Subgroups — C₈×Q₈×#6157 #28318620

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×#6157	C₂	1	1	yes	maximal, characteristic
3	#20290508	C₂	1	3	yes	maximal
4	C₈×C₄⋊#6157	C₂	1	3	yes	maximal
5	C₄⋊C₈×#6157	C₂	1	3	yes	maximal
6	C₄×C₈×#6157	C₂	1	3	yes	maximal
7	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×#6157	C₄	1	2	yes	—
9	#16811672	C₄	1	6	yes	complemented
10	Q₈×#88324	C₄	1	2	yes	complemented
11	D₅₃⋊(C₄⋊C₈)	C₂²	1	3	yes	—
12	C₄×D₅₃⋊C₈	C₂²	1	3	yes	—
13	C₂×C₈×#6157	C₂²	1	3	yes	—
14	C₂×C₈×#6157	C₂²	1	3	yes	—
15	C₂×C₈×#6157	C₂²	1	3	yes	—
16	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×#6157	C₄	1	6	yes	—
18	C₄×C₄⋊#6157	C₂²	1	3	yes	—
19	C₄²×#6157	C₂²	1	3	yes	—
20	(C₂×#6157)⋊C₈	C₂²	1	6	yes	—
21	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
22	Q₈×#88322	C₄	1	1	yes	characteristic, complemented
23	C₈×Dic₁₀₆	C₄	1	3	yes	complemented
24	#16810747	C₄	1	3	yes	complemented
25	C₄⋊C₈×D₅₃	C₂²	1	3	yes	—
26	C₄×C₈×D₅₃	C₂²	1	3	yes	—
27	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₂×D₅₃⋊C₈	C₂³	1	3	yes	—
29	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
30	Q₈×#6157	C₈	1	16	yes	direct product factor
31	C₂×C₄⋊#6157	C₂×C₄	1	6	yes	—
32	C₂×C₄×#6157	C₂×C₄	1	6	yes	—
33	C₂×C₄×#6157	C₂×C₄	1	6	yes	—
34	C₂×C₄×#6157	C₂³	1	3	yes	—
35	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
36	#13133284	C₂×C₄	1	6	yes	—
37	C₄⋊#88324	C₂×C₄	1	6	yes	—
38	C₄×#88324	C₂×C₄	1	6	yes	—
39	C₄×#88324	C₂×C₄	1	6	yes	—
40	C₈×#6157	—	2	6	no	—
41	C₈×#6157	—	2	6	no	—
42	C₈×#6157	—	2	6	no	—
43	C₈×#6157	Q₈	1	4	yes	direct product factor
44	(C₂×#6157)⋊C₄	C₂×C₄	1	6	yes	—
45	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	3	yes	—
47	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
48	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₀₆	C₂×C₄	1	3	yes	—
50	#13133182	C₂×C₄	1	3	yes	—
51	C₄⋊C₄×D₅₃	C₂×C₄	1	3	yes	—
52	C₄²×D₅₃	C₂³	1	3	yes	—
53	C₄⋊#88322	C₂×C₄	1	3	yes	—
54	#13133146	C₂×C₄	1	6	yes	—
55	C₄×#88322	C₂×C₄	1	3	yes	—
56	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
58	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
61	C₂²×#6157	C₂²×C₄	1	6	yes	—
62	C₂²×#6157	C₂²×C₄	1	2	yes	—
63	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
64	Q₈×D₅₃	C₂×C₈	1	8	yes	—
65	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
66	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
67	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₀₆	C₄²	1	3	yes	—
69	C₂×#88324	C₂²×C₄	1	6	yes	—
70	C₂×#88324	C₂²×C₄	1	2	yes	—
71	C₄⋊#6157	C₂×C₈	1	24	yes	—
72	C₄×#6157	—	2	12	no	—
73	C₄×#6157	—	2	3	no	—
74	C₄×#6157	—	2	3	no	—
75	C₄×#6157	C₂×C₈	1	24	yes	—
76	C₄×#6157	C₂×Q₈	1	2	yes	—
77	C₄×#6157	#41	1	2	yes	—
78	D₅₃⋊C₈	—	2	3	no	—
79	D₅₃⋊C₈	—	2	3	no	—
80	D₅₃⋊C₈	C₂×Q₈	1	2	yes	—
81	D₅₃⋊C₈	#41	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	C₄²	1	3	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
86	C₄⋊Dic₅₃	C₄²	1	3	yes	—
87	#11719919	C₄²	1	6	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
90	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
91	C₂×#88322	C₂²×C₄	1	3	yes	—
92	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₅₃	—	2	3	no	—
94	C₈×D₅₃	—	2	3	no	—
95	C₈×D₅₃	C₂×Q₈	1	2	yes	—
96	C₈×D₅₃	#41	1	2	yes	—
