Subgroups — #499×C₃₆₇ #37553817

Order 23488.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×C₃₆₇	C₂	1	2	yes	maximal
2	#107×C₃₆₇	C₂	1	2	yes	maximal
3	#106×C₃₆₇	C₂	1	2	yes	maximal
4	C₄×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₃₆₇	—	2	4	no	—
8	C₄⋊C₄×C₃₆₇	C₂²	1	4	yes	—
9	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₃₆₇	—	2	4	no	—
11	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
12	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
13	C₈×C₃₆₇	Q₈	1	2	yes	—
14	C₈×C₃₆₇	Q₈	1	2	yes	—
15	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₃₆₇	—	2	2	no	—
17	C₄×C₃₆₇	—	4	4	no	—
18	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₃₆₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₃₆₇	#135	1	2	yes	—
22	C₃₆₇	#499	1	1	yes	characteristic, direct product factor, Sylow
23	#499	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	#125	C₂×C₃₆₇	1	2	yes	—
25	#107	C₂×C₃₆₇	1	2	yes	—
26	#106	C₂×C₃₆₇	1	2	yes	—
27	C₄×C₈	C₂×C₃₆₇	1	1	yes	characteristic
28	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
29	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
30	C₄⋊C₄	—	2	4	no	—
31	C₄⋊C₄	C₂²×C₃₆₇	1	4	yes	—
32	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
35	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
36	C₈	Q₈×C₃₆₇	1	2	yes	—
37	C₈	Q₈×C₃₆₇	1	2	yes	—
38	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	4	4	no	—
41	C₄	C₂×Q₈×C₃₆₇	1	1	yes	characteristic
42	C₄	C₂×Q₈×C₃₆₇	1	1	yes	characteristic
43	C₂	C₄⋊Q₈×C₃₆₇	1	1	yes	characteristic
44	C₂	#135×C₃₆₇	1	2	yes	—

	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
22. C₃₆₇	#499	direct (N × H)
23. #499	C₃₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#125×C₃₆₇	2
#107×C₃₆₇	2
#106×C₃₆₇	2
C₄×C₈×C₃₆₇	2
#499	367

Subgroups — #499×C367 #37553817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #499×C₃₆₇ #37553817