Group #17995742 — subgroups

Subgroups — #17995742

Order 4736.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#11924528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924457	C₂	1	2	yes	maximal
3	#14557295	C₂	1	2	yes	maximal, complemented
4	#14557288	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	#11924528	—	2	4	no	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	C₂×#10596936	—	2	2	no	—
10	#11924457	—	2	4	no	—
11	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₃₇	—	2	1	no	—
16	C₂×D₄×C₃₇	—	2	2	no	—
17	C₂×D₄×C₃₇	—	2	2	no	—
18	C₂×D₄×C₃₇	—	2	1	no	—
19	C₂×D₄×C₃₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₇	—	2	1	no	—
21	C₂²×C₄×C₃₇	D₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₂²⋊C₄×C₃₇	—	4	4	no	—
26	C₄²×C₃₇	—	2	2	no	—
27	C₄⋊Dic₃₇	—	2	2	no	—
28	C₄⋊Dic₃₇	D₄	1	4	yes	—
29	#10596936	—	2	8	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂³×C₃₇	—	2	2	no	—
34	C₂³×C₃₇	—	4	1	no	—
35	C₂³×C₃₇	—	4	1	no	—
36	C₂³×C₃₇	—	4	2	no	—
37	C₂³×C₃₇	—	4	2	no	—
38	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₃₇	—	2	4	no	—
40	D₄×C₃₇	—	4	2	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	—	2	2	no	—
43	C₂×C₄×C₃₇	—	2	2	no	—
44	C₂×C₄×C₃₇	—	4	1	no	—
45	C₂×C₄×C₃₇	—	4	2	no	—
46	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₃₇	#41	1	2	yes	—
50	C₂×Dic₃₇	—	2	4	no	—
51	C₂×Dic₃₇	—	4	2	no	—
52	C₂×Dic₃₇	—	4	2	no	—
53	#64436	—	4	2	no	—
54	C₂²×C₃₇	—	4	2	no	—
55	C₂²×C₃₇	—	4	4	no	—
56	C₂²×C₃₇	—	8	2	no	—
57	C₂²×C₃₇	—	8	1	no	—
58	C₂²×C₃₇	—	8	1	no	—
59	C₂²×C₃₇	#120	1	1	yes	characteristic
60	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₃₇	#124	1	1	yes	characteristic
62	C₂²×C₃₇	#135	1	2	yes	—
63	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₃₇	—	4	2	no	—
65	C₄×C₃₇	—	8	1	no	—
66	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₃₇	#124	1	2	yes	—
68	Dic₃₇	—	4	4	no	—
69	#2025	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	8	2	no	—
71	C₂×C₃₇	#393	1	1	yes	characteristic
72	C₂×C₃₇	#449	1	2	yes	—
73	C₂×C₃₇	#478	1	2	yes	—
74	C₂×C₃₇	#488	1	1	yes	characteristic
75	C₂×C₃₇	#490	1	1	yes	characteristic
76	C₂×#125	—	37	1	no	—
77	C₂×#102	—	37	2	no	—
78	#390	D₃₇	1	1	yes	characteristic
79	#340	—	37	2	no	—
80	#336	—	37	1	no	—
81	C₃₇	#2025	1	1	yes	characteristic, complemented, Sylow
82	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
83	C₂²×C₈	—	37	2	no	—
84	#125	—	74	4	no	—
85	C₂×C₄⋊C₄	—	37	2	no	—
86	C₂×C₄⋊C₄	—	74	2	no	—
87	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
90	#102	—	74	4	no	—
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	Dic₇₄	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	74	2	no	—
98	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
100	C₂×C₈	—	74	2	no	—
101	C₂×C₈	—	74	2	no	—
102	C₂×C₈	—	74	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	37	4	no	—
105	C₄⋊C₄	—	74	8	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	74	4	no	—
123	C₂×C₄	—	148	2	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
126	C₂×C₄	#10596962	1	1	yes	characteristic
127	C₂×C₄	D₄×D₃₇	1	2	yes	—
128	C₂×C₄	#10596964	1	2	yes	—
129	C₈	—	148	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#11924504	1	1	yes	characteristic
136	C₂²	#11924515	1	1	yes	characteristic
137	C₂²	#11924523	1	1	yes	characteristic
138	C₂²	#11924556	1	2	yes	—
139	C₂²	#11924562	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	148	4	no	—
143	C₄	#11924513	1	2	yes	—
144	C₄	#11924522	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#14557445	1	1	yes	characteristic
147	C₂	#14557676	1	2	yes	—
148	C₂	#14557833	1	2	yes	—
149	C₂	#14557887	1	1	yes	characteristic
150	C₂	#14557897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic, complemented

C₂×#11924457

C₂

yes

maximal

#14557295

C₂

yes

maximal, complemented

#14557288

C₂

yes

maximal, characteristic, complemented

C₃₇×#390

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

#11924528

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924457

—

C₂²×D₄×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

—

—

—

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#124

yes

—

Dic₃₇

—

#2025

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#393

yes

characteristic

C₂×C₃₇

#449

yes

—

C₂×C₃₇

#478

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

D₃₇

yes

characteristic

#340

—

#336

—

C₃₇

#2025

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#102

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₇₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

148

—

124

C₂×C₄

—

148

—

125

C₂×C₄

C₂×Dic₇₄

yes

characteristic

126

C₂×C₄

#10596962

yes

characteristic

127

C₂×C₄

D₄×D₃₇

yes

—

128

C₂×C₄

#10596964

yes

—

129

C₈

—

148

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#11924504

yes

characteristic

136

C₂²

#11924515

yes

characteristic

137

C₂²

#11924523

yes

characteristic

138

C₂²

#11924556

yes

—

139

C₂²

#11924562

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

148

—

143

C₄

#11924513

yes

—

144

C₄

#11924522

yes

—

145

C₂

—

146

C₂

#14557445

yes

characteristic

147

C₂

#14557676

yes

—

148

C₂

#14557833

yes

—

149

C₂

#14557887

yes

characteristic

150

C₂

#14557897

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#390	C₂
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂²×D₄×C₃₇	C₂²
hypercenter	#390	D₃₇
hypocenter	C₃₇	#2025
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#390

C₂

socle

C₂³×C₃₇

C₂×D₄

prime radical

C₂²×D₄×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#11924528	C₂	semidirect
3. #14557295	C₂	semidirect
4. #14557288	C₂	semidirect
81. C₃₇	#2025	semidirect

H ≅ G/N

type

1. C₂×#11924528

C₂

semidirect

3. #14557295

C₂

semidirect

4. #14557288

C₂

semidirect

81. C₃₇

#2025

semidirect

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

subgroup	index
C₂×#11924528	2
C₂×#11924457	2
#14557295	2
#14557288	2
C₃₇×#390	2
#2025	37

subgroup

index

C₃₇×#390

Subgroups — #17995742

Conjugacy classes of subgroups

Complements

Maximal subgroups