Group #17948928 — subgroups

Subgroups — #17948928

Order 4352.

222 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14545123	C₂	1	1	yes	maximal, characteristic
2	#14544721	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#364	C₂	1	1	yes	maximal, characteristic
4	#11763865	—	2	1	no	—
5	C₂×#10587534	—	2	1	no	—
6	C₂×#10587533	—	2	1	no	—
7	#11763753	C₄	1	2	yes	—
8	#11763732	C₄	1	2	yes	complemented
9	#11763731	—	2	1	no	—
10	M₃₂×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762630	C₄	1	1	yes	characteristic, complemented
13	#11762629	—	2	1	no	—
14	C₁₇×#364	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#595	—	2	2	no	—
16	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
19	(C₂×#595)⋊C₄	—	4	2	no	—
20	#10587538	—	2	2	no	—
21	#10587534	—	4	1	no	—
22	#10587534	—	4	1	no	—
23	#10587533	—	4	1	no	—
24	#10587533	—	4	1	no	—
25	#10587530	—	4	1	no	—
26	#10587529	—	4	1	no	—
27	C₁₇×M₃₂	—	2	1	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	#10587483	—	2	1	no	—
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	#10587469	—	4	1	no	—
34	C₁₆×D₁₇	—	4	1	no	—
35	C₂²×#595	—	4	2	no	—
36	C₂×C₄×D₁₇	—	2	1	no	—
37	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄⋊#595	—	4	2	no	—
39	C₄⋊#595	—	4	2	no	—
40	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₄²	1	1	yes	characteristic
43	C₄×Dic₁₇	—	2	1	no	—
44	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₁₇	D₈	1	1	yes	characteristic
48	C₈×D₁₇	D₈	1	1	yes	characteristic
49	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
50	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
51	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
52	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
53	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
54	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
55	#63216	—	4	2	no	—
56	C₁₆×C₁₇	—	4	1	no	—
57	#63214	—	4	1	no	—
58	#7271	—	17	1	no	maximal, Sylow
59	C₂²×D₁₇	#100	1	1	yes	characteristic
60	C₂×#595	—	8	4	no	—
61	C₂×#595	—	8	2	no	—
62	C₂×C₄×C₁₇	—	2	1	no	—
63	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
64	C₂×Dic₁₇	—	2	1	no	—
65	C₂×Dic₁₇	#101	1	1	yes	characteristic
66	C₄×D₁₇	—	4	2	no	—
67	C₄×D₁₇	#102	1	1	yes	characteristic
68	C₄×D₁₇	#102	1	1	yes	characteristic
69	C₄×D₁₇	#103	1	1	yes	characteristic
70	C₄×D₁₇	#103	1	1	yes	characteristic
71	C₈×C₁₇	#102	1	1	yes	characteristic
72	C₈×C₁₇	#103	1	1	yes	characteristic
73	C₈×C₁₇	#106	1	1	yes	characteristic
74	C₈×C₁₇	#107	1	1	yes	characteristic
75	#3622	#102	1	1	yes	characteristic
76	#3622	#103	1	1	yes	characteristic
77	#3622	#106	1	1	yes	characteristic
78	#3622	#107	1	1	yes	characteristic
79	C₂×#364	—	17	1	no	—
80	C₂×#364	—	17	1	no	—
81	#1846	—	17	1	no	—
82	C₂²×C₁₇	#343	1	1	yes	characteristic
83	C₂×D₁₇	—	2	2	no	—
84	C₂×D₁₇	#331	1	1	yes	characteristic
85	C₂×D₁₇	#333	1	1	yes	characteristic
86	#595	—	16	2	no	—
87	C₄×C₁₇	—	4	1	no	—
88	C₄×C₁₇	#340	1	1	yes	characteristic
89	C₄×C₁₇	#340	1	1	yes	characteristic
90	Dic₁₇	—	4	1	no	—
91	Dic₁₇	#332	1	1	yes	characteristic
92	Dic₁₇	#332	1	1	yes	characteristic
93	C₂×M₃₂	—	34	1	no	—
94	C₂×M₃₂	—	34	1	no	—
95	C₂×#107	—	34	1	no	—
96	C₂×#106	—	34	1	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	#382	—	17	2	no	—
99	#364	—	17	2	no	—
100	#364	—	17	1	no	—
101	#364	—	34	1	no	—
102	#364	—	34	1	no	—
103	#364	#595	1	1	yes	characteristic, complemented
104	C₂×C₁₇	—	2	1	no	—
105	C₂×C₁₇	#1296	1	1	yes	characteristic
106	D₁₇	—	2	1	no	—
107	D₁₇	#1366	1	2	yes	—
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	—	17	1	no	—
110	C₂×C₄⋊C₄	—	34	2	no	—
111	C₂×C₄²	—	17	1	no	—
112	M₃₂	—	2	1	no	—
113	M₃₂	—	34	2	no	—
114	M₃₂	—	34	1	no	—
115	M₃₂	—	68	1	no	—
116	M₃₂	—	68	1	no	—
117	C₂×C₁₆	—	68	1	no	—
118	C₂×C₁₆	—	68	1	no	—
119	#107	—	68	1	no	—
120	#107	—	68	1	no	—
121	#106	—	68	1	no	—
122	#106	—	68	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	C₂×#595	1	1	yes	characteristic
127	#95	—	68	2	no	—
128	C₁₇	#7271	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	—	17	1	no	—
130	C₂²×C₄	—	34	1	no	—
131	C₂²×C₄	—	68	2	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	1	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
143	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
144	C₄⋊C₄	—	68	2	no	—
145	C₄⋊C₄	—	68	2	no	—
146	C₄²	—	17	1	no	—
147	C₄²	—	34	1	no	—
148	C₄²	C₄×#595	1	1	yes	characteristic
149	C₁₆	—	4	1	no	—
150	C₁₆	—	68	2	no	—
151	C₁₆	—	68	1	no	—
152	C₂³	—	17	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	17	1	no	—
155	C₂×C₄	—	17	1	no	—
156	C₂×C₄	—	17	1	no	—
157	C₂×C₄	—	17	1	no	—
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	34	1	no	—
160	C₂×C₄	—	68	2	no	—
161	C₂×C₄	—	136	2	no	—
162	C₂×C₄	—	136	4	no	—
163	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
164	C₈	—	17	1	no	—
165	C₈	—	17	1	no	—
166	C₈	—	17	1	no	—
167	C₈	—	17	1	no	—
168	C₈	#10587533	1	1	yes	characteristic
169	C₈	#10587534	1	1	yes	characteristic
170	C₈	#10587547	1	1	yes	characteristic
171	C₈	#10587549	1	1	yes	characteristic
172	C₂²	—	17	1	no	—
173	C₂²	—	17	1	no	—
174	C₂²	—	34	2	no	—
175	C₂²	#11763688	1	1	yes	characteristic
176	C₄	—	4	1	no	—
177	C₄	—	272	2	no	—
178	C₄	—	17	1	no	—
179	C₄	—	17	1	no	—
180	C₄	—	68	1	no	—
181	C₄	#11763682	1	1	yes	characteristic
182	C₄	#11763683	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	17	2	no	—
185	C₂	—	34	1	no	—
186	C₂	#14543663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545123

