Subgroups — #28319494

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291162	C₂	1	2	yes	maximal
2	#20290652	C₂	1	2	yes	maximal, complemented
3	#20290637	C₂	1	2	yes	maximal, complemented
4	#20290592	C₂	1	2	yes	maximal
5	#20290591	C₂	1	2	yes	maximal
6	C₄×#13133286	C₂	1	2	yes	maximal, complemented
7	C₄×#13133285	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133286	C₂²	1	4	yes	—
10	C₂×#13133285	C₂²	1	4	yes	—
11	#16811568	—	2	2	no	—
12	#16811567	C₂²	1	4	yes	—
13	#16811566	—	2	2	no	—
14	#16811565	C₂²	1	4	yes	—
15	C₄×C₄⋊#6157	C₂²	1	4	yes	—
16	C₄×C₂²⋊#6157	—	2	2	no	—
17	#16811559	—	2	2	no	—
18	#16811497	C₂²	1	4	yes	—
19	#16811497	C₂²	1	4	yes	—
20	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
24	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
25	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
26	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
27	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
28	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
29	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
30	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
35	C₂×C₄⋊#6157	C₂³	1	8	yes	—
36	C₂×C₂²⋊#6157	—	2	4	no	—
37	C₂×C₄×#6157	—	2	4	no	—
38	#13133286	—	2	8	no	—
39	#13133285	—	2	8	no	—
40	(C₂×#6157)⋊C₄	—	2	4	no	—
41	(C₂×#6157)⋊C₄	—	2	4	no	—
42	(C₂×#6157)⋊C₄	—	2	4	no	—
43	(C₂×#6157)⋊C₄	—	2	4	no	—
44	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
48	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	2	4	no	—
50	C₂×C₈×D₅₃	—	2	4	no	—
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
62	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	D₄	1	1	yes	characteristic
65	C₄²×D₅₃	D₄	1	1	yes	characteristic
66	C₄²×D₅₃	Q₈	1	1	yes	characteristic
67	C₄²×D₅₃	Q₈	1	1	yes	characteristic
68	C₄²×D₅₃	C₂³	1	1	yes	characteristic
69	C₄²×D₅₃	C₂³	1	1	yes	characteristic
70	C₄×#88322	C₂×C₄	1	1	yes	characteristic
71	C₄×#88322	C₂×C₄	1	1	yes	characteristic
72	C₄×#88322	C₂×C₄	1	1	yes	characteristic
73	C₄×#88322	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×#6157	—	2	8	no	—
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	4	no	—
98	C₂×C₄×D₅₃	—	2	4	no	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×C₄×D₅₃	—	2	2	no	—
101	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊#6157	—	4	8	no	—
114	C₄⋊#6157	—	2	8	no	—
115	C₄⋊#6157	—	2	8	no	—
116	C₄×#6157	—	4	8	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
136	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
142	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
143	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
144	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
145	C₈×D₅₃	—	2	2	no	—
146	C₈×D₅₃	—	2	2	no	—
147	C₈×D₅₃	—	2	2	no	—
