Subgroups — #14558086

Order 2368.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924535	C₂	1	2	yes	maximal, complemented
2	#11924527	C₂	1	2	yes	maximal, complemented
3	#11924513	C₂	1	4	yes	maximal
4	C₄×Dic₇₄	C₂	1	4	yes	maximal, complemented
5	#11924495	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
7	#117×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃₇	C₂²	1	2	yes	—
10	C₂×Dic₇₄	C₂²	1	4	yes	—
11	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
13	C₄²×C₃₇	C₂²	1	2	yes	—
14	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
15	C₄⋊Dic₃₇	C₂²	1	2	yes	—
16	C₄⋊Dic₃₇	C₂²	1	4	yes	—
17	#10596936	C₂²	1	4	yes	—
18	#10596936	C₂²	1	4	yes	—
19	C₄×Dic₃₇	C₂²	1	2	yes	—
20	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₇	C₂²	1	2	yes	—
23	C₂³×C₃₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₇	Q₈	1	4	yes	complemented
25	C₂×C₄×C₃₇	C₂³	1	4	yes	—
26	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₃₇	—	2	4	no	—
29	C₂×Dic₃₇	C₂³	1	2	yes	—
30	C₂×Dic₃₇	C₂³	1	4	yes	—
31	C₂×Dic₃₇	C₂³	1	2	yes	—
32	Dic₇₄	—	2	8	no	—
33	C₂²×C₃₇	—	2	2	no	—
34	C₂²×C₃₇	C₂×Q₈	1	2	yes	—
35	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
36	C₄×C₃₇	—	2	4	no	—
37	C₄×C₃₇	C₂×Q₈	1	2	yes	—
38	C₄×C₃₇	C₂×Q₈	1	2	yes	—
39	Dic₃₇	—	2	4	no	—
40	Dic₃₇	—	2	2	no	—
41	Dic₃₇	#41	1	4	yes	—
42	C₂×C₃₇	—	2	2	no	—
43	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₃₇	C₂×#41	1	2	yes	—
45	#533	—	37	1	no	maximal, Sylow
46	C₃₇	#533	1	1	yes	characteristic, complemented, Sylow
47	C₄⋊Q₈	—	37	2	no	—
48	#125	—	37	2	no	—
49	C₂²⋊Q₈	—	37	4	no	—
50	C₄×Q₈	—	37	4	no	—
51	#117	—	37	1	no	—
52	#117	D₃₇	1	1	yes	characteristic
53	C₂×C₄²	—	37	1	no	—
54	C₂×Q₈	—	37	4	no	—
55	C₂²×C₄	—	37	2	no	—
56	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
57	C₄⋊C₄	—	37	4	no	—
58	C₄⋊C₄	—	37	4	no	—
59	C₄⋊C₄	—	37	2	no	—
60	C₄⋊C₄	—	37	4	no	—
61	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
62	C₂²⋊C₄	—	37	2	no	—
63	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
64	C₄²	—	37	1	no	—
65	C₄²	—	37	1	no	—
66	C₄²	—	37	2	no	—
67	C₄²	—	37	2	no	—
68	C₄²	C₂×D₃₇	1	2	yes	—
69	C₂³	C₂²×D₃₇	1	1	yes	characteristic
70	Q₈	—	74	8	no	—
71	C₂×C₄	—	37	2	no	—
72	C₂×C₄	—	37	2	no	—
73	C₂×C₄	—	37	4	no	—
74	C₂×C₄	—	74	4	no	—
75	C₂×C₄	Dic₇₄	1	4	yes	complemented
76	C₂×C₄	C₂²×D₃₇	1	4	yes	—
77	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₇₄	1	2	yes	—
81	C₂²	C₂³×D₃₇	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	37	4	no	—
84	C₄	—	74	2	no	—
85	C₄	—	74	4	no	—
86	C₄	C₂×Dic₇₄	1	2	yes	—
87	C₄	C₂×Dic₇₄	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₇₄	1	1	yes	characteristic
90	C₂	#41×D₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Frattini	C₂²	C₂³×D₃₇
Fitting	#117×C₃₇	C₂
prime radical	C₂³×C₃₇	C₂³
hypercenter	#117	D₃₇
hypocenter	C₃₇	#533
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #11924535	C₂	semidirect
2. #11924527	C₂	semidirect
4. C₄×Dic₇₄	C₂	semidirect
24. C₂×C₄×C₃₇	Q₈	semidirect
46. C₃₇	#533	semidirect
75. C₂×C₄	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#11924535	2
#11924527	2
#11924513	2
C₄×Dic₇₄	2
#11924495	2
C₂×C₄×Dic₃₇	2
#117×C₃₇	2
#533	37