Subgroups — #477×C₃₆₇ #37553795

Order 23488.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
3	#107×C₃₆₇	C₂	1	2	yes	maximal
4	C₄⋊C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	#103×C₃₆₇	C₂	1	2	yes	maximal
6	C₂×Q₈×C₃₆₇	—	2	1	no	—
7	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
9	C₄⋊C₄×C₃₆₇	—	2	1	no	—
10	C₄⋊C₄×C₃₆₇	—	2	1	no	—
11	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
12	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
13	C₄²×C₃₆₇	—	2	1	no	—
14	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
15	Q₈×C₃₆₇	—	2	1	no	—
16	Q₈×C₃₆₇	—	4	2	no	—
17	Q₈×C₃₆₇	Q₈	1	2	yes	complemented
18	C₂×C₄×C₃₆₇	—	2	1	no	—
19	C₂×C₄×C₃₆₇	—	2	1	no	—
20	C₂×C₄×C₃₆₇	—	2	2	no	—
21	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₆₇	—	2	2	no	—
25	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₃₆₇	—	2	2	no	—
27	C₄×C₃₆₇	—	2	1	no	—
28	C₄×C₃₆₇	—	4	1	no	—
29	C₄×C₃₆₇	—	4	2	no	—
30	C₄×C₃₆₇	Q₁₆	1	2	yes	—
31	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
32	C₄×C₃₆₇	#41	1	1	yes	characteristic
33	C₂×C₃₆₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₃₆₇	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₃₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₃₆₇	#477	1	1	yes	characteristic, direct product factor, Sylow
37	#477	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₄⋊Q₈	C₂×C₃₆₇	1	1	yes	characteristic
39	C₄×Q₈	C₂×C₃₆₇	1	1	yes	characteristic
40	#107	C₂×C₃₆₇	1	2	yes	—
41	C₄⋊C₈	C₂×C₃₆₇	1	1	yes	characteristic
42	#103	C₂×C₃₆₇	1	2	yes	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	C₂²×C₃₆₇	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
49	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	4	2	no	—
54	Q₈	Q₈×C₃₆₇	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
59	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	—	4	1	no	—
67	C₄	Q₁₆×C₃₆₇	1	2	yes	—
68	C₄	C₂×Q₈×C₃₆₇	1	1	yes	characteristic
69	C₄	#41×C₃₆₇	1	1	yes	characteristic
70	C₂	C₂²⋊Q₈×C₃₆₇	1	1	yes	characteristic
71	C₂	C₂×Q₁₆×C₃₆₇	1	1	yes	characteristic
72	C₂	AGL(1,ℤ/8)×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ C₄²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. Q₈×C₃₆₇	Q₈	semidirect
36. C₃₆₇	#477	direct (N × H)
37. #477	C₃₆₇	direct (N × H)
54. Q₈	Q₈×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₃₆₇	2
C₄×Q₈×C₃₆₇	2
#107×C₃₆₇	2
C₄⋊C₈×C₃₆₇	2
#103×C₃₆₇	2
#477	367

Subgroups — #477×C367 #37553795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #477×C₃₆₇ #37553795