Group #18004999 — subgroups

Subgroups — #18004999

Order 4736.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#14558083	C₂	1	1	yes	maximal, characteristic
2	#14557903	C₂	1	2	yes	maximal, complemented
3	#14557899	C₂	1	2	yes	maximal, complemented
4	#14557890	C₂	1	4	yes	maximal, complemented
5	#14557532	C₂	1	4	yes	maximal, complemented
6	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596962	C₂²	1	2	yes	—
9	C₂×#10596934	C₂²	1	2	yes	—
10	#11924536	C₂²	1	1	yes	characteristic
11	#11924533	C₂²	1	1	yes	characteristic
12	#11924524	C₂²	1	2	yes	—
13	#11924510	—	2	2	no	—
14	C₄×D₁₄₈	—	2	2	no	—
15	#11924459	C₂²	1	8	yes	—
16	#11924456	C₂²	1	8	yes	—
17	#11924447	C₂²	1	2	yes	—
18	C₄×#64436	C₂²	1	2	yes	—
19	#125×C₃₇	—	2	2	no	—
20	#125×C₃₇	C₂²	1	2	yes	—
21	#125×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	—	2	2	no	—
26	C₂²×C₄×C₃₇	D₄	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₃₇	—	2	2	no	—
29	#10596962	—	2	8	no	—
30	C₂×D₁₄₈	—	2	2	no	—
31	C₂×D₁₄₈	C₂³	1	2	yes	—
32	C₂×C₄×D₃₇	—	2	2	no	—
33	C₂×Dic₇₄	C₂³	1	2	yes	—
34	C₄⋊C₄×C₃₇	—	2	4	no	—
35	C₄⋊C₄×C₃₇	—	2	4	no	—
36	C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
38	C₄²×C₃₇	D₄	1	2	yes	—
39	C₄²×C₃₇	C₂³	1	1	yes	characteristic
40	C₄²×C₃₇	C₂³	1	1	yes	characteristic
41	#10596938	—	2	4	no	—
42	C₄⋊Dic₃₇	—	2	2	no	—
43	#10596934	—	2	4	no	—
44	C₂×#64436	C₂³	1	4	yes	—
45	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	2	4	no	—
48	C₂×C₄×C₃₇	—	4	2	no	—
49	C₂×C₄×C₃₇	—	4	4	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	#41	1	4	yes	—
54	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₃₇	—	4	4	no	—
56	C₂×Dic₃₇	—	2	2	no	—
57	D₁₄₈	—	2	4	no	—
58	D₁₄₈	—	4	4	no	—
59	C₄×D₃₇	—	4	4	no	—
60	Dic₇₄	—	2	4	no	—
61	#64436	—	2	4	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	#124	1	2	yes	—
64	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₃₇	—	4	2	no	—
66	C₂×D₃₇	—	4	4	no	—
67	C₄×C₃₇	—	2	2	no	—
68	C₄×C₃₇	—	2	2	no	—
69	C₄×C₃₇	—	4	4	no	—
70	C₄×C₃₇	#124	1	2	yes	—
71	C₄×C₃₇	C₂×#41	1	2	yes	—
72	Dic₃₇	—	4	2	no	—
73	#3142	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	—	2	2	no	—
75	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₃₇	#575	1	2	yes	—
77	D₃₇	—	8	2	no	—
78	#532	—	37	1	no	—
79	C₂×#125	D₃₇	1	1	yes	characteristic, complemented
80	#491	—	37	2	no	—
81	#490	—	37	2	no	—
82	#488	—	37	4	no	—
83	#417	—	37	4	no	—
84	C₄×M₁₆	—	37	1	no	—
85	C₃₇	#3142	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	37	2	no	—
87	C₂×M₁₆	—	37	2	no	—
88	C₄⋊Q₈	—	37	1	no	—
89	#127	—	37	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₃₇	1	2	yes	—
92	#125	C₂×D₃₇	1	2	yes	—
93	#124	—	37	2	no	—
94	#121	—	74	2	no	—
95	C₄×D₄	—	74	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
98	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
99	#103	—	37	8	no	—
100	#102	—	37	8	no	—
101	#97	—	37	2	no	—
102	C₄×C₈	—	37	2	no	—
103	#41	—	74	8	no	—
104	C₂×Q₈	—	37	2	no	—
105	C₂×D₄	—	37	2	no	—
106	C₂×D₄	—	74	2	no	—
107	C₂×D₄	—	74	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	74	2	no	—
110	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	74	4	no	—
113	C₂×C₈	—	37	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	74	2	no	—
118	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
119	C₂²⋊C₄	—	74	4	no	—
120	C₄²	C₂²⋊D₃₇	1	2	yes	—
121	C₄²	C₂²×D₃₇	1	1	yes	characteristic
122	C₄²	C₂²×D₃₇	1	1	yes	characteristic
123	C₂³	—	74	2	no	—
124	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
125	Q₈	—	74	4	no	—
126	D₄	—	74	4	no	—
127	D₄	—	148	4	no	—
128	D₄	—	148	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	74	2	no	—
133	C₂×C₄	—	148	4	no	—
134	C₂×C₄	#10596966	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
139	C₈	—	74	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	148	2	no	—
142	C₂²	—	148	4	no	—
143	C₂²	#11924525	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	148	2	no	—
149	C₄	#11924525	1	2	yes	—
150	C₄	C₂×#10596966	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	296	2	no	—
153	C₂	C₂×#11924525	1	1	yes	characteristic
154	C₂	#14558331	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558083

