Subgroups — #474×C₂₇₇ #31799608

Order 17728.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic
3	#107×C₂₇₇	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#102×C₂₇₇	C₂	1	2	yes	maximal
6	C₂×Q₈×C₂₇₇	—	2	1	no	—
7	C₂×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₇₇	—	2	1	no	—
9	C₂×C₈×C₂₇₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₇₇	—	2	1	no	—
11	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
12	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₂₇₇	—	2	1	no	—
14	C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂³×C₂₇₇	—	2	1	no	—
16	Q₈×C₂₇₇	—	4	2	no	—
17	D₄×C₂₇₇	—	2	1	no	—
18	D₄×C₂₇₇	Q₈	1	2	yes	complemented
19	C₂×C₄×C₂₇₇	—	2	1	no	—
20	C₂×C₄×C₂₇₇	—	2	2	no	—
21	C₂×C₄×C₂₇₇	—	4	2	no	—
22	C₂×C₄×C₂₇₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₇₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
25	C₈×C₂₇₇	—	2	2	no	—
26	C₂²×C₂₇₇	—	2	2	no	—
27	C₂²×C₂₇₇	—	4	1	no	—
28	C₂²×C₂₇₇	—	4	1	no	—
29	C₂²×C₂₇₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₇₇	—	2	1	no	—
31	C₄×C₂₇₇	—	4	1	no	—
32	C₄×C₂₇₇	—	4	2	no	—
33	C₄×C₂₇₇	D₈	1	2	yes	—
34	C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₇₇	#41	1	1	yes	characteristic
36	C₂×C₂₇₇	—	4	2	no	—
37	C₂×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₂₇₇	C₂×D₈	1	1	yes	characteristic
39	C₂×C₂₇₇	#137	1	1	yes	characteristic
40	C₂₇₇	#474	1	1	yes	characteristic, direct product factor, Sylow
41	#474	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₄⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
43	C₄×D₄	C₂×C₂₇₇	1	1	yes	characteristic
44	#107	C₂×C₂₇₇	1	2	yes	complemented
45	C₄⋊C₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
46	#102	C₂×C₂₇₇	1	2	yes	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	C₂²×C₂₇₇	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	C₂²×C₂₇₇	1	2	yes	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
53	C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Q₈×C₂₇₇	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	D₄×C₂₇₇	1	1	yes	characteristic
64	C₂×C₄	D₄×C₂₇₇	1	1	yes	characteristic
65	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×D₄×C₂₇₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	2	no	—
74	C₄	D₈×C₂₇₇	1	2	yes	—
75	C₄	C₂×Q₈×C₂₇₇	1	1	yes	characteristic
76	C₄	#41×C₂₇₇	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	C₂²⋊Q₈×C₂₇₇	1	1	yes	characteristic
79	C₂	C₂×D₈×C₂₇₇	1	1	yes	characteristic
80	C₂	#137×C₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₇₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₇₇
prime radical	C₂×D₄×C₂₇₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇₇ ◁ C₄²×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₂₇₇	C₂	semidirect
3. #107×C₂₇₇	C₂	semidirect
4. C₄⋊C₈×C₂₇₇	C₂	semidirect
18. D₄×C₂₇₇	Q₈	semidirect
40. C₂₇₇	#474	direct (N × H)
41. #474	C₂₇₇	direct (N × H)
42. C₄⋊Q₈	C₂×C₂₇₇	semidirect
44. #107	C₂×C₂₇₇	semidirect
45. C₄⋊C₈	C₂×C₂₇₇	semidirect
59. D₄	Q₈×C₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₂₇₇	2
C₄×D₄×C₂₇₇	2
#107×C₂₇₇	2
C₄⋊C₈×C₂₇₇	2
#102×C₂₇₇	2
#474	277

Subgroups — #474×C277 #31799608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #474×C₂₇₇ #31799608