Subgroups — #17994210

Order 4736.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557910	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557909	C₂	1	4	yes	maximal, complemented
3	C₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₉₆	—	2	2	no	—
6	C₂×#10596939	—	2	8	no	—
7	#11924533	C₂²	1	2	yes	—
8	C₄×#64436	C₂²	1	2	yes	—
9	C₂×D₈×C₃₇	—	2	2	no	—
10	#127×C₃₇	C₂²	1	2	yes	—
11	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₃₇	—	2	4	no	—
13	C₂×D₄×C₃₇	—	2	2	no	—
14	C₂×D₁₄₈	—	2	4	no	—
15	C₂×D₁₄₈	—	2	2	no	—
16	D₈×C₃₇	—	4	4	no	—
17	C₂×C₈×C₃₇	D₄	1	2	yes	—
18	C₄²×C₃₇	C₂³	1	1	yes	characteristic
19	#10596939	—	4	16	no	—
20	C₂×#64436	D₄	1	4	yes	—
21	D₂₉₆	—	4	4	no	—
22	C₂³×C₃₇	—	4	2	no	—
23	C₂²×D₃₇	—	4	2	no	—
24	D₄×C₃₇	—	4	8	no	—
25	D₄×C₃₇	—	4	4	no	—
26	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
28	D₁₄₈	—	4	4	no	—
29	D₁₄₈	—	4	8	no	—
30	C₈×C₃₇	D₈	1	4	yes	complemented
31	#64436	D₈	1	8	yes	complemented
32	C₂²×C₃₇	—	8	4	no	—
33	C₂²×C₃₇	—	8	2	no	—
34	C₂²×C₃₇	#127	1	1	yes	characteristic
35	C₂×D₃₇	—	8	4	no	—
36	C₂×D₃₇	—	8	2	no	—
37	C₄×C₃₇	C₂×D₈	1	4	yes	—
38	C₄×C₃₇	C₂×D₈	1	2	yes	—
39	#1712	—	37	1	no	maximal, Sylow
40	C₂×C₃₇	—	16	2	no	—
41	C₂×C₃₇	#493	1	2	yes	—
42	C₂×C₃₇	#493	1	1	yes	characteristic
43	D₃₇	—	16	2	no	—
44	#493	—	37	4	no	—
45	#493	—	37	1	no	—
46	#493	D₃₇	1	1	yes	characteristic, complemented
47	C₈²	—	37	1	no	—
48	C₃₇	#1712	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	74	8	no	—
51	C₂×D₈	—	74	2	no	—
52	#127	—	37	2	no	—
53	#127	C₂×D₃₇	1	2	yes	—
54	C₄×C₈	—	37	2	no	—
55	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	74	4	no	—
59	C₂×D₄	—	74	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	148	16	no	—
62	D₈	—	148	4	no	—
63	C₂×C₈	—	37	4	no	—
64	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
65	C₄²	C₂²×D₃₇	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	148	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	148	8	no	—
71	D₄	—	148	4	no	—
72	C₂×C₄	D₄×D₃₇	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
74	C₈	—	37	8	no	—
75	C₈	#10596939	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	296	4	no	—
79	C₂²	—	296	2	no	—
80	C₂²	#11924532	1	1	yes	characteristic
81	C₄	D₈×D₃₇	1	4	yes	—
82	C₄	C₂×#10596939	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	592	2	no	—
85	C₂	#14557908	1	2	yes	—
86	C₂	#14557909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂²×C₃₇	#127
hypercenter	#493	D₃₇
hypocenter	C₃₇	#1712
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #14557910	C₂	semidirect
2. #14557909	C₂	semidirect
3. C₈×#64436	C₂	semidirect
4. C₃₇×#493	C₂	semidirect
30. C₈×C₃₇	D₈	semidirect
31. #64436	D₈	semidirect
46. #493	D₃₇	semidirect
48. C₃₇	#1712	semidirect
75. C₈	#10596939	semidirect

Maximal subgroups

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

subgroup	index
#14557910	2
#14557909	2
C₈×#64436	2
C₃₇×#493	2
#1712	37