Group #14528124 — subgroups

Subgroups — #14528124

Order 2176.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#11763181	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763090	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763001	C₂	1	1	yes	maximal, characteristic
4	#11762874	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762559	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762556	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#127	C₂²	1	1	yes	characteristic
9	C₂×#63229	—	2	1	no	—
10	C₂×#63229	—	2	1	no	—
11	C₂×#63228	—	2	1	no	—
12	C₂×C₈×D₁₇	—	2	1	no	—
13	C₄×D₆₈	C₂²	1	1	yes	characteristic
14	#10587555	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	#10587503	—	2	1	no	—
18	#10587492	—	2	1	no	—
19	#10587491	—	2	1	no	—
20	#10587489	—	2	1	no	—
21	C₄⋊#3622	C₂²	1	1	yes	characteristic
22	C₄×#3622	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₁₇	—	2	1	no	—
24	C₂×D₄×C₁₇	—	2	1	no	—
25	C₂×D₄×C₁₇	—	2	1	no	—
26	C₂×D₆₈	D₄	1	1	yes	characteristic
27	C₂×C₄×D₁₇	—	2	1	no	—
28	C₂×Dic₃₄	—	2	1	no	—
29	C₂×C₈×C₁₇	—	2	1	no	—
30	C₂×C₈×C₁₇	—	2	1	no	—
31	C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	#63229	—	4	2	no	—
33	#63229	—	4	2	no	—
34	#63228	—	4	2	no	—
35	#63227	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
39	C₂×#3622	—	2	1	no	—
40	C₂×#3622	D₄	1	1	yes	characteristic
41	C₂×#3622	D₄	1	1	yes	characteristic
42	C₈×D₁₇	—	4	2	no	—
43	C₂³×C₁₇	—	4	1	no	—
44	C₂²×D₁₇	—	2	1	no	—
45	D₄×C₁₇	—	4	1	no	—
46	D₄×C₁₇	—	4	1	no	—
47	D₄×C₁₇	—	4	1	no	—
48	D₄×C₁₇	—	4	1	no	—
49	D₄×C₁₇	—	8	1	no	—
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	1	no	—
54	C₂×Dic₁₇	—	4	1	no	—
55	D₆₈	—	2	1	no	—
56	D₆₈	SD₁₆	1	2	yes	complemented
57	C₄×D₁₇	—	4	1	no	—
58	C₄×D₁₇	—	4	1	no	—
59	Dic₃₄	—	4	1	no	—
60	Dic₃₄	—	4	1	no	—
61	C₈×C₁₇	—	4	1	no	—
62	C₈×C₁₇	—	4	1	no	—
63	#3622	—	2	1	no	—
64	#3622	—	4	1	no	—
65	#3622	D₈	1	2	yes	complemented
66	#1665	—	17	1	no	maximal, Sylow
67	C₂²×C₁₇	—	8	1	no	—
68	C₂²×C₁₇	—	8	1	no	—
69	C₂²×C₁₇	—	8	1	no	—
70	C₂²×C₁₇	#121	1	1	yes	characteristic
71	C₂×D₁₇	—	2	2	no	—
72	C₂×D₁₇	—	4	1	no	—
73	C₂×D₁₇	—	4	1	no	—
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
76	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₁₇	#135	1	1	yes	characteristic
78	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₁₇	—	4	1	no	—
80	Dic₁₇	—	8	1	no	—
81	#492	—	17	1	no	—
82	#474	—	17	1	no	—
83	#459	—	17	1	no	—
84	C₈×D₄	—	17	1	no	—
85	#334	—	17	1	no	—
86	#332	—	17	1	no	—
87	#331	D₁₇	1	1	yes	characteristic, complemented
88	C₂×C₁₇	—	16	1	no	—
89	C₂×C₁₇	#459	1	1	yes	characteristic
90	C₂×C₁₇	#465	1	1	yes	characteristic
91	C₂×C₁₇	#471	1	1	yes	characteristic
92	D₁₇	—	4	2	no	—
93	C₂×SD₁₆	—	34	1	no	—
94	C₂×SD₁₆	—	34	1	no	—
95	C₂×D₈	—	34	1	no	—
96	C₂²×C₈	—	34	1	no	—
97	C₄⋊Q₈	—	17	1	no	—
98	#127	C₂×D₁₇	1	1	yes	characteristic
99	C₄×D₄	—	17	1	no	—
100	#107	—	34	1	no	—
101	C₄⋊C₈	—	17	1	no	—
102	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
103	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
104	#102	—	34	1	no	—
105	#102	—	34	1	no	—
106	C₂²⋊C₈	—	34	1	no	—
107	C₄×C₈	—	17	1	no	—
108	C₁₇	#1665	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	34	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	17	1	no	—
114	C₂²×C₄	—	34	1	no	—
115	SD₁₆	—	68	2	no	—
116	SD₁₆	—	68	2	no	—
117	D₈	—	68	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	34	1	no	—
123	C₂×C₈	—	68	2	no	—
124	C₄⋊C₄	—	17	1	no	—
125	C₄⋊C₄	—	34	1	no	—
126	C₄⋊C₄	—	34	1	no	—
127	C₂²⋊C₄	—	34	1	no	—
128	C₄²	C₂²×D₁₇	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	34	1	no	—
131	Q₈	—	68	1	no	—
132	Q₈	—	68	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	17	2	no	—
139	D₄	—	34	1	no	—
140	C₂×C₄	—	34	1	no	—
141	C₂×C₄	—	68	1	no	—
142	C₂×C₄	—	68	1	no	—
143	C₂×C₄	—	68	1	no	—
144	C₂×C₄	#63252	1	1	yes	characteristic
145	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
146	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	17	2	no	—
150	C₈	—	34	1	no	—
151	C₈	—	68	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	34	2	no	—
156	C₂²	—	68	1	no	—
157	C₂²	—	68	1	no	—
158	C₂²	#10587570	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	68	1	no	—
161	C₄	—	136	1	no	—
162	C₄	D₈×D₁₇	1	1	yes	characteristic
163	C₄	#10587598	1	1	yes	characteristic
164	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
165	C₄	#10587600	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	68	2	no	—
168	C₂	#11763003	1	1	yes	characteristic
169	C₂	#11763037	1	1	yes	characteristic
170	C₂	#11763071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763181

