Group #14558103 — subgroups

Subgroups — #14558103

Order 2368.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#10596964	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924536	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924534	C₂	1	1	yes	maximal, characteristic, complemented
5	#11924523	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924518	C₂	1	4	yes	maximal, complemented
7	#11924513	C₂	1	4	yes	maximal, complemented
8	#11924493	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
13	C₂²×Dic₃₇	C₂²	1	2	yes	—
14	Q₈×D₃₇	—	2	4	no	—
15	#10596964	—	2	4	no	—
16	C₂×C₄×D₃₇	C₂²	1	2	yes	—
17	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₇₄	C₂²	1	2	yes	—
19	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₃₇	C₂²	1	4	yes	—
22	C₄²×C₃₇	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₃₇	C₂²	1	4	yes	—
24	#10596938	C₂²	1	2	yes	—
25	C₄⋊Dic₃₇	C₂²	1	4	yes	—
26	#10596936	C₂²	1	2	yes	—
27	#10596936	C₂²	1	4	yes	—
28	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
29	C₂³×C₃₇	C₂³	1	2	yes	—
30	C₂²×D₃₇	C₂³	1	1	yes	characteristic
31	Q₈×C₃₇	—	2	2	no	—
32	D₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	C₂³	1	2	yes	—
34	C₂×C₄×C₃₇	C₂³	1	2	yes	—
35	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²⋊D₃₇	—	2	4	no	—
37	C₂×Dic₃₇	—	2	4	no	—
38	C₂×Dic₃₇	C₂³	1	2	yes	—
39	C₂×Dic₃₇	C₂³	1	2	yes	—
40	C₂×Dic₃₇	C₂³	1	2	yes	—
41	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
42	C₄×D₃₇	—	2	4	no	—
43	C₄×D₃₇	D₄	1	4	yes	—
44	Dic₇₄	—	2	2	no	—
45	Dic₇₄	—	2	2	no	—
46	Dic₇₄	—	2	4	no	—
47	C₂²×C₃₇	—	2	2	no	—
48	C₂²×C₃₇	—	2	4	no	—
49	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₃₇	—	2	2	no	—
51	C₂×D₃₇	C₂×D₄	1	2	yes	—
52	C₄×C₃₇	—	2	2	no	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	C₂×D₄	1	2	yes	—
55	Dic₃₇	—	2	2	no	—
56	Dic₃₇	—	2	2	no	—
57	Dic₃₇	—	2	2	no	—
58	Dic₃₇	C₂×D₄	1	2	yes	—
59	C₂×C₃₇	—	4	2	no	—
60	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₃₇	2₋¹⁺⁴	1	2	yes	—
62	D₃₇	—	2	2	no	—
63	#536	—	37	1	no	maximal, Sylow
64	C₃₇	#536	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	37	1	no	—
66	C₂²×Q₈	—	37	1	no	—
67	C₄⋊Q₈	—	37	1	no	—
68	C₄⋊Q₈	—	37	1	no	—
69	#124	—	37	1	no	—
70	#124	D₃₇	1	1	yes	characteristic, complemented
71	#123	—	37	4	no	—
72	C₂²⋊Q₈	—	37	4	no	—
73	#117	—	37	1	no	—
74	#41	—	74	4	no	—
75	C₂×Q₈	—	37	2	no	—
76	C₂×Q₈	—	37	1	no	—
77	C₂×Q₈	—	37	1	no	—
78	C₂×Q₈	—	74	4	no	—
79	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
80	C₂×D₄	—	37	2	no	—
81	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
82	C₂²×C₄	—	37	1	no	—
83	C₂²×C₄	—	37	2	no	—
84	C₂²×C₄	—	37	2	no	—
85	C₄⋊C₄	—	37	2	no	—
86	C₄⋊C₄	—	37	4	no	—
87	C₄⋊C₄	—	37	4	no	—
88	C₂²⋊C₄	—	37	4	no	—
89	C₂²⋊C₄	—	37	2	no	—
90	C₂²⋊C₄	C₂×D₃₇	1	4	yes	—
91	C₄²	—	37	1	no	—
92	C₄²	C₂×D₃₇	1	1	yes	characteristic
93	C₂³	—	37	1	no	—
94	C₂³	C₂²×D₃₇	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	74	4	no	—
97	Q₈	—	74	2	no	—
98	Q₈	—	74	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	74	4	no	—
101	C₂×C₄	—	37	1	no	—
102	C₂×C₄	—	37	4	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	37	2	no	—
105	C₂×C₄	—	37	2	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	74	4	no	—
108	C₂×C₄	C₂²×D₃₇	1	2	yes	—
109	C₂×C₄	C₂²×D₃₇	1	2	yes	—
110	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	37	2	no	—
114	C₂²	—	74	2	no	—
115	C₂²	C₂³×D₃₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	37	2	no	—
119	C₄	—	74	2	no	—
120	C₄	—	74	2	no	—
121	C₄	—	74	2	no	—
122	C₄	D₄×D₃₇	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	74	2	no	—
125	C₂	C₂×D₄×D₃₇	1	1	yes	characteristic
126	C₂	#11924583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10596964

