Group #16933821 — subgroups

Subgroups — #16933821

Order 4352.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#14528922	C₂	1	1	yes	maximal, characteristic
2	#14527469	C₂	1	2	yes	maximal
3	#14527465	C₂	1	2	yes	maximal
4	#14527076	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	1	yes	characteristic
7	#11762862	C₂²	1	1	yes	characteristic
8	#11762833	C₂²	1	2	yes	—
9	#11762574	—	2	2	no	—
10	#11762558	—	2	4	no	—
11	#11762555	—	2	4	no	—
12	#11762531	—	2	2	no	—
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₇×#500	—	2	2	no	—
15	C₁₇×#500	C₄	1	4	yes	—
16	C₂×C₁₇×#107	—	2	2	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	2	no	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×#63223	—	2	2	no	—
28	C₁₇×#107	—	2	8	no	—
29	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₇	D₄	1	2	yes	—
31	C₂²⋊#3622	—	2	1	no	—
32	C₂²⋊#3622	—	2	1	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄⋊#3622	D₄	1	2	yes	—
36	C₄⋊#3622	D₄	1	2	yes	—
37	C₄×#3622	—	2	1	no	—
38	C₄×#3622	—	2	1	no	—
39	C₂×Q₈×C₁₇	—	2	4	no	—
40	C₂×Q₈×C₁₇	—	4	2	no	—
41	C₂×Q₈×C₁₇	—	8	4	no	—
42	C₂²×C₄×C₁₇	—	4	2	no	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₇	—	2	4	no	—
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
48	C₄⋊C₄×C₁₇	—	2	4	no	—
49	C₄⋊C₄×C₁₇	—	2	4	no	—
50	C₄⋊C₄×C₁₇	—	4	2	no	—
51	C₄⋊C₄×C₁₇	—	4	2	no	—
52	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
56	#63223	—	4	4	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	—	2	2	no	—
59	C₂×#3622	—	4	2	no	—
60	#9778	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	#124	1	1	yes	characteristic
62	Q₈×C₁₇	—	8	4	no	—
63	Q₈×C₁₇	—	8	4	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	4	4	no	—
67	C₂×C₄×C₁₇	—	8	4	no	—
68	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#137	1	4	yes	—
75	C₈×C₁₇	—	2	4	no	—
76	#3622	—	4	2	no	—
77	#3622	—	4	2	no	—
78	#3622	—	8	2	no	—
79	C₂×#500	D₁₇	1	1	yes	characteristic
80	#1837	—	17	1	no	—
81	#1549	—	17	2	no	—
82	#1548	—	17	2	no	—
83	#1452	—	17	1	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	#390	1	1	yes	characteristic
87	C₂²×C₁₇	#491	1	2	yes	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	8	4	no	—
91	C₄×C₁₇	#419	1	2	yes	—
92	C₄×C₁₇	#472	1	2	yes	—
93	C₄×C₁₇	#473	1	2	yes	—
94	C₄×C₁₇	#497	1	2	yes	—
95	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
96	#500	—	2	2	no	—
97	#500	Dic₁₇	1	4	yes	—
98	#431	—	17	1	no	—
99	#431	—	17	1	no	—
100	C₂×#107	—	2	2	no	—
101	#423	—	17	2	no	—
102	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
103	#341	—	34	2	no	—
104	#333	—	34	4	no	—
105	#332	—	34	4	no	—
106	#322	—	34	2	no	—
107	C₂×C₁₇	—	2	2	no	—
108	C₂×C₁₇	#1960	1	1	yes	characteristic
109	C₂×C₁₇	#1978	1	1	yes	characteristic
110	C₂×C₁₇	#1979	1	1	yes	characteristic
111	C₂²×Q₈	—	2	2	no	—
112	C₂×M₁₆	—	34	2	no	—
113	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
114	C₄⋊Q₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
121	#107	—	2	8	no	—
122	C₄⋊C₈	—	17	2	no	—
123	C₄⋊C₈	—	17	2	no	—
124	C₄⋊C₈	—	34	2	no	—
125	C₄⋊C₈	—	34	2	no	—
126	C₂²⋊C₈	—	34	1	no	—
127	C₂²⋊C₈	—	34	1	no	—
128	C₄×C₈	—	34	1	no	—
129	C₄×C₈	—	34	1	no	—
130	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
131	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
132	C₁₇	#9778	1	1	yes	characteristic, complemented, Sylow
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	#63254	1	1	yes	characteristic
138	C₂²×C₄	#63254	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	M₁₆	—	68	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	34	2	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	—	68	2	no	—
145	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	D₄×D₁₇	1	1	yes	characteristic
151	C₄²	D₄×D₁₇	1	1	yes	characteristic
152	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
154	C₂³	#10587622	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
163	C₂×C₄	#10587624	1	1	yes	characteristic
164	C₂×C₄	#10587624	1	1	yes	characteristic
165	C₂×C₄	#10587626	1	1	yes	characteristic
166	C₂×C₄	#10587628	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	68	2	no	—
170	C₈	—	68	2	no	—
171	C₈	—	136	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#11762712	1	1	yes	characteristic
175	C₂²	#11763177	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#11762818	1	2	yes	—
180	C₄	#11763076	1	2	yes	—
181	C₄	#11763081	1	2	yes	—
182	C₄	#11763201	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#14529605	1	1	yes	characteristic
185	C₂	#14529695	1	1	yes	characteristic
186	C₂	#14529700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528922

