Subgroups — #27278614

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal, complemented
4	#20272556	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	4	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×C₂²⋊#88322	—	2	1	no	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	#16810378	—	2	1	no	—
13	Q₈⋊#88322	—	2	4	no	—
14	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#391	C₄	1	1	yes	characteristic
18	C₅₃×#389	C₄	1	1	yes	characteristic
19	C₅₃×#387	—	2	1	no	—
20	C₅₃×#385	—	2	1	no	—
21	C₅₃×#380	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	—	4	2	no	—
29	C₂²×#88322	—	4	2	no	—
30	C₄⋊#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	D₄	1	4	yes	—
33	C₂²⋊#88322	—	2	4	no	—
34	C₄×#88322	—	2	2	no	—
35	C₄×#88322	—	2	2	no	—
36	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²⋊Q₈×C₅₃	—	2	2	no	—
41	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	complemented
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	—	2	1	no	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#95×C₅₃	—	2	1	no	—
61	#95×C₅₃	—	2	1	no	—
62	#95×C₅₃	—	2	1	no	—
63	#95×C₅₃	—	2	1	no	—
64	#95×C₅₃	—	2	1	no	—
65	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×Q₈×C₅₃	—	2	1	no	—
67	C₂×Q₈×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	complemented
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	—	2	1	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	1	no	—
78	C₂²×C₄×C₅₃	—	2	1	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
88	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	4	no	—
94	C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	complemented
100	C₂²⋊C₄×C₅₃	—	2	4	no	—
101	C₂²⋊C₄×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	C₄²×C₅₃	#41	1	2	yes	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	2	4	no	—
114	C₂×#88322	—	4	2	no	—
115	C₂×#88322	—	4	8	no	—
116	C₂×#88322	—	4	2	no	—
117	C₂×#88322	—	4	2	no	—
118	C₂×#88322	—	4	2	no	—
119	C₂³×C₅₃	—	2	1	no	—
120	C₂³×C₅₃	—	2	1	no	—
121	C₂³×C₅₃	—	2	1	no	—
122	C₂³×C₅₃	—	2	1	no	—
123	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
124	C₂³×C₅₃	#103	1	2	yes	—
125	C₂³×C₅₃	#104	1	2	yes	—
126	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	Q₈×C₅₃	—	2	4	no	—
128	Q₈×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	—	4	1	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
158	C₂×C₄×C₅₃	#103	1	2	yes	—
159	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
161	C₂×C₄×C₅₃	#121	1	2	yes	—
162	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#131	1	2	yes	—
166	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
168	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
170	#88322	—	4	4	no	—
171	#88322	—	4	4	no	—
172	#8937	—	53	1	no	maximal, Sylow
173	C₂²×C₅₃	—	2	2	no	—
174	C₂²×C₅₃	—	2	2	no	—
175	C₂²×C₅₃	—	2	2	no	—
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	Q₈⋊C₈	1	4	yes	complemented
180	C₂²×C₅₃	#386	1	1	yes	characteristic
181	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₂²×C₅₃	#408	1	1	yes	characteristic
183	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
184	C₂²×C₅₃	#417	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
186	C₂²×C₅₃	#421	1	1	yes	characteristic
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	2	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	—	4	1	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	8	1	no	—
194	C₄×C₅₃	—	8	1	no	—
195	C₄×C₅₃	C₈×D₄	1	2	yes	—
196	C₄×C₅₃	#435	1	2	yes	—
197	C₄×C₅₃	#450	1	1	yes	characteristic
198	C₄×C₅₃	#451	1	1	yes	characteristic
199	C₄×C₅₃	#483	1	1	yes	characteristic
200	C₄×C₅₃	#484	1	1	yes	characteristic
201	C₂²×C₄⋊C₈	—	53	1	no	—
202	C₄×C₂²⋊Q₈	D₅₃	1	1	yes	characteristic
203	C₄×C₂²⋊C₈	—	53	1	no	—
204	C₂×Q₈⋊C₈	—	53	2	no	—
205	#1279	—	53	2	no	—
206	C₂×C₅₃	—	2	4	no	—
207	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
208	C₂×C₅₃	#1477	1	1	yes	characteristic
209	C₂×C₅₃	#1481	1	1	yes	characteristic
210	C₂×C₅₃	#1483	1	1	yes	characteristic
211	C₂×C₅₃	#1869	1	1	yes	characteristic
212	C₂×C₅₃	#1871	1	1	yes	characteristic
213	C₂×C₅₃	#1872	1	1	yes	characteristic
214	C₂³×C₈	—	106	1	no	—
215	C₂×C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic
216	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
217	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
218	C₂×C₄⋊C₈	—	53	2	no	—
219	C₂×C₄⋊C₈	—	106	4	no	—
220	C₂×C₄⋊C₈	—	106	1	no	—
221	C₂×C₂²⋊C₈	—	106	1	no	—
222	C₂×C₄×C₈	—	53	2	no	—
