Group #14527513 — subgroups

Subgroups — #14527513

Order 2176.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11762865	C₂	1	1	yes	maximal, characteristic
3	#11762831	C₂	1	1	yes	maximal, characteristic
4	#11762557	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762557	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762555	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	#10587583	C₄	1	2	yes	—
13	#10587578	—	2	1	no	—
14	#10587576	—	2	1	no	—
15	#10587573	C₄	1	2	yes	—
16	C₄²×D₁₇	C₂²	1	1	yes	characteristic
17	#10587555	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	#10587491	—	2	1	no	—
21	#10587486	—	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	C₂×C₄	1	2	yes	—
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	1	no	—
38	C₄⋊Dic₁₇	—	2	2	no	—
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	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	Q₈×C₁₇	—	4	1	no	—
49	Q₈×C₁₇	—	4	1	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₇	—	2	2	no	—
55	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₁₇	—	2	1	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	4	1	no	—
63	Dic₃₄	—	4	2	no	—
64	C₈×C₁₇	—	4	1	no	—
65	C₈×C₁₇	—	4	1	no	—
66	#3622	—	4	1	no	—
67	#3622	—	4	1	no	—
68	#1559	—	17	1	no	maximal, Sylow
69	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₁₇	—	2	1	no	—
71	C₂×D₁₇	—	2	1	no	—
72	C₂×D₁₇	—	2	1	no	—
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	—	4	2	no	—
75	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₁₇	#137	1	1	yes	characteristic
77	C₄×C₁₇	#137	1	1	yes	characteristic
78	C₄×C₁₇	#137	1	1	yes	characteristic
79	Dic₁₇	—	2	1	no	—
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	—	4	2	no	—
83	#533	—	17	1	no	—
84	#432	—	17	1	no	—
85	#423	—	17	1	no	—
86	#333	—	17	1	no	—
87	#333	—	17	1	no	—
88	#332	—	17	1	no	—
89	#332	D₁₇	1	1	yes	characteristic, complemented
90	C₂×C₁₇	#413	1	1	yes	characteristic
91	C₂×C₁₇	#417	1	1	yes	characteristic
92	C₂×C₁₇	#419	1	1	yes	characteristic
93	D₁₇	—	4	1	no	—
94	C₂×M₁₆	—	34	1	no	—
95	C₄⋊Q₈	—	17	1	no	—
96	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
97	#125	—	34	1	no	—
98	C₂²⋊Q₈	—	17	2	no	—
99	C₂²⋊Q₈	—	34	1	no	—
100	C₄×Q₈	—	17	2	no	—
101	C₄×Q₈	—	34	1	no	—
102	#117	—	34	1	no	—
103	C₂×C₄²	—	17	1	no	—
104	C₄⋊C₈	—	17	1	no	—
105	C₄⋊C₈	—	17	1	no	—
106	C₄⋊C₈	—	34	1	no	—
107	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
108	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
109	C₂²⋊C₈	—	34	1	no	—
110	#97	—	34	1	no	—
111	C₁₇	#1559	1	1	yes	characteristic, complemented, Sylow
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	17	2	no	—
114	C₂²×C₄	—	17	1	no	—
115	C₂²×C₄	—	17	1	no	—
116	C₂²×C₄	—	17	1	no	—
117	M₁₆	—	68	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	34	1	no	—
121	C₂×C₈	—	34	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	34	2	no	—
124	C₄⋊C₄	—	34	2	no	—
125	C₄⋊C₄	—	34	1	no	—
126	C₄⋊C₄	—	34	1	no	—
127	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
128	C₂²⋊C₄	—	34	2	no	—
129	C₄²	—	17	1	no	—
130	C₄²	—	17	1	no	—
131	C₄²	—	17	1	no	—
132	C₄²	—	34	2	no	—
133	C₄²	C₂²×D₁₇	1	1	yes	characteristic
134	C₂³	—	17	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	68	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	—	34	1	no	—
144	C₂×C₄	—	34	1	no	—
145	C₂×C₄	—	34	1	no	—
146	C₂×C₄	—	34	2	no	—
147	C₂×C₄	—	68	1	no	—
148	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
149	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
150	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	68	1	no	—
154	C₈	—	68	1	no	—
155	C₂²	—	34	1	no	—
156	C₂²	—	34	1	no	—
157	C₂²	—	34	1	no	—
158	C₂²	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	—	34	1	no	—
162	C₄	—	34	1	no	—
163	C₄	—	34	1	no	—
164	C₄	—	68	2	no	—
165	C₄	#10587597	1	1	yes	characteristic
166	C₄	#10587601	1	1	yes	characteristic
167	C₄	#10587601	1	1	yes	characteristic
168	C₄	#10587604	1	1	yes	characteristic
169	C₂	—	68	1	no	—
170	C₂	#11762792	1	1	yes	characteristic
171	C₂	#11762807	1	1	yes	characteristic
172	C₂	#11762816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763364

C₂

yes

maximal, characteristic

#11762865

C₂

yes

maximal, characteristic

#11762831

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, characteristic, complemented

#11762557

C₂

yes

maximal, characteristic, complemented

#11762555

C₂

yes

maximal, characteristic, complemented

C₁₇×#332

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

#10587632

—

Q₈×Dic₁₇

—

C₂×#63218

—

#10587583

C₄

yes

—

#10587578

—

#10587576

—

#10587573

C₄

yes

—

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×C₄

yes

—

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

—

#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

—

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

—

—

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

C₂×D₁₇

yes

characteristic

108

C₄⋊C₈

C₂×D₁₇

yes

characteristic

109

C₂²⋊C₈

—

110

#97

—

111

C₁₇

#1559

yes

characteristic, complemented, Sylow

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₄×D₁₇

yes

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×D₁₇

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

149

C₂×C₄

D₄×D₁₇

yes

characteristic

150

C₂×C₄

D₄×D₁₇

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂²⋊C₄×D₁₇

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#10587597

yes

characteristic

166

C₄

#10587601

yes

characteristic

167

C₄

#10587601

yes

characteristic

168

C₄

#10587604

yes

characteristic

169

C₂

—

170

C₂

#11762792

yes

characteristic

171

C₂

#11762807

yes

characteristic

172

C₂

#11762816

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	#1559
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. #11762557	C₂	semidirect
5. #11762557	C₂	semidirect
6. #11762555	C₂	semidirect
7. C₁₇×#332	C₂	semidirect
89. #332	D₁₇	semidirect
111. C₁₇	#1559	semidirect

H ≅ G/N

type

4. #11762557

C₂

semidirect

5. #11762557

C₂

semidirect

6. #11762555

C₂

semidirect

7. C₁₇×#332

C₂

semidirect

89. #332

D₁₇

semidirect

111. C₁₇

#1559

semidirect

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

subgroup	index
#11763364	2
#11762865	2
#11762831	2
#11762557	2
#11762555	2
C₁₇×#332	2
#1559	17

subgroup

index

C₁₇×#332

Subgroups — #14527513

Conjugacy classes of subgroups

Complements

Maximal subgroups