Subgroups — #14558249

Order 2368.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924525	C₂	1	4	yes	maximal, complemented
2	#11924511	C₂	1	4	yes	maximal, complemented
3	C₄×D₁₄₈	C₂	1	2	yes	maximal, complemented
4	#11924494	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
7	C₂×D₁₄₈	C₂²	1	2	yes	—
8	C₂×D₁₄₈	C₂²	1	4	yes	—
9	C₂×C₄×D₃₇	C₂²	1	4	yes	—
10	C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
11	C₄²×C₃₇	C₂²	1	1	yes	characteristic
12	#10596938	C₂²	1	8	yes	—
13	#10596938	C₂²	1	4	yes	—
14	C₄⋊Dic₃₇	C₂²	1	2	yes	—
15	C₄×Dic₃₇	C₂²	1	4	yes	—
16	C₂²×D₃₇	C₂³	1	4	yes	—
17	Q₈×C₃₇	—	4	2	no	—
18	C₂×C₄×C₃₇	C₂³	1	4	yes	—
19	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₃₇	C₂³	1	2	yes	—
21	C₂×Dic₃₇	C₂³	1	4	yes	—
22	D₁₄₈	—	2	4	no	—
23	D₁₄₈	—	4	4	no	—
24	C₄×D₃₇	—	2	8	no	—
25	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₃₇	—	2	4	no	—
27	C₂×D₃₇	—	2	4	no	—
28	C₂×D₃₇	—	2	4	no	—
29	C₄×C₃₇	—	2	4	no	—
30	C₄×C₃₇	—	2	1	no	—
31	C₄×C₃₇	#41	1	4	yes	—
32	Dic₃₇	—	2	4	no	—
33	C₂×C₃₇	C₂×#41	1	2	yes	—
34	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
35	D₃₇	—	4	4	no	—
36	#555	—	37	1	no	maximal, Sylow
37	C₃₇	#555	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
39	#124	—	37	4	no	—
40	#121	—	37	4	no	—
41	C₄×D₄	—	37	2	no	—
42	#117	—	37	4	no	—
43	C₂×Q₈	C₂×D₃₇	1	2	yes	—
44	C₂×D₄	—	37	2	no	—
45	C₂×D₄	—	37	4	no	—
46	C₂²×C₄	—	37	4	no	—
47	C₄⋊C₄	—	37	2	no	—
48	C₄⋊C₄	C₂×D₃₇	1	4	yes	—
49	C₂²⋊C₄	—	37	8	no	—
50	C₂²⋊C₄	—	37	4	no	—
51	C₄²	—	37	4	no	—
52	C₄²	C₂×D₃₇	1	1	yes	characteristic
53	C₂³	—	37	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	74	4	no	—
56	D₄	—	148	4	no	—
57	C₂×C₄	—	37	4	no	—
58	C₂×C₄	—	74	8	no	—
59	C₂×C₄	C₂²×D₃₇	1	4	yes	—
60	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₃₇	1	2	yes	—
62	C₂²	—	74	4	no	—
63	C₂²	—	74	4	no	—
64	C₂²	—	74	4	no	—
65	C₂²	C₂³×D₃₇	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	74	4	no	—
69	C₄	#10596966	1	4	yes	—
70	C₂	—	148	4	no	—
71	C₂	C₂×#10596966	1	2	yes	—
72	C₂	#11924581	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄⋊Q₈×C₃₇	C₂
hypercenter	C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#555
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
1. #11924525	C₂	semidirect
2. #11924511	C₂	semidirect
3. C₄×D₁₄₈	C₂	semidirect
4. #11924494	C₂	semidirect
5. C₄⋊Q₈×C₃₇	C₂	semidirect
37. C₃₇	#555	semidirect
38. C₄⋊Q₈	D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924525	2
#11924511	2
C₄×D₁₄₈	2
#11924494	2
C₄⋊Q₈×C₃₇	2
#555	37