223	#391	Dic₅₃	1	1	yes	characteristic
224	#389	Dic₅₃	1	1	yes	characteristic
225	#387	—	2	1	no	—
226	#385	—	2	1	no	—
227	#380	Dic₅₃	1	1	yes	characteristic
228	C₄×C₄⋊C₄	—	2	1	no	—
229	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
230	C₄×C₂²⋊C₄	—	2	1	no	—
231	#336	—	106	1	no	—
232	Q₈⋊C₈	—	106	4	no	—
233	C₅₃	#8937	1	1	yes	characteristic, complemented, Sylow
234	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
235	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂²×C₈	—	106	4	no	—
239	C₂²×C₈	—	106	2	no	—
240	C₂²×C₈	—	106	2	no	—
241	C₂²×C₈	—	212	2	no	—
242	C₂²×C₈	—	212	2	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	#88322	1	4	yes	complemented
245	C₄×Q₈	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₂²⋊C₄	—	2	1	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
260	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
263	C₄⋊C₈	—	53	4	no	—
264	C₄⋊C₈	—	106	4	no	—
265	C₄⋊C₈	—	106	2	no	—
266	C₂²⋊C₈	—	106	4	no	—
267	C₄×C₈	—	106	2	no	—
268	C₄×C₈	—	106	2	no	—
269	#95	—	2	1	no	—
270	#95	—	2	1	no	—
271	#95	—	2	1	no	—
272	#95	—	2	1	no	—
273	#95	—	2	1	no	—
274	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	C₂×#88322	1	2	yes	complemented
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	C₂×#88322	1	2	yes	—
293	C₂²×C₄	#11719917	1	2	yes	—
294	C₂²×C₄	#11719924	1	2	yes	—
295	C₂²×C₄	#11719925	1	2	yes	—
296	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	—	106	4	no	—
304	C₂×C₈	—	212	8	no	—
305	C₂×C₈	—	212	2	no	—
306	C₂×C₈	—	212	2	no	—
307	C₂×C₈	—	212	2	no	—
308	C₂×C₈	—	212	2	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂×#88322	1	2	yes	complemented
316	C₂²⋊C₄	—	2	4	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	1	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	1	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	4	1	no	—
325	C₄²	—	4	1	no	—
326	C₄²	D₄×D₅₃	1	2	yes	—
327	C₄²	#11719947	1	2	yes	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	1	no	—
331	C₂³	—	2	1	no	—
332	C₂³	C₂²⋊#88322	1	2	yes	—
333	C₂³	#13133141	1	2	yes	—
334	C₂³	#13133144	1	2	yes	—
335	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
336	Q₈	—	2	4	no	—
337	Q₈	—	4	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	C₂²⋊#88322	1	2	yes	—
367	C₂×C₄	#13133141	1	2	yes	—
368	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
369	C₂×C₄	#13133185	1	1	yes	characteristic
370	C₂×C₄	#13133189	1	2	yes	—
371	C₂×C₄	#13133198	1	1	yes	characteristic
372	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
373	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
374	C₂×C₄	#13133224	1	2	yes	—
375	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
376	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
377	C₂×C₄	#13133242	1	1	yes	characteristic
378	C₂×C₄	#13133247	1	1	yes	characteristic
379	C₈	—	212	4	no	—
380	C₈	—	212	4	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	Q₈⋊#88322	1	4	yes	complemented
388	C₂²	#16810503	1	1	yes	characteristic
389	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
390	C₂²	#16810590	1	1	yes	characteristic
391	C₂²	C₂×#13133141	1	1	yes	characteristic
392	C₂²	#16810624	1	1	yes	characteristic
393	C₂²	C₂×#13133144	1	1	yes	characteristic
394	C₂²	#16810640	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	8	1	no	—
402	C₄	—	8	1	no	—
403	C₄	D₄×#88322	1	2	yes	—
404	C₄	#16810695	1	2	yes	—
405	C₄	#16810773	1	1	yes	characteristic
406	C₄	#16810778	1	1	yes	characteristic
407	C₄	#16810951	1	1	yes	characteristic
408	C₄	#16810956	1	1	yes	characteristic
409	C₂	—	2	4	no	—
410	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
411	C₂	#20273102	1	1	yes	characteristic
412	C₂	#20273118	1	1	yes	characteristic
413	C₂	#20273128	1	1	yes	characteristic
414	C₂	#20275008	1	1	yes	characteristic
415	C₂	#20275018	1	1	yes	characteristic
416	C₂	#20275023	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₂²⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#8937
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈⋊#88322	C₂	semidirect
4. #20272556	C₂	semidirect
41. C₂²⋊Q₈×C₅₃	C₈	semidirect
69. C₂×Q₈×C₅₃	C₂×C₈	semidirect
99. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
179. C₂²×C₅₃	Q₈⋊C₈	semidirect
233. C₅₃	#8937	semidirect
244. C₂²⋊Q₈	#88322	semidirect
278. C₂×Q₈	C₂×#88322	semidirect
315. C₄⋊C₄	C₂×#88322	semidirect
387. C₂²	Q₈⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
C₄×C₂²⋊#88322	2
C₂×Q₈⋊#88322	2
#20272556	2
C₄×C₂²⋊Q₈×C₅₃	2
#8937	53