Group #14558204 — subgroups

Subgroups — #14558204

Order 2368.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#10596962	C₂	1	2	yes	maximal, complemented
2	#11924533	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924532	C₂	1	2	yes	maximal, complemented
4	#11924510	C₂	1	2	yes	maximal, complemented
5	#11924509	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#11924498	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×C₂²⋊D₃₇	C₂²	1	4	yes	—
13	#10596962	—	2	8	no	—
14	C₂×D₁₄₈	C₂²	1	2	yes	—
15	C₂×D₁₄₈	C₂²	1	2	yes	—
16	C₂×D₁₄₈	C₂²	1	2	yes	—
17	C₂×C₄×D₃₇	C₂²	1	4	yes	—
18	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
21	C₄²×C₃₇	C₂²	1	1	yes	characteristic
22	#10596938	C₂²	1	4	yes	—
23	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
24	#10596936	C₂²	1	2	yes	—
25	C₄×Dic₃₇	C₂²	1	2	yes	—
26	C₂³×C₃₇	C₂³	1	2	yes	—
27	C₂²×D₃₇	C₂³	1	4	yes	—
28	D₄×C₃₇	—	2	2	no	—
29	C₂×C₄×C₃₇	—	2	4	no	—
30	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂³	1	2	yes	—
32	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²⋊D₃₇	—	2	8	no	—
35	C₂²⋊D₃₇	—	4	4	no	—
36	C₂×Dic₃₇	C₂³	1	2	yes	—
37	C₂×Dic₃₇	C₂³	1	2	yes	—
38	D₁₄₈	—	2	4	no	—
39	D₁₄₈	—	2	4	no	—
40	D₁₄₈	—	4	2	no	—
41	C₄×D₃₇	—	2	8	no	—
42	Dic₇₄	D₄	1	4	yes	complemented
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	—	2	2	no	—
45	C₂²×C₃₇	—	2	2	no	—
46	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₃₇	—	2	4	no	—
48	C₂×D₃₇	—	2	4	no	—
49	C₂×D₃₇	—	2	4	no	—
50	C₄×C₃₇	—	2	2	no	—
51	C₄×C₃₇	—	2	1	no	—
52	C₄×C₃₇	C₂×D₄	1	2	yes	—
53	C₄×C₃₇	#41	1	2	yes	complemented
54	Dic₃₇	—	2	2	no	—
55	Dic₃₇	C₂×D₄	1	4	yes	—
56	C₂×C₃₇	—	4	2	no	—
57	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
60	D₃₇	—	4	4	no	—
61	#550	—	37	1	no	maximal, Sylow
62	C₃₇	#550	1	1	yes	characteristic, complemented, Sylow
63	C₂×#41	—	37	2	no	—
64	#127	—	37	2	no	—
65	#127	—	37	1	no	—
66	#121	—	37	2	no	—
67	#121	—	37	4	no	—
68	C₄×Q₈	—	37	1	no	—
69	C₄×D₄	—	37	2	no	—
70	C₄×D₄	D₃₇	1	1	yes	characteristic, complemented
71	#41	—	74	8	no	—
72	C₂×Q₈	—	37	1	no	—
73	C₂×D₄	—	37	4	no	—
74	C₂×D₄	—	37	4	no	—
75	C₂×D₄	—	37	2	no	—
76	C₂×D₄	—	37	2	no	—
77	C₂×D₄	—	37	2	no	—
78	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
79	C₂²×C₄	—	37	4	no	—
80	C₂²×C₄	C₂×D₃₇	1	2	yes	—
81	C₄⋊C₄	—	37	2	no	—
82	C₄⋊C₄	—	37	1	no	—
83	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	37	4	no	—
85	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
86	C₄²	—	37	2	no	—
87	C₄²	C₂×D₃₇	1	1	yes	characteristic
88	C₂³	—	37	4	no	—
89	C₂³	C₂²×D₃₇	1	2	yes	—
90	Q₈	—	37	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	74	8	no	—
93	D₄	—	74	4	no	—
94	D₄	—	74	4	no	—
95	D₄	—	148	2	no	—
96	D₄	—	148	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	37	2	no	—
99	C₂×C₄	—	37	2	no	—
100	C₂×C₄	—	74	8	no	—
101	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₇	1	2	yes	—
103	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	74	4	no	—
109	C₂²	—	74	4	no	—
110	C₂²	—	74	4	no	—
111	C₂²	C₂³×D₃₇	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	37	4	no	—
115	C₄	—	74	2	no	—
116	C₄	#10596962	1	2	yes	complemented
117	C₄	D₄×D₃₇	1	2	yes	—
118	C₂	—	4	2	no	—
119	C₂	—	148	4	no	—
120	C₂	C₂×D₄×D₃₇	1	1	yes	characteristic
121	C₂	C₂×#10596962	1	1	yes	characteristic
122	C₂	#11924582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10596962

