Subgroups — #14558574

Order 2368.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#3866	C₂	1	2	yes	maximal, complemented
2	#11924604	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924598	C₂	1	2	yes	maximal, complemented
4	#11924590	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#3866	—	2	2	no	—
7	D₄×D₃₇	C₂²	1	2	yes	—
8	C₂²⋊#3866	—	2	2	no	—
9	C₄⋊#3866	C₂²	1	2	yes	—
10	C₄×#3866	C₂²	1	1	yes	characteristic
11	D₃₇⋊C₈	C₂²	1	1	yes	characteristic
12	D₈×C₃₇	C₄	1	1	yes	characteristic, complemented
13	#10596939	C₄	1	2	yes	complemented
14	D₂₉₆	C₄	1	1	yes	characteristic, complemented
15	C₈×D₃₇	C₂²	1	1	yes	characteristic
16	C₂²×D₃₇	—	2	2	no	—
17	C₂×#3866	—	2	2	no	—
18	C₂×#3866	—	4	4	no	—
19	C₂×#3866	D₄	1	2	yes	—
20	D₄×C₃₇	C₂×C₄	1	2	yes	complemented
21	C₂²⋊D₃₇	—	2	2	no	—
22	D₁₄₈	C₂×C₄	1	2	yes	complemented
23	C₄×D₃₇	C₂³	1	1	yes	characteristic
24	#64438	—	2	1	no	—
25	C₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
26	#64436	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₃₇	—	2	2	no	—
28	C₂×D₃₇	—	2	2	no	—
29	C₂×D₃₇	—	4	2	no	—
30	C₂×D₃₇	—	4	2	no	—
31	C₂×D₃₇	C₂×D₄	1	1	yes	characteristic
32	#3866	—	2	2	no	—
33	#3866	—	4	2	no	—
34	C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	Dic₃₇	#41	1	1	yes	characteristic
36	C₂×C₃₇	—	4	2	no	—
37	C₂×C₃₇	C₄×D₄	1	1	yes	characteristic
38	D₃₇	—	4	2	no	—
39	D₃₇	AGL(1,ℤ/8)	1	2	yes	—
40	#442	—	37	1	no	maximal, Sylow
41	C₃₇	#442	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₈	—	37	1	no	—
43	C₄×D₄	—	37	2	no	—
44	#106	—	37	1	no	—
45	#102	—	37	2	no	—
46	#97	—	37	1	no	—
47	C₂×D₄	—	37	2	no	—
48	C₂²×C₄	—	74	2	no	—
49	D₈	—	37	2	no	—
50	D₈	—	37	1	no	—
51	D₈	#3866	1	1	yes	characteristic, complemented
52	C₂×C₈	—	37	1	no	—
53	C₂×C₈	—	37	1	no	—
54	C₄⋊C₄	—	37	2	no	—
55	C₂²⋊C₄	—	74	2	no	—
56	C₄²	—	37	1	no	—
57	C₂³	—	74	2	no	—
58	D₄	—	37	2	no	—
59	D₄	—	74	2	no	—
60	D₄	C₂×#3866	1	2	yes	complemented
61	C₂×C₄	—	37	1	no	—
62	C₂×C₄	—	37	2	no	—
63	C₂×C₄	—	74	2	no	—
64	C₂×C₄	—	148	4	no	—
65	C₈	—	37	1	no	—
66	C₈	—	74	1	no	—
67	C₈	C₂×#3866	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	37	1	no	—
70	C₂²	—	74	2	no	—
71	C₂²	—	148	2	no	—
72	C₂²	—	148	2	no	—
73	C₄	—	37	1	no	—
74	C₄	—	74	2	no	—
75	C₄	—	148	2	no	—
76	C₄	C₂²×#3866	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	37	2	no	—
79	C₂	—	148	2	no	—
80	C₂	D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#3866
commutator	C₄×C₃₇	C₂²×C₄
Frattini	C₄	C₂²×#3866
Fitting	D₈×C₃₇	C₄
socle	C₂×C₃₇	C₄×D₄
prime radical	D₈×D₃₇	C₂
cosocle	C₄×D₃₇	C₂³
hypercenter	D₈	#3866
hypocenter	C₃₇	#442
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#3866	C₂	semidirect
2. #11924604	C₂	semidirect
3. #11924598	C₂	semidirect
4. #11924590	C₂	semidirect
12. D₈×C₃₇	C₄	semidirect
13. #10596939	C₄	semidirect
14. D₂₉₆	C₄	semidirect
20. D₄×C₃₇	C₂×C₄	semidirect
22. D₁₄₈	C₂×C₄	semidirect
25. C₈×C₃₇	C₂×C₄	semidirect
26. #64436	C₂×C₄	semidirect
41. C₃₇	#442	semidirect
51. D₈	#3866	semidirect
60. D₄	C₂×#3866	semidirect
67. C₈	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
D₄×#3866	2
#11924604	2
#11924598	2
#11924590	2
D₈×D₃₇	2
#442	37