148	C₈×D₅₃	—	2	2	no	—
149	C₈×D₅₃	—	2	2	no	—
150	C₈×D₅₃	—	2	2	no	—
151	C₈×D₅₃	#41	1	4	yes	—
152	C₈×D₅₃	#41	1	4	yes	—
153	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×D₅₃	—	2	2	no	—
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×D₅₃	#135	1	4	yes	—
158	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×#6157	—	4	8	no	—
160	C₂×#6157	—	4	8	no	—
161	C₂×#6157	—	4	8	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	2	2	no	—
174	C₂×Dic₅₃	—	2	2	no	—
175	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	2	no	—
183	C₄×D₅₃	—	2	2	no	—
184	C₄×D₅₃	—	2	2	no	—
185	C₄×D₅₃	—	2	2	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	2	1	no	—
188	C₄×D₅₃	—	2	1	no	—
189	C₄×D₅₃	—	2	2	no	—
190	C₄×D₅₃	—	2	2	no	—
191	C₄×D₅₃	—	2	2	no	—
192	C₄×D₅₃	#135	1	4	yes	—
193	C₄×D₅₃	C₂×#41	1	2	yes	—
194	C₄×D₅₃	C₂×#41	1	2	yes	—
195	C₈×C₅₃	—	2	1	no	—
196	C₈×C₅₃	—	2	1	no	—
197	C₈×C₅₃	#117	1	2	yes	—
198	C₈×C₅₃	#117	1	2	yes	—
199	#88322	—	2	1	no	—
200	#88322	—	2	1	no	—
201	#88322	#117	1	2	yes	—
202	#88322	#117	1	2	yes	—
203	#20818	—	53	1	no	maximal, Sylow
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	#427	1	2	yes	—
207	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
208	C₂×D₅₃	—	2	1	no	—
209	C₂×D₅₃	—	2	1	no	—
210	C₂×D₅₃	—	2	8	no	—
211	C₂×D₅₃	#427	1	2	yes	—
212	C₂×D₅₃	#532	1	2	yes	—
213	C₂×D₅₃	C₂×#135	1	2	yes	—
214	C₂×D₅₃	C₂×#135	1	2	yes	—
215	#6157	—	8	8	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	#427	1	2	yes	—
221	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
222	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
223	Dic₅₃	—	2	1	no	—
224	Dic₅₃	—	2	1	no	—
225	Dic₅₃	—	2	1	no	—
226	Dic₅₃	—	2	1	no	—
227	Dic₅₃	#427	1	2	yes	—
228	Dic₅₃	#533	1	2	yes	—
229	C₂²×C₄×C₈	—	53	1	no	—
230	#2283	—	53	2	no	—
231	#1846	—	53	2	no	—
232	#1835	—	53	2	no	—
233	#1808	—	53	2	no	—
234	#1807	—	53	2	no	—
235	C₄×#107	—	53	2	no	—
236	C₄×#106	—	53	2	no	—
237	C₂×C₅₃	—	2	2	no	—
238	C₂×C₅₃	#2285	1	1	yes	characteristic
239	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
241	D₅₃	—	2	2	no	—
242	D₅₃	#3168	1	4	yes	—
243	C₂³×C₈	—	53	1	no	—
244	C₂³×C₈	—	53	1	no	—
245	C₂²×C₄²	—	53	1	no	—
246	C₂×#107	—	53	4	no	—
247	C₂×#106	—	53	4	no	—
248	C₂×C₄×C₈	—	53	1	no	—
249	C₂×C₄×C₈	—	53	1	no	—
250	C₂×C₄×C₈	—	53	1	no	—
251	C₂×C₄×C₈	—	53	1	no	—
252	C₂×C₄×C₈	—	53	1	no	—
253	C₂×C₄×C₈	—	53	1	no	—
254	C₂×C₄×C₈	—	53	1	no	—
255	C₂×C₄×C₈	—	53	1	no	—
256	C₂×C₄×C₈	—	53	1	no	—
257	C₂×C₄×C₈	—	53	1	no	—
258	C₂×C₄×C₈	—	53	1	no	—
259	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
260	#383	—	106	2	no	—
261	#382	—	53	4	no	—
262	#381	—	106	2	no	—
263	#380	—	53	4	no	—
264	C₄×C₄⋊C₄	—	53	4	no	—
265	C₄×C₂²⋊C₄	—	106	2	no	—
266	#376	—	106	2	no	—
267	#340	—	53	4	no	—
268	#340	—	53	4	no	—
