₂²×#13133175	C₂	semidirect
12. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
15. C₂²×#11719949	C₂²	semidirect
18. C₂³×D₂₁₂	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. C₂³×C₄×D₅₃	C₂²	semidirect
28. #16811136	C₂²	direct (N × H)
29. C₂×#13133202	C₂²	semidirect
30. C₂×#13133193	C₂²	semidirect
31. C₂×#13133192	C₂²	semidirect
32. C₂×#13133191	C₂²	semidirect
33. C₂×#13133187	C₂²	semidirect
34. C₂×#13133179	C₂²	semidirect
35. C₂×#13133177	C₂²	semidirect
36. C₂×#13133175	C₂²	semidirect
37. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
38. C₂²×#11719919	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
43. C₂²×C₄×Dic₅₃	C₂²	semidirect
44. C₂³×Q₈×C₅₃	C₂²	semidirect
46. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
58. C₂×#11719949	C₂³	semidirect
64. C₂²×D₂₁₂	C₂³	semidirect
77. C₂²×C₄×D₅₃	D₄	semidirect
82. C₂²×C₄×D₅₃	C₂³	semidirect
83. C₂²×C₄×D₅₃	C₂³	semidirect
84. #13133202	C₂³	semidirect
85. #13133193	C₂³	semidirect
86. #13133192	C₂³	semidirect
87. #13133191	C₂³	semidirect
88. #13133187	C₂³	semidirect
89. #13133179	C₂³	semidirect
90. #13133177	C₂³	semidirect
91. #13133175	C₂³	semidirect
92. C₂×C₄⋊Dic₅₃	C₂³	semidirect
93. C₂×#11719919	C₂³	semidirect
96. C₂×#11719921	C₂³	semidirect
97. C₂×#11719921	C₂³	semidirect
98. C₂×C₄×Dic₅₃	C₂³	semidirect
101. C₂²×Q₈×C₅₃	C₂³	semidirect
108. C₂²⋊Q₈×C₅₃	C₂³	semidirect
109. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
110. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
114. C₂⁴×C₅₃	#41	semidirect
129. C₂³×D₅₃	#41	semidirect
135. C₂×Q₈×C₅₃	C₂⁴	semidirect
162. C₂×D₂₁₂	C₂⁴	semidirect
168. C₂×C₄×D₅₃	C₂×D₄	semidirect
173. C₂×C₄×D₅₃	C₂⁴	semidirect
174. C₄⋊C₄×C₅₃	C₂⁴	semidirect
175. C₄⋊C₄×C₅₃	C₂⁴	semidirect
179. #11719921	C₂⁴	semidirect
180. #11719921	C₂⁴	semidirect
181. C₄⋊Dic₅₃	C₂⁴	semidirect
182. #11719919	C₂⁴	semidirect
183. C₄×Dic₅₃	C₂⁴	semidirect
190. C₂³×C₅₃	C₂×#41	semidirect
208. C₂²×D₅₃	C₂×#41	semidirect
237. C₄×D₅₃	C₂²×D₄	semidirect
242. C₂²×C₅₃	#525	semidirect
246. C₂²×C₅₃	C₂²×#41	semidirect
260. C₂×D₅₃	C₂²×#41	semidirect
273. C₂²×C₂²⋊Q₈	D₅₃	semidirect
281. C₂×C₅₃	C₂×#525	semidirect
291. C₂³×Q₈	C₂×D₅₃	semidirect
308. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
317. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
318. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
325. C₅₃	C₂²×#525	semidirect
337. C₂²×Q₈	C₂²×D₅₃	semidirect
371. C₂²⋊Q₈	C₂²×D₅₃	semidirect
380. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
381. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
407. C₂⁴	#11719949	semidirect
412. C₂×Q₈	C₂³×D₅₃	semidirect
452. C₄⋊C₄	C₂³×D₅₃	semidirect
453. C₄⋊C₄	C₂³×D₅₃	semidirect
485. C₂³	C₂×#11719949	semidirect
530. C₂²	#16811136	direct (N × H)
534. C₂²	C₂²×#11719949	semidirect
547. C₂	C₂×#16811136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719949	2
C₂⁴×C₄×D₅₃	2
C₂×#16811136	2
C₂²×#13133202	2
C₂²×#13133193	2
C₂²×#13133192	2
C₂²×#13133191	2
C₂²×#13133187	2
C₂²×#13133179	2
C₂²×#13133177	2
C₂²×#13133175	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×#525	53

Subgroups — C22×#16811136 #28290751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811136 #28290751