Subgroups — #27402912

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	2	yes	maximal
2	#20275439	C₂	1	2	yes	maximal
3	#20275439	C₂	1	2	yes	maximal
4	#20275434	C₂	1	2	yes	maximal
5	#20274869	C₂	1	2	yes	maximal
6	#20274867	C₂	1	1	yes	maximal, characteristic
7	#20274572	C₂	1	2	yes	maximal
8	#20274568	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
10	#16810754	—	2	8	no	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133128	C₂²	1	2	yes	—
22	C₂×#13133128	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	2	yes	—
24	#16810378	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#391	C₄	1	1	yes	characteristic
29	C₅₃×#391	C₄	1	1	yes	characteristic
30	C₅₃×#389	C₄	1	1	yes	characteristic
31	C₅₃×#389	C₄	1	1	yes	characteristic
32	C₅₃×#387	C₄	1	2	yes	—
33	C₅₃×#387	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#376	C₂²	1	2	yes	—
37	C₄³×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	2	yes	—
39	C₂²×#88322	C₂³	1	2	yes	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₄⋊#88322	—	2	4	no	—
43	C₄⋊#88322	—	2	4	no	—
44	C₄⋊#88322	—	2	4	no	—
45	C₄⋊#88322	—	2	2	no	—
46	C₄⋊#88322	—	2	4	no	—
47	C₄⋊#88322	—	4	1	no	—
48	C₄⋊#88322	—	4	1	no	—
49	C₄⋊#88322	—	4	2	no	—
50	C₄⋊#88322	—	4	2	no	—
51	C₄⋊#88322	Q₈	1	4	yes	complemented
52	#13133128	—	2	4	no	—
53	#13133128	—	2	4	no	—
54	C₄×#88322	—	2	4	no	—
55	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₅₃	D₄	1	4	yes	—
57	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	#95×C₅₃	C₂×C₄	1	2	yes	—
75	#95×C₅₃	C₂×C₄	1	2	yes	—
76	#95×C₅₃	C₂×C₄	1	2	yes	—
77	#95×C₅₃	C₂×C₄	1	2	yes	—
78	#95×C₅₃	C₂×C₄	1	2	yes	—
79	#95×C₅₃	C₂×C₄	1	2	yes	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	4	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
99	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	4	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	C₂×D₄	1	4	yes	—
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
111	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
112	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
113	C₄²×C₅₃	#41	1	2	yes	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	—	2	2	no	—
123	C₂×#88322	—	2	2	no	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	C₂×Q₈	1	4	yes	—
126	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	Q₈×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	2	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	2	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
157	C₂×C₄×C₅₃	#131	1	2	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
164	#88322	—	2	4	no	—
165	#88322	—	4	2	no	—
166	#88322	—	4	2	no	—
167	#88322	—	4	2	no	—
168	#22723	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₂²×C₅₃	#520	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
173	C₂²×C₅₃	#568	1	1	yes	characteristic
174	C₂²×C₅₃	#568	1	1	yes	characteristic
175	C₂²×C₅₃	#568	1	1	yes	characteristic
176	C₄×C₅₃	—	2	4	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	2	2	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	—	4	1	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
187	C₄×C₅₃	#550	1	1	yes	characteristic
188	C₄×C₅₃	#557	1	1	yes	characteristic
189	C₄×C₅₃	Q₈²	1	1	yes	characteristic
190	C₂×#446	—	53	2	no	—
191	#2295	D₅₃	1	1	yes	characteristic
192	#1940	—	53	2	no	—
193	#1940	—	53	2	no	—
194	#1939	—	53	2	no	—
195	#1843	—	53	2	no	—
196	#1842	—	53	1	no	—
197	#1768	—	53	2	no	—
198	#1767	—	53	1	no	—
199	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₅₃	#2905	1	1	yes	characteristic
201	C₂×C₅₃	#2906	1	1	yes	characteristic
202	C₂×C₅₃	#2961	1	1	yes	characteristic
203	C₂×C₅₃	#2962	1	1	yes	characteristic
204	C₂×C₅₃	#2992	1	1	yes	characteristic
205	C₂×C₅₃	#2995	1	1	yes	characteristic
206	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
207	#446	—	106	8	no	—
208	C₂×C₄⋊C₈	—	53	1	no	—
209	C₂×C₄⋊C₈	—	53	2	no	—
210	C₂×C₄⋊C₈	—	53	2	no	—
211	C₂×C₄⋊C₈	—	53	1	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₂×C₄⋊C₈	—	53	2	no	—
214	C₂×C₄⋊C₈	—	53	1	no	—
215	C₂×C₄⋊C₈	—	53	1	no	—
216	C₂×C₄⋊C₈	—	53	2	no	—
217	C₂×C₄⋊C₈	—	53	2	no	—
218	C₂×#97	—	53	2	no	—
219	C₂×#97	—	53	2	no	—
220	C₂×C₄×C₈	—	53	2	no	—
221	#391	Dic₅₃	1	1	yes	characteristic
222	#391	Dic₅₃	1	1	yes	characteristic
