Group #16939317 — subgroups

Subgroups — #16939317

Order 4352.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14528959	C₂	1	1	yes	maximal, characteristic
2	#14527554	C₂	1	2	yes	maximal
3	#14527213	C₂	1	2	yes	maximal
4	#14526615	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1985	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×#10587490	—	2	2	no	—
8	#11762833	C₂²	1	2	yes	—
9	#11762559	—	2	4	no	—
10	Q₈⋊#3622	—	2	4	no	—
11	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
14	C₁₇×#336	—	2	2	no	—
15	C₂²×Q₈×C₁₇	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	—	2	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	2	yes	—
18	C₄×Q₈×C₁₇	—	2	2	no	—
19	C₄×Q₈×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×#97	—	2	2	no	—
27	C₂²⋊#3622	—	2	2	no	—
28	#10587490	—	4	4	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄×#3622	D₄	1	4	yes	—
33	C₂×Q₈×C₁₇	—	2	2	no	—
34	C₂×Q₈×C₁₇	—	4	2	no	—
35	C₂²×C₄×C₁₇	—	2	1	no	—
36	C₂²×C₄×C₁₇	—	2	1	no	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	4	1	no	—
43	C₂×C₈×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
45	C₄⋊C₄×C₁₇	—	2	2	no	—
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄²×C₁₇	—	4	1	no	—
48	C₄²×C₁₇	—	4	1	no	—
49	C₄²×C₁₇	C₂×D₄	1	2	yes	—
50	C₄²×C₁₇	#41	1	2	yes	—
51	#63223	—	4	4	no	—
52	C₂×#3622	—	2	4	no	—
53	C₂×#3622	—	4	2	no	—
54	#10663	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
56	Q₈×C₁₇	—	2	4	no	—
57	Q₈×C₁₇	—	4	2	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	4	1	no	—
60	C₂×C₄×C₁₇	—	4	1	no	—
61	C₂×C₄×C₁₇	—	4	1	no	—
62	C₂×C₄×C₁₇	—	4	1	no	—
63	C₂×C₄×C₁₇	—	4	2	no	—
64	C₂×C₄×C₁₇	—	4	1	no	—
65	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₁₇	#123	1	2	yes	—
67	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
70	C₂×C₄×C₁₇	#137	1	2	yes	—
71	C₈×C₁₇	—	2	4	no	—
72	C₈×C₁₇	—	8	1	no	—
73	#3622	—	4	4	no	—
74	#3622	—	8	2	no	—
75	#1985	D₁₇	1	1	yes	characteristic
76	#1849	—	17	1	no	—
77	#1570	—	17	2	no	—
78	#1487	—	17	2	no	—
79	#1275	—	17	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	—	2	1	no	—
82	C₂²×C₁₇	#399	1	1	yes	characteristic
83	C₂²×C₁₇	#475	1	2	yes	—
84	C₄×C₁₇	—	2	2	no	—
85	C₄×C₁₇	—	2	2	no	—
86	C₄×C₁₇	—	4	2	no	—
87	C₄×C₁₇	—	8	1	no	—
88	C₄×C₁₇	#483	1	2	yes	—
89	C₄×C₁₇	#492	1	1	yes	characteristic
90	C₄×C₁₇	#497	1	1	yes	characteristic
91	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
92	#431	—	17	2	no	—
93	C₂×#108	—	34	2	no	—
94	#423	—	17	2	no	—
95	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
96	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
97	#336	—	2	2	no	—
98	#334	—	34	4	no	—
99	Q₈⋊C₈	—	34	4	no	—
100	C₂×C₁₇	—	2	2	no	—
101	C₂×C₁₇	#2091	1	1	yes	characteristic
102	C₂×C₁₇	#2096	1	1	yes	characteristic
103	C₂×C₁₇	#2098	1	1	yes	characteristic
104	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
105	C₂×M₁₆	—	34	2	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	Dic₃₄	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
114	#108	—	68	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	34	2	no	—
117	C₄⋊C₈	—	34	2	no	—
118	C₄⋊C₈	—	34	2	no	—
119	C₂²⋊C₈	—	34	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	—	17	4	no	—
122	C₁₇	#10663	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
128	C₂²×C₄	#63252	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	M₁₆	—	68	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	34	4	no	—
136	C₂×C₈	—	68	2	no	—
137	C₂×C₈	#63230	1	4	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	D₄×D₁₇	1	2	yes	—
143	C₄²	#63254	1	2	yes	—
144	C₂³	#10587610	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	#10587564	1	2	yes	—
155	C₂×C₄	#10587569	1	2	yes	—
156	C₂×C₄	#10587621	1	1	yes	characteristic
157	C₂×C₄	#10587626	1	1	yes	characteristic
158	C₂×C₄	C₂×#63230	1	2	yes	—
159	C₂×C₄	#10587637	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	1	no	—
162	C₈	—	68	4	no	—
163	C₈	—	136	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#11762745	1	1	yes	characteristic
167	C₂²	#11763094	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	#11763137	1	2	yes	—
173	C₄	#11763183	1	1	yes	characteristic
174	C₄	#11763204	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#14530300	1	1	yes	characteristic
177	C₂	#14530327	1	1	yes	characteristic
178	C₂	#14530337	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528959

