Subgroups — #27293897

Order 13568.

362 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276469	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
4	#20272695	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133156	—	2	1	no	—
10	C₂×#13133156	—	2	1	no	—
11	Q₈×#88322	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₆	—	2	1	no	—
13	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	#16810747	—	2	1	no	—
15	#16810747	C₄	1	1	yes	characteristic
16	C₄⋊C₈×D₅₃	—	2	1	no	—
17	C₄⋊C₈×D₅₃	C₄	1	1	yes	characteristic
18	C₄×C₈×D₅₃	—	2	1	no	—
19	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊#11719908	C₂²	1	1	yes	characteristic
21	#16810441	—	2	1	no	—
22	#16810398	—	2	1	no	—
23	C₄×#11719908	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
32	C₄×Dic₁₀₆	—	2	1	no	—
33	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	#13133182	—	2	1	no	—
35	#13133182	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₅₃	—	2	1	no	—
37	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
38	C₄²×D₅₃	—	2	1	no	—
39	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
40	#13133156	—	2	4	no	—
41	#13133156	—	4	2	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	—	2	1	no	—
44	C₄⋊#88322	—	2	1	no	—
45	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
46	#13133146	—	2	1	no	—
47	#13133146	—	2	1	no	—
48	#13133146	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₅₃	—	2	1	no	—
52	C₈×Dic₅₃	—	2	1	no	—
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
63	Q₈×D₅₃	—	2	2	no	—
64	Q₈×D₅₃	—	2	2	no	—
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
70	C₂×Dic₁₀₆	—	2	1	no	—
71	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₅₃	—	2	1	no	—
80	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
81	#11719919	—	2	1	no	—
82	#11719919	—	2	1	no	—
83	#11719919	—	2	1	no	—
84	C₄×Dic₅₃	—	2	1	no	—
85	C₄×Dic₅₃	—	2	1	no	—
86	C₄×Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
93	C₈×D₅₃	—	2	2	no	—
94	C₈×D₅₃	—	4	2	no	—
95	C₈×D₅₃	SD₁₆	1	2	yes	—
96	C₈×D₅₃	Q₁₆	1	2	yes	—
97	C₁₆×C₅₃	—	2	2	no	—
98	C₁₆×C₅₃	—	4	1	no	—
99	#11719908	—	2	2	no	—
100	#11719908	—	4	1	no	—
101	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	1	no	—
103	Q₈×C₅₃	M₃₂	1	2	yes	complemented
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
114	C₄×D₅₃	—	2	4	no	—
115	C₄×D₅₃	—	2	1	no	—
116	C₄×D₅₃	—	2	1	no	—
117	C₄×D₅₃	—	4	1	no	—
118	C₄×D₅₃	—	4	2	no	—
119	C₄×D₅₃	#103	1	2	yes	—
120	C₄×D₅₃	#104	1	2	yes	—
121	Dic₁₀₆	—	2	4	no	—
122	Dic₁₀₆	—	2	1	no	—
123	Dic₁₀₆	M₃₂	1	2	yes	complemented
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	4	1	no	—
126	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
128	#88322	—	2	1	no	—
129	#88322	—	4	1	no	—
130	#88322	#103	1	2	yes	—
131	#11607	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂×D₅₃	—	2	1	no	—
134	C₂×D₅₃	—	2	1	no	—
135	C₂×D₅₃	Q₈⋊C₈	1	2	yes	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
141	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
142	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
143	Dic₅₃	—	2	1	no	—
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	—	2	1	no	—
146	Dic₅₃	—	4	1	no	—
147	Dic₅₃	Q₈⋊C₈	1	2	yes	—
148	C₂×C₈×Q₈	—	53	1	no	—
149	#2150	—	53	1	no	—
150	C₄×M₃₂	—	53	1	no	—
151	Q₈⋊C₁₆	—	53	1	no	—
152	Q₈⋊C₁₆	—	53	1	no	—
153	Q₈⋊C₁₆	—	53	1	no	—
154	Q₈⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
155	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
156	C₂×C₅₃	#2112	1	1	yes	characteristic
157	C₂×C₅₃	#2117	1	1	yes	characteristic
158	D₅₃	—	2	2	no	—
159	C₂×C₄×Q₈	—	53	1	no	—
160	C₂×M₃₂	—	106	1	no	—
161	C₂×M₃₂	—	106	1	no	—
162	C₈×Q₈	—	53	1	no	—
163	C₈×Q₈	—	53	1	no	—
