Group #17949008 — subgroups

Subgroups — #17949008

Order 4352.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₄×#10587536	C₂	1	1	yes	maximal, characteristic
2	#14544342	C₂	1	1	yes	maximal, characteristic
3	#14530582	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587536	—	2	1	no	—
5	#11763813	—	2	1	no	—
6	C₄×#63243	—	2	1	no	—
7	#11763685	C₄	1	2	yes	—
8	#11763672	—	2	1	no	—
9	#11763672	C₄	1	2	yes	—
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊#63214	C₄	1	1	yes	characteristic
12	#11762588	—	2	1	no	—
13	C₁₆×Dic₁₇	—	2	1	no	—
14	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
15	C₂×#63243	—	2	2	no	—
16	C₂×#63243	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	#10587542	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₄⋊#3624	—	4	1	no	—
23	C₄×#3624	—	4	1	no	—
24	#10587536	—	2	2	no	—
25	#10587536	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	C₂×#3624	—	4	2	no	—
37	#63243	—	2	4	no	—
38	#63243	—	4	2	no	—
39	#63243	—	8	2	no	—
40	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	D₈	1	1	yes	characteristic
45	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₁₇	—	2	1	no	—
49	C₈×D₁₇	—	2	1	no	—
50	C₈×D₁₇	—	4	1	no	—
51	C₁₆×C₁₇	—	4	1	no	—
52	#63214	—	4	1	no	—
53	#7298	—	17	1	no	maximal, Sylow
54	C₂²×D₁₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
58	C₂×Dic₁₇	#102	1	1	yes	characteristic
59	C₂×Dic₁₇	#102	1	1	yes	characteristic
60	C₂×Dic₁₇	#103	1	1	yes	characteristic
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	2	1	no	—
65	C₄×D₁₇	—	2	1	no	—
66	C₄×D₁₇	—	2	1	no	—
67	#3624	—	4	4	no	—
68	#3624	—	8	2	no	—
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	#104	1	1	yes	characteristic
71	C₈×C₁₇	#104	1	1	yes	characteristic
72	#3622	—	2	1	no	—
73	#3622	—	2	1	no	—
74	#2162	—	17	1	no	—
75	C₄×#108	—	17	1	no	—
76	#1567	—	17	1	no	—
77	C₂²×C₁₇	#340	1	1	yes	characteristic
78	C₂×D₁₇	—	2	1	no	—
79	C₂×D₁₇	—	2	1	no	—
80	C₂×D₁₇	—	2	1	no	—
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	#339	1	1	yes	characteristic
83	C₄×C₁₇	#344	1	1	yes	characteristic
84	C₄×C₁₇	#368	1	1	yes	characteristic
85	C₄×C₁₇	#368	1	1	yes	characteristic
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	#361	1	1	yes	characteristic
89	Dic₁₇	#362	1	1	yes	characteristic
90	C₂×#108	—	34	1	no	—
91	#423	—	34	1	no	—
92	C₄×M₁₆	—	34	1	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₄⋊C₁₆	—	17	1	no	—
95	C₄⋊C₁₆	#595	1	1	yes	characteristic
96	C₂²⋊C₁₆	—	34	1	no	—
97	C₄×C₁₆	—	34	1	no	—
98	#341	—	17	2	no	—
99	#335	—	17	2	no	—
100	#335	—	34	1	no	—
101	C₂×C₁₇	#1303	1	1	yes	characteristic
102	C₂×C₁₇	#1381	1	1	yes	characteristic
103	C₂×C₁₇	#1387	1	1	yes	characteristic
104	D₁₇	—	4	1	no	—
105	C₂×M₁₆	—	34	2	no	—
106	C₂×M₁₆	—	68	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂×C₄²	—	17	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	34	1	no	—
112	#108	—	34	2	no	—
113	#108	—	34	2	no	—
114	C₄⋊C₈	—	34	2	no	—
115	C₄⋊C₈	—	68	1	no	—
116	#97	—	68	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	68	1	no	—
121	C₄×C₈	C₂×#595	1	1	yes	characteristic
122	C₁₇	#7298	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	—	17	1	no	—
124	C₂²×C₄	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	M₁₆	—	34	4	no	—
127	M₁₆	—	68	2	no	—
128	M₁₆	—	136	2	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	34	1	no	—
132	C₂×C₈	—	34	1	no	—
133	C₂×C₈	—	68	1	no	—
134	C₂×C₈	—	68	2	no	—
135	C₂×C₈	—	68	2	no	—
136	C₂×C₈	C₄×#595	1	1	yes	characteristic
137	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
138	C₄²	—	17	1	no	—
139	C₄²	—	17	1	no	—
140	C₄²	—	17	1	no	—
141	C₄²	C₄⋊#595	1	1	yes	characteristic
142	C₁₆	—	4	1	no	—
143	C₁₆	—	68	1	no	—
144	C₂³	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	34	1	no	—
151	C₂×C₄	—	34	1	no	—
152	C₂×C₄	—	34	1	no	—
153	C₂×C₄	—	34	1	no	—
154	C₂×C₄	#10587533	1	1	yes	characteristic
155	C₂×C₄	#10587534	1	1	yes	characteristic
156	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	34	1	no	—
159	C₈	—	34	1	no	—
160	C₈	—	68	4	no	—
161	C₈	—	136	2	no	—
162	C₈	#10587548	1	1	yes	characteristic
163	C₈	#10587550	1	1	yes	characteristic
164	C₂²	—	34	1	no	—
165	C₂²	—	34	1	no	—
166	C₂²	—	34	1	no	—
167	C₂²	#11763683	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	17	1	no	—
170	C₄	—	17	1	no	—
171	C₄	—	34	1	no	—
172	C₄	—	34	1	no	—
173	C₄	#11763679	1	1	yes	characteristic
174	C₄	#11763692	1	1	yes	characteristic
175	C₄	#11763737	1	1	yes	characteristic
176	C₄	#11763738	1	1	yes	characteristic
177	C₂	—	68	1	no	—
178	C₂	#14543681	1	1	yes	characteristic
179	C₂	#14543844	1	1	yes	characteristic
180	C₂	#14543859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587536