97	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×#6157	—	2	6	no	—
99	C₂×#6157	—	2	6	no	—
100	C₂×#6157	C₄×Q₈	1	4	yes	—
101	C₂×#6157	C₂²×C₈	1	12	yes	—
102	C₂×#6157	C₂²×C₈	1	4	yes	—
103	C₂×#6157	#131	1	4	yes	—
104	Q₈×C₅₃	C₄×C₈	1	4	yes	complemented
105	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
109	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
110	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₄×D₅₃	—	2	6	no	—
112	C₄×D₅₃	C₂²×C₈	1	12	yes	—
113	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
114	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
115	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
116	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
117	Dic₁₀₆	C₄×C₈	1	12	yes	complemented
118	#88324	—	2	6	no	—
119	#88324	C₄×Q₈	1	4	yes	complemented
120	C₈×C₅₃	—	2	3	no	—
121	C₈×C₅₃	C₄×Q₈	1	2	yes	complemented
122	#88322	—	2	3	no	—
123	#88322	C₄×Q₈	1	2	yes	complemented
124	C₄×C₈×Q₈	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
126	C₂×D₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₂×D₅₃	C₄×#41	1	1	yes	characteristic
128	C₂×D₅₃	C₂³×C₈	1	2	yes	—
129	C₂×D₅₃	C₂×#131	1	2	yes	—
130	#6157	—	2	12	no	—
131	#6157	C₈×Q₈	1	8	yes	direct product factor
132	C₄×C₅₃	—	2	3	no	—
133	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
134	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
136	Dic₅₃	—	2	3	no	—
137	Dic₅₃	C₂×C₄×C₈	1	6	yes	—
138	Dic₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
139	Dic₅₃	C₄×#41	1	1	yes	characteristic
140	C₂×C₈×Q₈	—	53	1	no	—
141	C₂×C₈×Q₈	—	53	1	no	—
142	C₄²×Q₈	—	53	1	no	—
143	C₈×C₄⋊C₄	—	53	3	no	—
144	C₈×C₄⋊C₄	—	53	3	no	—
145	C₄×C₄⋊C₈	—	53	3	no	—
146	C₄²×C₈	—	53	3	no	—
147	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
148	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
149	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
150	D₅₃	C₂×C₈×Q₈	1	2	yes	—
151	D₅₃	C₈×#41	1	2	yes	—
152	C₂×C₄×Q₈	—	53	2	no	—
153	C₂×C₄×Q₈	—	53	1	no	—
154	C₈×Q₈	—	53	2	no	—
155	C₈×Q₈	—	53	6	no	—
156	C₈×Q₈	—	53	1	no	—
157	C₈×Q₈	—	53	3	no	—
158	C₈×Q₈	—	53	3	no	—
159	C₈×Q₈	#6157	1	1	yes	characteristic, direct product factor
160	C₂×C₄⋊C₈	—	53	3	no	—
161	C₂×C₄⋊C₈	—	53	3	no	—
162	C₂×C₄×C₈	—	53	1	no	—
163	C₂×C₄×C₈	—	53	3	no	—
164	C₂×C₄×C₈	—	53	3	no	—
165	C₂×C₄×C₈	—	53	3	no	—
166	C₂×C₄×C₈	—	53	3	no	—
167	C₂×C₄×C₈	—	53	3	no	—
168	C₄×C₄⋊C₄	—	53	3	no	—
169	C₄×C₄⋊C₄	—	53	6	no	—
170	C₄³	—	53	3	no	—
171	#336	—	53	6	no	—
172	C₅₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
173	C₂²×Q₈	—	53	1	no	—
174	C₂²×C₈	—	53	1	no	—
175	C₂²×C₈	—	53	1	no	—
176	C₂²×C₈	—	53	3	no	—
177	C₂²×C₈	—	53	3	no	—
178	C₄×Q₈	—	53	16	no	—
179	C₄×Q₈	—	53	1	no	—
180	C₄×Q₈	—	53	3	no	—
181	C₄×Q₈	—	53	3	no	—
182	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	53	6	no	—
184	C₂×C₄⋊C₄	—	53	3	no	—
185	C₂×C₄²	—	53	6	no	—
186	C₂×C₄²	—	53	1	no	—
187	C₂×C₄²	—	53	3	no	—
188	C₂×C₄²	—	53	3	no	—
189	C₂×C₄²	—	53	6	no	—
190	C₄⋊C₈	—	53	6	no	—
191	C₄⋊C₈	—	53	6	no	—
192	C₄⋊C₈	—	53	6	no	—
193	C₄⋊C₈	—	53	3	no	—
194	C₄⋊C₈	C₂×#6157	1	3	yes	—
195	C₄×C₈	—	53	6	no	—
196	C₄×C₈	—	53	6	no	—
197	C₄×C₈	—	53	4	no	—
198	C₄×C₈	—	53	3	no	—
199	C₄×C₈	—	53	3	no	—
200	C₄×C₈	—	53	3	no	—
201	C₄×C₈	—	106	6	no	—
202	C₄×C₈	—	106	6	no	—
