Group #14530030 — subgroups

Subgroups — #14530030

Order 2176.

298 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#10587555	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587481	C₂	1	2	yes	maximal
3	C₂×#10587481	C₂	1	2	yes	maximal
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#389	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	—	2	1	no	—
12	C₂×C₄⋊Dic₁₇	—	2	1	no	—
13	C₂²×#3622	C₂²	1	2	yes	—
14	#10587555	—	2	4	no	—
15	C₁₇×#95	—	2	1	no	—
16	#10587481	—	2	4	no	—
17	#10587481	—	2	4	no	—
18	C₂²×C₄×C₁₇	—	2	1	no	—
19	C₂²×C₄×C₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₇	—	2	2	no	—
24	C₂×Dic₃₄	—	2	2	no	—
25	C₂×Dic₃₄	—	2	2	no	—
26	C₂×Dic₃₄	—	2	2	no	—
27	C₂×Dic₃₄	—	2	2	no	—
28	C₂×Dic₃₄	—	2	2	no	—
29	C₂×Dic₃₄	D₄	1	4	yes	—
30	C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₄⋊C₄×C₁₇	—	2	2	no	—
32	C₄⋊C₄×C₁₇	—	4	2	no	—
33	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
34	C₄²×C₁₇	—	2	2	no	—
35	C₄⋊Dic₁₇	—	2	4	no	—
36	C₄⋊Dic₁₇	—	2	4	no	—
37	C₂×#3622	—	2	2	no	—
38	C₂×#3622	—	2	2	no	—
39	C₂×#3622	—	2	2	no	—
40	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	—	2	1	no	—
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	—	4	1	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	—	4	1	no	—
49	C₂×C₄×C₁₇	—	4	1	no	—
50	C₂×C₄×C₁₇	—	4	1	no	—
51	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#41	1	2	yes	—
57	C₂×Dic₁₇	—	2	4	no	—
58	C₂×Dic₁₇	—	4	2	no	—
59	C₂×Dic₁₇	—	4	2	no	—
60	Dic₃₄	—	2	4	no	—
61	Dic₃₄	—	2	4	no	—
62	Dic₃₄	—	4	2	no	—
63	Dic₃₄	—	4	2	no	—
64	#3622	—	4	2	no	—
65	#2043	—	17	1	no	maximal, Sylow
66	C₂²×C₁₇	#120	1	1	yes	characteristic
67	C₂²×C₁₇	#121	1	1	yes	characteristic
68	C₂²×C₁₇	#124	1	1	yes	characteristic
69	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂²×C₁₇	#137	1	1	yes	characteristic
72	C₂²×C₁₇	#137	1	1	yes	characteristic
73	C₄×C₁₇	—	2	2	no	—
74	C₄×C₁₇	—	2	2	no	—
75	C₄×C₁₇	—	4	2	no	—
76	C₄×C₁₇	—	8	1	no	—
77	C₄×C₁₇	#121	1	2	yes	—
78	C₄×C₁₇	#123	1	2	yes	—
79	Dic₁₇	—	4	4	no	—
80	C₂×C₄⋊Q₈	—	17	1	no	—
81	C₂×#103	—	17	2	no	—
82	C₂×#103	—	17	2	no	—
83	#389	D₁₇	1	1	yes	characteristic
84	#336	—	17	1	no	—
85	C₂×C₁₇	#394	1	1	yes	characteristic
86	C₂×C₁₇	#448	1	1	yes	characteristic
87	C₂×C₁₇	#451	1	1	yes	characteristic
88	C₂×C₁₇	#461	1	1	yes	characteristic
89	C₂×C₁₇	#462	1	1	yes	characteristic
90	C₂×C₁₇	#487	1	1	yes	characteristic
91	C₂×C₁₇	#491	1	1	yes	characteristic
92	C₂²×Q₈	—	17	2	no	—
93	C₂²×C₈	—	17	2	no	—
94	C₄⋊Q₈	—	34	4	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	—	34	1	no	—
97	C₂×C₄⋊C₄	—	34	1	no	—
98	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
101	#103	—	34	4	no	—
102	#103	—	34	4	no	—
103	#95	—	2	1	no	—
104	C₁₇	#2043	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	17	4	no	—
106	C₂×Q₈	—	34	2	no	—
107	C₂×Q₈	—	34	2	no	—
108	C₂×Q₈	—	34	2	no	—
109	C₂×Q₈	—	34	2	no	—
110	C₂×Q₈	—	34	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	34	2	no	—
114	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
117	C₂×C₈	—	34	2	no	—
118	C₂×C₈	—	34	2	no	—
119	C₂×C₈	—	34	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	34	4	no	—
124	C₄⋊C₄	—	34	4	no	—
125	C₄⋊C₄	D₆₈	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
128	Q₈	—	34	4	no	—
129	Q₈	—	34	4	no	—
130	Q₈	—	68	2	no	—
131	Q₈	—	68	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	34	4	no	—
143	C₂×C₄	—	68	2	no	—
144	C₂×C₄	—	68	2	no	—
145	C₂×C₄	#63229	1	2	yes	—
146	C₂×C₄	#63231	1	2	yes	—
147	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
148	C₂×C₄	#63252	1	1	yes	characteristic
149	C₂×C₄	D₄×D₁₇	1	2	yes	—
150	C₂×C₄	#63256	1	2	yes	—
151	C₈	—	68	2	no	—
152	C₂²	#10587616	1	1	yes	characteristic
153	C₂²	C₂×#63229	1	1	yes	characteristic
154	C₂²	#10587623	1	1	yes	characteristic
155	C₂²	C₂×#63231	1	1	yes	characteristic
156	C₂²	#10587633	1	1	yes	characteristic
157	C₂²	#10587637	1	1	yes	characteristic
158	C₂²	#10587637	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	68	4	no	—
164	C₄	#10587580	1	2	yes	—
165	C₄	#10587581	1	2	yes	—
166	C₂	#11762726	1	1	yes	characteristic
167	C₂	#11762947	1	1	yes	characteristic
168	C₂	#11762964	1	1	yes	characteristic
169	C₂	#11763014	1	1	yes	characteristic
170	C₂	#11763019	1	1	yes	characteristic
171	C₂	#11763161	1	1	yes	characteristic
172	C₂	#11763179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587555

