Subgroups — #472×C₃₆₇ #37553790

Order 23488.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,ℤ/8)×C₃₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#108×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#100×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	#41×C₃₆₇	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₃₆₇	C₂²	1	2	yes	—
8	SD₁₆×C₃₆₇	—	2	2	no	—
9	D₈×C₃₆₇	—	2	2	no	—
10	D₈×C₃₆₇	—	2	1	no	—
11	D₈×C₃₆₇	—	2	1	no	—
12	M₁₆×C₃₆₇	—	2	1	no	—
13	M₁₆×C₃₆₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₃₆₇	C₂²	1	2	yes	—
15	C₂×C₈×C₃₆₇	—	2	1	no	—
16	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₃₆₇	—	2	2	no	—
18	Q₈×C₃₆₇	D₄	1	1	yes	characteristic
19	D₄×C₃₆₇	—	2	1	no	—
20	D₄×C₃₆₇	—	2	2	no	—
21	D₄×C₃₆₇	—	2	2	no	—
22	D₄×C₃₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₆₇	—	2	1	no	—
24	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
25	C₈×C₃₆₇	—	2	1	no	—
26	C₈×C₃₆₇	—	2	2	no	—
27	C₈×C₃₆₇	D₄	1	1	yes	characteristic
28	C₈×C₃₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₃₆₇	—	2	1	no	—
30	C₂²×C₃₆₇	—	4	2	no	—
31	C₂²×C₃₆₇	—	8	2	no	—
32	C₂²×C₃₆₇	#41	1	1	yes	characteristic
33	C₄×C₃₆₇	—	2	1	no	—
34	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃₆₇	—	2	1	no	—
37	C₂×C₃₆₇	—	4	1	no	—
38	C₂×C₃₆₇	—	8	2	no	—
39	C₂×C₃₆₇	#121	1	1	yes	characteristic
40	C₃₆₇	#472	1	1	yes	characteristic, direct product factor, Sylow
41	#472	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	AGL(1,ℤ/8)	C₂×C₃₆₇	1	2	yes	complemented
43	C₂×D₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
44	#131	C₂×C₃₆₇	1	1	yes	characteristic, complemented
45	#108	C₂×C₃₆₇	1	1	yes	characteristic, complemented
46	#100	C₂×C₃₆₇	1	2	yes	complemented
47	#41	C₂²×C₃₆₇	1	1	yes	characteristic
48	C₂×D₄	C₂²×C₃₆₇	1	2	yes	—
49	SD₁₆	—	2	2	no	—
50	D₈	—	2	2	no	—
51	D₈	—	2	1	no	—
52	D₈	—	2	1	no	—
53	M₁₆	—	2	1	no	—
54	M₁₆	C₂²×C₃₆₇	1	1	yes	characteristic
55	M₁₆	C₂²×C₃₆₇	1	2	yes	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	D₄×C₃₆₇	1	1	yes	characteristic
60	D₄	—	2	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	D₄×C₃₆₇	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	D₄×C₃₆₇	1	1	yes	characteristic
69	C₈	D₄×C₃₆₇	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	8	2	no	—
73	C₂²	#41×C₃₆₇	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×D₄×C₃₆₇	1	1	yes	characteristic
76	C₄	C₂×D₄×C₃₆₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	8	2	no	—
80	C₂	#121×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₆₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₆₇ ◁ C₂×C₄×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. AGL(1,ℤ/8)×C₃₆₇	C₂	semidirect
2. C₂×D₈×C₃₆₇	C₂	semidirect
3. #131×C₃₆₇	C₂	semidirect
4. #108×C₃₆₇	C₂	semidirect
5. #100×C₃₆₇	C₂	semidirect
40. C₃₆₇	#472	direct (N × H)
41. #472	C₃₆₇	direct (N × H)
42. AGL(1,ℤ/8)	C₂×C₃₆₇	semidirect
43. C₂×D₈	C₂×C₃₆₇	semidirect
44. #131	C₂×C₃₆₇	semidirect
45. #108	C₂×C₃₆₇	semidirect
46. #100	C₂×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
AGL(1,ℤ/8)×C₃₆₇	2
C₂×D₈×C₃₆₇	2
#131×C₃₆₇	2
#108×C₃₆₇	2
#100×C₃₆₇	2
#472	367

Subgroups — #472×C367 #37553790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #472×C₃₆₇ #37553790