Subgroups — #27402987

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810754	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	2	yes	maximal
4	#20275490	C₂	1	1	yes	maximal, characteristic
5	#20275488	C₂	1	1	yes	maximal, characteristic
6	#20275488	C₂	1	2	yes	maximal
7	#20274889	C₂	1	2	yes	maximal
8	#20274886	C₂	1	1	yes	maximal, characteristic
9	#20274622	C₂	1	2	yes	maximal
10	C₄×#13133152	C₂	1	1	yes	maximal, characteristic
11	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16810754	—	2	1	no	—
13	#16810754	—	2	2	no	—
14	#16810754	C₄	1	2	yes	—
15	#16810754	C₄	1	2	yes	—
16	#16810754	C₄	1	4	yes	—
17	Q₈×#88322	—	2	1	no	—
18	C₂×#13133152	C₂²	1	1	yes	characteristic
19	C₂×#13133152	C₂²	1	2	yes	—
20	C₂×#13133152	C₂²	1	2	yes	—
21	C₂×#13133152	C₂²	1	2	yes	—
22	C₂×#13133152	C₂²	1	2	yes	—
23	C₂×#13133152	C₂²	1	1	yes	characteristic
24	#16810647	C₂²	1	1	yes	characteristic
25	#16810647	C₂²	1	2	yes	—
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	2	yes	—
28	C₄×#11719917	C₂²	1	1	yes	characteristic
29	C₄×#11719917	C₂²	1	2	yes	—
30	C₂×#13133128	C₂²	1	2	yes	—
31	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
32	#16810387	C₂²	1	1	yes	characteristic
33	#16810387	C₂²	1	2	yes	—
34	#16810387	C₂²	1	1	yes	characteristic
35	#16810387	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#446	—	2	2	no	—
38	C₅₃×#446	—	2	1	no	—
39	C₈×Q₈×C₅₃	—	2	1	no	—
40	C₅₃×#423	C₂²	1	1	yes	characteristic
41	C₅₃×#423	C₂²	1	2	yes	—
42	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
44	C₂×#11719917	C₂³	1	1	yes	characteristic
45	C₂×#11719917	C₂³	1	2	yes	—
46	C₂×#11719917	C₂³	1	1	yes	characteristic
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	C₂²×#88322	C₂³	1	2	yes	—
49	C₂²×#88322	C₂³	1	1	yes	characteristic
50	#13133152	—	2	2	no	—
51	#13133152	—	2	4	no	—
52	#13133152	—	4	2	no	—
53	#13133152	—	4	2	no	—
54	#13133152	—	4	2	no	—
55	#13133152	Q₈	1	4	yes	complemented
56	C₄⋊#88322	—	2	2	no	—
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	—	2	1	no	—
59	C₄⋊#88322	—	2	2	no	—
60	C₄⋊#88322	C₂×C₄	1	4	yes	—
61	C₄⋊#88322	C₂×C₄	1	4	yes	—
62	C₄⋊#88322	C₂×C₄	1	2	yes	—
63	C₄⋊#88322	C₂×C₄	1	2	yes	—
64	#13133128	—	2	2	no	—
65	#13133128	—	2	1	no	—
66	#13133128	C₂×C₄	1	4	yes	—
67	#13133128	C₂×C₄	1	4	yes	—
68	C₄×#88322	—	2	1	no	—
69	C₄×#88322	—	2	2	no	—
70	C₄×#88322	C₂×C₄	1	2	yes	—
71	C₄×#88322	C₂×C₄	1	2	yes	—
72	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
73	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
75	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	D₄	1	2	yes	—
81	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	—	2	2	no	—
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	#97×C₅₃	—	2	2	no	—
91	#97×C₅₃	—	2	1	no	—
92	C₄×C₈×C₅₃	—	2	1	no	—
93	C₄×C₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	4	2	no	—
95	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₅₃	—	2	1	no	—
104	M₁₆×C₅₃	—	2	1	no	—
105	M₁₆×C₅₃	—	2	2	no	—
106	M₁₆×C₅₃	—	2	1	no	—
107	M₁₆×C₅₃	—	2	2	no	—
108	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
117	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
123	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂×D₄	1	2	yes	—
125	C₄²×C₅₃	C₂×D₄	1	2	yes	—
126	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
127	#11719917	—	2	1	no	—
128	#11719917	—	2	1	no	—
129	#11719917	—	2	2	no	—
130	#11719917	—	2	2	no	—
131	#11719917	—	2	1	no	—
132	#11719917	C₂×Q₈	1	2	yes	—
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	1	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	—	2	1	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	C₂²×C₄	1	2	yes	—
