Group #16950538 — subgroups

Subgroups — #16950538

Order 4352.

454 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₂²⋊#63214	C₂	1	2	yes	maximal
2	#14526862	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₇×#405	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63214	C₂²	1	4	yes	—
5	C₂²⋊#63214	—	2	4	no	—
6	C₂²×M₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
9	C₁₇×#405	—	2	4	no	—
10	C₂×C₁₇×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇	—	2	2	no	—
14	C₂×M₁₆×C₁₇	—	2	4	no	—
15	C₂²×C₈×C₁₇	—	2	4	no	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	D₄	1	4	yes	—
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₁₇×#117	—	2	2	no	—
21	C₁₇×#117	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
25	C₁₇×#97	—	2	4	no	—
26	C₄×C₈×C₁₇	—	2	4	no	—
27	C₂×#63214	—	2	4	no	—
28	C₂×#63214	—	2	8	no	—
29	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	—	2	2	no	—
31	C₂²×C₄×C₁₇	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₁₇	—	2	8	no	—
38	C₂×C₈×C₁₇	—	2	2	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	—	2	8	no	—
42	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
44	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	4	yes	—
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₂²⋊C₄×C₁₇	—	2	4	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	—	2	2	no	—
50	#63214	—	4	4	no	—
51	#12845	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	—	2	2	no	—
53	C₂³×C₁₇	—	2	4	no	—
54	C₂³×C₁₇	C₂²⋊C₈	1	2	yes	—
55	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	2	4	no	—
60	C₂×C₄×C₁₇	—	2	4	no	—
61	C₂×C₄×C₁₇	—	2	8	no	—
62	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#117	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₇	—	2	2	no	—
68	C₈×C₁₇	—	2	4	no	—
69	C₈×C₁₇	#123	1	4	yes	—
70	C₂×#405	D₁₇	1	1	yes	characteristic
71	C₂×C₂²⋊C₁₆	—	17	2	no	—
72	#1374	—	17	4	no	—
73	C₂²×C₁₇	—	2	2	no	—
74	C₂²×C₁₇	—	2	4	no	—
75	C₂²×C₁₇	—	4	4	no	—
76	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₁₇	#431	1	2	yes	—
78	C₂²×C₁₇	#504	1	4	yes	—
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	—	4	4	no	—
81	C₄×C₁₇	#381	1	1	yes	characteristic
82	C₄×C₁₇	#407	1	1	yes	characteristic
83	C₄×C₁₇	#432	1	2	yes	—
84	C₂²×M₁₆	Dic₁₇	1	1	yes	characteristic
85	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂×#117	Dic₁₇	1	1	yes	characteristic
87	C₂²×C₁₆	—	17	4	no	—
88	#405	—	2	4	no	—
89	C₂×#97	Dic₁₇	1	2	yes	—
90	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
91	C₂²⋊C₁₆	—	34	4	no	—
92	C₂×C₁₇	—	4	2	no	—
93	C₂×C₁₇	#1817	1	1	yes	characteristic
94	C₂×C₁₇	#2113	1	2	yes	—
95	C₂×C₁₇	#2164	1	4	yes	—
96	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	4	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
102	C₂²×C₈	C₂²⋊D₁₇	1	4	yes	—
103	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	#117	—	2	2	no	—
106	C₂×C₄⋊C₄	#3622	1	2	yes	—
107	C₂×C₂²⋊C₄	#3622	1	2	yes	—
108	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
109	C₂×C₁₆	—	34	8	no	—
110	C₂×C₁₆	—	34	4	no	—
111	#97	—	2	4	no	—
112	C₄×C₈	—	2	4	no	—
113	C₁₇	#12845	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	C₂×#3622	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×#3622	1	4	yes	—
118	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
119	C₂²×C₄	#63223	1	2	yes	—
120	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	#63223	1	4	yes	—
128	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
129	C₂×C₈	#63252	1	4	yes	—
130	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	68	4	no	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂²⋊#3622	1	2	yes	—
139	C₂³	C₂²×#3622	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊#3622	1	2	yes	—
147	C₂×C₄	C₂×#63223	1	2	yes	—
148	C₂×C₄	C₂×#63223	1	1	yes	characteristic
149	C₂×C₄	#10587612	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	2	2	no	—
153	C₈	#10587615	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
158	C₂²	#11762862	1	2	yes	—
159	C₂²	#11763226	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#11762672	1	1	yes	characteristic
163	C₄	#11762773	1	1	yes	characteristic
164	C₄	#11762866	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#14528852	1	1	yes	characteristic
167	C₂	#14530398	1	2	yes	—
168	C₂	#14530591	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#63214