C₂

yes

maximal, characteristic, complemented

#11763090

C₂

yes

maximal, characteristic, complemented

#11763001

C₂

yes

maximal, characteristic

#11762874

C₂

yes

maximal, characteristic, complemented

#11762559

C₂

yes

maximal, characteristic, complemented

#11762556

C₂

yes

maximal, characteristic, complemented

C₁₇×#331

C₂

yes

maximal, characteristic, complemented

C₁₇×#127

C₂²

yes

—

—

—

C₂×C₈×D₁₇

—

C₄×D₆₈

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×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₆₈

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂³

yes

—

—

—

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

C₈×D₁₇

—

C₂³×C₁₇

—

C₂²×D₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

SD₁₆

yes

complemented

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

—

#3622

D₈

yes

complemented

#1665

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

C₂×SD₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

#492

—

#474

—

#459

—

C₈×D₄

—

#334

—

#332

—

#331

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

C₂×D₁₇

yes

characteristic

C₄×D₄

—

100

#107

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₁₇

yes

characteristic

103

C₄⋊C₈

C₂×D₁₇

yes

characteristic

104

#102

—

105

#102

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₁₇

#1665

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

SD₁₆

—

116

SD₁₆

—

117

D₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₁₇

yes

characteristic

129

C₂³

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#63252

yes

characteristic

145

C₂×C₄

D₄×D₁₇

yes

characteristic

146

C₂×C₄

D₄×D₁₇

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#10587570

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

136

—

162

C₄

D₈×D₁₇

yes

characteristic

163

C₄

#10587598

yes

characteristic

164

C₄

SD₁₆×D₁₇

yes

characteristic

165

C₄

#10587600

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#11763003

yes

characteristic

169

C₂

#11763037

yes

characteristic

170

C₂

#11763071

yes

characteristic

	subgroup	G/H
center	C₂²	#10587570
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#331	C₂
socle	C₂²×C₁₇	#121
prime radical	#11763001	C₂
hypercenter	#331	D₁₇
hypocenter	C₁₇	#1665
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#10587570

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#331

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #11763181	C₂	semidirect
2. #11763090	C₂	semidirect
4. #11762874	C₂	semidirect
5. #11762559	C₂	semidirect
6. #11762556	C₂	semidirect
7. C₁₇×#331	C₂	semidirect
56. D₆₈	SD₁₆	semidirect
65. #3622	D₈	semidirect
87. #331	D₁₇	semidirect
108. C₁₇	#1665	semidirect

H ≅ G/N

type

1. #11763181

C₂

semidirect

2. #11763090

C₂

semidirect

4. #11762874

C₂

semidirect

5. #11762559

C₂

semidirect

6. #11762556

C₂

semidirect

7. C₁₇×#331

C₂

semidirect

56. D₆₈

SD₁₆

semidirect

65. #3622

D₈

semidirect

87. #331

D₁₇

semidirect

108. C₁₇

#1665

semidirect

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

subgroup	index
#11763181	2
#11763090	2
#11763001	2
#11762874	2
#11762559	2
#11762556	2
C₁₇×#331	2
#1665	17

subgroup

index

C₁₇×#331

Subgroups — #14528124

Conjugacy classes of subgroups

Complements

Maximal subgroups