Subgroups — #485×C₃₆₇ #37553803

Order 23488.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#117×C₃₆₇	C₂	1	1	yes	maximal, characteristic
3	#107×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#106×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#103×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#103×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₆₇	—	2	1	no	—
13	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₃₆₇	—	2	1	no	—
17	C₂²⋊C₄×C₃₆₇	—	2	1	no	—
18	C₄²×C₃₆₇	—	2	1	no	—
19	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
20	Q₈×C₃₆₇	—	2	1	no	—
21	Q₈×C₃₆₇	—	2	1	no	—
22	C₂×C₄×C₃₆₇	—	2	1	no	—
23	C₂×C₄×C₃₆₇	—	2	1	no	—
24	C₂×C₄×C₃₆₇	—	2	1	no	—
25	C₂×C₄×C₃₆₇	—	2	1	no	—
26	C₂×C₄×C₃₆₇	—	2	1	no	—
27	C₂×C₄×C₃₆₇	—	2	1	no	—
28	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
30	C₈×C₃₆₇	—	2	1	no	—
31	C₈×C₃₆₇	—	2	1	no	—
32	C₂²×C₃₆₇	—	2	1	no	—
33	C₂²×C₃₆₇	—	2	1	no	—
34	C₂²×C₃₆₇	—	2	1	no	—
35	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₆₇	—	2	1	no	—
37	C₄×C₃₆₇	—	4	1	no	—
38	C₄×C₃₆₇	—	4	1	no	—
39	C₄×C₃₆₇	—	4	1	no	—
40	C₄×C₃₆₇	—	4	1	no	—
41	C₄×C₃₆₇	#41	1	1	yes	characteristic
42	C₄×C₃₆₇	#41	1	1	yes	characteristic
43	C₂×C₃₆₇	—	4	1	no	—
44	C₂×C₃₆₇	#123	1	1	yes	characteristic
45	C₂×C₃₆₇	#135	1	1	yes	characteristic
46	C₂×C₃₆₇	#137	1	1	yes	characteristic
47	C₃₆₇	#485	1	1	yes	characteristic, direct product factor, Sylow
48	#485	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²⋊Q₈	C₂×C₃₆₇	1	1	yes	characteristic
50	#117	C₂×C₃₆₇	1	1	yes	characteristic
51	#107	C₂×C₃₆₇	1	1	yes	characteristic, complemented
52	#106	C₂×C₃₆₇	1	1	yes	characteristic, complemented
53	#103	C₂×C₃₆₇	1	1	yes	characteristic, complemented
54	#103	C₂×C₃₆₇	1	1	yes	characteristic, complemented
55	C₂²⋊C₈	C₂×C₃₆₇	1	1	yes	characteristic
56	C₂×Q₈	C₂²×C₃₆₇	1	1	yes	characteristic
57	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
58	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
62	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
63	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	2	1	no	—
66	C₄²	—	2	1	no	—
67	C₂³	D₄×C₃₆₇	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
77	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#41×C₃₆₇	1	1	yes	characteristic
90	C₄	#41×C₃₆₇	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#123×C₃₆₇	1	1	yes	characteristic
93	C₂	#135×C₃₆₇	1	1	yes	characteristic
94	C₂	#137×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
prime radical	C₂³×C₃₆₇	D₄
LCS	G ▷ C₂×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
3. #107×C₃₆₇	C₂	semidirect
4. #106×C₃₆₇	C₂	semidirect
5. #103×C₃₆₇	C₂	semidirect
6. #103×C₃₆₇	C₂	semidirect
47. C₃₆₇	#485	direct (N × H)
48. #485	C₃₆₇	direct (N × H)
51. #107	C₂×C₃₆₇	semidirect
52. #106	C₂×C₃₆₇	semidirect
53. #103	C₂×C₃₆₇	semidirect
54. #103	C₂×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×C₃₆₇	2
#117×C₃₆₇	2
#107×C₃₆₇	2
#106×C₃₆₇	2
#103×C₃₆₇	2
C₂²⋊C₈×C₃₆₇	2
#485	367

Subgroups — #485×C367 #37553803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #485×C₃₆₇ #37553803