Subgroups — #17992651

Order 4736.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#10596925	C₂	1	1	yes	maximal, characteristic
2	#14557286	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#500	C₂	1	1	yes	maximal, characteristic
4	C₂×#10596925	—	2	1	no	—
5	C₄⋊#64436	—	2	1	no	—
6	C₄⋊Q₈×C₃₇	—	2	1	no	—
7	#107×C₃₇	—	2	1	no	—
8	#107×C₃₇	C₄	1	2	yes	—
9	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×Q₈×C₃₇	—	4	1	no	—
11	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
13	C₄⋊C₄×C₃₇	—	2	2	no	—
14	C₄⋊C₄×C₃₇	—	4	1	no	—
15	C₄²×C₃₇	D₄	1	1	yes	characteristic
16	C₂×#64436	—	4	1	no	—
17	#10596925	—	4	1	no	—
18	Q₈×C₃₇	—	8	1	no	—
19	Q₈×C₃₇	—	8	1	no	—
20	C₂×C₄×C₃₇	—	4	2	no	—
21	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₃₇	SD₁₆	1	1	yes	characteristic
24	C₈×C₃₇	—	2	1	no	—
25	C₈×C₃₇	SD₁₆	1	1	yes	characteristic
26	C₈×C₃₇	Q₁₆	1	1	yes	characteristic
27	#64436	—	8	1	no	—
28	C₂²×C₃₇	#102	1	1	yes	characteristic
29	C₄×C₃₇	—	2	1	no	—
30	C₄×C₃₇	—	8	2	no	—
31	C₄×C₃₇	#101	1	1	yes	characteristic
32	C₄×C₃₇	#103	1	1	yes	characteristic
33	C₄×C₃₇	D₁₆	1	1	yes	characteristic
34	C₄×C₃₇	SD₃₂	1	1	yes	characteristic
35	#1364	—	37	1	no	maximal, Sylow
36	C₂×C₃₇	#332	1	1	yes	characteristic
37	C₂×C₃₇	#357	1	1	yes	characteristic
38	C₂×C₃₇	#362	1	1	yes	characteristic
39	#500	D₃₇	1	1	yes	characteristic
40	C₄⋊C₁₆	—	37	1	no	—
41	#335	—	37	1	no	—
42	C₃₇	#1364	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊Q₈	—	2	1	no	—
44	C₂×C₁₆	—	74	1	no	—
45	#107	—	2	1	no	—
46	#107	Dic₃₇	1	2	yes	—
47	C₄⋊C₈	—	74	1	no	—
48	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	148	1	no	—
51	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₃₇	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
56	C₁₆	—	148	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#10596939	1	1	yes	characteristic
61	C₂×C₄	#10596940	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	296	1	no	—
65	C₈	#10596941	1	1	yes	characteristic
66	C₈	#10596942	1	1	yes	characteristic
67	C₂²	#11924457	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#11924455	1	1	yes	characteristic
71	C₄	#11924460	1	1	yes	characteristic
72	C₄	#11924477	1	1	yes	characteristic
73	C₄	#11924480	1	1	yes	characteristic
74	C₂	#14557279	1	1	yes	characteristic
75	C₂	#14557334	1	1	yes	characteristic
76	C₂	#14557349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924457
commutator	C₂×C₈×C₃₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₃₇
Fitting	C₃₇×#500	C₂
socle, prime radical	C₂²×C₃₇	#102
cosocle	C₄×C₈×C₃₇	C₂²
hypercenter	#500	D₃₇
hypocenter	C₃₇	#1364
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
42. C₃₇	#1364	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#10596925	2
#14557286	2
C₃₇×#500	2
#1364	37