Subgroups — C₂×Q₈×#11764308 #31801067

Order 17728.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11764308	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11764308	C₂	1	3	yes	maximal
3	C₂×C₄×#11764308	C₂	1	3	yes	maximal
4	C₂×Q₈×D₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764308	C₂²	1	3	yes	—
6	C₂²×#11764308	C₂²	1	1	yes	characteristic
7	C₄⋊#11764308	C₂²	1	12	yes	—
8	C₄×#11764308	C₂²	1	12	yes	—
9	C₂×Dic₅₅₄	C₄	1	3	yes	complemented
10	Q₈×D₂₇₇	C₄	1	4	yes	complemented
11	Q₈×D₂₇₇	C₂²	1	4	yes	—
12	C₂×C₄×D₂₇₇	C₂²	1	3	yes	—
13	C₂×Q₈×C₂₇₇	C₄	1	1	yes	characteristic, complemented
14	C₂×#11764308	—	2	6	no	—
15	C₂×#11764308	Q₈	1	4	yes	direct product factor
16	C₂×#11764308	C₂³	1	6	yes	—
17	C₂×#11764308	C₂³	1	2	yes	—
18	C₂²×D₂₇₇	C₂³	1	1	yes	characteristic
19	Dic₅₅₄	C₂×C₄	1	12	yes	complemented
20	C₂×Dic₂₇₇	C₂×C₄	1	3	yes	—
21	C₄×D₂₇₇	C₂×C₄	1	6	yes	—
22	C₄×D₂₇₇	C₂³	1	6	yes	—
23	Q₈×C₂₇₇	C₂×C₄	1	4	yes	complemented
24	C₂×C₄×C₂₇₇	C₂×C₄	1	3	yes	—
25	C₂²×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×D₂₇₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₇₇	C₂×Q₈	1	2	yes	—
28	C₂×D₂₇₇	#41	1	2	yes	—
29	C₂×D₂₇₇	C₂⁴	1	1	yes	characteristic
30	#11764308	—	2	6	no	—
31	#11764308	C₂×Q₈	1	4	yes	direct product factor
32	C₄×C₂₇₇	C₂²×C₄	1	6	yes	—
33	Dic₂₇₇	C₂²×C₄	1	6	yes	—
34	C₂×C₂₇₇	C₄×Q₈	1	2	yes	complemented
35	C₂×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
36	D₂₇₇	C₄×Q₈	1	2	yes	complemented
37	D₂₇₇	C₂²×Q₈	1	1	yes	characteristic
38	D₂₇₇	C₂×#41	1	1	yes	characteristic
39	C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄×Q₈	—	277	1	no	maximal, Sylow
41	C₂²×Q₈	—	277	1	no	—
42	C₄×Q₈	—	277	8	no	—
43	C₂×C₄⋊C₄	—	277	3	no	—
44	C₂×C₄²	—	277	3	no	—
45	C₂×Q₈	—	277	4	no	—
46	C₂×Q₈	—	277	4	no	—
47	C₂×Q₈	—	277	3	no	—
48	C₂×Q₈	#11764308	1	1	yes	characteristic, direct product factor
49	C₂²×C₄	—	277	1	no	—
50	C₂²×C₄	—	277	3	no	—
51	C₂²×C₄	—	277	3	no	—
52	C₄⋊C₄	—	277	12	no	—
53	C₄²	—	277	12	no	—
54	C₂³	—	277	1	no	—
55	Q₈	—	277	12	no	—
56	Q₈	C₂×#11764308	1	4	yes	direct product factor
57	C₂×C₄	—	277	4	no	—
58	C₂×C₄	—	277	2	no	—
59	C₂×C₄	—	277	3	no	—
60	C₂×C₄	—	277	6	no	—
61	C₂×C₄	—	277	6	no	—
62	C₂×C₄	—	277	6	no	—
63	C₂×C₄	—	554	6	no	—
64	C₂×C₄	C₂×#11764308	1	3	yes	—
65	C₂²	—	277	1	no	—
66	C₂²	—	277	1	no	—
67	C₂²	—	277	2	no	—
68	C₂²	—	277	2	no	—
69	C₂²	C₂²×#11764308	1	1	yes	characteristic
70	C₄	—	277	4	no	—
71	C₄	—	277	6	no	—
72	C₄	—	554	6	no	—
73	C₄	C₂²×#11764308	1	6	yes	—
74	C₂	—	277	1	no	—
75	C₂	—	277	1	no	—
76	C₂	—	277	2	no	—
77	C₂	Q₈×#11764308	1	2	yes	direct product factor
78	C₂	C₂³×#11764308	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11764308
commutator	C₂×C₂₇₇	C₂³×C₄
Frattini	C₂	C₂³×#11764308
Fitting	C₂×Q₈×C₂₇₇	C₄
socle	C₂²×C₂₇₇	C₂²×C₄
prime radical	C₂²×D₂₇₇	C₂³
cosocle	C₂×D₂₇₇	C₂⁴
hypercenter	C₂×Q₈	#11764308
hypocenter	C₂₇₇	C₂×C₄×Q₈
LCS	G ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11764308	C₂	direct (N × H)
9. C₂×Dic₅₅₄	C₄	semidirect
10. Q₈×D₂₇₇	C₄	semidirect
13. C₂×Q₈×C₂₇₇	C₄	semidirect
15. C₂×#11764308	Q₈	direct (N × H)
19. Dic₅₅₄	C₂×C₄	semidirect
23. Q₈×C₂₇₇	C₂×C₄	semidirect
31. #11764308	C₂×Q₈	direct (N × H)
34. C₂×C₂₇₇	C₄×Q₈	semidirect
36. D₂₇₇	C₄×Q₈	semidirect
39. C₂₇₇	C₂×C₄×Q₈	semidirect
48. C₂×Q₈	#11764308	direct (N × H)
56. Q₈	C₂×#11764308	direct (N × H)
77. C₂	Q₈×#11764308	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×#11764308	2
C₂×C₄⋊#11764308	2
C₂×C₄×#11764308	2
C₂×Q₈×D₂₇₇	2
C₂×C₄×Q₈	277

Subgroups — C2×Q8×#11764308 #31801067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11764308 #31801067