Subgroups — C₈×#13133135 #27274656

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133135	C₂	1	1	yes	maximal, characteristic
3	#20273038	C₂	1	1	yes	maximal, characteristic
4	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
5	#20272566	C₂	1	1	yes	maximal, characteristic
6	#20272564	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	—	2	1	no	—
11	C₈×Dic₁₀₆	—	2	1	no	—
12	C₈×Dic₁₀₆	—	2	1	no	—
13	#16810676	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×#13133146	C₄	1	2	yes	—
16	C₂×#13133135	C₄	1	2	yes	—
17	C₄×#11719917	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
19	#16810391	C₄	1	2	yes	—
20	#16810348	C₄	1	4	yes	—
21	#16810345	—	2	1	no	—
22	C₈×#88322	—	2	1	no	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#322	—	2	1	no	—
27	C₈²×C₅₃	—	2	1	no	—
28	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719917	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₀₆	—	2	1	no	—
32	C₄×Dic₁₀₆	—	2	1	no	—
33	C₄×Dic₁₀₆	—	2	1	no	—
34	C₄×Dic₁₀₆	—	2	1	no	—
35	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719919	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₄⋊#88322	—	2	1	no	—
39	C₄⋊#88322	—	2	1	no	—
40	C₄⋊#88322	—	2	1	no	—
41	#13133146	—	2	2	no	—
42	#13133146	—	2	2	no	—
43	#13133135	C₈	1	8	yes	direct product factor
44	C₂²⋊#88322	C₂×C₄	1	2	yes	—
45	#13133128	—	2	1	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₈×Dic₅₃	—	2	2	no	—
49	C₈×Dic₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	1	no	—
55	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
56	#97×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₄²	1	2	yes	—
67	C₂²×Dic₅₃	C₄²	1	2	yes	—
68	C₂×Dic₁₀₆	—	2	1	no	—
69	C₂×Dic₁₀₆	—	2	1	no	—
70	C₂×Dic₁₀₆	—	2	2	no	—
71	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	#41	1	1	yes	characteristic
86	C₄⋊Dic₅₃	—	2	1	no	—
87	C₄⋊Dic₅₃	—	2	1	no	—
88	#11719919	—	2	2	no	—
89	#11719919	—	2	2	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₄×Dic₅₃	—	2	2	no	—
92	#11719917	—	2	2	no	—
93	#11719917	C₂×C₈	1	4	yes	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	#131	1	2	yes	—
110	C₂×Dic₅₃	—	2	2	no	—
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
115	Dic₁₀₆	—	2	4	no	—
116	Dic₁₀₆	—	2	4	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	#101	1	2	yes	complemented
123	#88322	—	2	4	no	—
124	#88322	—	4	2	no	—
125	#88322	—	4	2	no	—
126	C₈×#101	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₅₃	#405	1	1	yes	characteristic
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
136	C₄×C₅₃	#413	1	1	yes	characteristic
137	C₄×C₅₃	C₈×D₄	1	2	yes	—
138	C₄×C₅₃	C₈×D₄	1	2	yes	—
139	Dic₅₃	—	2	4	no	—
140	Dic₅₃	—	4	2	no	—
141	C₂×C₈×Q₈	—	53	1	no	—
142	C₄×#101	—	53	1	no	—
143	#1461	—	53	1	no	—
144	C₈×M₁₆	—	53	1	no	—
145	C₈×M₁₆	D₅₃	1	1	yes	characteristic
146	#1283	—	53	1	no	—
147	#1283	—	53	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
152	C₂×C₅₃	#1757	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	53	1	no	—
154	C₈×Q₈	—	106	1	no	—
155	C₈×Q₈	—	106	1	no	—
156	C₈×Q₈	—	106	1	no	—
157	C₈×Q₈	—	106	1	no	—
158	#431	—	53	1	no	—
159	#431	C₂×D₅₃	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×C₄⋊C₈	—	53	1	no	—
162	C₂×#101	—	53	2	no	—
163	C₄×M₁₆	—	53	1	no	—
164	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
165	C₂×C₄×C₈	—	53	2	no	—
166	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	#343	—	53	2	no	—
168	#324	—	53	4	no	—
169	#322	—	2	1	no	—
