Group #18010750 — subgroups

Subgroups — #18010750

Order 4736.

292 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 70 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×#11924606	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924601	C₂	1	2	yes	maximal, complemented
3	#14558508	C₂	1	2	yes	maximal, complemented
4	C₂×#11924590	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11924559	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	2	yes	—
7	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
9	#11924606	—	2	2	no	—
10	#11924601	—	2	4	no	—
11	#11924593	—	2	4	no	—
12	#11924590	—	2	2	no	—
13	C₂×#10596966	C₂²	1	2	yes	—
14	#11924559	—	2	2	no	—
15	#11924559	—	2	2	no	—
16	C₂×#10596941	C₄	1	2	yes	complemented
17	C₂×D₂₉₆	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂²×#3866	—	2	1	no	—
21	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
22	#10596966	—	2	2	no	—
23	#10596966	—	2	2	no	—
24	#10596966	—	2	2	no	—
25	#10596966	—	2	2	no	—
26	C₂×D₁₄₈	—	2	2	no	—
27	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₃₇	—	2	2	no	—
29	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
30	C₂×#64438	—	2	2	no	—
31	C₂×#64438	D₄	1	2	yes	—
32	C₄×#3866	—	2	1	no	—
33	C₄×#3866	D₄	1	2	yes	—
34	#10596953	—	2	2	no	—
35	#10596953	—	2	2	no	—
36	D₃₇⋊C₈	—	2	1	no	—
37	D₃₇⋊C₈	—	2	1	no	—
38	Q₁₆×C₃₇	—	2	2	no	—
39	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
40	#10596941	—	2	4	no	—
41	C₂×#64436	C₂×C₄	1	1	yes	characteristic
42	D₂₉₆	—	2	2	no	—
43	C₈×D₃₇	D₄	1	2	yes	—
44	C₈×D₃₇	D₄	1	2	yes	—
45	C₂²×D₃₇	—	2	2	no	—
46	C₂²×D₃₇	#41	1	1	yes	characteristic
47	C₂×#3866	—	2	2	no	—
48	C₂×#3866	—	4	2	no	—
49	Q₈×C₃₇	—	2	2	no	—
50	Q₈×C₃₇	—	2	2	no	—
51	C₂×C₄×C₃₇	—	2	2	no	—
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
54	D₁₄₈	—	2	2	no	—
55	D₁₄₈	—	2	2	no	—
56	D₁₄₈	—	4	2	no	—
57	D₁₄₈	—	4	2	no	—
58	C₄×D₃₇	—	4	2	no	—
59	C₄×D₃₇	—	4	2	no	—
60	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
61	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
62	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
63	C₄×D₃₇	#41	1	1	yes	characteristic
64	#64438	—	2	2	no	—
65	#64438	—	4	2	no	—
66	C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
67	#64436	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
69	C₂×D₃₇	—	2	2	no	—
70	C₂×D₃₇	—	4	2	no	—
71	C₂×D₃₇	—	4	4	no	—
72	C₂×D₃₇	#121	1	1	yes	characteristic
73	C₂×D₃₇	#124	1	1	yes	characteristic
74	#3866	—	4	2	no	—
75	C₄×C₃₇	—	4	2	no	—
76	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
78	Dic₃₇	#121	1	1	yes	characteristic
79	Dic₃₇	#127	1	1	yes	characteristic
80	#1975	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	#390	1	1	yes	characteristic
82	C₂×C₃₇	#444	1	2	yes	—
83	D₃₇	—	2	1	no	—
84	D₃₇	—	2	1	no	—
85	D₃₇	—	8	2	no	—
86	C₂×#135	—	37	1	no	—
87	C₂×#108	—	37	1	no	—
88	C₂×#104	—	37	2	no	—
89	#408	—	37	2	no	—
90	C₂×#97	—	37	1	no	—
91	C₃₇	#1975	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	—	37	2	no	—
93	#135	—	74	2	no	—
94	#135	—	74	2	no	—
95	C₂×Q₁₆	#3866	1	1	yes	characteristic, complemented
96	C₂×SD₁₆	—	37	2	no	—
97	C₂×D₈	—	37	1	no	—
98	C₂×M₁₆	—	37	2	no	—
99	C₂²×C₈	—	37	1	no	—
100	C₂²×C₈	—	37	1	no	—
101	C₂×C₄²	—	37	1	no	—
102	#108	—	74	2	no	—
103	#104	—	74	4	no	—
104	C₂²⋊C₈	—	74	4	no	—
105	#97	—	74	2	no	—
106	#41	—	74	2	no	—
107	#41	—	74	2	no	—
108	#41	—	74	2	no	—
109	#41	—	74	2	no	—
110	C₂×Q₈	C₂×#3866	1	2	yes	—
111	C₂×D₄	—	37	2	no	—
112	C₂×D₄	—	74	2	no	—
113	C₂²×C₄	—	37	1	no	—
114	C₂²×C₄	—	74	2	no	—
115	C₂²×C₄	—	74	1	no	—
116	Q₁₆	—	2	2	no	—
117	SD₁₆	—	74	4	no	—
118	D₈	—	74	2	no	—
119	M₁₆	—	74	2	no	—
120	M₁₆	—	74	2	no	—
121	C₂×C₈	—	37	1	no	—
122	C₂×C₈	—	37	2	no	—
123	C₂×C₈	—	37	2	no	—
124	C₂×C₈	—	37	2	no	—
125	C₂×C₈	—	74	1	no	—
126	C₂×C₈	—	74	1	no	—
127	C₂×C₈	—	74	2	no	—
128	C₂×C₈	C₂×#3866	1	1	yes	characteristic
129	C₄²	—	37	2	no	—
130	C₄²	—	74	1	no	—
131	C₂³	—	37	1	no	—
132	C₂³	—	74	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	D₄	—	74	2	no	—
136	D₄	—	74	2	no	—
137	D₄	—	148	2	no	—
138	D₄	—	148	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	37	1	no	—
141	C₂×C₄	—	37	1	no	—
142	C₂×C₄	—	37	1	no	—
143	C₂×C₄	—	37	1	no	—
144	C₂×C₄	—	37	1	no	—
145	C₂×C₄	—	74	2	no	—
146	C₂×C₄	—	148	2	no	—
147	C₂×C₄	—	148	2	no	—
148	C₂×C₄	—	148	2	no	—
149	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
150	C₈	—	37	2	no	—
151	C₈	—	74	2	no	—
152	C₈	—	148	2	no	—
153	C₈	C₂²⋊#3866	1	2	yes	—
154	C₂²	—	37	1	no	—
155	C₂²	—	37	1	no	—
156	C₂²	—	74	2	no	—
157	C₂²	—	148	2	no	—
158	C₂²	—	148	4	no	—
159	C₂²	D₄×#3866	1	1	yes	characteristic
160	C₄	—	4	2	no	—
161	C₄	—	37	1	no	—
162	C₄	—	37	1	no	—
163	C₄	—	148	2	no	—
164	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
165	C₄	D₄×#3866	1	1	yes	characteristic
166	C₂	—	296	2	no	—
167	C₂	—	74	1	no	—
168	C₂	—	74	1	no	—
169	C₂	#14558485	1	1	yes	characteristic
170	C₂	#14558579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924606