C₂

yes

maximal, characteristic

#14557903

C₂

yes

maximal, complemented

#14557899

C₂

yes

maximal, complemented

#14557890

C₂

yes

maximal, complemented

#14557532

C₂

yes

maximal, complemented

C₄×#10596934

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

C₂×#10596934

C₂²

yes

—

#11924536

C₂²

yes

characteristic

#11924533

C₂²

yes

characteristic

#11924524

C₂²

yes

—

#11924510

—

C₄×D₁₄₈

—

#11924459

C₂²

yes

—

#11924456

C₂²

yes

—

#11924447

C₂²

yes

—

C₄×#64436

C₂²

yes

—

#125×C₃₇

—

#125×C₃₇

C₂²

yes

—

#125×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

#10596962

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₄×D₃₇

—

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

C₄⋊Dic₃₇

—

#10596934

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

#3142

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#124

yes

characteristic

C₂×C₃₇

#575

yes

—

D₃₇

—

#532

—

C₂×#125

D₃₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₃₇

#3142

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₃₇

yes

—

#125

C₂×D₃₇

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₃₇

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₃₇

yes

—

121

C₄²

C₂²×D₃₇

yes

characteristic

122

C₄²

C₂²×D₃₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

148

—

128

D₄

—

148

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

#10596966

yes

—

135

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

138

C₂×C₄

C₂³×D₃₇

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

148

—

142

C₂²

—

148

—

143

C₂²

#11924525

yes

—

144

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

148

—

149

C₄

#11924525

yes

—

150

C₄

C₂×#10596966

yes

—

151

C₂

—

152

C₂

—

296

—

153

C₂

C₂×#11924525

yes

characteristic

154

C₂

#14558331

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#125×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	#14558083	C₂
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	#3142
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#125×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

C₂

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #14557903	C₂	semidirect
3. #14557899	C₂	semidirect
4. #14557890	C₂	semidirect
5. #14557532	C₂	semidirect
6. C₄×#10596934	C₂	semidirect
7. C₂×#125×C₃₇	C₂	semidirect
79. C₂×#125	D₃₇	semidirect
85. C₃₇	#3142	semidirect

H ≅ G/N

type

2. #14557903

C₂

semidirect

3. #14557899

C₂

semidirect

4. #14557890

C₂

semidirect

5. #14557532

C₂

semidirect

6. C₄×#10596934

C₂

semidirect

7. C₂×#125×C₃₇

C₂

semidirect

79. C₂×#125

D₃₇

semidirect

85. C₃₇

#3142

semidirect

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

subgroup	index
#14558083	2
#14557903	2
#14557899	2
#14557890	2
#14557532	2
C₄×#10596934	2
C₂×#125×C₃₇	2
#3142	37

subgroup

index

C₂×#125×C₃₇

Subgroups — #18004999

Conjugacy classes of subgroups

Complements

Maximal subgroups