164	C₈×Q₈	—	53	1	no	—
165	C₈×Q₈	—	106	1	no	—
166	C₈×Q₈	—	106	1	no	—
167	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	53	1	no	—
169	C₂×C₄⋊C₈	—	106	1	no	—
170	C₂×C₄×C₈	—	53	1	no	—
171	C₂×C₄×C₈	—	106	1	no	—
172	C₄⋊C₁₆	—	53	1	no	—
173	C₄⋊C₁₆	—	106	1	no	—
174	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
175	#346	—	106	1	no	—
176	C₄×C₁₆	—	53	1	no	—
177	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
178	C₅₃	#11607	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	53	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	106	1	no	—
183	C₄×Q₈	—	53	1	no	—
184	C₄×Q₈	—	53	1	no	—
185	C₄×Q₈	—	53	1	no	—
186	C₄×Q₈	—	106	1	no	—
187	C₄×Q₈	—	106	1	no	—
188	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	53	1	no	—
190	C₂×C₄⋊C₄	—	106	1	no	—
191	C₂×C₄²	—	53	1	no	—
192	C₂×C₄²	—	106	1	no	—
193	M₃₂	—	106	4	no	—
194	M₃₂	—	212	2	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	106	1	no	—
198	C₂×C₁₆	—	106	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	53	1	no	—
201	C₄⋊C₈	—	106	1	no	—
202	C₄⋊C₈	—	106	1	no	—
203	C₄⋊C₈	—	106	1	no	—
204	C₄⋊C₈	—	106	1	no	—
205	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	1	no	—
209	C₄×C₈	—	53	1	no	—
210	C₄×C₈	—	106	1	no	—
211	C₄×C₈	—	106	1	no	—
212	C₄×C₈	—	106	1	no	—
213	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
214	C₂×Q₈	—	53	1	no	—
215	C₂×Q₈	—	106	1	no	—
216	C₂×Q₈	—	106	2	no	—
217	C₂×Q₈	—	106	2	no	—
218	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	—	53	1	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	—	106	1	no	—
223	C₂²×C₄	—	106	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	1	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	53	1	no	—
236	C₄⋊C₄	—	106	1	no	—
237	C₄⋊C₄	—	106	1	no	—
238	C₄⋊C₄	—	106	1	no	—
239	C₄⋊C₄	—	106	1	no	—
240	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	—	53	1	no	—
243	C₄²	—	53	1	no	—
244	C₄²	—	53	1	no	—
245	C₄²	—	106	1	no	—
246	C₄²	—	106	1	no	—
247	C₄²	—	106	1	no	—
248	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₁₆	—	2	2	no	—
250	C₁₆	—	4	1	no	—
251	C₁₆	—	106	2	no	—
252	C₁₆	—	212	1	no	—
253	C₂³	—	53	1	no	—
254	Q₈	—	2	1	no	—
255	Q₈	—	53	2	no	—
256	Q₈	—	106	4	no	—
257	Q₈	—	106	1	no	—
258	Q₈	#13133156	1	2	yes	complemented
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	106	1	no	—
268	C₂×C₄	—	106	1	no	—
269	C₂×C₄	—	106	1	no	—
270	C₂×C₄	—	106	1	no	—
271	C₂×C₄	—	212	1	no	—
272	C₂×C₄	—	212	2	no	—
273	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	53	2	no	—
279	C₈	—	106	1	no	—
280	C₈	—	212	1	no	—
281	C₈	SD₁₆×D₅₃	1	1	yes	characteristic
282	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
283	C₂²	—	53	2	no	—
284	C₂²	—	106	1	no	—
285	C₂²	—	106	1	no	—
286	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	53	2	no	—
291	C₄	—	106	1	no	—
292	C₄	—	106	2	no	—
293	C₄	—	106	1	no	—
294	C₄	—	212	1	no	—
295	C₄	#103×D₅₃	1	1	yes	characteristic
296	C₄	#104×D₅₃	1	1	yes	characteristic
297	C₄	M₃₂×D₅₃	1	1	yes	characteristic
298	C₄	C₂×#13133156	1	1	yes	characteristic
299	C₂	—	106	2	no	—
300	C₂	Q₈⋊C₈×D₅₃	1	1	yes	characteristic
301	C₂	#20276324	1	1	yes	characteristic
302	C₂	#20276341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	Q₈⋊C₁₆	D₅₃
hypocenter	C₅₃	#11607
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #20272695	C₂	semidirect
5. Q₈⋊#11719908	C₂	semidirect
6. #20272693	C₂	semidirect
7. C₅₃×Q₈⋊C₁₆	C₂	semidirect
103. Q₈×C₅₃	M₃₂	semidirect
123. Dic₁₀₆	M₃₂	semidirect
154. Q₈⋊C₁₆	D₅₃	semidirect
178. C₅₃	#11607	semidirect
258. Q₈	#13133156	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₅₃	2
#20276469	2
C₄×#13133156	2
#20272695	2
Q₈⋊#11719908	2
#20272693	2
C₅₃×Q₈⋊C₁₆	2
#11607	53