Subgroups — #14558010

Order 2368.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₇₄	C₂	1	4	yes	maximal, complemented
3	#11924501	C₂	1	4	yes	maximal, complemented
4	#11924495	C₂	1	1	yes	maximal, characteristic
5	#11924492	C₂	1	4	yes	maximal
6	#117×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃₇	C₂²	1	2	yes	—
9	C₂×Dic₇₄	C₄	1	8	yes	complemented
10	C₂×Dic₇₄	C₂²	1	2	yes	—
11	C₂×Dic₇₄	C₂²	1	2	yes	—
12	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
14	C₄²×C₃₇	C₂²	1	2	yes	—
15	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
16	C₄⋊Dic₃₇	C₂²	1	2	yes	—
17	#10596936	C₂²	1	8	yes	—
18	C₄×Dic₃₇	C₂²	1	8	yes	—
19	C₄×Dic₃₇	C₂²	1	2	yes	—
20	C₂³×C₃₇	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₇	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₃₇	C₂³	1	4	yes	—
23	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₇	C₂×C₄	1	8	yes	—
26	C₂×Dic₃₇	C₂³	1	4	yes	—
27	C₂×Dic₃₇	C₂³	1	4	yes	—
28	Dic₇₄	C₂×C₄	1	8	yes	—
29	Dic₇₄	C₂×C₄	1	8	yes	—
30	C₂²×C₃₇	—	2	2	no	—
31	C₂²×C₃₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
33	C₄×C₃₇	—	2	4	no	—
34	C₄×C₃₇	C₂²×C₄	1	2	yes	—
35	C₄×C₃₇	C₂²×C₄	1	2	yes	—
36	Dic₃₇	—	2	4	no	—
37	Dic₃₇	C₂²×C₄	1	8	yes	—
38	C₂×C₃₇	—	2	2	no	—
39	C₂×C₃₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₃₇	2₋¹⁺⁴	1	2	yes	—
41	#519	—	37	1	no	maximal, Sylow
42	C₃₇	#519	1	1	yes	characteristic, complemented, Sylow
43	C₂²×Q₈	—	37	1	no	—
44	C₄×Q₈	—	37	4	no	—
45	C₄×Q₈	—	37	4	no	—
46	#117	—	37	1	no	—
47	#117	—	37	4	no	—
48	#117	D₃₇	1	1	yes	characteristic
49	C₂×Q₈	—	37	8	no	—
50	C₂×Q₈	—	37	2	no	—
51	C₂×Q₈	—	37	2	no	—
52	C₂²×C₄	—	37	2	no	—
53	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
54	C₄⋊C₄	—	37	8	no	—
55	C₄⋊C₄	—	37	2	no	—
56	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
57	C₂²⋊C₄	—	37	2	no	—
58	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
59	C₄²	—	37	8	no	—
60	C₄²	—	37	2	no	—
61	C₄²	C₂×D₃₇	1	2	yes	—
62	C₂³	C₂²×D₃₇	1	1	yes	characteristic
63	Q₈	—	37	8	no	—
64	Q₈	—	37	8	no	—
65	C₂×C₄	—	37	8	no	—
66	C₂×C₄	—	37	4	no	—
67	C₂×C₄	—	37	4	no	—
68	C₂×C₄	C₄×D₃₇	1	4	yes	—
69	C₂×C₄	C₂²×D₃₇	1	4	yes	—
70	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₃₇	1	2	yes	—
74	C₂²	C₂³×D₃₇	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	37	8	no	—
77	C₄	—	74	4	no	—
78	C₄	C₂×C₄×D₃₇	1	2	yes	—
79	C₄	C₂×C₄×D₃₇	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₃₇	1	1	yes	characteristic
82	C₂	#11924583	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator	C₂×C₃₇	C₂³×C₄
Fitting	#117×C₃₇	C₂
socle, cosocle	C₂²×C₃₇	C₂⁴
prime radical	C₂³×C₃₇	C₂³
hypercenter	#117	D₃₇
hypocenter	C₃₇	#519
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₇₄	C₂	semidirect
3. #11924501	C₂	semidirect
9. C₂×Dic₇₄	C₄	semidirect
42. C₃₇	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₄	2
C₄×Dic₇₄	2
#11924501	2
#11924495	2
#11924492	2
#117×C₃₇	2
#519	37