Subgroups — C₂²×D₈×C₃₆₇ #37553887

Order 23488.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₃₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×C₃₆₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₃₆₇	—	2	2	no	—
5	C₂×D₄×C₃₆₇	—	2	6	no	—
6	C₂×D₄×C₃₆₇	C₂²	1	12	yes	complemented
7	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
8	D₈×C₃₆₇	C₂²	1	16	yes	direct product factor
9	C₂×C₈×C₃₆₇	C₂²	1	6	yes	complemented
10	C₂³×C₃₆₇	—	2	12	no	—
11	C₂³×C₃₆₇	—	4	8	no	—
12	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
13	D₄×C₃₆₇	—	2	12	no	—
14	D₄×C₃₆₇	C₂³	1	8	yes	complemented
15	C₂×C₄×C₃₆₇	D₄	1	3	yes	—
16	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
17	C₈×C₃₆₇	C₂³	1	4	yes	complemented
18	C₂²×C₃₆₇	—	2	8	no	—
19	C₂²×C₃₆₇	—	4	24	no	—
20	C₂²×C₃₆₇	D₈	1	4	yes	direct product factor
21	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
22	C₄×C₃₆₇	C₂×D₄	1	3	yes	—
23	C₄×C₃₆₇	C₂⁴	1	1	yes	characteristic
24	C₂×C₃₆₇	—	4	8	no	—
25	C₂×C₃₆₇	C₂×D₈	1	6	yes	direct product factor
26	C₂×C₃₆₇	C₂²×D₄	1	1	yes	characteristic
27	C₃₆₇	C₂²×D₈	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²×D₈	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×D₄	C₂×C₃₆₇	1	2	yes	complemented
30	C₂×D₈	C₂×C₃₆₇	1	12	yes	direct product factor
31	C₂²×C₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
32	C₂⁴	—	2	2	no	—
33	C₂×D₄	—	2	6	no	—
34	C₂×D₄	C₂²×C₃₆₇	1	12	yes	complemented
35	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
36	D₈	C₂²×C₃₆₇	1	16	yes	direct product factor
37	C₂×C₈	C₂²×C₃₆₇	1	6	yes	complemented
38	C₂³	—	2	12	no	—
39	C₂³	—	4	8	no	—
40	C₂³	D₄×C₃₆₇	1	1	yes	characteristic
41	D₄	—	2	12	no	—
42	D₄	C₂³×C₃₆₇	1	8	yes	complemented
43	C₂×C₄	D₄×C₃₆₇	1	3	yes	—
44	C₂×C₄	C₂³×C₃₆₇	1	3	yes	—
45	C₈	C₂³×C₃₆₇	1	4	yes	complemented
46	C₂²	—	2	8	no	—
47	C₂²	—	4	24	no	—
48	C₂²	D₈×C₃₆₇	1	4	yes	direct product factor
49	C₂²	C₂×D₄×C₃₆₇	1	3	yes	—
50	C₄	C₂×D₄×C₃₆₇	1	3	yes	—
51	C₄	C₂⁴×C₃₆₇	1	1	yes	characteristic
52	C₂	—	4	8	no	—
53	C₂	C₂×D₈×C₃₆₇	1	6	yes	direct product factor
54	C₂	C₂²×D₄×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₆₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₆₇ ◁ C₂²×C₄×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×C₃₆₇	C₂	semidirect
2. C₂×D₈×C₃₆₇	C₂	direct (N × H)
3. C₂²×C₈×C₃₆₇	C₂	semidirect
6. C₂×D₄×C₃₆₇	C₂²	semidirect
8. D₈×C₃₆₇	C₂²	direct (N × H)
9. C₂×C₈×C₃₆₇	C₂²	semidirect
14. D₄×C₃₆₇	C₂³	semidirect
17. C₈×C₃₆₇	C₂³	semidirect
20. C₂²×C₃₆₇	D₈	direct (N × H)
25. C₂×C₃₆₇	C₂×D₈	direct (N × H)
27. C₃₆₇	C₂²×D₈	direct (N × H)
28. C₂²×D₈	C₃₆₇	direct (N × H)
29. C₂²×D₄	C₂×C₃₆₇	semidirect
30. C₂×D₈	C₂×C₃₆₇	direct (N × H)
31. C₂²×C₈	C₂×C₃₆₇	semidirect
34. C₂×D₄	C₂²×C₃₆₇	semidirect
36. D₈	C₂²×C₃₆₇	direct (N × H)
37. C₂×C₈	C₂²×C₃₆₇	semidirect
42. D₄	C₂³×C₃₆₇	semidirect
45. C₈	C₂³×C₃₆₇	semidirect
48. C₂²	D₈×C₃₆₇	direct (N × H)
53. C₂	C₂×D₈×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
C₂²×D₄×C₃₆₇	2
C₂×D₈×C₃₆₇	2
C₂²×C₈×C₃₆₇	2
C₂²×D₈	367

Subgroups — C22×D8×C367 #37553887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₈×C₃₆₇ #37553887