Subgroups — #17992649

Order 4736.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557935	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#10596925	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#334	C₂	1	1	yes	maximal, characteristic
4	#11924528	—	2	1	no	—
5	C₂×#10596925	—	2	1	no	—
6	#11924470	C₄	1	1	yes	characteristic
7	#11924470	C₄	1	1	yes	characteristic
8	#11924468	—	2	1	no	—
9	C₄⋊C₈×C₃₇	—	2	1	no	—
10	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₃₇	—	4	1	no	—
12	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
14	C₄²×C₃₇	D₄	1	1	yes	characteristic
15	C₄⋊Dic₃₇	—	2	1	no	—
16	C₄⋊Dic₃₇	—	2	1	no	—
17	#10596936	—	4	1	no	—
18	#10596936	—	4	1	no	—
19	#10596925	—	4	1	no	—
20	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₃₇	SD₁₆	1	1	yes	characteristic
23	C₂×Dic₃₇	—	4	1	no	—
24	C₂×Dic₃₇	—	4	1	no	—
25	C₈×C₃₇	—	2	1	no	—
26	C₈×C₃₇	—	8	1	no	—
27	C₈×C₃₇	SD₁₆	1	1	yes	characteristic
28	C₈×C₃₇	Q₁₆	1	1	yes	characteristic
29	C₂²×C₃₇	#102	1	1	yes	characteristic
30	C₄×C₃₇	—	2	1	no	—
31	C₄×C₃₇	—	2	1	no	—
32	C₄×C₃₇	#101	1	1	yes	characteristic
33	C₄×C₃₇	#103	1	1	yes	characteristic
34	Dic₃₇	—	8	1	no	—
35	Dic₃₇	—	8	1	no	—
36	#1363	—	37	1	no	maximal, Sylow
37	C₂×C₃₇	#332	1	1	yes	characteristic
38	C₂×C₃₇	#359	1	1	yes	characteristic
39	C₂×C₃₇	#360	1	1	yes	characteristic
40	#499	—	37	1	no	—
41	C₄⋊C₁₆	—	37	1	no	—
42	#334	D₃₇	1	1	yes	characteristic
43	C₃₇	#1363	1	1	yes	characteristic, complemented, Sylow
44	#125	—	74	1	no	—
45	C₂×C₁₆	—	74	1	no	—
46	#107	—	37	1	no	—
47	#107	—	37	1	no	—
48	#106	—	74	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
53	C₂×C₈	D₁₄₈	1	1	yes	characteristic
54	C₄⋊C₄	—	74	1	no	—
55	C₄⋊C₄	—	74	1	no	—
56	C₄⋊C₄	—	148	1	no	—
57	C₄⋊C₄	—	148	1	no	—
58	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
59	C₁₆	—	148	1	no	—
60	C₂×C₄	—	148	1	no	—
61	C₂×C₄	—	148	1	no	—
62	C₂×C₄	#10596938	1	1	yes	characteristic
63	C₂×C₄	#10596939	1	1	yes	characteristic
64	C₂×C₄	#10596941	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	8	1	no	—
67	C₈	#10596930	1	1	yes	characteristic
68	C₈	Dic₁₄₈	1	1	yes	characteristic
69	C₂²	#11924456	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	296	1	no	—
73	C₄	—	296	1	no	—
74	C₄	#11924454	1	1	yes	characteristic
75	C₄	#11924461	1	1	yes	characteristic
76	C₂	#14557280	1	1	yes	characteristic
77	C₂	#14557339	1	1	yes	characteristic
78	C₂	#14557342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924456
commutator	C₂×C₈×C₃₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₃₇
Fitting	C₃₇×#334	C₂
socle, prime radical	C₂²×C₃₇	#102
cosocle	C₄×C₈×C₃₇	C₂²
hypercenter	#334	D₃₇
hypocenter	C₃₇	#1363
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
43. C₃₇	#1363	semidirect

Maximal subgroups

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

subgroup	index
#14557935	2
C₄⋊#10596925	2
C₃₇×#334	2
#1363	37