Subgroups — D₄²×C₃₆₇ #37553863

Order 23488.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₃₆₇	C₂	1	4	yes	maximal, complemented
2	#127×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×C₃₆₇	C₂	1	4	yes	maximal, complemented
4	#120×C₃₆₇	C₂	1	4	yes	maximal, complemented
5	C₄×D₄×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₃₆₇	C₂²	1	4	yes	complemented
7	C₂×D₄×C₃₆₇	—	2	16	no	—
8	C₂×D₄×C₃₆₇	C₂²	1	2	yes	—
9	C₂×D₄×C₃₆₇	C₂²	1	8	yes	complemented
10	C₂×D₄×C₃₆₇	C₂²	1	4	yes	complemented
11	C₂×D₄×C₃₆₇	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₃₆₇	C₂²	1	4	yes	complemented
13	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₃₆₇	C₂²	1	4	yes	complemented
15	C₂²⋊C₄×C₃₆₇	C₂²	1	4	yes	complemented
16	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₃₆₇	—	2	16	no	—
18	C₂³×C₃₆₇	—	4	4	no	—
19	C₂³×C₃₆₇	C₂³	1	4	yes	—
20	C₂³×C₃₆₇	C₂³	1	4	yes	—
21	D₄×C₃₆₇	—	2	16	no	—
22	D₄×C₃₆₇	—	2	8	no	—
23	D₄×C₃₆₇	—	4	2	no	—
24	D₄×C₃₆₇	D₄	1	8	yes	direct product factor
25	C₂×C₄×C₃₆₇	—	2	8	no	—
26	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₃₆₇	C₂³	1	4	yes	—
28	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
29	C₂²×C₃₆₇	—	2	4	no	—
30	C₂²×C₃₆₇	—	2	4	no	—
31	C₂²×C₃₆₇	—	2	8	no	—
32	C₂²×C₃₆₇	—	2	4	no	—
33	C₂²×C₃₆₇	—	4	16	no	—
34	C₂²×C₃₆₇	C₂×D₄	1	8	yes	complemented
35	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
36	C₄×C₃₆₇	—	2	4	no	—
37	C₄×C₃₆₇	—	2	1	no	—
38	C₄×C₃₆₇	C₂×D₄	1	4	yes	complemented
39	C₂×C₃₆₇	—	2	8	no	—
40	C₂×C₃₆₇	—	4	4	no	—
41	C₂×C₃₆₇	C₂²×D₄	1	2	yes	—
42	C₂×C₃₆₇	H(5,𝔽₂)	1	1	yes	characteristic
43	C₃₆₇	D₄²	1	1	yes	characteristic, direct product factor, Sylow
44	D₄²	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×D₄	C₂×C₃₆₇	1	4	yes	complemented
46	#127	C₂×C₃₆₇	1	1	yes	characteristic, complemented
47	#121	C₂×C₃₆₇	1	4	yes	complemented
48	#120	C₂×C₃₆₇	1	4	yes	complemented
49	C₄×D₄	C₂×C₃₆₇	1	2	yes	complemented
50	C₂⁴	C₂²×C₃₆₇	1	4	yes	complemented
51	C₂×D₄	—	2	16	no	—
52	C₂×D₄	C₂²×C₃₆₇	1	2	yes	—
53	C₂×D₄	C₂²×C₃₆₇	1	8	yes	complemented
54	C₂×D₄	C₂²×C₃₆₇	1	4	yes	complemented
55	C₂×D₄	C₂²×C₃₆₇	1	2	yes	complemented
56	C₂²×C₄	C₂²×C₃₆₇	1	4	yes	complemented
57	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	complemented
58	C₂²⋊C₄	C₂²×C₃₆₇	1	4	yes	complemented
59	C₂²⋊C₄	C₂²×C₃₆₇	1	4	yes	complemented
60	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic, complemented
61	C₂³	—	2	16	no	—
62	C₂³	—	4	4	no	—
63	C₂³	C₂³×C₃₆₇	1	4	yes	—
64	C₂³	C₂³×C₃₆₇	1	4	yes	—
65	D₄	—	2	16	no	—
66	D₄	—	2	8	no	—
67	D₄	—	4	2	no	—
68	D₄	D₄×C₃₆₇	1	8	yes	direct product factor
69	C₂×C₄	—	2	8	no	—
70	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
71	C₂×C₄	C₂³×C₃₆₇	1	4	yes	—
72	C₂×C₄	C₂³×C₃₆₇	1	2	yes	—
73	C₂²	—	2	4	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	8	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	4	16	no	—
78	C₂²	C₂×D₄×C₃₆₇	1	8	yes	complemented
79	C₂²	C₂⁴×C₃₆₇	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	2	1	no	—
82	C₄	C₂×D₄×C₃₆₇	1	4	yes	complemented
83	C₂	—	2	8	no	—
84	C₂	—	4	4	no	—
85	C₂	C₂²×D₄×C₃₆₇	1	2	yes	—
86	C₂	H(5,𝔽₂)×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₆₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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. #127×C₃₆₇	C₂	semidirect
3. #121×C₃₆₇	C₂	semidirect
4. #120×C₃₆₇	C₂	semidirect
5. C₄×D₄×C₃₆₇	C₂	semidirect
6. C₂⁴×C₃₆₇	C₂²	semidirect
9. C₂×D₄×C₃₆₇	C₂²	semidirect
10. C₂×D₄×C₃₆₇	C₂²	semidirect
11. C₂×D₄×C₃₆₇	C₂²	semidirect
12. C₂²×C₄×C₃₆₇	C₂²	semidirect
13. C₄⋊C₄×C₃₆₇	C₂²	semidirect
14. C₂²⋊C₄×C₃₆₇	C₂²	semidirect
15. C₂²⋊C₄×C₃₆₇	C₂²	semidirect
16. C₄²×C₃₆₇	C₂²	semidirect
24. D₄×C₃₆₇	D₄	direct (N × H)
34. C₂²×C₃₆₇	C₂×D₄	semidirect
38. C₄×C₃₆₇	C₂×D₄	semidirect
43. C₃₆₇	D₄²	direct (N × H)
44. D₄²	C₃₆₇	direct (N × H)
45. C₂²×D₄	C₂×C₃₆₇	semidirect
46. #127	C₂×C₃₆₇	semidirect
47. #121	C₂×C₃₆₇	semidirect
48. #120	C₂×C₃₆₇	semidirect
49. C₄×D₄	C₂×C₃₆₇	semidirect
50. C₂⁴	C₂²×C₃₆₇	semidirect
53. C₂×D₄	C₂²×C₃₆₇	semidirect
54. C₂×D₄	C₂²×C₃₆₇	semidirect
55. C₂×D₄	C₂²×C₃₆₇	semidirect
56. C₂²×C₄	C₂²×C₃₆₇	semidirect
57. C₄⋊C₄	C₂²×C₃₆₇	semidirect
58. C₂²⋊C₄	C₂²×C₃₆₇	semidirect
59. C₂²⋊C₄	C₂²×C₃₆₇	semidirect
60. C₄²	C₂²×C₃₆₇	semidirect
68. D₄	D₄×C₃₆₇	direct (N × H)
78. C₂²	C₂×D₄×C₃₆₇	semidirect
82. C₄	C₂×D₄×C₃₆₇	semidirect

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
#127×C₃₆₇	2
#121×C₃₆₇	2
#120×C₃₆₇	2
C₄×D₄×C₃₆₇	2
D₄²	367

Subgroups — D42×C367 #37553863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄²×C₃₆₇ #37553863