Subgroups — #14558114

Order 2368.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10596962	C₂	1	2	yes	maximal, complemented
2	#11924516	C₂	1	4	yes	maximal, complemented
3	#11924511	C₂	1	4	yes	maximal, complemented
4	#11924510	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₃₇	C₂²	1	4	yes	—
9	#10596962	—	2	8	no	—
10	C₂×D₁₄₈	C₂²	1	2	yes	—
11	C₂×D₁₄₈	C₂²	1	4	yes	—
12	C₂×C₄×D₃₇	C₂²	1	4	yes	—
13	C₂×Dic₇₄	C₂²	1	2	yes	—
14	C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
15	#10596938	C₂²	1	8	yes	—
16	C₄⋊Dic₃₇	C₂²	1	4	yes	—
17	C₂³×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×D₃₇	C₂³	1	4	yes	—
19	C₂×C₄×C₃₇	—	4	2	no	—
20	C₂×C₄×C₃₇	D₄	1	4	yes	complemented
21	C₂×C₄×C₃₇	C₂³	1	4	yes	—
22	C₂×C₄×C₃₇	C₂³	1	2	yes	—
23	C₂²⋊D₃₇	—	2	8	no	—
24	C₂×Dic₃₇	C₂³	1	4	yes	—
25	D₁₄₈	—	2	4	no	—
26	D₁₄₈	—	4	4	no	—
27	C₄×D₃₇	—	2	8	no	—
28	Dic₇₄	—	2	4	no	—
29	C₂²×C₃₇	—	2	1	no	—
30	C₂²×C₃₇	—	2	1	no	—
31	C₂²×C₃₇	C₂×D₄	1	2	yes	—
32	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
33	C₂×D₃₇	—	2	4	no	—
34	C₂×D₃₇	—	2	4	no	—
35	C₂×D₃₇	—	2	4	no	—
36	C₄×C₃₇	—	2	4	no	—
37	C₄×C₃₇	C₂×D₄	1	4	yes	—
38	Dic₃₇	—	2	4	no	—
39	C₂×C₃₇	—	2	2	no	—
40	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
43	D₃₇	—	4	4	no	—
44	#537	—	37	1	no	maximal, Sylow
45	C₃₇	#537	1	1	yes	characteristic, complemented, Sylow
46	C₂×#41	—	37	2	no	—
47	C₂²⋊Q₈	—	37	4	no	—
48	#121	—	37	4	no	—
49	#121	—	37	4	no	—
50	C₂×C₄⋊C₄	D₃₇	1	1	yes	characteristic, complemented
51	#41	—	74	8	no	—
52	C₂×Q₈	—	37	2	no	—
53	C₂×D₄	—	37	4	no	—
54	C₂×D₄	—	37	2	no	—
55	C₂×D₄	—	37	4	no	—
56	C₂²×C₄	—	37	4	no	—
57	C₂²×C₄	C₂×D₃₇	1	2	yes	—
58	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
59	C₄⋊C₄	—	37	4	no	—
60	C₄⋊C₄	C₂×D₃₇	1	4	yes	—
61	C₂²⋊C₄	—	37	8	no	—
62	C₂³	—	37	4	no	—
63	C₂³	C₂²×D₃₇	1	1	yes	characteristic
64	Q₈	—	74	4	no	—
65	D₄	—	74	8	no	—
66	D₄	—	74	4	no	—
67	D₄	—	148	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	37	4	no	—
70	C₂×C₄	—	74	8	no	—
71	C₂×C₄	D₁₄₈	1	4	yes	complemented
72	C₂×C₄	C₂²×D₃₇	1	4	yes	—
73	C₂×C₄	C₂²×D₃₇	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	74	4	no	—
77	C₂²	—	74	4	no	—
78	C₂²	—	74	4	no	—
79	C₂²	C₂×D₁₄₈	1	2	yes	—
80	C₂²	C₂³×D₃₇	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	74	4	no	—
83	C₄	C₂×D₁₄₈	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	148	4	no	—
86	C₂	C₂²×D₁₄₈	1	1	yes	characteristic
87	C₂	#11924581	1	1	yes	characteristic
88	C₂	#11924584	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₃₇	C₂
hypercenter	C₂×C₄⋊C₄	D₃₇
hypocenter	C₃₇	#537
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10596962	C₂	semidirect
2. #11924516	C₂	semidirect
3. #11924511	C₂	semidirect
4. #11924510	C₂	semidirect
5. C₂×C₄⋊C₄×C₃₇	C₂	semidirect
20. C₂×C₄×C₃₇	D₄	semidirect
45. C₃₇	#537	semidirect
50. C₂×C₄⋊C₄	D₃₇	semidirect
71. C₂×C₄	D₁₄₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10596962	2
#11924516	2
#11924511	2
#11924510	2
C₂×C₄⋊C₄×C₃₇	2
#537	37