170	#322	—	106	1	no	—
171	C₈²	—	2	1	no	—
172	C₈²	—	106	1	no	—
173	C₅₃	C₈×#101	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	53	1	no	—
175	C₂×M₁₆	—	53	2	no	—
176	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
180	C₄×Q₈	—	106	1	no	—
181	C₄×Q₈	—	106	1	no	—
182	C₄×Q₈	—	106	1	no	—
183	C₄×Q₈	—	106	1	no	—
184	C₂×C₄⋊C₄	—	53	2	no	—
185	C₂×C₄⋊C₄	—	53	1	no	—
186	C₂×C₄²	—	53	2	no	—
187	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	106	2	no	—
190	C₄⋊C₈	—	106	2	no	—
191	C₄⋊C₈	—	106	1	no	—
192	C₄⋊C₈	—	106	1	no	—
193	C₄⋊C₈	—	106	1	no	—
194	#101	—	53	8	no	—
195	C₂²⋊C₈	—	53	2	no	—
196	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
197	#97	—	2	1	no	—
198	#97	—	106	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	106	2	no	—
202	C₄×C₈	—	106	2	no	—
203	C₄×C₈	—	106	1	no	—
204	C₄×C₈	—	106	1	no	—
205	C₄×C₈	D₂₁₂	1	1	yes	characteristic
206	C₄×C₈	D₂₁₂	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂×Q₈	—	53	4	no	—
210	C₂×Q₈	—	106	1	no	—
211	C₂×Q₈	—	106	1	no	—
212	C₂×Q₈	—	106	2	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	53	4	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	C₈×D₅₃	1	4	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	—	106	2	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	212	2	no	—
230	C₂×C₈	#11719921	1	1	yes	characteristic
231	C₂×C₈	#11719921	1	1	yes	characteristic
232	C₂×C₈	#11719921	1	1	yes	characteristic
233	C₂×C₈	#11719921	1	1	yes	characteristic
234	C₄⋊C₄	—	106	2	no	—
235	C₄⋊C₄	—	106	1	no	—
236	C₄⋊C₄	—	106	2	no	—
237	C₄⋊C₄	—	106	1	no	—
238	C₄²	—	106	2	no	—
239	C₄²	—	106	2	no	—
240	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
241	C₄²	#11719945	1	1	yes	characteristic
242	C₄²	#11719945	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
244	C₂³	C₄²×D₅₃	1	1	yes	characteristic
245	Q₈	—	106	4	no	—
246	Q₈	—	106	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	53	4	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	106	2	no	—
253	C₂×C₄	—	106	2	no	—
254	C₂×C₄	—	212	2	no	—
255	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
256	C₂×C₄	#13133174	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
262	C₂×C₄	#13133223	1	2	yes	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	106	4	no	—
269	C₈	—	212	2	no	—
270	C₈	—	212	2	no	—
271	C₈	#13133135	1	2	yes	direct product factor
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₄×#11719921	1	1	yes	characteristic
275	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
276	C₂²	#16810577	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	106	4	no	—
281	C₄	—	212	2	no	—
282	C₄	C₂×#13133135	1	1	yes	characteristic
283	C₄	#16810605	1	1	yes	characteristic
284	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
285	C₄	C₈×D₂₁₂	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₈×#11719921	1	1	yes	characteristic
289	C₂	C₄×#13133135	1	1	yes	characteristic
290	C₂	#20274516	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11719921
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	C₈×#101
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
43. #13133135	C₈	direct (N × H)
122. C₈×C₅₃	#101	semidirect
173. C₅₃	C₈×#101	semidirect
271. C₈	#13133135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
C₄×#13133135	2
#20273038	2
C₈×#11719917	2
#20272566	2
#20272564	2
C₈×M₁₆×C₅₃	2
C₈×#101	53

Subgroups — C8×#13133135 #27274656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133135 #27274656