C₂

yes

maximal, characteristic

C₂×Q₈×D₃₇

C₂

yes

maximal, characteristic, complemented

#11924536

C₂

yes

maximal, characteristic, complemented

#11924534

C₂

yes

maximal, characteristic, complemented

#11924523

C₂

yes

maximal, characteristic, complemented

#11924518

C₂

yes

maximal, complemented

#11924513

C₂

yes

maximal, complemented

#11924493

C₂

yes

maximal, characteristic, complemented

#124×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

characteristic

C₂×D₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂²×Dic₃₇

C₂²

yes

—

Q₈×D₃₇

—

#10596964

—

C₂×C₄×D₃₇

C₂²

yes

—

C₂×C₄×D₃₇

C₂²

yes

characteristic

C₂×Dic₇₄

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₂×Dic₇₄

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

C₂²⋊Dic₃₇

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

characteristic

C₂³×C₃₇

C₂³

yes

—

C₂²×D₃₇

C₂³

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

D₄

yes

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₂×D₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₄

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×D₄

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₄

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

—

D₃₇

—

#536

—

maximal, Sylow

C₃₇

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

D₃₇

yes

characteristic, complemented

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₃₇

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×D₃₇

yes

—

109

C₂×C₄

C₂²×D₃₇

yes

—

110

C₂×C₄

C₂²×D₃₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₃₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₃₇

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₂×D₄×D₃₇

yes

characteristic

126

C₂

#11924583

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	#124×C₃₇	C₂
prime radical	C₂×#10596964	C₂
hypercenter	#124	D₃₇
hypocenter	C₃₇	#536
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

#124×C₃₇

C₂

C₂

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₃₇	C₂	semidirect
3. #11924536	C₂	semidirect
4. #11924534	C₂	semidirect
5. #11924523	C₂	semidirect
6. #11924518	C₂	semidirect
7. #11924513	C₂	semidirect
8. #11924493	C₂	semidirect
9. #124×C₃₇	C₂	semidirect
64. C₃₇	#536	semidirect
70. #124	D₃₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₃₇

C₂

semidirect

3. #11924536

C₂

semidirect

4. #11924534

C₂

semidirect

5. #11924523

C₂

semidirect

6. #11924518

C₂

semidirect

7. #11924513

C₂

semidirect

8. #11924493

C₂

semidirect

9. #124×C₃₇

C₂

semidirect

64. C₃₇

#536

semidirect

70. #124

D₃₇

semidirect

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

subgroup	index
C₂×#10596964	2
C₂×Q₈×D₃₇	2
#11924536	2
#11924534	2
#11924523	2
#11924518	2
#11924513	2
#11924493	2
#124×C₃₇	2
#536	37

subgroup

index

C₂×Q₈×D₃₇

#124×C₃₇

Subgroups — #14558103

Conjugacy classes of subgroups

Complements

Maximal subgroups