Subgroups — #14557815

Order 2368.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	#11924467	C₂	1	2	yes	maximal
3	C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
4	#11924459	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₃₇	—	2	1	no	—
7	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₃₇	—	2	1	no	—
9	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
10	C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
12	#10596936	—	2	1	no	—
13	C₄×Dic₃₇	—	2	1	no	—
14	C₂×#64436	C₂²	1	2	yes	—
15	Q₈×C₃₇	—	4	2	no	—
16	C₂×C₄×C₃₇	—	2	2	no	—
17	C₂×C₄×C₃₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₃₇	—	2	1	no	—
21	C₂×Dic₃₇	—	2	1	no	—
22	Dic₇₄	—	2	1	no	—
23	Dic₇₄	Q₈	1	2	yes	complemented
24	#64436	—	2	2	no	—
25	C₂²×C₃₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₃₇	—	2	1	no	—
27	C₄×C₃₇	—	4	2	no	—
28	C₄×C₃₇	SD₁₆	1	2	yes	—
29	C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₃₇	#41	1	1	yes	characteristic
31	Dic₃₇	—	2	2	no	—
32	Dic₃₇	—	4	1	no	—
33	C₂×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₃₇	#137	1	1	yes	characteristic
36	#475	—	37	1	no	maximal, Sylow
37	C₃₇	#475	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	D₃₇	1	1	yes	characteristic
39	C₄×Q₈	—	37	1	no	—
40	#106	—	37	2	no	—
41	C₄⋊C₈	—	37	1	no	—
42	#103	—	37	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	37	1	no	—
45	C₂×C₈	—	37	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	37	1	no	—
48	C₄⋊C₄	—	74	1	no	—
49	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
50	C₄²	—	74	1	no	—
51	C₄²	C₂×D₃₇	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	37	2	no	—
54	Q₈	—	74	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	74	1	no	—
57	C₂×C₄	—	74	1	no	—
58	C₂×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
61	C₈	—	74	2	no	—
62	C₂²	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	74	2	no	—
66	C₄	—	148	1	no	—
67	C₄	#10596940	1	2	yes	—
68	C₄	Q₈×D₃₇	1	1	yes	characteristic
69	C₄	#10596966	1	1	yes	characteristic
70	C₂	#11924512	1	1	yes	characteristic
71	C₂	C₂×#10596940	1	1	yes	characteristic
72	C₂	#11924567	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₇
Fitting	C₄⋊Q₈×C₃₇	C₂
socle, prime radical	C₂²×C₃₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#475
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. Dic₇₄	Q₈	semidirect
37. C₃₇	#475	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₄	2
#11924467	2
C₄⋊#64436	2
#11924459	2
C₄⋊Q₈×C₃₇	2
#475	37