Subgroups — #458×C₃₆₇ #37553776

Order 23488.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#123×C₃₆₇	C₂	1	3	yes	maximal, complemented
3	#99×C₃₆₇	C₂	1	3	yes	maximal, complemented
4	#41×C₃₆₇	—	2	3	no	—
5	C₂×D₄×C₃₆₇	—	2	3	no	—
6	C₂×D₄×C₃₆₇	C₂²	1	3	yes	—
7	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic, complemented
8	C₄⋊C₄×C₃₆₇	—	2	3	no	—
9	C₂²⋊C₄×C₃₆₇	—	2	3	no	—
10	C₂²⋊C₄×C₃₆₇	C₂²	1	3	yes	complemented
11	C₂³×C₃₆₇	—	2	1	no	—
12	C₂³×C₃₆₇	D₄	1	3	yes	—
13	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
14	Q₈×C₃₆₇	—	2	1	no	—
15	D₄×C₃₆₇	—	2	3	no	—
16	D₄×C₃₆₇	—	2	6	no	—
17	C₂×C₄×C₃₆₇	—	2	3	no	—
18	C₂×C₄×C₃₆₇	—	2	3	no	—
19	C₂×C₄×C₃₆₇	—	2	3	no	—
20	C₂×C₄×C₃₆₇	D₄	1	3	yes	—
21	C₂²×C₃₆₇	—	2	3	no	—
22	C₂²×C₃₆₇	—	4	1	no	—
23	C₂²×C₃₆₇	—	4	3	no	—
24	C₂²×C₃₆₇	—	8	1	no	—
25	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
26	C₄×C₃₆₇	—	2	3	no	—
27	C₄×C₃₆₇	—	4	3	no	—
28	C₄×C₃₆₇	—	4	1	no	—
29	C₂×C₃₆₇	—	2	3	no	—
30	C₂×C₃₆₇	—	4	3	no	—
31	C₂×C₃₆₇	#120	1	1	yes	characteristic
32	C₃₆₇	#458	1	1	yes	characteristic, direct product factor, Sylow
33	#458	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	H(5,𝔽₂)	C₂×C₃₆₇	1	1	yes	characteristic
35	#123	C₂×C₃₆₇	1	3	yes	complemented
36	#99	C₂×C₃₆₇	1	3	yes	complemented
37	#41	—	2	3	no	—
38	C₂×D₄	—	2	3	no	—
39	C₂×D₄	C₂²×C₃₆₇	1	3	yes	—
40	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic, complemented
41	C₄⋊C₄	—	2	3	no	—
42	C₂²⋊C₄	—	2	3	no	—
43	C₂²⋊C₄	C₂²×C₃₆₇	1	3	yes	complemented
44	C₂³	—	2	1	no	—
45	C₂³	D₄×C₃₆₇	1	3	yes	—
46	C₂³	C₂³×C₃₆₇	1	1	yes	characteristic
47	Q₈	—	2	1	no	—
48	D₄	—	2	3	no	—
49	D₄	—	2	6	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	—	2	3	no	—
53	C₂×C₄	D₄×C₃₆₇	1	3	yes	—
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×D₄×C₃₆₇	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	4	3	no	—
61	C₄	—	4	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#120×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₆₇	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₃₆₇
prime radical	H(5,𝔽₂)×C₃₆₇	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₆₇ ◁ C₂³×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #123×C₃₆₇	C₂	semidirect
3. #99×C₃₆₇	C₂	semidirect
7. C₂²×C₄×C₃₆₇	C₂²	semidirect
10. C₂²⋊C₄×C₃₆₇	C₂²	semidirect
32. C₃₆₇	#458	direct (N × H)
33. #458	C₃₆₇	direct (N × H)
35. #123	C₂×C₃₆₇	semidirect
36. #99	C₂×C₃₆₇	semidirect
40. C₂²×C₄	C₂²×C₃₆₇	semidirect
43. C₂²⋊C₄	C₂²×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×C₃₆₇	2
#123×C₃₆₇	2
#99×C₃₆₇	2
#458	367

Subgroups — #458×C367 #37553776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #458×C₃₆₇ #37553776