Group #14558088 — subgroups

Subgroups — #14558088

Order 2368.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924536	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924527	C₂	1	2	yes	maximal, complemented
3	#11924516	C₂	1	2	yes	maximal
4	#11924512	C₂	1	2	yes	maximal
5	Q₈×Dic₃₇	C₂	1	2	yes	maximal, complemented
6	#11924501	C₂	1	2	yes	maximal, complemented
7	#11924494	C₂	1	2	yes	maximal
8	C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₄	C₂²	1	2	yes	—
15	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
16	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	#10596938	C₂²	1	4	yes	—
19	C₄⋊Dic₃₇	C₂²	1	2	yes	—
20	C₄⋊Dic₃₇	C₂²	1	2	yes	—
21	C₄⋊Dic₃₇	C₂²	1	4	yes	—
22	#10596936	C₂²	1	4	yes	—
23	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
24	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₇	C₂²	1	4	yes	—
27	C₂²×D₃₇	C₂³	1	1	yes	characteristic
28	Q₈×C₃₇	—	2	4	no	—
29	C₂×C₄×C₃₇	C₂³	1	4	yes	—
30	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
34	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
35	C₂×Dic₃₇	C₂³	1	4	yes	—
36	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
37	C₄×D₃₇	—	2	2	no	—
38	C₄×D₃₇	—	2	2	no	—
39	C₄×D₃₇	Q₈	1	4	yes	complemented
40	Dic₇₄	—	2	4	no	—
41	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₃₇	—	2	1	no	—
43	C₂×D₃₇	—	2	1	no	—
44	C₂×D₃₇	C₂×Q₈	1	2	yes	—
45	C₄×C₃₇	—	2	4	no	—
46	C₄×C₃₇	C₂×Q₈	1	2	yes	—
47	C₄×C₃₇	#41	1	2	yes	—
48	C₄×C₃₇	#41	1	2	yes	—
49	Dic₃₇	—	2	4	no	—
50	Dic₃₇	—	2	1	no	—
51	Dic₃₇	—	2	1	no	—
52	Dic₃₇	C₂×Q₈	1	2	yes	—
53	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
56	D₃₇	—	2	2	no	—
57	#533	—	37	1	no	maximal, Sylow
58	C₃₇	#533	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	37	1	no	—
60	C₄⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
61	#125	—	37	2	no	—
62	C₂²⋊Q₈	—	37	2	no	—
63	C₂²⋊Q₈	—	37	2	no	—
64	C₄×Q₈	—	37	2	no	—
65	C₄×Q₈	—	37	2	no	—
66	#117	—	37	2	no	—
67	C₂×C₄²	—	37	1	no	—
68	C₂×Q₈	—	37	2	no	—
69	C₂×Q₈	C₂×D₃₇	1	2	yes	—
70	C₂²×C₄	—	37	1	no	—
71	C₂²×C₄	—	37	1	no	—
72	C₂²×C₄	—	37	1	no	—
73	C₄⋊C₄	—	37	4	no	—
74	C₄⋊C₄	—	37	2	no	—
75	C₄⋊C₄	—	37	4	no	—
76	C₄⋊C₄	—	37	2	no	—
77	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
78	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
79	C₂²⋊C₄	—	37	4	no	—
80	C₄²	—	37	1	no	—
81	C₄²	—	37	1	no	—
82	C₄²	—	37	1	no	—
83	C₄²	—	37	4	no	—
84	C₄²	C₂×D₃₇	1	1	yes	characteristic
85	C₂³	—	37	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	74	4	no	—
88	C₂×C₄	—	37	1	no	—
89	C₂×C₄	—	37	4	no	—
90	C₂×C₄	—	37	1	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	4	no	—
93	C₂×C₄	—	74	2	no	—
94	C₂×C₄	—	74	2	no	—
95	C₂×C₄	C₂²×D₃₇	1	4	yes	—
96	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	C₂²	—	37	2	no	—
100	C₂²	—	74	1	no	—
101	C₂²	—	74	1	no	—
102	C₂²	C₂³×D₃₇	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	37	2	no	—
105	C₄	—	74	1	no	—
106	C₄	—	74	1	no	—
107	C₄	—	74	4	no	—
108	C₄	#10596964	1	2	yes	—
109	C₄	Q₈×D₃₇	1	2	yes	—
110	C₄	#10596966	1	2	yes	—
111	C₂	—	74	2	no	—
112	C₂	C₂×Q₈×D₃₇	1	1	yes	characteristic
113	C₂	C₂×#10596964	1	1	yes	characteristic
114	C₂	C₂×#10596966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924536

C₂

yes

maximal, characteristic, complemented

#11924527

C₂

yes

maximal, complemented

#11924516

C₂

yes

maximal

#11924512

C₂

yes

maximal

Q₈×Dic₃₇

C₂

yes

maximal, complemented

#11924501

C₂

yes

maximal, complemented

#11924494

C₂

yes

maximal

C₄²×D₃₇

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

—

C₂×C₄×D₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

characteristic

C₂×Dic₇₄

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

characteristic

C₄×Dic₃₇

C₂²

yes

characteristic

C₄×Dic₃₇

C₂²

yes

characteristic

C₄×Dic₃₇

C₂²

yes

—

C₂²×D₃₇

C₂³

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

Q₈

yes

complemented

Dic₇₄

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₈

yes

—

C₄×C₃₇

#41

yes

—

C₄×C₃₇

#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×Q₈

yes

—

C₂×C₃₇

C₂²×Q₈

yes

characteristic

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

C₂×#41

yes

characteristic

D₃₇

—

#533

—

maximal, Sylow

C₃₇

#533

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₃₇

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

—

C₄⋊C₄

C₂×D₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₃₇

yes

—

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂³×D₃₇

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#10596964

yes

—

109

C₄

Q₈×D₃₇

yes

—

110

C₄

#10596966

yes

—

111

C₂

—

112

C₂

C₂×Q₈×D₃₇

yes

characteristic

113

C₂

C₂×#10596964

yes

characteristic

114

C₂

C₂×#10596966

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄⋊Q₈×C₃₇	C₂
prime radical	C₂²×D₃₇	C₂³
hypercenter	C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#533
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₄⋊Q₈×C₃₇

C₂

prime radical

C₂²×D₃₇

C₂³

hypercenter

C₄⋊Q₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #11924536	C₂	semidirect
2. #11924527	C₂	semidirect
5. Q₈×Dic₃₇	C₂	semidirect
6. #11924501	C₂	semidirect
9. C₄⋊Q₈×C₃₇	C₂	semidirect
39. C₄×D₃₇	Q₈	semidirect
58. C₃₇	#533	semidirect
60. C₄⋊Q₈	D₃₇	semidirect

H ≅ G/N

type

1. #11924536

C₂

semidirect

2. #11924527

C₂

semidirect

5. Q₈×Dic₃₇

C₂

semidirect

6. #11924501

C₂

semidirect

9. C₄⋊Q₈×C₃₇

C₂

semidirect

39. C₄×D₃₇

Q₈

semidirect

58. C₃₇

#533

semidirect

60. C₄⋊Q₈

D₃₇

semidirect

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

subgroup	index
#11924536	2
#11924527	2
#11924516	2
#11924512	2
Q₈×Dic₃₇	2
#11924501	2
#11924494	2
C₄²×D₃₇	2
C₄⋊Q₈×C₃₇	2
#533	37

subgroup

index

Q₈×Dic₃₇

C₄²×D₃₇

C₄⋊Q₈×C₃₇

#533

Subgroups — #14558088

Conjugacy classes of subgroups

Complements

Maximal subgroups