Group #17949009 — subgroups

Subgroups — #17949009

Order 4352.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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	#14545142	C₂	1	1	yes	maximal, characteristic
2	#14544341	C₂	1	1	yes	maximal, characteristic
3	#14528685	C₂	1	1	yes	maximal, characteristic
4	#11763812	—	2	1	no	—
5	#11763730	C₄	1	2	yes	—
6	C₂²⋊#63216	—	2	1	no	—
7	#11763695	—	2	1	no	—
8	#11763673	—	2	1	no	—
9	#11763672	C₄	1	2	yes	—
10	C₂×#10587493	—	2	1	no	—
11	M₁₆×Dic₁₇	—	2	1	no	—
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762574	C₄	1	1	yes	characteristic
14	C₁₇×#341	C₄	1	1	yes	characteristic
15	C₂×#63247	—	4	1	no	—
16	C₂×M₁₆×C₁₇	—	2	1	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
19	C₂×#63223	—	2	1	no	—
20	C₂²×#3622	D₄	1	1	yes	characteristic
21	#10587544	—	4	1	no	—
22	C₄⋊#3624	—	2	2	no	—
23	C₂×#63216	—	2	2	no	—
24	#10587493	—	2	4	no	—
25	#10587486	—	4	1	no	—
26	C₈×Dic₁₇	—	4	1	no	—
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	SD₁₆	1	1	yes	characteristic
33	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
34	#63247	—	8	2	no	—
35	C₂×#3624	—	4	2	no	—
36	M₁₆×C₁₇	—	2	2	no	—
37	M₁₆×C₁₇	—	4	1	no	—
38	C₂×C₈×C₁₇	—	4	1	no	—
39	C₂×C₈×C₁₇	—	4	1	no	—
40	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	D₈	1	1	yes	characteristic
45	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
46	#63223	—	2	2	no	—
47	#63223	—	4	1	no	—
48	C₂×#3622	—	2	2	no	—
49	C₂×#3622	—	4	1	no	—
50	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
52	#63216	—	4	2	no	—
53	#7298	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	1	no	—
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
59	C₂×Dic₁₇	—	2	1	no	—
60	C₂×Dic₁₇	—	2	1	no	—
61	C₂×Dic₁₇	—	2	2	no	—
62	C₂×Dic₁₇	#102	1	1	yes	characteristic
63	C₂×Dic₁₇	#103	1	1	yes	characteristic
64	C₂×Dic₁₇	#106	1	1	yes	characteristic
65	C₂×Dic₁₇	#107	1	1	yes	characteristic
66	#3624	—	8	2	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	—	4	2	no	—
70	#3622	—	2	1	no	—
71	#3622	—	4	2	no	—
72	#3622	#104	1	2	yes	—
73	#2162	—	17	1	no	—
74	C₄×#108	—	17	1	no	—
75	#1567	—	17	1	no	—
76	C₂²×C₁₇	—	2	1	no	—
77	C₂²×C₁₇	—	2	1	no	—
78	C₂²×C₁₇	—	2	1	no	—
79	C₂²×C₁₇	#340	1	1	yes	characteristic
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#339	1	1	yes	characteristic
82	C₄×C₁₇	#344	1	1	yes	characteristic
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	—	2	1	no	—
85	Dic₁₇	#361	1	1	yes	characteristic
86	Dic₁₇	#362	1	1	yes	characteristic
87	Dic₁₇	#368	1	2	yes	—
88	C₂×#108	—	34	1	no	—
89	#423	—	34	1	no	—
90	C₄×M₁₆	—	34	1	no	—
91	C₂×C₄×C₈	—	17	1	no	—
92	C₄⋊C₁₆	—	17	2	no	—
93	C₂²⋊C₁₆	—	34	1	no	—
94	C₄×C₁₆	—	34	1	no	—
95	#341	—	17	1	no	—
96	#341	#595	1	1	yes	characteristic
97	#335	—	17	2	no	—
98	#335	—	34	1	no	—
99	C₂×C₁₇	—	4	1	no	—
100	C₂×C₁₇	#1303	1	1	yes	characteristic
101	C₂×C₁₇	#1381	1	1	yes	characteristic
102	C₂×C₁₇	#1387	1	1	yes	characteristic
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	34	1	no	—
105	C₂×M₁₆	—	68	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	C₂×#595	1	1	yes	characteristic
108	C₂×C₄²	—	17	1	no	—
109	C₂×C₁₆	—	34	2	no	—
110	#108	—	34	4	no	—
111	C₄⋊C₈	—	34	2	no	—
112	C₄⋊C₈	—	68	1	no	—
113	#97	—	68	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	68	1	no	—
119	C₁₇	#7298	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	—	17	1	no	—
121	C₂²×C₄	—	17	1	no	—
122	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
123	M₁₆	—	2	2	no	—
124	M₁₆	—	4	1	no	—
125	M₁₆	—	34	2	no	—
126	M₁₆	—	68	1	no	—
127	M₁₆	—	136	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	34	2	no	—
133	C₂×C₈	—	68	1	no	—
134	C₂×C₈	—	68	2	no	—
135	C₂×C₈	C₄×#595	1	1	yes	characteristic
136	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
137	C₄²	—	17	1	no	—
138	C₄²	—	17	1	no	—
139	C₄²	—	17	1	no	—
140	C₄²	—	17	1	no	—
141	C₁₆	—	68	2	no	—
142	C₂³	#10587549	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	34	2	no	—
151	C₂×C₄	—	34	1	no	—
152	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
153	C₂×C₄	#10587547	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	4	2	no	—
157	C₈	—	17	2	no	—
158	C₈	—	34	1	no	—
159	C₈	—	68	2	no	—
160	C₈	—	136	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11763682	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	17	2	no	—
167	C₄	—	17	1	no	—
168	C₄	—	17	1	no	—
169	C₄	—	34	1	no	—
170	C₄	—	34	1	no	—
171	C₄	#11763680	1	1	yes	characteristic
172	C₄	#11763690	1	1	yes	characteristic
173	C₂	—	4	1	no	—
174	C₂	#14543682	1	1	yes	characteristic
175	C₂	#14543843	1	1	yes	characteristic
176	C₂	#14543860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545142