269	C₅₃	#20818	1	1	yes	characteristic, complemented, Sylow
270	C₂³×C₄	—	53	1	no	—
271	C₂³×C₄	—	53	1	no	—
272	C₂³×C₄	—	53	1	no	—
273	C₂²×C₈	—	53	1	no	—
274	C₂²×C₈	—	53	1	no	—
275	C₂²×C₈	—	53	1	no	—
276	C₂²×C₈	—	53	1	no	—
277	C₂²×C₈	—	53	1	no	—
278	C₂²×C₈	—	53	1	no	—
279	C₂²×C₈	—	53	1	no	—
280	C₂²×C₈	—	53	1	no	—
281	C₂²×C₈	—	53	1	no	—
282	C₂²×C₈	—	53	1	no	—
283	C₂²×C₈	—	106	4	no	—
284	C₂²×C₈	—	106	4	no	—
285	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
286	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
287	C₂×C₄⋊C₄	—	53	8	no	—
288	C₂×C₂²⋊C₄	—	106	4	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	53	1	no	—
294	C₂×C₄²	—	53	1	no	—
295	C₂×C₄²	—	53	1	no	—
296	C₂×C₄²	—	53	1	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	—	53	1	no	—
300	C₂×C₄²	—	106	4	no	—
301	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
302	#107	—	106	8	no	—
303	#106	—	106	8	no	—
304	C₄×C₈	—	53	1	no	—
305	C₄×C₈	—	53	1	no	—
306	C₄×C₈	—	53	1	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	53	1	no	—
314	C₄×C₈	—	53	1	no	—
315	C₄×C₈	—	53	1	no	—
316	C₄×C₈	C₂×#6157	1	1	yes	characteristic
317	C₄×C₈	C₂×#6157	1	1	yes	characteristic
318	C₄×C₈	C₂×#6157	1	1	yes	characteristic
319	C₄×C₈	C₂×#6157	1	1	yes	characteristic
320	#95	—	106	4	no	—
321	#95	—	106	4	no	—
322	#95	—	106	4	no	—
323	#95	—	106	4	no	—
324	C₂⁴	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	53	1	no	—
336	C₂²×C₄	—	53	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	53	1	no	—
340	C₂²×C₄	—	106	2	no	—
341	C₂²×C₄	—	106	2	no	—
342	C₂²×C₄	—	106	4	no	—
343	C₂²×C₄	—	106	4	no	—
344	C₂²×C₄	—	106	8	no	—
345	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
346	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
347	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	53	1	no	—
351	C₂×C₈	—	53	1	no	—
352	C₂×C₈	—	53	4	no	—
353	C₂×C₈	—	53	4	no	—
354	C₂×C₈	—	53	1	no	—
355	C₂×C₈	—	53	1	no	—
356	C₂×C₈	—	106	2	no	—
357	C₂×C₈	—	106	2	no	—
358	C₂×C₈	—	106	2	no	—
359	C₂×C₈	—	106	2	no	—
360	C₂×C₈	—	106	2	no	—
361	C₂×C₈	—	106	2	no	—
362	C₂×C₈	—	106	2	no	—
363	C₂×C₈	—	106	2	no	—
364	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
365	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
366	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
367	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
368	C₄⋊C₄	—	106	8	no	—
369	C₄⋊C₄	—	106	8	no	—
370	C₂²⋊C₄	—	212	8	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₄²	—	53	1	no	—
376	C₄²	—	53	1	no	—
377	C₄²	—	53	1	no	—
378	C₄²	—	53	1	no	—
379	C₄²	—	53	1	no	—
380	C₄²	—	53	1	no	—
381	C₄²	—	53	1	no	—
382	C₄²	—	53	1	no	—
383	C₄²	—	212	8	no	—
384	C₄²	C₄⋊#6157	1	1	yes	characteristic
385	C₄²	C₄⋊#6157	1	1	yes	characteristic
386	C₄²	C₂²×#6157	1	1	yes	characteristic