141	C₂×#88322	C₂²×C₄	1	4	yes	—
142	C₂×#88322	C₂²×C₄	1	2	yes	—
143	C₂×#88322	C₂×Q₈	1	2	yes	—
144	C₂×#88322	#41	1	2	yes	—
145	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
146	Q₈×C₅₃	—	2	2	no	—
147	Q₈×C₅₃	—	2	2	no	—
148	Q₈×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	4	1	no	—
169	C₈×C₅₃	—	4	2	no	—
170	#88322	—	2	2	no	—
171	#88322	—	2	4	no	—
172	#88322	—	2	2	no	—
173	#88322	—	4	1	no	—
174	#88322	—	4	2	no	—
175	#88322	C₄×Q₈	1	4	yes	—
176	#22733	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₅₃	#557	1	1	yes	characteristic
181	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	2	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	#430	1	2	yes	—
190	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₄×C₅₃	#520	1	1	yes	characteristic
192	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
193	C₄×C₅₃	#550	1	1	yes	characteristic
194	Q₈×M₁₆	—	53	1	no	—
195	Q₈×M₁₆	D₅₃	1	1	yes	characteristic
196	C₂×#446	—	53	1	no	—
197	#1950	—	53	2	no	—
198	#1950	—	53	1	no	—
199	#1950	—	53	2	no	—
200	#1950	—	53	1	no	—
201	#1849	—	53	2	no	—
202	#1848	—	53	1	no	—
203	#1778	—	53	2	no	—
204	C₄×#108	—	53	1	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
209	C₂×C₅₃	#2916	1	1	yes	characteristic
210	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
211	#446	—	2	1	no	—
212	#446	—	2	2	no	—
213	#446	—	53	2	no	—
214	#446	—	53	2	no	—
215	#446	—	53	4	no	—
216	#446	—	106	1	no	—
217	#446	—	106	2	no	—
218	C₈×Q₈	—	2	1	no	—
219	C₈×Q₈	—	106	1	no	—
220	C₂×#108	—	53	1	no	—
221	C₂×#108	—	53	2	no	—
222	C₂×#108	—	53	1	no	—
223	C₂×#108	—	53	2	no	—
224	C₂×#108	—	53	2	no	—
225	C₂×#108	—	53	2	no	—
226	#423	—	53	1	no	—
227	#423	—	53	2	no	—
228	#423	C₂×D₅₃	1	1	yes	characteristic
229	#423	C₂×D₅₃	1	2	yes	—
230	C₂×C₄⋊C₈	—	53	1	no	—
231	C₂×C₄⋊C₈	—	53	2	no	—
232	C₄×M₁₆	—	53	1	no	—
233	C₄×M₁₆	—	53	2	no	—
234	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
235	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
236	C₂×#97	—	53	2	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	#341	—	53	1	no	—
239	#341	—	53	1	no	—
240	#341	—	53	2	no	—
241	#341	—	53	2	no	—
242	C₅₃	#22733	1	1	yes	characteristic, complemented, Sylow
243	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
244	C₂×M₁₆	—	53	1	no	—
245	C₂×M₁₆	—	53	1	no	—
246	C₂×M₁₆	—	53	2	no	—
247	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
248	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
249	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
250	C₂²×C₈	—	53	1	no	—
251	C₂²×C₈	—	53	1	no	—
252	C₂²×C₈	—	53	2	no	—
253	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
254	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
255	C₄×Q₈	C₄×D₅₃	1	2	yes	—
256	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₄×Q₈	C₂²⋊D₅₃	1	2	yes	—
259	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
260	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
261	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
263	#108	—	53	4	no	—
264	#108	—	106	4	no	—
265	#108	—	106	2	no	—
266	#108	—	212	2	no	—
267	#108	—	212	2	no	—
268	#108	—	212	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	53	2	no	—
274	C₄⋊C₈	—	53	2	no	—
275	C₄⋊C₈	—	53	4	no	—
276	C₄⋊C₈	—	53	4	no	—
277	C₄⋊C₈	—	106	1	no	—
278	C₄⋊C₈	—	106	2	no	—
279	C₄⋊C₈	—	106	2	no	—
280	C₄⋊C₈	—	106	1	no	—
281	#97	—	2	2	no	—
282	#97	—	2	1	no	—
283	#97	—	53	4	no	—
284	#97	—	53	4	no	—
285	#97	—	106	2	no	—
286	#97	—	106	1	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	—	2	2	no	—
289	C₄×C₈	—	53	2	no	—
290	C₄×C₈	—	53	2	no	—