C₂

yes

maximal

#14526862

C₂

yes

maximal, complemented

C₂×C₁₇×#405

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

—

C₂²⋊#63214

—

C₂²×M₁₆×C₁₇

C₄

yes

characteristic

C₂³×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#117

C₄

yes

characteristic

C₁₇×#405

—

C₂×C₁₇×#97

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₂³×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

—

C₁₇×#117

—

C₂×C₄⋊C₄×C₁₇

C₈

yes

—

C₂×C₂²⋊C₄×C₁₇

C₈

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

C₂×#63214

—

C₂×#63214

—

C₂⁴×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₈

yes

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

#63214

—

#12845

—

maximal, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂²⋊C₈

yes

—

C₂³×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#123

yes

—

C₂×#405

D₁₇

yes

characteristic

C₂×C₂²⋊C₁₆

—

#1374

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₇

#431

yes

—

C₂²×C₁₇

#504

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂²×M₁₆

Dic₁₇

yes

characteristic

C₂³×C₈

C₂×D₁₇

yes

characteristic

C₂×#117

Dic₁₇

yes

characteristic

C₂²×C₁₆

—

#405

—

C₂×#97

Dic₁₇

yes

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

C₂²⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1817

yes

characteristic

C₂×C₁₇

#2113

yes

—

C₂×C₁₇

#2164

yes

—

C₂³×C₄

C₂×Dic₁₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

101

C₂²×C₈

C₂×Dic₁₇

yes

—

102

C₂²×C₈

C₂²⋊D₁₇

yes

—

103

C₂²×C₈

C₂²×D₁₇

yes

characteristic

104

#117

—

105

#117

—

106

C₂×C₄⋊C₄

#3622

yes

—

107

C₂×C₂²⋊C₄

#3622

yes

—

108

C₂×C₄²

C₂×Dic₁₇

yes

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

#97

—

112

C₄×C₈

—

113

C₁₇

#12845

yes

characteristic, complemented, Sylow

114

C₂⁴

C₂×#3622

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×#3622

yes

—

118

C₂²×C₄

C₂×#3622

yes

characteristic

119

C₂²×C₄

#63223

yes

—

120

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

121

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#63223

yes

—

128

C₂×C₈

C₂²⋊Dic₁₇

yes

—

129

C₂×C₈

#63252

yes

—

130

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊#3622

yes

—

139

C₂³

C₂²×#3622

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#3622

yes

—

147

C₂×C₄

C₂×#63223

yes

—

148

C₂×C₄

C₂×#63223

yes

characteristic

149

C₂×C₄

#10587612

yes

—

150

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#10587615

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#3622

yes

characteristic

158

C₂²

#11762862

yes

—

159

C₂²

#11763226

yes

—

160

C₄

—

161

C₄

—

162

C₄

#11762672

yes

characteristic

163

C₄

#11762773

yes

characteristic

164

C₄

#11762866

yes

—

165

C₂

—

166

C₂

#14528852

yes

characteristic

167

C₂

#14530398

yes

—

168

C₂

#14530591

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle	C₂³×C₁₇	C₂²×C₈
Fitting	C₂×C₁₇×#405	C₂
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#405	D₁₇
hypocenter	C₁₇	#12845
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₁₇

commutator, socle

C₂³×C₁₇

C₂²×C₈

Fitting

C₂×C₁₇×#405

C₂

prime radical

C₂⁴×C₁₇

C₂×C₈

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂²×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #14526862	C₂	semidirect
113. C₁₇	#12845	semidirect

H ≅ G/N

type

2. #14526862

C₂

semidirect

113. C₁₇

#12845

semidirect

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

subgroup	index
C₂×C₂²⋊#63214	2
#14526862	2
C₂×C₁₇×#405	2
#12845	17

subgroup

index

C₂×C₂²⋊#63214

#14526862

C₂×C₁₇×#405

#12845

Subgroups — #16950538

Conjugacy classes of subgroups

Complements

Maximal subgroups