Subgroups — #497×C₃₆₇ #37553815

Order 23488.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₃₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×SD₁₆×C₃₆₇	C₂	1	2	yes	maximal
3	C₄⋊Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#124×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	#97×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
7	C₂×Q₈×C₃₆₇	C₂²	1	2	yes	—
8	C₂×D₄×C₃₆₇	C₂²	1	1	yes	characteristic
9	Q₁₆×C₃₆₇	—	2	4	no	—
10	SD₁₆×C₃₆₇	—	2	4	no	—
11	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
12	C₄⋊C₄×C₃₆₇	—	2	2	no	—
13	C₂²⋊C₄×C₃₆₇	—	2	2	no	—
14	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₃₆₇	—	2	1	no	—
16	Q₈×C₃₆₇	—	2	2	no	—
17	Q₈×C₃₆₇	—	2	2	no	—
18	Q₈×C₃₆₇	—	2	2	no	—
19	D₄×C₃₆₇	—	2	2	no	—
20	C₂×C₄×C₃₆₇	—	2	1	no	—
21	C₂×C₄×C₃₆₇	—	2	2	no	—
22	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
23	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₆₇	D₄	1	4	yes	—
25	C₂²×C₃₆₇	—	4	1	no	—
26	C₂²×C₃₆₇	—	4	2	no	—
27	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₃₆₇	—	2	2	no	—
29	C₄×C₃₆₇	—	4	1	no	—
30	C₄×C₃₆₇	—	4	2	no	—
31	C₄×C₃₆₇	C₂×D₄	1	2	yes	—
32	C₂×C₃₆₇	—	8	1	no	—
33	C₂×C₃₆₇	#127	1	1	yes	characteristic
34	C₂×C₃₆₇	#137	1	2	yes	—
35	C₃₆₇	#497	1	1	yes	characteristic, direct product factor, Sylow
36	#497	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×Q₁₆	C₂×C₃₆₇	1	2	yes	complemented
38	C₂×SD₁₆	C₂×C₃₆₇	1	2	yes	—
39	C₄⋊Q₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
40	#124	C₂×C₃₆₇	1	1	yes	characteristic
41	#97	C₂×C₃₆₇	1	1	yes	characteristic, complemented
42	C₂×Q₈	C₂²×C₃₆₇	1	1	yes	characteristic
43	C₂×Q₈	C₂²×C₃₆₇	1	2	yes	—
44	C₂×D₄	C₂²×C₃₆₇	1	1	yes	characteristic
45	Q₁₆	—	2	4	no	—
46	SD₁₆	—	2	4	no	—
47	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	2	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
59	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
60	C₈	D₄×C₃₆₇	1	4	yes	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	4	2	no	—
67	C₄	C₂×D₄×C₃₆₇	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#127×C₃₆₇	1	1	yes	characteristic
70	C₂	#137×C₃₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
prime radical	C₂×D₄×C₃₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ C₄²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×C₃₆₇	C₂	semidirect
3. C₄⋊Q₈×C₃₆₇	C₂	semidirect
5. #97×C₃₆₇	C₂	semidirect
35. C₃₆₇	#497	direct (N × H)
36. #497	C₃₆₇	direct (N × H)
37. C₂×Q₁₆	C₂×C₃₆₇	semidirect
39. C₄⋊Q₈	C₂×C₃₆₇	semidirect
41. #97	C₂×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×C₃₆₇	2
C₂×SD₁₆×C₃₆₇	2
C₄⋊Q₈×C₃₆₇	2
#124×C₃₆₇	2
#97×C₃₆₇	2
#497	367

Subgroups — #497×C367 #37553815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #497×C₃₆₇ #37553815