Subgroups — #466×C₃₆₇ #37553784

Order 23488.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×C₃₆₇	C₂	1	2	yes	maximal, complemented
4	#107×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#102×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×C₃₆₇	—	2	2	no	—
7	C₂×D₄×C₃₆₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
9	D₈×C₃₆₇	—	2	2	no	—
10	C₂×C₈×C₃₆₇	—	2	2	no	—
11	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₃₆₇	—	2	2	no	—
15	C₂³×C₃₆₇	—	2	2	no	—
16	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
17	D₄×C₃₆₇	—	2	2	no	—
18	D₄×C₃₆₇	—	2	2	no	—
19	D₄×C₃₆₇	—	4	4	no	—
20	C₂×C₄×C₃₆₇	—	2	1	no	—
21	C₂×C₄×C₃₆₇	—	2	1	no	—
22	C₂×C₄×C₃₆₇	—	2	2	no	—
23	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
25	C₈×C₃₆₇	—	2	1	no	—
26	C₈×C₃₆₇	D₄	1	2	yes	complemented
27	C₂²×C₃₆₇	—	2	1	no	—
28	C₂²×C₃₆₇	—	2	1	no	—
29	C₂²×C₃₆₇	—	4	2	no	—
30	C₂²×C₃₆₇	—	4	2	no	—
31	C₂²×C₃₆₇	—	4	2	no	—
32	C₂²×C₃₆₇	D₈	1	2	yes	complemented
33	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₆₇	—	2	1	no	—
35	C₄×C₃₆₇	—	4	2	no	—
36	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₃₆₇	#41	1	1	yes	characteristic
38	C₂×C₃₆₇	—	2	2	no	—
39	C₂×C₃₆₇	—	8	2	no	—
40	C₂×C₃₆₇	#121	1	1	yes	characteristic
41	C₂×C₃₆₇	C₂×D₈	1	1	yes	characteristic
42	C₂×C₃₆₇	#135	1	1	yes	characteristic
43	C₃₆₇	#466	1	1	yes	characteristic, direct product factor, Sylow
44	#466	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×D₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
46	C₂²×C₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
47	#121	C₂×C₃₆₇	1	2	yes	complemented
48	#107	C₂×C₃₆₇	1	1	yes	characteristic, complemented
49	#102	C₂×C₃₆₇	1	2	yes	complemented
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	C₂²×C₃₆₇	1	2	yes	—
52	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
53	D₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
56	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
57	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
58	C₂²⋊C₄	—	2	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	D₄×C₃₆₇	1	1	yes	characteristic
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	—	4	4	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
68	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	D₄×C₃₆₇	1	2	yes	complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	D₈×C₃₆₇	1	2	yes	complemented
77	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	4	2	no	—
80	C₄	C₂×D₄×C₃₆₇	1	1	yes	characteristic
81	C₄	#41×C₃₆₇	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	8	2	no	—
84	C₂	#121×C₃₆₇	1	1	yes	characteristic
85	C₂	C₂×D₈×C₃₆₇	1	1	yes	characteristic
86	C₂	#135×C₃₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×C₃₆₇	C₂	semidirect
2. C₂²×C₈×C₃₆₇	C₂	semidirect
3. #121×C₃₆₇	C₂	semidirect
4. #107×C₃₆₇	C₂	semidirect
5. #102×C₃₆₇	C₂	semidirect
26. C₈×C₃₆₇	D₄	semidirect
32. C₂²×C₃₆₇	D₈	semidirect
43. C₃₆₇	#466	direct (N × H)
44. #466	C₃₆₇	direct (N × H)
45. C₂×D₈	C₂×C₃₆₇	semidirect
46. C₂²×C₈	C₂×C₃₆₇	semidirect
47. #121	C₂×C₃₆₇	semidirect
48. #107	C₂×C₃₆₇	semidirect
49. #102	C₂×C₃₆₇	semidirect
70. C₈	D₄×C₃₆₇	semidirect
76. C₂²	D₈×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×C₃₆₇	2
C₂²×C₈×C₃₆₇	2
#121×C₃₆₇	2
#107×C₃₆₇	2
#102×C₃₆₇	2
#466	367

Subgroups — #466×C367 #37553784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #466×C₃₆₇ #37553784