C₂

yes

maximal, characteristic

#14544721

C₂

yes

maximal, characteristic

D₁₇×#364

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11763732

C₄

yes

complemented

#11763731

—

M₃₂×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762630

C₄

yes

characteristic, complemented

#11762629

—

C₁₇×#364

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

(C₂×#595)⋊C₄

—

—

—

—

—

—

—

—

C₁₇×M₃₂

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

#10587483

—

C₄×#3622

C₂×C₄

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂²×#595

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₄⋊#595

—

C₄⋊#595

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

characteristic

#63216

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#101

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

yes

yes

yes

yes

—

—

—

C₂²×C₁₇

#343

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

#382

—

#364

—

100

#364

—

101

#364

—

102

#364

—

103

#364

#595

yes

characteristic, complemented

104

C₂×C₁₇

—

105

C₂×C₁₇

#1296

yes

characteristic

106

D₁₇

—

107

D₁₇

#1366

yes

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄²

—

112

M₃₂

—

113

M₃₂

—

114

M₃₂

—

115

M₃₂

—

116

M₃₂

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

#107

—

120

#107

—

121

#106

—

122

#106

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×#595

yes

characteristic

127

#95

—

128

C₁₇

#7271

yes

characteristic, complemented, Sylow

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

C₄⋊#595

yes

characteristic

143

C₂×C₈

C₂²⋊#595

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₄×#595

yes

characteristic

149

C₁₆

—

150

C₁₆

—

151

C₁₆

—

152

C₂³

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

136

—

162

C₂×C₄

—

136

—

163

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

#10587533

yes

characteristic

169

C₈

#10587534

yes

characteristic

170

C₈

#10587547

yes

characteristic

171

C₈

#10587549

yes

characteristic

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#11763688

yes

characteristic

176

C₄

—

177

C₄

—

272

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#11763682

yes

characteristic

182

C₄

#11763683

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

—

186

C₂

#14543663

yes

characteristic

	subgroup	G/H
center	C₂	#14543663
commutator	C₄²×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#364	C₄
socle	C₂×C₁₇	#1296
prime radical	C₂²×D₁₇	#100
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#364	#595
hypocenter	C₁₇	#7271
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#14543663

commutator

C₄²×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#364

C₄

C₂×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
8. #11763732	C₄	semidirect
12. #11762630	C₄	semidirect
14. C₁₇×#364	C₄	semidirect
103. #364	#595	semidirect
128. C₁₇	#7271	semidirect

H ≅ G/N

type

8. #11763732

C₄

semidirect

12. #11762630

C₄

semidirect

14. C₁₇×#364

C₄

semidirect

103. #364

#595

semidirect

128. C₁₇

#7271

semidirect

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

subgroup	index
#14545123	2
#14544721	2
D₁₇×#364	2
#7271	17

subgroup

index

#14545123

#14544721

D₁₇×#364

#7271

Subgroups — #17948928

Conjugacy classes of subgroups

Complements

Maximal subgroups