291	C₄×C₈	—	106	1	no	—
292	C₄×C₈	—	106	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	#11719919	1	1	yes	characteristic
295	C₂×Q₈	#11719919	1	1	yes	characteristic
296	C₂×Q₈	#11719919	1	2	yes	—
297	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
298	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
299	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
300	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
301	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	1	no	—
305	M₁₆	—	2	2	no	—
306	M₁₆	—	2	1	no	—
307	M₁₆	—	53	2	no	—
308	M₁₆	—	106	1	no	—
309	M₁₆	—	106	2	no	—
310	M₁₆	—	106	1	no	—
311	M₁₆	—	106	2	no	—
312	M₁₆	—	106	1	no	—
313	M₁₆	Q₈×D₅₃	1	2	yes	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	53	2	no	—
318	C₂×C₈	—	53	2	no	—
319	C₂×C₈	—	53	2	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	53	4	no	—
322	C₂×C₈	—	106	1	no	—
323	C₂×C₈	—	106	1	no	—
324	C₂×C₈	—	106	2	no	—
325	C₂×C₈	—	106	1	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	—	106	1	no	—
328	C₂×C₈	—	106	2	no	—
329	C₄⋊C₄	#11719919	1	2	yes	—
330	C₄⋊C₄	#11719919	1	2	yes	—
331	C₄⋊C₄	#11719919	1	2	yes	—
332	C₄⋊C₄	#11719919	1	2	yes	—
333	C₄⋊C₄	#11719919	1	2	yes	—
334	C₄⋊C₄	#11719919	1	1	yes	characteristic
335	C₄⋊C₄	#11719919	1	1	yes	characteristic
336	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
337	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
338	C₄²	C₂×C₄×D₅₃	1	2	yes	—
339	C₄²	C₂×C₄×D₅₃	1	2	yes	—
340	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
341	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
342	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
343	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
344	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	4	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
353	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
354	C₂×C₄	C₂×#11719919	1	2	yes	—
355	C₂×C₄	C₂×#11719919	1	2	yes	—
356	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719919	1	2	yes	—
358	C₂×C₄	C₂×#11719919	1	2	yes	—
359	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
360	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
361	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
362	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
363	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
364	C₂×C₄	#13133263	1	1	yes	characteristic
365	C₂×C₄	#13133265	1	1	yes	characteristic
366	C₈	—	2	2	no	—
367	C₈	—	4	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	53	4	no	—
370	C₈	—	106	2	no	—
371	C₈	—	106	2	no	—
372	C₈	—	106	4	no	—
373	C₈	—	212	1	no	—
374	C₈	—	212	2	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	C₂²×#11719919	1	1	yes	characteristic
378	C₂²	#16811349	1	1	yes	characteristic
379	C₂²	Q₈×Dic₁₀₆	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	#16810671	1	2	yes	—
388	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
389	C₄	#16811105	1	1	yes	characteristic
390	C₄	Q₈×C₂²⋊D₅₃	1	1	yes	characteristic
391	C₄	#16811292	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	Q₈×#11719919	1	1	yes	characteristic
395	C₂	C₂×#16810671	1	1	yes	characteristic
396	C₂	#20283324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	Q₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	Q₈×M₁₆	D₅₃
hypocenter	C₅₃	#22733
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
55. #13133152	Q₈	semidirect
242. C₅₃	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719917	2
C₂×#16810754	2
#20275490	2
#20275488	2
#20274889	2
#20274886	2
#20274622	2
C₄×#13133152	2
Q₈×M₁₆×C₅₃	2
#22733	53