Subgroups — #14558262

Order 2368.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924536	C₂	1	3	yes	maximal
2	#11924527	C₂	1	6	yes	maximal
3	Q₈×Dic₃₇	C₂	1	2	yes	maximal
4	C₄×Dic₇₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₇₄	C₂²	1	3	yes	—
8	C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
9	C₄²×C₃₇	C₂²	1	3	yes	—
10	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₇	C₂²	1	6	yes	—
12	C₄⋊Dic₃₇	C₂²	1	6	yes	—
13	#10596936	C₂²	1	6	yes	—
14	C₄×Dic₃₇	C₂²	1	6	yes	—
15	Q₈×C₃₇	Q₈	1	4	yes	complemented
16	C₂×C₄×C₃₇	C₂³	1	3	yes	—
17	C₂×C₄×C₃₇	C₂³	1	3	yes	—
18	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₃₇	C₂³	1	2	yes	—
20	C₂×Dic₃₇	C₂³	1	6	yes	—
21	Dic₇₄	—	2	6	no	—
22	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₃₇	—	2	3	no	—
24	C₄×C₃₇	C₂×Q₈	1	6	yes	—
25	C₄×C₃₇	#41	1	2	yes	—
26	Dic₃₇	—	2	6	no	—
27	Dic₃₇	—	2	2	no	—
28	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
31	#557	—	37	1	no	maximal, Sylow
32	C₃₇	#557	1	1	yes	characteristic, complemented, Sylow
33	C₄⋊Q₈	—	37	3	no	—
34	#125	—	37	6	no	—
35	C₄×Q₈	—	37	2	no	—
36	C₄×Q₈	—	37	3	no	—
37	C₄×Q₈	D₃₇	1	1	yes	characteristic
38	C₂×Q₈	—	37	3	no	—
39	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
40	C₄⋊C₄	—	37	6	no	—
41	C₄⋊C₄	—	37	1	no	—
42	C₄⋊C₄	—	37	6	no	—
43	C₄⋊C₄	—	37	6	no	—
44	C₄⋊C₄	C₂×D₃₇	1	3	yes	—
45	C₄²	—	37	6	no	—
46	C₄²	C₂×D₃₇	1	3	yes	—
47	Q₈	—	74	6	no	—
48	Q₈	Dic₇₄	1	4	yes	complemented
49	C₂×C₄	—	37	2	no	—
50	C₂×C₄	—	37	6	no	—
51	C₂×C₄	C₂²×D₃₇	1	3	yes	—
52	C₂×C₄	C₂²×D₃₇	1	3	yes	—
53	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
54	C₂²	C₂³×D₃₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	74	6	no	—
57	C₄	—	74	2	no	—
58	C₄	C₂×Dic₇₄	1	6	yes	—
59	C₄	#10596966	1	2	yes	—
60	C₂	C₂²×Dic₇₄	1	1	yes	characteristic
61	C₂	C₂×#10596966	1	1	yes	characteristic
62	C₂	#11924583	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, prime radical, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄×Q₈×C₃₇	C₂
hypercenter	C₄×Q₈	D₃₇
hypocenter	C₃₇	#557
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
15. Q₈×C₃₇	Q₈	semidirect
32. C₃₇	#557	semidirect
48. Q₈	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#11924536	2
#11924527	2
Q₈×Dic₃₇	2
C₄×Dic₇₄	2
C₄×Q₈×C₃₇	2
#557	37