Group #17982749 — subgroups

Subgroups — #17982749

Order 4352.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#11763917	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10587528	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545877	C₂	1	4	yes	maximal, complemented
4	C₂×#11763868	C₂	1	1	yes	maximal, characteristic
5	#11763917	—	2	2	no	—
6	C₂×#10587528	—	2	2	no	—
7	C₂×#10587528	C₂²	1	2	yes	—
8	C₂×#10587528	C₂²	1	2	yes	—
9	#11763868	—	2	2	no	—
10	#11763868	—	2	2	no	—
11	C₂×#10587530	C₄	1	2	yes	—
12	C₂×#10587538	C₄	1	1	yes	characteristic
13	C₂×#10587529	C₂²	1	2	yes	—
14	C₂²×#63216	C₂²	1	1	yes	characteristic
15	C₂×#10587593	C₄	1	1	yes	characteristic
16	C₂×#63252	C₈	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
18	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
19	#10587593	—	2	2	no	—
20	#10587593	—	2	2	no	—
21	C₂×#63218	C₈	1	2	yes	—
22	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
23	#10587538	—	2	2	no	—
24	C₂×#63216	C₂×C₄	1	2	yes	—
25	C₂×#63216	C₂×C₄	1	1	yes	characteristic
26	C₂×#63216	D₄	1	2	yes	—
27	C₂×#63216	D₄	1	2	yes	—
28	C₂×#63216	C₂³	1	1	yes	characteristic
29	#10587530	—	2	4	no	—
30	#10587529	—	2	2	no	—
31	#10587529	—	2	2	no	—
32	#10587528	—	2	2	no	—
33	#10587528	—	2	2	no	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₇	C₁₆	1	2	yes	—
36	#63252	—	2	2	no	—
37	#63252	—	2	2	no	—
38	C₂×D₆₈	C₁₆	1	1	yes	characteristic
39	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
40	C₂×Dic₃₄	C₁₆	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₂×#3622	C₂²⋊C₄	1	2	yes	—
44	C₂×#3622	C₂×C₈	1	1	yes	characteristic
45	C₂×#3622	M₁₆	1	2	yes	—
46	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
47	#63218	—	2	4	no	—
48	C₈×D₁₇	—	2	2	no	—
49	C₈×D₁₇	—	2	2	no	—
50	#63216	—	2	2	no	—
51	#63216	C₂²⋊C₄	1	2	yes	—
52	#63216	C₂×D₄	1	1	yes	characteristic
53	#63216	C₂×D₄	1	1	yes	characteristic
54	#13554	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂²×D₁₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
60	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
61	C₂²⋊D₁₇	—	2	4	no	—
62	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
63	D₆₈	—	2	2	no	—
64	C₄×D₁₇	—	2	2	no	—
65	C₄×D₁₇	—	2	2	no	—
66	Dic₃₄	—	2	2	no	—
67	C₈×C₁₇	—	2	2	no	—
68	#3622	C₂²⋊C₈	1	2	yes	—
69	#3622	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	#3622	C₂×M₁₆	1	1	yes	characteristic
71	C₂×#504	—	17	1	no	—
72	C₂×M₆₄	—	17	1	no	—
73	C₂²×C₃₂	—	17	1	no	—
74	C₂²⋊C₃₂	—	17	4	no	—
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
77	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
78	C₂×D₁₇	—	2	2	no	—
79	C₂×D₁₇	—	2	4	no	—
80	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
81	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
83	Dic₁₇	—	2	2	no	—
84	C₂×#131	—	17	1	no	—
85	#504	—	34	2	no	—
86	#504	—	34	2	no	—
87	C₂×M₃₂	—	17	1	no	—
88	C₂×M₃₂	—	17	2	no	—
89	C₂²×C₁₆	—	17	1	no	—
90	C₂²×C₁₆	—	17	2	no	—
91	M₆₄	—	34	2	no	—
92	C₂×C₃₂	—	17	2	no	—
93	C₂×C₃₂	—	17	2	no	—
94	C₂×C₃₂	—	34	2	no	—
95	C₂×C₁₇	—	2	2	no	—
96	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂×C₁₇	#2258	1	2	yes	—
98	D₁₇	—	4	2	no	—
99	C₂×#41	—	17	1	no	—
100	#131	—	34	2	no	—
101	#131	—	34	2	no	—
102	C₂×M₁₆	—	17	2	no	—
103	C₂×M₁₆	#3633	1	1	yes	characteristic
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	2	no	—
106	M₃₂	—	34	4	no	—
107	M₃₂	—	34	2	no	—
108	C₂×C₁₆	—	17	1	no	—
109	C₂×C₁₆	—	17	2	no	—
110	C₂×C₁₆	—	17	2	no	—
111	C₂×C₁₆	—	17	1	no	—
112	C₂×C₁₆	—	17	2	no	—
113	C₂×C₁₆	—	34	2	no	—
114	C₂×C₁₆	—	34	2	no	—
115	C₃₂	—	34	2	no	—
116	C₃₂	—	34	2	no	—
117	C₁₇	#13554	1	1	yes	characteristic, complemented, Sylow
118	#41	—	34	2	no	—
119	#41	—	34	2	no	—
120	C₂×Q₈	—	17	1	no	—
121	C₂×D₄	—	17	2	no	—
122	C₂×D₄	—	17	1	no	—
123	C₂²×C₄	—	17	2	no	—
124	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	M₁₆	—	34	4	no	—
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	34	2	no	—
132	C₂×C₈	—	34	2	no	—
133	C₂×C₈	C₂×#3633	1	2	yes	—
134	C₁₆	—	17	1	no	—
135	C₁₆	—	17	1	no	—
136	C₁₆	—	17	2	no	—
137	C₁₆	—	34	2	no	—
138	C₂³	—	17	2	no	—
139	C₂³	#10587665	1	1	yes	characteristic
140	Q₈	—	34	2	no	—
141	D₄	—	34	4	no	—
142	D₄	—	34	2	no	—
143	C₂×C₄	—	17	2	no	—
144	C₂×C₄	—	34	2	no	—
145	C₂×C₄	—	34	2	no	—
146	C₂×C₄	#10587665	1	1	yes	characteristic
147	C₂×C₄	#10587666	1	2	yes	—
148	C₂×C₄	C₂²⋊#3633	1	2	yes	—
149	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	17	1	no	—
152	C₈	—	17	1	no	—
153	C₈	—	17	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	34	2	no	—
156	C₂²	—	34	4	no	—
157	C₂²	#11763906	1	2	yes	—
158	C₂²	C₂×#10587665	1	1	yes	characteristic
159	C₄	—	34	2	no	—
160	C₄	#11763906	1	2	yes	—
161	C₄	C₂×C₂²⋊#3633	1	1	yes	characteristic
162	C₄	C₂×#10587666	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	68	2	no	—
165	C₂	C₂×#11763906	1	1	yes	characteristic
166	C₂	#14546018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763917