C₂

yes

maximal, characteristic

#14544342

C₂

yes

maximal, characteristic

#14530582

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11763672

—

#11763672

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊#63214

C₄

yes

characteristic

#11762588

—

C₁₆×Dic₁₇

—

C₁₇×C₄⋊C₁₆

C₄

yes

characteristic

C₂×#63243

—

C₂×#63243

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₄×#3622

Q₈

yes

characteristic

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

—

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

#595

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#335

—

100

#335

—

101

C₂×C₁₇

#1303

yes

characteristic

102

C₂×C₁₇

#1381

yes

characteristic

103

C₂×C₁₇

#1387

yes

characteristic

104

D₁₇

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

#108

—

113

#108

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₂×#595

yes

characteristic

122

C₁₇

#7298

yes

characteristic, complemented, Sylow

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

136

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₄×#595

yes

characteristic

137

C₂×C₈

C₂²⋊#595

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₄⋊#595

yes

characteristic

142

C₁₆

—

143

C₁₆

—

144

C₂³

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#10587533

yes

characteristic

155

C₂×C₄

#10587534

yes

characteristic

156

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

136

—

162

C₈

#10587548

yes

characteristic

163

C₈

#10587550

yes

characteristic

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#11763683

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#11763679

yes

characteristic

174

C₄

#11763692

yes

characteristic

175

C₄

#11763737

yes

characteristic

176

C₄

#11763738

yes

characteristic

177

C₂

—

178

C₂

#14543681

yes

characteristic

179

C₂

#14543844

yes

characteristic

180

C₂

#14543859

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×C₄⋊C₁₆	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#103
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#7298
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×C₄⋊C₁₆

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₄⋊C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
122. C₁₇	#7298	semidirect

H ≅ G/N

type

122. C₁₇

#7298

semidirect

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

subgroup	index
C₄×#10587536	2
#14544342	2
#14530582	2
#7298	17

subgroup

index

Subgroups — #17949008

Conjugacy classes of subgroups

Complements

Maximal subgroups