Subgroups — #14558404

Order 2368.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#11924615	C₂	1	1	yes	maximal, characteristic
3	C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×D₃₇	—	2	1	no	—
5	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
6	C₂²⋊#3866	—	2	1	no	—
7	#10596957	—	2	1	no	—
8	C₄⋊#3866	C₄	1	2	yes	complemented
9	C₄×#3866	—	2	1	no	—
10	#10596953	C₄	1	2	yes	complemented
11	D₃₇⋊C₈	—	2	1	no	—
12	C₄²×C₃₇	C₄	1	1	yes	characteristic, complemented
13	C₄×Dic₃₇	—	2	1	no	—
14	C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₃₇	D₄	1	1	yes	characteristic
16	C₂×#3866	—	2	2	no	—
17	C₂×C₄×C₃₇	—	2	1	no	—
18	C₂×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₃₇	—	2	1	no	—
20	C₂×Dic₃₇	D₄	1	1	yes	characteristic
21	C₄×D₃₇	—	2	4	no	—
22	C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₇	D₄	1	1	yes	characteristic
25	C₄×D₃₇	Q₈	1	1	yes	characteristic
26	#64438	—	2	2	no	—
27	C₂²×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₃₇	—	2	2	no	—
29	C₂×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₇	C₄⋊C₄	1	1	yes	characteristic
31	#3866	—	4	2	no	—
32	C₄×C₃₇	—	2	2	no	—
33	C₄×C₃₇	C₄²	1	1	yes	characteristic
34	C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₇	—	2	2	no	—
36	Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₇	—	2	1	no	—
39	C₂×C₃₇	#95	1	1	yes	characteristic
40	D₃₇	—	2	1	no	—
41	D₃₇	#104	1	2	yes	—
42	#339	—	37	1	no	maximal, Sylow
43	C₃₇	#339	1	1	yes	characteristic, complemented, Sylow
44	C₂×M₁₆	—	37	1	no	—
45	#117	—	37	1	no	—
46	C₂×C₄²	—	37	1	no	—
47	C₂²×C₄	—	37	1	no	—
48	C₂²×C₄	—	74	1	no	—
49	M₁₆	—	37	2	no	—
50	M₁₆	—	74	1	no	—
51	C₂×C₈	—	74	1	no	—
52	C₄⋊C₄	—	37	2	no	—
53	C₂²⋊C₄	—	74	1	no	—
54	C₄²	—	37	1	no	—
55	C₄²	—	74	1	no	—
56	C₄²	—	74	1	no	—
57	C₄²	#3866	1	1	yes	characteristic, complemented
58	C₂³	—	37	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	37	1	no	—
61	C₂×C₄	—	37	1	no	—
62	C₂×C₄	—	37	1	no	—
63	C₂×C₄	—	37	1	no	—
64	C₂×C₄	—	37	1	no	—
65	C₂×C₄	—	74	4	no	—
66	C₂×C₄	—	74	1	no	—
67	C₂×C₄	—	74	2	no	—
68	C₂×C₄	C₂×#3866	1	1	yes	characteristic
69	C₈	—	74	2	no	—
70	C₂²	—	37	1	no	—
71	C₂²	—	37	1	no	—
72	C₂²	—	74	2	no	—
73	C₂²	C₂²⋊#3866	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	37	1	no	—
76	C₄	—	37	1	no	—
77	C₄	—	74	2	no	—
78	C₄	—	148	2	no	—
79	C₄	C₄×#3866	1	1	yes	characteristic
80	C₄	C₄⋊#3866	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	37	2	no	—
83	C₂	—	74	1	no	—
84	C₂	(C₂×#3866)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#3866
commutator	C₄×C₃₇	C₄²
Frattini	C₂×C₄	C₂×#3866
Fitting	C₄²×C₃₇	C₄
socle	C₂×C₃₇	#95
prime radical	C₂²×D₃₇	D₄
cosocle	C₂×C₄×D₃₇	C₂²
hypercenter	C₄²	#3866
hypocenter	C₃₇	#339
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
8. C₄⋊#3866	C₄	semidirect
10. #10596953	C₄	semidirect
12. C₄²×C₃₇	C₄	semidirect
14. C₄×Dic₃₇	C₄	semidirect
43. C₃₇	#339	semidirect
57. C₄²	#3866	semidirect

Maximal subgroups

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

subgroup	index
D₃₇⋊M₁₆	2
#11924615	2
C₄²×D₃₇	2
#339	37