C₂

yes

maximal, characteristic, complemented

C₂×#11924601

C₂

yes

maximal, complemented

#14558508

C₂

yes

maximal, complemented

C₂×#11924590

C₂

yes

maximal, characteristic, complemented

C₂×#11924559

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×#3866

C₂²

yes

—

—

—

—

C₂²

yes

—

#11924559

—

#11924559

—

C₂×#10596941

C₄

yes

complemented

C₂×D₂₉₆

C₄

yes

characteristic, complemented

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

—

—

—

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

—

C₂×#64438

D₄

yes

—

C₄×#3866

—

C₄×#3866

D₄

yes

—

#10596953

—

#10596953

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

#10596941

—

C₂×#64436

C₂×C₄

yes

characteristic

D₂₉₆

—

C₈×D₃₇

D₄

yes

—

C₈×D₃₇

D₄

yes

—

C₂²×D₃₇

—

C₂²×D₃₇

yes

—

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

C₄×D₃₇

yes

—

—

C₈×C₃₇

C₂²⋊C₄

yes

—

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

yes

C₂×D₃₇

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

D₃₇

—

—

—

—

—

C₂×#97

—

C₃₇

#1975

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

#3866

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

#108

—

103

#104

—

104

C₂²⋊C₈

—

105

#97

—

106

#41

—

107

#41

—

108

#41

—

109

#41

—

110

C₂×Q₈

C₂×#3866

yes

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

Q₁₆

—

117

SD₁₆

—

118

D₈

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×#3866

yes

characteristic

129

C₄²

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

148

—

138

D₄

—

148

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

148

—

147

C₂×C₄

—

148

—

148

C₂×C₄

—

148

—

149

C₂×C₄

C₂²×#3866

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

148

—

153

C₈

C₂²⋊#3866

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

148

—

158

C₂²

—

148

—

159

C₂²

D₄×#3866

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

148

—

164

C₄

C₂×C₂²⋊#3866

yes

characteristic

165

C₄

D₄×#3866

yes

characteristic

166

C₂

—

296

—

167

C₂

—

168

C₂

—

169

C₂

#14558485

yes

characteristic

170

C₂

#14558579

yes

—

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×Q₁₆×C₃₇	C₄
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂×#11924559	C₂
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₂×Q₁₆	#3866
hypocenter	C₃₇	#1975
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×Q₁₆×C₃₇

C₄

socle

C₂²×C₃₇

C₄×D₄

prime radical

C₂×#11924559

C₂

cosocle

C₂×C₄×D₃₇

C₂³

C₂×Q₁₆

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. C₂×#11924606	C₂	semidirect
2. C₂×#11924601	C₂	semidirect
3. #14558508	C₂	semidirect
4. C₂×#11924590	C₂	semidirect
16. C₂×#10596941	C₄	semidirect
17. C₂×D₂₉₆	C₄	semidirect
19. C₂×Q₁₆×C₃₇	C₄	semidirect
91. C₃₇	#1975	semidirect
95. C₂×Q₁₆	#3866	semidirect

H ≅ G/N

type

1. C₂×#11924606

C₂

semidirect

2. C₂×#11924601

C₂

semidirect

3. #14558508

C₂

semidirect

4. C₂×#11924590

C₂

semidirect

16. C₂×#10596941

C₄

semidirect

17. C₂×D₂₉₆

C₄

semidirect

19. C₂×Q₁₆×C₃₇

C₄

semidirect

91. C₃₇

#1975

semidirect

95. C₂×Q₁₆

#3866

semidirect

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

subgroup	index
C₂×#11924606	2
C₂×#11924601	2
#14558508	2
C₂×#11924590	2
C₂×#11924559	2
#1975	37

subgroup

index

Subgroups — #18010750

Conjugacy classes of subgroups

Complements

Maximal subgroups