Group #14527520 — subgroups

Subgroups — #14527520

Order 2176.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#11763364	C₂	1	1	yes	maximal, characteristic
2	#11762831	C₂	1	1	yes	maximal, characteristic
3	#11762831	C₂	1	1	yes	maximal, characteristic
4	#11762556	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762555	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762554	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#10587632	—	2	1	no	—
10	Q₈×Dic₁₇	—	2	1	no	—
11	C₂×#63218	—	2	1	no	—
12	C₂×#63218	—	2	1	no	—
13	#10587583	—	2	1	no	—
14	#10587578	C₄	1	2	yes	—
15	#10587576	C₄	1	2	yes	—
16	#10587573	—	2	1	no	—
17	C₄²×D₁₇	C₂²	1	1	yes	characteristic
18	#10587555	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	#10587485	—	2	1	no	—
22	#10587485	—	2	1	no	—
23	C₄⋊#3622	C₂²	1	1	yes	characteristic
24	C₄⋊#3622	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₇	—	2	1	no	—
26	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
29	C₂×Dic₃₄	—	2	1	no	—
30	C₂×C₈×C₁₇	—	2	1	no	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	#63227	—	2	2	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	2	no	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	#63225	—	2	2	no	—
40	C₄×Dic₁₇	—	2	2	no	—
41	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	D₄	1	1	yes	characteristic
43	C₄×Dic₁₇	D₄	1	1	yes	characteristic
44	C₂×#3622	—	2	1	no	—
45	C₂×#3622	—	2	1	no	—
46	#63218	—	4	2	no	—
47	#63218	—	4	2	no	—
48	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₁₇	—	4	1	no	—
50	Q₈×C₁₇	—	4	1	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₁₇	—	2	2	no	—
56	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₁₇	C₂²⋊C₄	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	2	no	—
64	Dic₃₄	—	4	1	no	—
65	Dic₃₄	—	4	1	no	—
66	C₈×C₁₇	—	4	1	no	—
67	C₈×C₁₇	—	4	1	no	—
68	#3622	—	4	1	no	—
69	#3622	—	4	1	no	—
70	#1561	—	17	1	no	maximal, Sylow
71	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×D₁₇	—	2	1	no	—
73	C₂×D₁₇	—	2	1	no	—
74	C₂×D₁₇	—	2	1	no	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	—	4	2	no	—
77	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₁₇	#137	1	1	yes	characteristic
80	C₄×C₁₇	#137	1	1	yes	characteristic
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	—	4	2	no	—
84	Dic₁₇	#101	1	2	yes	—
85	#533	—	17	1	no	—
86	#423	—	17	1	no	—
87	#423	—	17	1	no	—
88	#332	—	17	1	no	—
89	#332	—	17	1	no	—
90	#332	—	17	1	no	—
91	#332	D₁₇	1	1	yes	characteristic, complemented
92	C₂×C₁₇	C₂×#101	1	1	yes	characteristic
93	C₂×C₁₇	#417	1	1	yes	characteristic
94	C₂×C₁₇	#417	1	1	yes	characteristic
95	D₁₇	—	4	1	no	—
96	C₂×M₁₆	—	34	1	no	—
97	C₂×M₁₆	—	34	1	no	—
98	C₄⋊Q₈	—	17	1	no	—
99	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
100	#125	—	17	2	no	—
101	C₂²⋊Q₈	—	34	1	no	—
102	C₂²⋊Q₈	—	34	1	no	—
103	C₄×Q₈	—	34	1	no	—
104	C₄×Q₈	—	34	1	no	—
105	#117	—	17	2	no	—
106	C₂×C₄²	—	17	1	no	—
107	C₄⋊C₈	—	17	1	no	—
108	C₄⋊C₈	—	17	1	no	—
109	C₄⋊C₈	—	34	1	no	—
110	C₄⋊C₈	—	34	1	no	—
111	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
112	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
113	C₁₇	#1561	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	34	1	no	—
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	1	no	—
119	M₁₆	—	68	2	no	—
120	M₁₆	—	68	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	34	1	no	—
124	C₂×C₈	—	34	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	17	2	no	—
127	C₄⋊C₄	—	34	2	no	—
128	C₄⋊C₄	—	34	2	no	—
129	C₄⋊C₄	—	34	1	no	—
130	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
131	C₂²⋊C₄	—	34	2	no	—
132	C₄²	—	17	1	no	—
133	C₄²	—	17	1	no	—
134	C₄²	—	17	1	no	—
135	C₄²	—	34	2	no	—
136	C₄²	C₂²×D₁₇	1	1	yes	characteristic
137	C₂³	—	17	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	68	1	no	—
141	Q₈	—	68	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	34	2	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	—	34	2	no	—
152	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
153	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	68	1	no	—
158	C₈	—	68	1	no	—
159	C₂²	—	34	1	no	—
160	C₂²	—	34	1	no	—
161	C₂²	—	34	1	no	—
162	C₂²	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	17	2	no	—
166	C₄	—	34	1	no	—
167	C₄	—	34	1	no	—
168	C₄	—	68	2	no	—
169	C₄	#10587597	1	1	yes	characteristic
170	C₄	#10587600	1	1	yes	characteristic
171	C₄	#10587601	1	1	yes	characteristic
172	C₄	#10587604	1	1	yes	characteristic
173	C₂	—	68	1	no	—
174	C₂	#101×D₁₇	1	1	yes	characteristic
175	C₂	#11762807	1	1	yes	characteristic
176	C₂	#11762809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763364