C₂

yes

maximal, characteristic

#14544341

C₂

yes

maximal, characteristic

#14528685

C₂

yes

maximal, characteristic

#11763812

—

#11763730

C₄

yes

—

—

—

—

C₄

yes

—

C₂×#10587493

—

M₁₆×Dic₁₇

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762574

C₄

yes

characteristic

C₁₇×#341

C₄

yes

characteristic

C₂×#63247

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂×#63223

—

C₂²×#3622

D₄

yes

—

—

—

—

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₇

SD₁₆

yes

characteristic

C₂²×Dic₁₇

Q₁₆

yes

characteristic

#63247

—

C₂×#3624

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

—

#2162

—

C₄×#108

—

#1567

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#340

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

Dic₁₇

yes

—

C₂×#108

—

#423

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

yes

—

—

C₂×C₁₇

—

100

C₂×C₁₇

#1303

yes

characteristic

101

C₂×C₁₇

#1381

yes

characteristic

102

C₂×C₁₇

#1387

yes

characteristic

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×#595

yes

characteristic

108

C₂×C₄²

—

109

C₂×C₁₆

—

110

#108

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₁₇

#7298

yes

characteristic, complemented, Sylow

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²⋊#595

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

136

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₄×#595

yes

characteristic

136

C₂×C₈

C₄⋊#595

yes

characteristic

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₁₆

—

142

C₂³

#10587549

yes

characteristic

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

—

152

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

153

C₂×C₄

#10587547

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

136

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

#11763682

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#11763680

yes

characteristic

172

C₄

#11763690

yes

characteristic

173

C₂

—

174

C₂

#14543682

yes

characteristic

175

C₂

#14543843

yes

characteristic

176

C₂

#14543860

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₁₇×#341	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂³×C₁₇	#103
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	#341	#595
hypocenter	C₁₇	#7298
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂²×C₈

C₂×#595

Fitting

C₁₇×#341

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

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

H ≅ G/N

type

119. C₁₇

#7298

semidirect

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

subgroup	index
#14545142	2
#14544341	2
#14528685	2
#7298	17

subgroup

index

Subgroups — #17949009

Conjugacy classes of subgroups

Complements

Maximal subgroups