Subgroups — #14558263

Order 2368.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924535	C₂	1	4	yes	maximal
2	#11924534	C₂	1	4	yes	maximal
3	C₄×Dic₇₄	C₂	1	2	yes	maximal
4	#11924501	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
7	C₂×Dic₇₄	C₂²	1	4	yes	—
8	C₂×Dic₇₄	C₂²	1	2	yes	—
9	C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
10	C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	#10596936	C₂²	1	8	yes	—
13	#10596936	C₂²	1	4	yes	—
14	C₄×Dic₃₇	C₂²	1	4	yes	—
15	C₄×Dic₃₇	C₂²	1	4	yes	—
16	Q₈×C₃₇	—	4	2	no	—
17	C₂×C₄×C₃₇	C₂³	1	4	yes	—
18	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₃₇	C₂³	1	2	yes	—
20	C₂×Dic₃₇	C₂³	1	4	yes	—
21	C₂×Dic₃₇	C₂³	1	4	yes	—
22	Dic₇₄	—	4	4	no	—
23	Dic₇₄	Q₈	1	8	yes	complemented
24	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₃₇	—	2	4	no	—
26	C₄×C₃₇	—	2	1	no	—
27	C₄×C₃₇	C₂×Q₈	1	4	yes	—
28	Dic₃₇	—	2	4	no	—
29	Dic₃₇	C₂×Q₈	1	8	yes	—
30	C₂×C₃₇	C₂²×Q₈	1	2	yes	—
31	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
32	Q₈²	—	37	1	no	maximal, Sylow
33	C₃₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
34	C₄⋊Q₈	—	37	4	no	—
35	C₄⋊Q₈	—	37	4	no	—
36	C₄⋊Q₈	D₃₇	1	1	yes	characteristic
37	C₄×Q₈	—	37	4	no	—
38	C₄×Q₈	—	37	2	no	—
39	C₂×Q₈	—	37	2	no	—
40	C₂×Q₈	—	37	4	no	—
41	C₂×Q₈	C₂×D₃₇	1	2	yes	—
42	C₄⋊C₄	—	37	8	no	—
43	C₄⋊C₄	—	37	4	no	—
44	C₄⋊C₄	—	37	2	no	—
45	C₄⋊C₄	C₂×D₃₇	1	4	yes	—
46	C₄²	—	37	4	no	—
47	C₄²	—	37	4	no	—
48	C₄²	C₂×D₃₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	37	8	no	—
51	Q₈	—	148	4	no	—
52	C₂×C₄	—	37	4	no	—
53	C₂×C₄	—	37	4	no	—
54	C₂×C₄	C₂²×D₃₇	1	4	yes	—
55	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃₇	1	2	yes	—
57	C₂²	C₂³×D₃₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	37	8	no	—
61	C₄	—	74	4	no	—
62	C₄	Q₈×D₃₇	1	4	yes	—
63	C₂	C₂×Q₈×D₃₇	1	2	yes	—
64	C₂	#11924581	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	Q₈²
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
23. Dic₇₄	Q₈	semidirect
33. C₃₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#11924535	2
#11924534	2
C₄×Dic₇₄	2
#11924501	2
C₄⋊Q₈×C₃₇	2
Q₈²	37