C₂

yes

maximal, characteristic

C₂×#10587481

C₂

yes

maximal

C₂×#10587481

C₂

yes

maximal

#11762564

C₂

yes

maximal, characteristic

C₁₇×#389

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

C₂²

yes

—

#10587555

—

C₁₇×#95

—

#10587481

—

#10587481

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#121

yes

—

C₄×C₁₇

#123

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#103

—

#389

D₁₇

yes

characteristic

#336

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

100

C₂×C₄²

C₂×D₁₇

yes

characteristic

101

#103

—

102

#103

—

103

#95

—

104

C₁₇

#2043

yes

characteristic, complemented, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

115

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

116

C₂²×C₄

C₂²×D₁₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

D₆₈

yes

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#63229

yes

—

146

C₂×C₄

#63231

yes

—

147

C₂×C₄

C₂×D₆₈

yes

characteristic

148

C₂×C₄

#63252

yes

characteristic

149

C₂×C₄

D₄×D₁₇

yes

—

150

C₂×C₄

#63256

yes

—

151

C₈

—

152

C₂²

#10587616

yes

characteristic

153

C₂²

C₂×#63229

yes

characteristic

154

C₂²

#10587623

yes

characteristic

155

C₂²

C₂×#63231

yes

characteristic

156

C₂²

#10587633

yes

characteristic

157

C₂²

#10587637

yes

characteristic

158

C₂²

#10587637

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#10587580

yes

—

165

C₄

#10587581

yes

—

166

C₂

#11762726

yes

characteristic

167

C₂

#11762947

yes

characteristic

168

C₂

#11762964

yes

characteristic

169

C₂

#11763014

yes

characteristic

170

C₂

#11763019

yes

characteristic

171

C₂

#11763161

yes

characteristic

172

C₂

#11763179

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#389	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#389	D₁₇
hypocenter	C₁₇	#2043
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#389

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
104. C₁₇	#2043	semidirect

H ≅ G/N

type

104. C₁₇

#2043

semidirect

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

subgroup	index
C₂×#10587555	2
C₂×#10587481	2
#11762564	2
C₁₇×#389	2
#2043	17

subgroup

index

C₁₇×#389

Subgroups — #14530030

Conjugacy classes of subgroups

Complements

Maximal subgroups