Group #14558242 — subgroups

Subgroups — #14558242

Order 2368.

228 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 113 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	#11924535	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924528	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924527	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924515	C₂	1	2	yes	maximal, complemented
5	#11924513	C₂	1	2	yes	maximal, complemented
6	#11924513	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
9	D₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#10596936	C₂	1	2	yes	maximal, complemented
11	C₄×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×D₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
14	C₂²×Dic₃₇	C₂²	1	2	yes	—
15	C₂²×Dic₃₇	C₂²	1	2	yes	—
16	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₄	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
20	C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₇	C₂²	1	2	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	2	yes	—
27	#10596936	C₂²	1	1	yes	characteristic
28	#10596936	C₂²	1	2	yes	—
29	#10596936	C₂²	1	2	yes	—
30	#10596936	C₂²	1	2	yes	—
31	#10596936	C₂²	1	1	yes	characteristic
32	#10596936	C₂²	1	2	yes	—
33	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	C₂³×C₃₇	C₂³	1	2	yes	—
36	D₄×C₃₇	Q₈	1	4	yes	complemented
37	C₂×C₄×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₇	—	2	4	no	—
43	C₂×Dic₃₇	—	4	2	no	—
44	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₇	C₂³	1	2	yes	—
48	C₂×Dic₃₇	C₂³	1	2	yes	—
49	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
50	Dic₇₄	—	2	2	no	—
51	Dic₇₄	—	4	2	no	—
52	C₂²×C₃₇	—	2	2	no	—
53	C₂²×C₃₇	—	2	2	no	—
54	C₂²×C₃₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
56	C₄×C₃₇	—	2	1	no	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	—	2	1	no	—
59	C₄×C₃₇	C₂×Q₈	1	2	yes	—
60	Dic₃₇	—	2	1	no	—
61	Dic₃₇	—	2	2	no	—
62	Dic₃₇	—	2	1	no	—
63	Dic₃₇	—	2	2	no	—
64	Dic₃₇	—	2	1	no	—
65	Dic₃₇	#41	1	2	yes	—
66	C₂×C₃₇	—	2	4	no	—
67	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
70	#554	—	37	1	no	maximal, Sylow
71	C₃₇	#554	1	1	yes	characteristic, complemented, Sylow
72	C₄⋊Q₈	—	37	1	no	—
73	#125	—	37	1	no	—
74	#125	—	37	1	no	—
75	C₂²⋊Q₈	—	37	2	no	—
76	C₂²⋊Q₈	—	37	2	no	—
77	C₂²⋊Q₈	—	37	2	no	—
78	C₄×Q₈	—	37	1	no	—
79	C₄×D₄	—	37	1	no	—
80	C₄×D₄	—	37	1	no	—
81	C₄×D₄	D₃₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	37	2	no	—
83	C₂×Q₈	—	37	1	no	—
84	C₂×Q₈	—	37	2	no	—
85	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	—	37	2	no	—
88	C₂²×C₄	C₂×D₃₇	1	2	yes	—
89	C₄⋊C₄	—	37	1	no	—
90	C₄⋊C₄	—	37	2	no	—
91	C₄⋊C₄	—	37	2	no	—
92	C₄⋊C₄	—	37	2	no	—
93	C₄⋊C₄	—	37	1	no	—
94	C₄⋊C₄	—	37	2	no	—
95	C₄⋊C₄	—	37	1	no	—
96	C₄⋊C₄	—	37	1	no	—
97	C₄⋊C₄	—	37	1	no	—
98	C₄⋊C₄	—	37	2	no	—
99	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
100	C₂²⋊C₄	—	37	2	no	—
101	C₂²⋊C₄	—	37	2	no	—
102	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
103	C₄²	—	37	1	no	—
104	C₄²	—	37	1	no	—
105	C₄²	C₂×D₃₇	1	1	yes	characteristic
106	C₂³	C₂²×D₃₇	1	2	yes	—
107	Q₈	—	74	2	no	—
108	Q₈	—	148	2	no	—
109	D₄	Dic₇₄	1	4	yes	complemented
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	37	1	no	—
112	C₂×C₄	—	37	1	no	—
113	C₂×C₄	—	37	1	no	—
114	C₂×C₄	—	37	1	no	—
115	C₂×C₄	—	37	2	no	—
116	C₂×C₄	—	37	2	no	—
117	C₂×C₄	—	74	4	no	—
118	C₂×C₄	—	148	2	no	—
119	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₃₇	1	2	yes	—
122	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×Dic₇₄	1	4	yes	—
126	C₂²	C₂³×D₃₇	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	37	2	no	—
131	C₄	—	74	2	no	—
132	C₄	—	74	1	no	—
133	C₄	—	74	1	no	—
134	C₄	—	74	2	no	—
135	C₄	—	74	1	no	—
136	C₄	C₂×Dic₇₄	1	2	yes	—
137	C₂	—	2	4	no	—
138	C₂	C₂²×Dic₇₄	1	1	yes	characteristic
139	C₂	#41×D₃₇	1	1	yes	characteristic
140	C₂	#11924581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924535