223	#389	Dic₅₃	1	1	yes	characteristic
224	#389	Dic₅₃	1	1	yes	characteristic
225	#387	Dic₅₃	1	2	yes	—
226	#387	Dic₅₃	1	2	yes	—
227	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
228	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
229	#376	C₂×D₅₃	1	2	yes	—
230	C₄³	C₂×D₅₃	1	1	yes	characteristic
231	#336	—	53	2	no	—
232	#336	—	53	2	no	—
233	#336	—	53	2	no	—
234	C₅₃	#22723	1	1	yes	characteristic, complemented, Sylow
235	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂²×C₈	—	53	2	no	—
237	C₂²×C₈	—	53	2	no	—
238	C₂²×C₈	—	53	2	no	—
239	C₂²×C₈	—	53	2	no	—
240	C₄×Q₈	Dic₁₀₆	1	4	yes	—
241	C₄×Q₈	D₂₁₂	1	4	yes	—
242	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
244	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
254	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
256	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
257	C₄⋊C₈	—	53	4	no	—
258	C₄⋊C₈	—	106	4	no	—
259	C₄⋊C₈	—	106	4	no	—
260	C₄⋊C₈	—	106	4	no	—
261	C₄⋊C₈	—	106	4	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	212	1	no	—
264	C₄⋊C₈	—	212	1	no	—
265	C₄⋊C₈	—	212	2	no	—
266	C₄⋊C₈	—	212	2	no	—
267	#97	—	106	4	no	—
268	#97	—	106	4	no	—
269	C₄×C₈	—	106	4	no	—
270	#95	C₂×Dic₅₃	1	2	yes	—
271	#95	C₂×Dic₅₃	1	2	yes	—
272	#95	C₂×Dic₅₃	1	2	yes	—
273	#95	C₂×Dic₅₃	1	2	yes	—
274	#95	C₂×Dic₅₃	1	2	yes	—
275	#95	C₂×Dic₅₃	1	2	yes	—
276	#95	C₂×Dic₅₃	1	2	yes	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
280	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
281	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
282	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
284	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
286	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
291	C₂×C₈	—	53	4	no	—
292	C₂×C₈	—	106	2	no	—
293	C₂×C₈	—	106	2	no	—
294	C₂×C₈	—	106	2	no	—
295	C₂×C₈	—	106	2	no	—
296	C₂×C₈	—	106	2	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	106	2	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	106	2	no	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
308	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	4	1	no	—
317	C₄²	C₂×Dic₁₀₆	1	2	yes	—
318	C₄²	C₂×Dic₁₀₆	1	4	yes	—
319	C₄²	C₂×D₂₁₂	1	4	yes	—
320	C₄²	C₂×D₂₁₂	1	2	yes	—
321	C₄²	Q₈×D₅₃	1	2	yes	—
322	C₄²	#11719949	1	2	yes	—
323	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
349	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
350	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
351	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
352	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
353	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
354	C₂×C₄	#13133224	1	2	yes	—
355	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
359	C₂×C₄	#13133263	1	1	yes	characteristic
360	C₂×C₄	#13133264	1	1	yes	characteristic
361	C₈	—	106	4	no	—
362	C₈	—	212	2	no	—
363	C₈	—	212	2	no	—
364	C₈	—	212	2	no	—
365	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
366	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
367	C₂²	#16811106	1	1	yes	characteristic
368	C₂²	C₂×#13133224	1	1	yes	characteristic
369	C₂²	#16811390	1	1	yes	characteristic
370	C₂²	#16811390	1	1	yes	characteristic
371	C₂²	#16811390	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	4	no	—
376	C₄	—	2	2	no	—
377	C₄	—	2	4	no	—
378	C₄	—	2	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	2	no	—
382	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
383	C₄	#16811295	1	1	yes	characteristic
384	C₄	#16811352	1	1	yes	characteristic
385	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
386	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
387	C₂	#20283279	1	1	yes	characteristic
388	C₂	#20283283	1	1	yes	characteristic
389	C₂	#20283656	1	1	yes	characteristic
390	C₂	#20283664	1	1	yes	characteristic
391	C₂	#20283854	1	1	yes	characteristic
392	C₂	#20283885	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2295	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#22723
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
51. C₄⋊#88322	Q₈	semidirect
234. C₅₃	#22723	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
#20275439	2
#20275434	2
#20274869	2
#20274867	2
#20274572	2
#20274568	2
C₅₃×#2295	2
#22723	53