Group #14558248 — subgroups

Subgroups — #14558248

Order 2368.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11924522	C₂	1	4	yes	maximal, complemented
3	#11924510	C₂	1	2	yes	maximal, complemented
4	#11924507	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924495	C₂	1	2	yes	maximal, complemented
7	#11924494	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₃₇	C₂²	1	4	yes	—
12	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₇	C₂²	1	2	yes	—
14	C₂×Dic₇₄	C₂²	1	2	yes	—
15	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₃₇	C₂²	1	4	yes	—
19	#10596938	C₂²	1	4	yes	—
20	#10596938	C₂²	1	2	yes	—
21	C₄⋊Dic₃₇	C₂²	1	2	yes	—
22	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₃₇	C₂²	1	2	yes	—
24	C₄×Dic₃₇	C₂²	1	4	yes	—
25	C₂³×C₃₇	C₂³	1	2	yes	—
26	C₂²×D₃₇	C₂³	1	2	yes	—
27	D₄×C₃₇	—	2	2	no	—
28	C₂×C₄×C₃₇	—	2	4	no	—
29	C₂×C₄×C₃₇	C₂³	1	2	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	C₂²⋊D₃₇	—	4	4	no	—
34	C₂×Dic₃₇	C₂³	1	2	yes	—
35	C₂×Dic₃₇	C₂³	1	4	yes	—
36	D₁₄₈	—	2	2	no	—
37	C₄×D₃₇	—	2	4	no	—
38	Dic₇₄	—	4	2	no	—
39	C₂²×C₃₇	—	2	2	no	—
40	C₂²×C₃₇	—	2	2	no	—
41	C₂²×C₃₇	—	2	2	no	—
42	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₃₇	—	2	2	no	—
44	C₂×D₃₇	—	2	2	no	—
45	C₂×D₃₇	—	2	2	no	—
46	C₄×C₃₇	—	2	2	no	—
47	C₄×C₃₇	—	2	1	no	—
48	C₄×C₃₇	#41	1	2	yes	—
49	C₄×C₃₇	#41	1	2	yes	—
50	Dic₃₇	—	2	2	no	—
51	Dic₃₇	—	2	4	no	—
52	C₂×C₃₇	—	4	2	no	—
53	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
54	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
56	D₃₇	—	4	2	no	—
57	#555	—	37	1	no	maximal, Sylow
58	C₃₇	#555	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	37	1	no	—
60	#124	—	37	4	no	—
61	#121	—	37	2	no	—
62	#121	—	37	2	no	—
63	C₄×D₄	—	37	1	no	—
64	C₄×D₄	D₃₇	1	1	yes	characteristic, complemented
65	#117	—	37	2	no	—
66	#117	—	37	2	no	—
67	C₂×Q₈	—	37	2	no	—
68	C₂×D₄	—	37	4	no	—
69	C₂×D₄	—	37	1	no	—
70	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
71	C₂²×C₄	—	37	2	no	—
72	C₂²×C₄	C₂×D₃₇	1	2	yes	—
73	C₄⋊C₄	—	37	1	no	—
74	C₄⋊C₄	—	37	2	no	—
75	C₄⋊C₄	—	37	2	no	—
76	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
77	C₂²⋊C₄	—	37	4	no	—
78	C₂²⋊C₄	—	37	4	no	—
79	C₂²⋊C₄	—	37	2	no	—
80	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
81	C₄²	—	37	4	no	—
82	C₄²	C₂×D₃₇	1	1	yes	characteristic
83	C₂³	—	37	2	no	—
84	C₂³	C₂²×D₃₇	1	2	yes	—
85	Q₈	—	148	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	74	2	no	—
88	D₄	—	148	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	37	2	no	—
91	C₂×C₄	—	37	4	no	—
92	C₂×C₄	—	74	4	no	—
93	C₂×C₄	C₂²×D₃₇	1	2	yes	—
94	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	74	2	no	—
101	C₂²	—	74	2	no	—
102	C₂²	—	74	2	no	—
103	C₂²	C₂³×D₃₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	74	2	no	—
107	C₄	—	74	4	no	—
108	C₄	#10596962	1	2	yes	—
109	C₄	#10596964	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	148	2	no	—
112	C₂	C₂×#10596964	1	1	yes	characteristic
113	C₂	C₂×#10596962	1	1	yes	characteristic
114	C₂	#11924582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924536

C₂

yes

maximal, characteristic, complemented

#11924522

C₂

yes

maximal, complemented

#11924510

C₂

yes

maximal, complemented

#11924507

C₂

yes

maximal, complemented

C₄×D₁₄₈

C₂

yes

maximal, characteristic, complemented

#11924495

C₂

yes

maximal, complemented

#11924494

C₂

yes

maximal, complemented

C₄×D₄×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂²

yes

—

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

C₂²⋊Dic₃₇

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

—

C₂²×D₃₇

C₂³

yes

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#41

yes

—

C₄×C₃₇

#41

yes

—

Dic₃₇

—

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

D₃₇

—

#555

—

maximal, Sylow

C₃₇

#555

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

D₃₇

yes

characteristic, complemented

#117

—

#117

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₃₇

yes

—

Q₈

—

148

—

D₄

—

D₄

—

D₄

—

148

—

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

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂³×D₃₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#10596962

yes

—

109

C₄

#10596964

yes

—

110

C₂

—

111

C₂

—

148

—

112

C₂

C₂×#10596964

yes

characteristic

113

C₂

C₂×#10596962

yes

characteristic

114

C₂

#11924582

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄×D₄×C₃₇	C₂
hypercenter	C₄×D₄	D₃₇
hypocenter	C₃₇	#555
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₄×D₄×C₃₇

C₂

hypercenter

C₄×D₄

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #11924536	C₂	semidirect
2. #11924522	C₂	semidirect
3. #11924510	C₂	semidirect
4. #11924507	C₂	semidirect
5. C₄×D₁₄₈	C₂	semidirect
6. #11924495	C₂	semidirect
7. #11924494	C₂	semidirect
8. C₄×D₄×C₃₇	C₂	semidirect
58. C₃₇	#555	semidirect
64. C₄×D₄	D₃₇	semidirect

H ≅ G/N

type

1. #11924536

C₂

semidirect

2. #11924522

C₂

semidirect

3. #11924510

C₂

semidirect

4. #11924507

C₂

semidirect

5. C₄×D₁₄₈

C₂

semidirect

6. #11924495

C₂

semidirect

7. #11924494

C₂

semidirect

8. C₄×D₄×C₃₇

C₂

semidirect

58. C₃₇

#555

semidirect

64. C₄×D₄

D₃₇

semidirect

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

subgroup	index
#11924536	2
#11924522	2
#11924510	2
#11924507	2
C₄×D₁₄₈	2
#11924495	2
#11924494	2
C₄×D₄×C₃₇	2
#555	37

subgroup

index

C₄×D₁₄₈

C₄×D₄×C₃₇

Subgroups — #14558248

Conjugacy classes of subgroups

Complements

Maximal subgroups