Subgroups — #18009727

Order 4736.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558438	C₂	1	2	yes	maximal, complemented
2	#14557909	C₂	1	1	yes	maximal, characteristic
3	#11924595	—	2	2	no	—
4	C₂×#10596939	—	2	2	no	—
5	#11924533	C₂²	1	1	yes	characteristic
6	C₄×#64436	C₄	1	1	yes	characteristic, complemented
7	#127×C₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₃₇	—	2	1	no	—
9	C₂×D₄×C₃₇	—	4	1	no	—
10	C₂×D₁₄₈	—	4	1	no	—
11	C₂×D₁₄₈	D₄	1	2	yes	—
12	C₂²⋊#3866	—	4	2	no	—
13	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
14	#10596939	—	4	4	no	—
15	C₂×#64436	—	2	1	no	—
16	C₂³×C₃₇	—	4	1	no	—
17	C₂²×D₃₇	—	2	2	no	—
18	C₂×#3866	—	8	2	no	—
19	D₄×C₃₇	—	4	2	no	—
20	D₄×C₃₇	—	8	2	no	—
21	C₂×C₄×C₃₇	—	2	1	no	—
22	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	D₁₄₈	—	4	2	no	—
24	D₁₄₈	—	8	2	no	—
25	#64436	—	2	2	no	—
26	C₂²×C₃₇	—	8	2	no	—
27	C₂²×C₃₇	—	8	1	no	—
28	C₂²×C₃₇	#99	1	1	yes	characteristic
29	C₂×D₃₇	—	4	2	no	—
30	C₂×D₃₇	—	8	2	no	—
31	#3866	—	16	2	no	—
32	C₄×C₃₇	—	2	2	no	—
33	C₄×C₃₇	—	2	1	no	—
34	#1408	—	37	1	no	maximal, Sylow
35	C₂×C₃₇	—	2	1	no	—
36	C₂×C₃₇	—	16	1	no	—
37	C₂×C₃₇	#353	1	1	yes	characteristic
38	D₃₇	—	8	2	no	—
39	#493	—	37	1	no	—
40	#353	—	37	2	no	—
41	C₃₇	#1408	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₈	—	74	2	no	—
43	#127	—	37	1	no	—
44	#127	#3866	1	1	yes	characteristic, complemented
45	#99	—	74	2	no	—
46	C₄×C₈	—	37	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	37	2	no	—
50	C₂×D₄	—	148	1	no	—
51	D₈	—	148	4	no	—
52	C₂×C₈	—	74	1	no	—
53	C₂²⋊C₄	—	148	2	no	—
54	C₄²	C₂×#3866	1	1	yes	characteristic
55	C₂³	—	4	1	no	—
56	C₂³	—	74	2	no	—
57	D₄	—	4	2	no	—
58	D₄	—	8	2	no	—
59	D₄	—	296	2	no	—
60	D₄	—	148	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	296	2	no	—
63	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
64	C₈	—	74	2	no	—
65	C₂²	—	8	2	no	—
66	C₂²	—	8	1	no	—
67	C₂²	—	296	2	no	—
68	C₂²	—	148	2	no	—
69	C₂²	#11924594	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	592	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	296	2	no	—
76	C₂	#14558437	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14558437
commutator	C₄²×C₃₇	C₂×C₄
Frattini	C₄²	C₂×#3866
Fitting	#127×C₃₇	C₄
socle	C₂×C₃₇	#353
prime radical	#14557909	C₂
cosocle	#11924533	C₂²
hypercenter	#127	#3866
hypocenter	C₃₇	#1408
LCS	G ▷ C₄²×C₃₇ ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #14558438	C₂	semidirect
6. C₄×#64436	C₄	semidirect
7. #127×C₃₇	C₄	semidirect
41. C₃₇	#1408	semidirect
44. #127	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558438	2
#14557909	2
#1408	37