387	C₄²	C₂²×#6157	1	1	yes	characteristic
388	C₂³	—	53	1	no	—
389	C₂³	—	53	4	no	—
390	C₂³	—	53	1	no	—
391	C₂³	—	106	2	no	—
392	C₂³	—	106	2	no	—
393	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	53	1	no	—
399	C₂×C₄	—	53	1	no	—
400	C₂×C₄	—	53	4	no	—
401	C₂×C₄	—	53	2	no	—
402	C₂×C₄	—	53	2	no	—
403	C₂×C₄	—	53	1	no	—
404	C₂×C₄	—	53	1	no	—
405	C₂×C₄	—	53	1	no	—
406	C₂×C₄	—	53	1	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	2	no	—
410	C₂×C₄	—	106	2	no	—
411	C₂×C₄	—	106	2	no	—
412	C₂×C₄	—	106	2	no	—
413	C₂×C₄	—	106	2	no	—
414	C₂×C₄	—	106	2	no	—
415	C₂×C₄	—	106	2	no	—
416	C₂×C₄	—	106	2	no	—
417	C₂×C₄	—	106	2	no	—
418	C₂×C₄	—	106	2	no	—
419	C₂×C₄	—	106	1	no	—
420	C₂×C₄	—	106	1	no	—
421	C₂×C₄	—	212	8	no	—
422	C₂×C₄	—	212	8	no	—
423	C₂×C₄	—	212	8	no	—
424	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
425	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
426	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
427	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
428	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
429	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
430	C₈	—	2	1	no	—
431	C₈	—	2	1	no	—
432	C₈	—	53	2	no	—
433	C₈	—	53	2	no	—
434	C₈	—	106	1	no	—
435	C₈	—	106	1	no	—
436	C₈	#13133296	1	2	yes	—
437	C₈	#13133296	1	2	yes	—
438	C₂²	—	2	1	no	—
439	C₂²	—	2	1	no	—
440	C₂²	—	53	2	no	—
441	C₂²	—	53	2	no	—
442	C₂²	—	53	2	no	—
443	C₂²	—	53	2	no	—
444	C₂²	—	106	8	no	—
445	C₂²	—	106	1	no	—
446	C₂²	—	106	1	no	—
447	C₂²	#16811633	1	2	yes	—
448	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	53	2	no	—
454	C₄	—	53	2	no	—
455	C₄	—	106	1	no	—
456	C₄	—	106	1	no	—
457	C₄	—	106	1	no	—
458	C₄	—	106	1	no	—
459	C₄	—	424	8	no	—
460	C₄	#16811633	1	2	yes	—
461	C₄	C₂×#13133296	1	1	yes	characteristic
462	C₄	C₂×#13133296	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	53	4	no	—
465	C₂	—	106	2	no	—
466	C₂	#20291164	1	1	yes	characteristic
467	C₂	C₂×#16811633	1	1	yes	characteristic
468	C₂	C₂×#16811633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#20818
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20290652	C₂	semidirect
3. #20290637	C₂	semidirect
6. C₄×#13133286	C₂	semidirect
7. C₄×#13133285	C₂	semidirect
23. C₂×C₄×#88322	C₄	semidirect
24. C₂×C₈×Dic₅₃	C₄	semidirect
25. C₂×C₈×Dic₅₃	C₄	semidirect
26. C₄×C₈×D₅₃	C₄	semidirect
27. C₄×C₈×D₅₃	C₄	semidirect
28. C₄×C₈×D₅₃	C₄	semidirect
29. C₄×C₈×D₅₃	C₄	semidirect
34. C₂×C₄×C₈×C₅₃	C₄	semidirect
259. C₂×C₄×C₈	#6157	semidirect
269. C₅₃	#20818	semidirect

Maximal subgroups

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

subgroup	index
#20291162	2
#20290652	2
#20290637	2
#20290592	2
#20290591	2
C₄×#13133286	2
C₄×#13133285	2
C₂×C₄×C₈×D₅₃	2
#20818	53