Group #17948841 — subgroups

Subgroups — #17948841

Order 4352.

200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14545117	C₂	1	1	yes	maximal, characteristic
2	#14544343	C₂	1	1	yes	maximal, characteristic
3	#14527541	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587530	C₄	1	2	yes	—
5	#11763813	—	2	1	no	—
6	#11763730	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	#11763673	—	2	1	no	—
9	#11763673	C₄	1	2	yes	—
10	#11762831	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762561	—	2	1	no	—
14	C₁₇×#335	C₄	1	1	yes	characteristic
15	C₂×#63243	—	4	1	no	—
16	C₂×#63218	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	C₄⋊#3624	—	2	2	no	—
21	C₄⋊#3624	—	4	1	no	—
22	C₂×#63216	—	2	2	no	—
23	#10587530	—	4	2	no	—
24	C₁₇×C₄⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	#10587485	—	4	1	no	—
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₄⋊#3622	—	2	1	no	—
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×#3624	—	4	2	no	—
35	#63243	—	8	2	no	—
36	C₂×C₈×C₁₇	—	4	1	no	—
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	D₈	1	1	yes	characteristic
42	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
43	C₂×#3622	—	4	1	no	—
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	#63218	—	8	2	no	—
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	2	1	no	—
49	C₈×D₁₇	—	4	1	no	—
50	#63216	—	4	2	no	—
51	#7245	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
56	C₂×Dic₁₇	#102	1	1	yes	characteristic
57	C₂×Dic₁₇	#103	1	1	yes	characteristic
58	C₂×Dic₁₇	#107	1	1	yes	characteristic
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
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	#3624	—	8	2	no	—
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	8	1	no	—
69	#3622	—	2	1	no	—
70	#3622	—	8	1	no	—
71	#3622	#100	1	1	yes	characteristic
72	#3622	#101	1	1	yes	characteristic
73	#2150	—	17	1	no	—
74	#1567	—	17	1	no	—
75	#1567	—	17	1	no	—
76	C₂²×C₁₇	#340	1	1	yes	characteristic
77	C₂×D₁₇	—	2	1	no	—
78	C₂×D₁₇	—	2	1	no	—
79	C₂×D₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#338	1	1	yes	characteristic
82	C₄×C₁₇	#343	1	1	yes	characteristic
83	C₄×C₁₇	#361	1	1	yes	characteristic
84	C₄×C₁₇	#362	1	1	yes	characteristic
85	Dic₁₇	—	2	1	no	—
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	#361	1	1	yes	characteristic
88	Dic₁₇	#362	1	1	yes	characteristic
89	C₂×M₃₂	—	17	2	no	—
90	#423	—	34	1	no	—
91	#423	—	34	1	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₄⋊C₁₆	—	34	1	no	—
94	C₄⋊C₁₆	—	34	1	no	—
95	#335	—	17	2	no	—
96	#335	—	17	1	no	—
97	#335	—	34	1	no	—
98	#335	—	34	1	no	—
99	#335	#595	1	1	yes	characteristic
100	C₂×C₁₇	#1292	1	1	yes	characteristic
101	C₂×C₁₇	#1387	1	1	yes	characteristic
102	C₂×C₁₇	#1387	1	1	yes	characteristic
103	D₁₇	—	4	1	no	—
104	C₂×M₁₆	—	68	1	no	—
105	C₂×M₁₆	—	68	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂×C₄²	—	17	1	no	—
109	M₃₂	—	68	2	no	—
110	C₂×C₁₆	—	34	2	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	34	2	no	—
113	C₄⋊C₈	—	34	1	no	—
114	C₄⋊C₈	—	68	1	no	—
115	C₄⋊C₈	—	68	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	C₂×#595	1	1	yes	characteristic
120	C₁₇	#7245	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	17	1	no	—
122	C₂²×C₄	—	17	1	no	—
123	C₂²×C₄	—	17	1	no	—
124	M₁₆	—	136	2	no	—
125	M₁₆	—	136	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	34	1	no	—
130	C₂×C₈	—	34	1	no	—
131	C₂×C₈	—	68	1	no	—
132	C₂×C₈	—	68	2	no	—
133	C₂×C₈	—	68	1	no	—
134	C₂×C₈	C₄×#595	1	1	yes	characteristic
135	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
136	C₄²	—	17	1	no	—
137	C₄²	—	17	1	no	—
138	C₄²	—	17	1	no	—
139	C₄²	C₂²⋊#595	1	1	yes	characteristic
140	C₁₆	—	68	2	no	—
141	C₂³	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	34	1	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	34	1	no	—
151	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	#10587547	1	1	yes	characteristic
153	C₂×C₄	#10587549	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	8	1	no	—
157	C₈	—	17	1	no	—
158	C₈	—	17	1	no	—
159	C₈	—	34	1	no	—
160	C₈	—	136	2	no	—
161	C₈	—	136	1	no	—
162	C₂²	—	34	1	no	—
163	C₂²	—	34	1	no	—
164	C₂²	—	34	1	no	—
165	C₂²	#11763682	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	17	1	no	—
168	C₄	—	17	1	no	—
169	C₄	—	34	1	no	—
170	C₄	—	34	1	no	—
171	C₄	#11763678	1	1	yes	characteristic
172	C₄	#11763688	1	1	yes	characteristic
173	C₄	#11763727	1	1	yes	characteristic
174	C₄	#11763728	1	1	yes	characteristic
175	C₂	—	68	1	no	—
176	C₂	#14543653	1	1	yes	characteristic
177	C₂	#14543858	1	1	yes	characteristic
178	C₂	#14543860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545117

C₂

yes

maximal, characteristic

#14544343

C₂

yes

maximal, characteristic

#14527541

C₂

yes

maximal, characteristic

C₂×#10587530

C₄

yes

—

—

—

—

—

C₄

yes

—

#11762831

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762561

—

C₁₇×#335

C₄

yes

characteristic

C₂×#63243

—

C₂×#63218

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×#3624

—

#63243

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

Q₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

—

—

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

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

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

C₂×M₃₂

—

#423

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

yes

100

C₂×C₁₇

#1292

yes

characteristic

101

C₂×C₁₇

#1387

yes

characteristic

102

C₂×C₁₇

#1387

yes

characteristic

103

D₁₇

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

M₃₂

—

110

C₂×C₁₆

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

C₂×#595

yes

characteristic

120

C₁₇

#7245

yes

characteristic, complemented, Sylow

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

M₁₆

—

136

—

125

M₁₆

—

136

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₄×#595

yes

characteristic

135

C₂×C₈

C₄⋊#595

yes

characteristic

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂²⋊#595

yes

characteristic

140

C₁₆

—

141

C₂³

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

152

C₂×C₄

#10587547

yes

characteristic

153

C₂×C₄

#10587549

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

136

—

161

C₈

—

136

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11763682

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#11763678

yes

characteristic

172

C₄

#11763688

yes

characteristic

173

C₄

#11763727

yes

characteristic

174

C₄

#11763728

yes

characteristic

175

C₂

—

176

C₂

#14543653

yes

characteristic

177

C₂

#14543858

yes

characteristic

178

C₂

#14543860

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#106
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#335	#595
hypocenter	C₁₇	#7245
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#335

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
120. C₁₇	#7245	semidirect

H ≅ G/N

type

120. C₁₇

#7245

semidirect

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

subgroup	index
#14545117	2
#14544343	2
#14527541	2
#7245	17

subgroup

index

Subgroups — #17948841

Conjugacy classes of subgroups

Complements

Maximal subgroups