C₂

yes

maximal, characteristic, complemented

#11924528

C₂

yes

maximal, characteristic, complemented

#11924527

C₂

yes

maximal, characteristic, complemented

#11924515

C₂

yes

maximal, complemented

#11924513

C₂

yes

maximal, complemented

#11924513

C₂

yes

maximal, complemented

C₄×Dic₇₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₃₇

C₂

yes

maximal, characteristic

D₄×Dic₃₇

C₂

yes

maximal, characteristic

C₂×#10596936

C₂

yes

maximal, complemented

C₄×D₄×C₃₇

C₂

yes

maximal, characteristic

C₂×D₄×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂²

yes

—

C₂²×Dic₃₇

C₂²

yes

—

C₂²×Dic₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₂×Dic₇₄

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

C₂²⋊Dic₃₇

C₂²

yes

—

C₂²⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

—

#10596936

C₂²

yes

characteristic

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

characteristic

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

characteristic

C₄×Dic₃₇

C₂²

yes

characteristic

C₂³×C₃₇

C₂³

yes

—

D₄×C₃₇

Q₈

yes

complemented

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

characteristic

Dic₇₄

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×Q₈

yes

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#41

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×Q₈

yes

characteristic

C₂×C₃₇

C₂×#41

yes

C₂×C₃₇

yes

—

C₃₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

C₂×D₃₇

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₃₇

yes

characteristic

106

C₂³

C₂²×D₃₇

yes

—

107

Q₈

—

108

Q₈

—

148

—

109

D₄

Dic₇₄

yes

complemented

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

148

—

119

C₂×C₄

C₂²×D₃₇

yes

characteristic

120

C₂×C₄

C₂²×D₃₇

yes

characteristic

121

C₂×C₄

C₂²×D₃₇

yes

—

122

C₂×C₄

C₂²×D₃₇

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×Dic₇₄

yes

—

126

C₂²

C₂³×D₃₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×Dic₇₄

yes

—

137

C₂

—

138

C₂

C₂²×Dic₇₄

yes

characteristic

139

C₂

#41×D₃₇

yes

characteristic

140

C₂

#11924581

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄×D₄×C₃₇	C₂
prime radical	C₂×D₄×C₃₇	C₂²
hypercenter	C₄×D₄	D₃₇
hypocenter	C₃₇	#554
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

prime radical

C₂×D₄×C₃₇

C₂²

hypercenter

C₄×D₄

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #11924535	C₂	semidirect
2. #11924528	C₂	semidirect
3. #11924527	C₂	semidirect
4. #11924515	C₂	semidirect
5. #11924513	C₂	semidirect
6. #11924513	C₂	semidirect
7. C₄×Dic₇₄	C₂	semidirect
10. C₂×#10596936	C₂	semidirect
36. D₄×C₃₇	Q₈	semidirect
71. C₃₇	#554	semidirect
109. D₄	Dic₇₄	semidirect

H ≅ G/N

type

1. #11924535

C₂

semidirect

2. #11924528

C₂

semidirect

3. #11924527

C₂

semidirect

4. #11924515

C₂

semidirect

5. #11924513

C₂

semidirect

6. #11924513

C₂

semidirect

7. C₄×Dic₇₄

C₂

semidirect

10. C₂×#10596936

C₂

semidirect

36. D₄×C₃₇

Q₈

semidirect

71. C₃₇

#554

semidirect

109. D₄

Dic₇₄

semidirect

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

subgroup	index
#11924535	2
#11924528	2
#11924527	2
#11924515	2
#11924513	2
C₄×Dic₇₄	2
D₄×Dic₃₇	2
C₂×#10596936	2
C₄×D₄×C₃₇	2
#554	37

subgroup

index

C₄×Dic₇₄

D₄×Dic₃₇

C₂×#10596936

C₄×D₄×C₃₇

#554

Subgroups — #14558242

Conjugacy classes of subgroups

Complements

Maximal subgroups