Subgroups — #428×C₃₆₇ #37553746

Order 23488.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#117×C₃₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₃₆₇	C₂	1	1	yes	maximal, characteristic
4	#107×C₃₆₇	C₂	1	2	yes	maximal, complemented
5	#106×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₃₆₇	—	2	1	no	—
7	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
8	M₁₆×C₃₆₇	C₄	1	4	yes	complemented
9	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₆₇	—	2	1	no	—
11	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₃₆₇	—	2	1	no	—
15	C₄²×C₃₆₇	—	2	1	no	—
16	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₃₆₇	—	2	1	no	—
18	C₂×C₄×C₃₆₇	—	2	1	no	—
19	C₂×C₄×C₃₆₇	—	2	2	no	—
20	C₂×C₄×C₃₆₇	—	4	2	no	—
21	C₂×C₄×C₃₆₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₆₇	Q₈	1	2	yes	—
24	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
25	C₈×C₃₆₇	C₂×C₄	1	4	yes	complemented
26	C₂²×C₃₆₇	—	2	1	no	—
27	C₂²×C₃₆₇	—	2	1	no	—
28	C₂²×C₃₆₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₃₆₇	—	4	1	no	—
31	C₄×C₃₆₇	—	4	2	no	—
32	C₄×C₃₆₇	—	4	1	no	—
33	C₄×C₃₆₇	C₄⋊C₄	1	2	yes	—
34	C₄×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₃₆₇	—	2	2	no	—
37	C₂×C₃₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₃₆₇	#137	1	1	yes	characteristic
40	C₃₆₇	#428	1	1	yes	characteristic, direct product factor, Sylow
41	#428	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×M₁₆	C₂×C₃₆₇	1	1	yes	characteristic
43	#117	C₂×C₃₆₇	1	1	yes	characteristic
44	C₂×C₄⋊C₄	C₂×C₃₆₇	1	1	yes	characteristic
45	#107	C₂×C₃₆₇	1	2	yes	complemented
46	#106	C₂×C₃₆₇	1	2	yes	complemented
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
49	M₁₆	C₄×C₃₆₇	1	4	yes	complemented
50	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
53	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
54	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
55	C₂²⋊C₄	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₂³	D₄×C₃₆₇	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	C₂×C₄×C₃₆₇	1	2	yes	—
63	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
64	C₂×C₄	Q₈×C₃₆₇	1	2	yes	—
65	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
66	C₈	C₂×C₄×C₃₆₇	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊C₄×C₃₆₇	1	2	yes	—
70	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	C₄⋊C₄×C₃₆₇	1	2	yes	—
75	C₄	C₂²×C₄×C₃₆₇	1	1	yes	characteristic
76	C₄	C₂×Q₈×C₃₆₇	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊C₄×C₃₆₇	1	1	yes	characteristic
79	C₂	AGL(1,ℤ/8)×C₃₆₇	1	1	yes	characteristic
80	C₂	#137×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₆₇	C₂×D₄
commutator	C₄	C₂²×C₄×C₃₆₇
Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
prime radical	C₂³×C₃₆₇	D₄
LCS	G ▷ 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
4. #107×C₃₆₇	C₂	semidirect
5. #106×C₃₆₇	C₂	semidirect
8. M₁₆×C₃₆₇	C₄	semidirect
25. C₈×C₃₆₇	C₂×C₄	semidirect
40. C₃₆₇	#428	direct (N × H)
41. #428	C₃₆₇	direct (N × H)
45. #107	C₂×C₃₆₇	semidirect
46. #106	C₂×C₃₆₇	semidirect
49. M₁₆	C₄×C₃₆₇	semidirect
66. C₈	C₂×C₄×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₃₆₇	2
#117×C₃₆₇	2
C₂×C₄⋊C₄×C₃₆₇	2
#107×C₃₆₇	2
#106×C₃₆₇	2
#428	367

Subgroups — #428×C367 #37553746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #428×C₃₆₇ #37553746