C₂

yes

maximal, characteristic

#14527554

C₂

yes

maximal

#14527213

C₂

yes

maximal

#14526615

C₂

yes

maximal, characteristic

C₁₇×#1985

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

C₂×#10587490

—

#11762833

C₂²

yes

—

#11762559

—

Q₈⋊#3622

—

C₂×C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#336

—

C₂²×Q₈×C₁₇

Q₈

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₁₇×C₄⋊C₈

—

C₁₇×#97

—

C₂²⋊#3622

—

—

—

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

—

C₂³×C₁₇

C₂²⋊Q₈

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

—

#1985

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#475

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#483

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂×C₄×Q₈

C₂×D₁₇

yes

—

—

—

C₂×C₄⋊C₈

C₂×D₁₇

yes

characteristic

C₂×#97

C₂×D₁₇

yes

characteristic

#336

—

#334

—

Q₈⋊C₈

—

100

C₂×C₁₇

—

101

C₂×C₁₇

#2091

yes

characteristic

102

C₂×C₁₇

#2096

yes

characteristic

103

C₂×C₁₇

#2098

yes

characteristic

104

C₂²×Q₈

Dic₃₄

yes

characteristic

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₁₇

yes

—

108

C₄×Q₈

—

109

C₄×Q₈

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

Dic₃₄

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

C₂²×D₁₇

yes

characteristic

114

#108

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₂²⋊C₈

—

120

#97

—

121

C₄×C₈

—

122

C₁₇

#10663

yes

characteristic, complemented, Sylow

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

128

C₂²×C₄

#63252

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

#63230

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

D₄×D₁₇

yes

—

143

C₄²

#63254

yes

—

144

C₂³

#10587610

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#10587564

yes

—

155

C₂×C₄

#10587569

yes

—

156

C₂×C₄

#10587621

yes

characteristic

157

C₂×C₄

#10587626

yes

characteristic

158

C₂×C₄

C₂×#63230

yes

—

159

C₂×C₄

#10587637

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

136

—

164

C₂²

—

165

C₂²

—

166

C₂²

#11762745

yes

characteristic

167

C₂²

#11763094

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#11763137

yes

—

173

C₄

#11763183

yes

characteristic

174

C₄

#11763204

yes

characteristic

175

C₂

—

176

C₂

#14530300

yes

characteristic

177

C₂

#14530327

yes

characteristic

178

C₂

#14530337

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1985	C₂
socle	C₂²×C₁₇	#399
prime radical	C₂³×C₁₇	C₂²⋊Q₈
hypercenter	#1985	D₁₇
hypocenter	C₁₇	#10663
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

subgroup

G/H

center

C₂×C₄

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1985

C₂

socle

C₂²×C₁₇

#399

prime radical

C₂³×C₁₇

C₂²⋊Q₈

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

N	H ≅ G/N	type
122. C₁₇	#10663	semidirect

H ≅ G/N

type

122. C₁₇

#10663

semidirect

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

subgroup	index
#14528959	2
#14527554	2
#14527213	2
#14526615	2
C₁₇×#1985	2
#10663	17

subgroup

index

C₁₇×#1985

Subgroups — #16939317

Conjugacy classes of subgroups

Complements

Maximal subgroups