C₂

yes

maximal, characteristic

#14527469

C₂

yes

maximal

#14527465

C₂

yes

maximal

#14527076

C₂

yes

maximal, characteristic

C₂×C₁₇×#500

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

characteristic

#11762862

C₂²

yes

characteristic

#11762833

C₂²

yes

—

—

—

—

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#500

—

C₁₇×#500

C₄

yes

—

C₂×C₁₇×#107

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₁₇×#107

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₂²⋊#3622

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

D₄

yes

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

C₂²×C₄×C₁₇

yes

C₂²×C₄×C₁₇

yes

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#491

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#419

yes

—

C₄×C₁₇

#472

yes

—

C₄×C₁₇

#473

yes

—

C₄×C₁₇

#497

yes

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

#500

—

#500

Dic₁₇

yes

—

#431

—

#431

—

100

C₂×#107

—

101

#423

—

102

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

103

#341

—

104

#333

—

105

#332

—

106

#322

—

107

C₂×C₁₇

—

108

C₂×C₁₇

#1960

yes

characteristic

109

C₂×C₁₇

#1978

yes

characteristic

110

C₂×C₁₇

#1979

yes

characteristic

111

C₂²×Q₈

—

112

C₂×M₁₆

—

113

C₂²×C₈

C₂²⋊D₁₇

yes

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

C₂×Dic₁₇

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

C₂²×D₁₇

yes

characteristic

121

#107

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

C₂×Dic₁₇

yes

—

131

C₄×C₈

C₂²⋊D₁₇

yes

—

132

C₁₇

#9778

yes

characteristic, complemented, Sylow

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

#63254

yes

characteristic

138

C₂²×C₄

#63254

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

C₂²⋊Dic₁₇

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

D₄×D₁₇

yes

characteristic

151

C₄²

D₄×D₁₇

yes

characteristic

152

C₄²

C₂²×Dic₁₇

yes

characteristic

153

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

154

C₂³

#10587622

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

D₄×Dic₁₇

yes

characteristic

162

C₂×C₄

D₄×Dic₁₇

yes

characteristic

163

C₂×C₄

#10587624

yes

characteristic

164

C₂×C₄

#10587624

yes

characteristic

165

C₂×C₄

#10587626

yes

characteristic

166

C₂×C₄

#10587628

yes

—

167

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

136

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11762712

yes

characteristic

175

C₂²

#11763177

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#11762818

yes

—

180

C₄

#11763076

yes

—

181

C₄

#11763081

yes

—

182

C₄

#11763201

yes

—

183

C₂

—

184

C₂

#14529605

yes

characteristic

185

C₂

#14529695

yes

characteristic

186

C₂

#14529700

yes

characteristic

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#9778
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#11762712

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#500

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
132. C₁₇	#9778	semidirect

H ≅ G/N

type

132. C₁₇

#9778

semidirect

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

subgroup	index
#14528922	2
#14527469	2
#14527465	2
#14527076	2
C₂×C₁₇×#500	2
#9778	17

subgroup

index

C₂×C₁₇×#500

Subgroups — #16933821

Conjugacy classes of subgroups

Complements

Maximal subgroups