Group #16939103 — subgroups

Subgroups — #16939103

Order 4352.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14529616	C₂	1	1	yes	maximal, characteristic
2	#14527434	C₂	1	2	yes	maximal
3	#14526681	C₂	1	2	yes	maximal, complemented
4	#14526613	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763359	C₂²	1	1	yes	characteristic
7	#11762862	C₂²	1	2	yes	—
8	#11762833	C₂²	1	1	yes	characteristic
9	C₂×#10587476	—	2	2	no	—
10	#11762556	—	2	2	no	—
11	#11762552	—	2	2	no	—
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₁₇×#376	C₂²	1	1	yes	characteristic
14	C₁₇×#336	—	2	2	no	—
15	C₂×#63252	—	2	1	no	—
16	C₂²×C₈×C₁₇	—	2	2	no	—
17	C₂×C₄²×C₁₇	D₄	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	#10587616	—	2	2	no	—
20	C₂×#63223	—	2	1	no	—
21	#10587561	—	2	1	no	—
22	#10587560	D₄	1	1	yes	characteristic
23	C₄×D₆₈	—	2	2	no	—
24	#10587555	D₄	1	1	yes	characteristic
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₁₇×#95	—	2	2	no	—
28	C₂²⋊#3622	—	2	2	no	—
29	#10587476	—	4	4	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂²×C₄×C₁₇	—	2	2	no	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	#41	1	2	yes	—
36	C₂×C₂²⋊D₁₇	—	4	1	no	—
37	#63252	—	4	2	no	—
38	#63252	—	4	2	no	—
39	C₂×D₆₈	—	2	1	no	—
40	C₂×D₆₈	—	2	2	no	—
41	C₂×C₄×D₁₇	—	4	1	no	—
42	C₂×Dic₃₄	—	2	1	no	—
43	C₂×C₈×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	—	4	2	no	—
46	C₄²×C₁₇	—	2	2	no	—
47	C₄²×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	2	yes	—
51	#63227	—	4	2	no	—
52	C₄⋊Dic₁₇	—	2	2	no	—
53	#63223	—	4	2	no	—
54	C₂×#3622	—	4	2	no	—
55	C₂×#3622	—	4	1	no	—
56	#10629	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	#123	1	1	yes	characteristic
58	C₂²×D₁₇	—	4	1	no	—
59	C₂×C₄×C₁₇	—	4	2	no	—
60	C₂×C₄×C₁₇	—	4	4	no	—
61	C₂×C₄×C₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#123	1	2	yes	—
64	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#126	1	2	yes	—
66	C₂×C₄×C₁₇	#135	1	2	yes	—
67	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₂²⋊D₁₇	—	8	2	no	—
69	C₂×Dic₁₇	—	4	1	no	—
70	D₆₈	—	4	2	no	—
71	D₆₈	—	8	2	no	—
72	C₄×D₁₇	—	8	1	no	—
73	C₄×D₁₇	—	8	1	no	—
74	Dic₃₄	—	4	2	no	—
75	C₈×C₁₇	—	8	2	no	—
76	#3622	—	8	2	no	—
77	#3622	—	8	1	no	—
78	#1963	—	17	1	no	—
79	#1832	D₁₇	1	1	yes	characteristic, complemented
80	#1540	—	17	2	no	—
81	#1301	—	17	2	no	—
82	#1274	—	17	1	no	—
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	#397	1	1	yes	characteristic
85	C₂²×C₁₇	#482	1	2	yes	—
86	C₂×D₁₇	—	8	1	no	—
87	C₂×D₁₇	—	8	2	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	8	2	no	—
91	C₄×C₁₇	#463	1	2	yes	—
92	C₄×C₁₇	#488	1	1	yes	characteristic
93	C₄×C₁₇	#490	1	1	yes	characteristic
94	Dic₁₇	—	8	1	no	—
95	#532	—	17	1	no	—
96	#431	—	17	2	no	—
97	#423	—	17	1	no	—
98	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
99	C₂×#104	—	34	2	no	—
100	#376	C₂×D₁₇	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#332	—	34	2	no	—
103	#331	—	34	2	no	—
104	C₂×C₁₇	—	2	2	no	—
105	C₂×C₁₇	#2073	1	1	yes	characteristic
106	C₂×C₁₇	#2080	1	2	yes	—
107	D₁₇	—	16	1	no	—
108	C₂×#41	—	34	1	no	—
109	C₂×M₁₆	—	34	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₄⋊Q₈	—	17	1	no	—
112	#127	—	17	1	no	—
113	#124	—	34	1	no	—
114	#121	—	34	2	no	—
115	C₄×D₄	—	34	2	no	—
116	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	34	2	no	—
121	C₄⋊C₈	—	34	2	no	—
122	#104	—	68	4	no	—
123	C₂²⋊C₈	—	34	2	no	—
124	C₄×C₈	—	34	2	no	—
125	#95	—	2	2	no	—
126	C₁₇	#10629	1	1	yes	characteristic, complemented, Sylow
127	#41	—	68	2	no	—
128	#41	—	68	2	no	—
129	C₂×Q₈	—	34	1	no	—
130	C₂×D₄	—	34	2	no	—
131	C₂×D₄	—	34	1	no	—
132	C₂×D₄	—	68	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	68	1	no	—
135	C₂²×C₄	#63252	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	M₁₆	—	68	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	68	2	no	—
142	C₂×C₈	—	68	1	no	—
143	C₄⋊C₄	—	34	2	no	—
144	C₂²⋊C₄	—	68	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	D₄×D₁₇	1	1	yes	characteristic
148	C₄²	#63254	1	1	yes	characteristic
149	C₄²	#63256	1	2	yes	—
150	C₂³	—	68	1	no	—
151	C₂³	#10587615	1	1	yes	characteristic
152	Q₈	—	68	2	no	—
153	D₄	—	68	2	no	—
154	D₄	—	136	2	no	—
155	D₄	—	136	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	68	1	no	—
160	C₂×C₄	—	136	1	no	—
161	C₂×C₄	—	136	1	no	—
162	C₂×C₄	#10587580	1	2	yes	—
163	C₂×C₄	#10587584	1	2	yes	—
164	C₂×C₄	#10587624	1	1	yes	characteristic
165	C₂×C₄	#10587633	1	1	yes	characteristic
166	C₂×C₄	#10587635	1	2	yes	—
167	C₂×C₄	#10587636	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	136	2	no	—
170	C₈	—	136	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	136	1	no	—
173	C₂²	—	136	2	no	—
174	C₂²	#11762735	1	1	yes	characteristic
175	C₂²	#11763133	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	136	1	no	—
180	C₄	#11763021	1	2	yes	—
181	C₄	#11763166	1	1	yes	characteristic
182	C₄	#11763175	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	272	1	no	—
185	C₂	#14530198	1	1	yes	characteristic
186	C₂	#14530235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529616