C₂

yes

maximal, characteristic

#11762831

C₂

yes

maximal, characteristic

#11762831

C₂

yes

maximal, characteristic

#11762556

C₂

yes

maximal, characteristic, complemented

#11762555

C₂

yes

maximal, characteristic, complemented

#11762554

C₂

yes

maximal, characteristic, complemented

C₁₇×#332

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

#10587632

—

Q₈×Dic₁₇

—

—

—

—

C₄

yes

—

#10587576

C₄

yes

—

#10587573

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

—

—

—

—

C₂²×D₁₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#101

yes

—

—

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

100

#125

—

101

C₂²⋊Q₈

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₁₇

yes

characteristic

112

C₄⋊C₈

C₂×D₁₇

yes

characteristic

113

C₁₇

#1561

yes

characteristic, complemented, Sylow

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₄×D₁₇

yes

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₁₇

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

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

—

152

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

153

C₂×C₄

D₄×D₁₇

yes

characteristic

154

C₂×C₄

D₄×D₁₇

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

C₂²⋊C₄×D₁₇

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#10587597

yes

characteristic

170

C₄

#10587600

yes

characteristic

171

C₄

#10587601

yes

characteristic

172

C₄

#10587604

yes

characteristic

173

C₂

—

174

C₂

#101×D₁₇

yes

characteristic

175

C₂

#11762807

yes

characteristic

176

C₂

#11762809

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#332	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂²⋊C₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	#332	D₁₇
hypocenter	C₁₇	#1561
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#332

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂²⋊C₄

cosocle

C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #11762556	C₂	semidirect
5. #11762555	C₂	semidirect
6. #11762554	C₂	semidirect
7. C₁₇×#332	C₂	semidirect
91. #332	D₁₇	semidirect
113. C₁₇	#1561	semidirect

H ≅ G/N

type

4. #11762556

C₂

semidirect

5. #11762555

C₂

semidirect

6. #11762554

C₂

semidirect

7. C₁₇×#332

C₂

semidirect

91. #332

D₁₇

semidirect

113. C₁₇

#1561

semidirect

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

subgroup	index
#11763364	2
#11762831	2
#11762556	2
#11762555	2
#11762554	2
C₁₇×#332	2
#1561	17

subgroup

index

C₁₇×#332

Subgroups — #14527520

Conjugacy classes of subgroups

Complements

Maximal subgroups