C₂

yes

maximal, characteristic, complemented

C₂²×#10587528

C₂

yes

maximal, characteristic, complemented

#14545877

C₂

yes

maximal, complemented

C₂×#11763868

C₂

yes

maximal, characteristic

#11763917

—

C₂×#10587528

—

C₂×#10587528

C₂²

yes

—

C₂×#10587528

C₂²

yes

—

#11763868

—

#11763868

—

C₂×#10587530

C₄

yes

—

C₂×#10587538

C₄

yes

characteristic

C₂×#10587529

C₂²

yes

—

C₂²×#63216

C₂²

yes

characteristic

C₂×#10587593

C₄

yes

characteristic

C₂×#63252

C₈

yes

characteristic

C₂×M₁₆×C₁₇

C₈

yes

characteristic

C₂²×#3622

C₂×C₄

yes

—

—

C₈

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

#10587538

—

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

D₄

yes

—

C₂×#63216

D₄

yes

—

C₂×#63216

C₂³

yes

—

—

—

—

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×C₂²⋊D₁₇

C₁₆

yes

—

#63252

—

#63252

—

C₂×D₆₈

C₁₆

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

—

C₂×Dic₃₄

C₁₆

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×#3622

C₂²⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

characteristic

C₂×#3622

M₁₆

yes

—

C₂×#3622

C₂²×C₄

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

#63216

—

#63216

C₂²⋊C₄

yes

—

#63216

C₂×D₄

yes

characteristic

#63216

C₂×D₄

yes

characteristic

#13554

—

maximal, Sylow

C₂³×C₁₇

C₂×C₁₆

yes

characteristic

C₂²×D₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

M₃₂

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₂×C₁₆

yes

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

#3622

C₂²⋊C₈

yes

—

#3622

C₂×C₂²⋊C₄

yes

characteristic

#3622

C₂×M₁₆

yes

characteristic

C₂×#504

—

C₂×M₆₄

—

C₂²×C₃₂

—

C₂²⋊C₃₂

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²⋊C₁₆

yes

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

C₂²⋊C₁₆

yes

—

C₄×C₁₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₁₇

C₂×M₃₂

yes

characteristic

Dic₁₇

—

C₂×#131

—

#504

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₁₇

#2258

yes

—

D₁₇

—

C₂×#41

—

100

#131

—

101

#131

—

102

C₂×M₁₆

—

103

C₂×M₁₆

#3633

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₃₂

—

116

C₃₂

—

117

C₁₇

#13554

yes

characteristic, complemented, Sylow

118

#41

—

119

#41

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×#3633

yes

characteristic

125

M₁₆

—

126

M₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

C₂×#3633

yes

—

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

—

138

C₂³

—

139

C₂³

#10587665

yes

characteristic

140

Q₈

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#10587665

yes

characteristic

147

C₂×C₄

#10587666

yes

—

148

C₂×C₄

C₂²⋊#3633

yes

—

149

C₂×C₄

C₂²×#3633

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#11763906

yes

—

158

C₂²

C₂×#10587665

yes

characteristic

159

C₄

—

160

C₄

#11763906

yes

—

161

C₄

C₂×C₂²⋊#3633

yes

characteristic

162

C₄

C₂×#10587666

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

C₂×#11763906

yes

characteristic

166

C₂

#14546018

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×M₁₆×C₁₇	C₈
prime radical	C₂×#63252	C₈
cosocle	C₂×#63216	C₂³
hypercenter	C₂×M₁₆	#3633
hypocenter	C₁₇	#13554
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3633

commutator, socle

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₂×M₁₆×C₁₇

C₈

C₈

C₂³

C₂×M₁₆

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#11763917	C₂	semidirect
2. C₂²×#10587528	C₂	semidirect
3. #14545877	C₂	semidirect
117. C₁₇	#13554	semidirect

H ≅ G/N

type

1. C₂×#11763917

C₂

semidirect

2. C₂²×#10587528

C₂

semidirect

3. #14545877

C₂

semidirect

117. C₁₇

#13554

semidirect

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

subgroup	index
C₂×#11763917	2
C₂²×#10587528	2
#14545877	2
C₂×#11763868	2
#13554	17

subgroup

index

Subgroups — #17982749

Conjugacy classes of subgroups

Complements

Maximal subgroups