C₂

yes

maximal, complemented

#11924533

C₂

yes

maximal, characteristic, complemented

#11924532

C₂

yes

maximal, complemented

#11924510

C₂

yes

maximal, complemented

#11924509

C₂

yes

maximal, complemented

C₄×Dic₇₄

C₂

yes

maximal, characteristic, complemented

#11924498

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×C₂²⋊D₃₇

C₂²

yes

—

#10596962

—

C₂×D₁₄₈

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

—

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

—

C₂²×D₃₇

C₂³

yes

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊D₃₇

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

D₄

yes

complemented

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₄

yes

—

C₄×C₃₇

#41

yes

complemented

Dic₃₇

—

Dic₃₇

C₂×D₄

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₄

yes

characteristic

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

D₃₇

—

#550

—

maximal, Sylow

C₃₇

#550

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₃₇

yes

characteristic, complemented

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₃₇

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²×D₃₇

yes

characteristic

102

C₂×C₄

C₂²×D₃₇

yes

—

103

C₂×C₄

C₂²×D₃₇

yes

characteristic

104

C₂×C₄

C₂²×D₃₇

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₃₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#10596962

yes

complemented

117

C₄

D₄×D₃₇

yes

—

118

C₂

—

119

C₂

—

148

—

120

C₂

C₂×D₄×D₃₇

yes

characteristic

121

C₂

C₂×#10596962

yes

characteristic

122

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	#550
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. C₂×#10596962	C₂	semidirect
2. #11924533	C₂	semidirect
3. #11924532	C₂	semidirect
4. #11924510	C₂	semidirect
5. #11924509	C₂	semidirect
6. C₄×Dic₇₄	C₂	semidirect
7. #11924498	C₂	semidirect
8. C₄×D₄×C₃₇	C₂	semidirect
42. Dic₇₄	D₄	semidirect
53. C₄×C₃₇	#41	semidirect
62. C₃₇	#550	semidirect
70. C₄×D₄	D₃₇	semidirect
116. C₄	#10596962	semidirect

H ≅ G/N

type

1. C₂×#10596962

C₂

semidirect

2. #11924533

C₂

semidirect

3. #11924532

C₂

semidirect

4. #11924510

C₂

semidirect

5. #11924509

C₂

semidirect

6. C₄×Dic₇₄

C₂

semidirect

7. #11924498

C₂

semidirect

8. C₄×D₄×C₃₇

C₂

semidirect

42. Dic₇₄

D₄

semidirect

53. C₄×C₃₇

#41

semidirect

62. C₃₇

#550

semidirect

70. C₄×D₄

D₃₇

semidirect

116. C₄

#10596962

semidirect

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

subgroup	index
C₂×#10596962	2
#11924533	2
#11924532	2
#11924510	2
#11924509	2
C₄×Dic₇₄	2
#11924498	2
C₄×D₄×C₃₇	2
#550	37

subgroup

index

C₄×Dic₇₄

C₄×D₄×C₃₇

Subgroups — #14558204

Conjugacy classes of subgroups

Complements

Maximal subgroups