203	C₄×C₈	—	106	6	no	—
204	C₄×C₈	C₂×#6157	1	3	yes	—
205	#95	—	53	6	no	—
206	C₂×Q₈	—	53	8	no	—
207	C₂×Q₈	—	53	3	no	—
208	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
209	C₂²×C₄	—	53	2	no	—
210	C₂²×C₄	—	53	3	no	—
211	C₂²×C₄	—	53	1	no	—
212	C₂²×C₄	—	53	6	no	—
213	C₂²×C₄	—	53	3	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	2	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	53	6	no	—
221	C₂×C₈	—	53	3	no	—
222	C₂×C₈	—	106	3	no	—
223	C₂×C₈	—	106	3	no	—
224	C₂×C₈	—	106	3	no	—
225	C₂×C₈	—	106	3	no	—
226	C₂×C₈	C₂²×#6157	1	3	yes	—
227	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
228	C₄⋊C₄	—	53	24	no	—
229	C₄⋊C₄	—	53	6	no	—
230	C₄⋊C₄	—	53	3	no	—
231	C₄⋊C₄	C₄×#6157	1	3	yes	—
232	C₄²	—	53	24	no	—
233	C₄²	—	53	2	no	—
234	C₄²	—	53	2	no	—
235	C₄²	—	53	3	no	—
236	C₄²	—	53	3	no	—
237	C₄²	—	53	3	no	—
238	C₄²	—	106	3	no	—
239	C₄²	—	106	3	no	—
240	C₄²	—	106	12	no	—
241	C₄²	C₂²×#6157	1	3	yes	—
242	C₂³	—	53	1	no	—
243	Q₈	—	53	12	no	—
244	Q₈	C₈×#6157	1	4	yes	direct product factor
245	C₂×C₄	—	53	4	no	—
246	C₂×C₄	—	53	4	no	—
247	C₂×C₄	—	53	4	no	—
248	C₂×C₄	—	53	3	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	53	12	no	—
251	C₂×C₄	—	53	1	no	—
252	C₂×C₄	—	53	1	no	—
253	C₂×C₄	—	53	1	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	12	no	—
256	C₂×C₄	—	53	3	no	—
257	C₂×C₄	—	106	6	no	—
258	C₂×C₄	—	106	6	no	—
259	C₂×C₄	—	106	6	no	—
260	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
261	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
262	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
263	C₈	—	2	3	no	—
264	C₈	—	53	4	no	—
265	C₈	—	53	2	no	—
266	C₈	—	106	6	no	—
267	C₈	—	106	3	no	—
268	C₈	Q₈×#6157	1	2	yes	direct product factor
269	C₂²	—	53	1	no	—
270	C₂²	—	53	1	no	—
271	C₂²	—	53	2	no	—
272	C₂²	—	53	2	no	—
273	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
274	C₄	—	2	3	no	—
275	C₄	—	53	8	no	—
276	C₄	—	53	1	no	—
277	C₄	—	53	1	no	—
278	C₄	—	53	6	no	—
279	C₄	—	106	12	no	—
280	C₄	—	106	3	no	—
281	C₄	C₂×C₈×#6157	1	6	yes	—
282	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
283	C₄	#41×#6157	1	1	yes	characteristic
284	C₂	—	53	2	no	—
285	C₂	—	53	2	no	—
286	C₂	C₄×Q₈×#6157	1	1	yes	characteristic
287	C₂	C₂²×C₈×#6157	1	1	yes	characteristic
288	C₂	#131×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂²×D₅₃	C₂³×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	C₄×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. #16811672	C₄	semidirect
10. Q₈×#88324	C₄	semidirect
22. Q₈×#88322	C₄	semidirect
23. C₈×Dic₁₀₆	C₄	semidirect
24. #16810747	C₄	semidirect
27. C₈×Q₈×C₅₃	C₄	semidirect
30. Q₈×#6157	C₈	direct (N × H)
43. C₈×#6157	Q₈	direct (N × H)
104. Q₈×C₅₃	C₄×C₈	semidirect
117. Dic₁₀₆	C₄×C₈	semidirect
119. #88324	C₄×Q₈	semidirect
121. C₈×C₅₃	C₄×Q₈	semidirect
123. #88322	C₄×Q₈	semidirect
131. #6157	C₈×Q₈	direct (N × H)
159. C₈×Q₈	#6157	direct (N × H)
172. C₅₃	C₄×C₈×Q₈	semidirect
244. Q₈	C₈×#6157	direct (N × H)
268. C₈	Q₈×#6157	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×D₅₃⋊C₈	2
C₄×Q₈×#6157	2
#20290508	2
C₈×C₄⋊#6157	2
C₄⋊C₈×#6157	2
C₄×C₈×#6157	2
C₈×Q₈×D₅₃	2
C₄×C₈×Q₈	53

Subgroups — C8×Q8×#6157 #28318620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₈×#6157 #28318620