C₂

yes

maximal, characteristic

#14527434

C₂

yes

maximal

#14526681

C₂

yes

maximal, complemented

#14526613

C₂

yes

maximal, characteristic, complemented

C₁₇×#1832

C₂

yes

maximal, characteristic, complemented

#11763359

C₂²

yes

characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

—

—

C₂×C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#376

C₂²

yes

characteristic

C₁₇×#336

—

C₂×#63252

—

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

D₄

yes

characteristic

C₄×D₆₈

—

#10587555

D₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#95

—

C₂²⋊#3622

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₂²⋊D₁₇

—

#63252

—

#63252

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

#63227

—

C₄⋊Dic₁₇

—

—

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

characteristic, complemented

#1540

—

#1301

—

#1274

—

C₂²×C₁₇

—

C₂²×C₁₇

#397

yes

characteristic

C₂²×C₁₇

#482

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#463

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#532

—

#431

—

#423

—

C₂×C₄⋊C₈

C₂×D₁₇

yes

—

C₂×#104

—

100

#376

C₂×D₁₇

yes

characteristic

101

#336

—

102

#332

—

103

#331

—

104

C₂×C₁₇

—

105

C₂×C₁₇

#2073

yes

characteristic

106

C₂×C₁₇

#2080

yes

—

107

D₁₇

—

108

C₂×#41

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₄⋊Q₈

—

112

#127

—

113

#124

—

114

#121

—

115

C₄×D₄

—

116

C₂×C₄²

C₂²⋊D₁₇

yes

—

117

C₂×C₄²

C₂²×D₁₇

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

#104

—

123

C₂²⋊C₈

—

124

C₄×C₈

—

125

#95

—

126

C₁₇

#10629

yes

characteristic, complemented, Sylow

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

#63252

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

D₄×D₁₇

yes

characteristic

148

C₄²

#63254

yes

characteristic

149

C₄²

#63256

yes

—

150

C₂³

—

151

C₂³

#10587615

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

136

—

155

D₄

—

136

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

136

—

161

C₂×C₄

—

136

—

162

C₂×C₄

#10587580

yes

—

163

C₂×C₄

#10587584

yes

—

164

C₂×C₄

#10587624

yes

characteristic

165

C₂×C₄

#10587633

yes

characteristic

166

C₂×C₄

#10587635

yes

—

167

C₂×C₄

#10587636

yes

—

168

C₈

—

169

C₈

—

136

—

170

C₈

—

136

—

171

C₂²

—

172

C₂²

—

136

—

173

C₂²

—

136

—

174

C₂²

#11762735

yes

characteristic

175

C₂²

#11763133

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

136

—

180

C₄

#11763021

yes

—

181

C₄

#11763166

yes

characteristic

182

C₄

#11763175

yes

characteristic

183

C₂

—

184

C₂

—

272

—

185

C₂

#14530198

yes

characteristic

186

C₂

#14530235

yes

—

	subgroup	G/H
center	C₂×C₄	#10587633
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1832	C₂
socle	C₂²×C₁₇	#397
prime radical	#11763359	C₂²
hypercenter	#1832	D₁₇
hypocenter	C₁₇	#10629
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

subgroup

G/H

center

C₂×C₄

#10587633

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1832

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

N	H ≅ G/N	type
3. #14526681	C₂	semidirect
4. #14526613	C₂	semidirect
5. C₁₇×#1832	C₂	semidirect
79. #1832	D₁₇	semidirect
126. C₁₇	#10629	semidirect

H ≅ G/N

type

3. #14526681

C₂

semidirect

4. #14526613

C₂

semidirect

5. C₁₇×#1832

C₂

semidirect

79. #1832

D₁₇

semidirect

126. C₁₇

#10629

semidirect

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

subgroup	index
#14529616	2
#14527434	2
#14526681	2
#14526613	2
C₁₇×#1832	2
#10629	17

subgroup

index

C₁₇×#1832

Subgroups — #16939103

Conjugacy classes of subgroups

Complements

Maximal subgroups