Group #14529922 — subgroups

Subgroups — #14529922

Order 2176.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#10587481	C₂	1	2	yes	maximal
2	#11762715	C₂	1	1	yes	maximal, characteristic
3	#11762571	C₂	1	2	yes	maximal
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	—	2	1	no	—
7	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
8	C₁₇×C₄⋊Q₈	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂²×#3622	C₂²	1	2	yes	—
14	#10587502	—	2	1	no	—
15	#10587502	—	2	1	no	—
16	#10587481	—	2	4	no	—
17	C₂×Q₈×C₁₇	—	2	4	no	—
18	C₂×Q₈×C₁₇	—	4	2	no	—
19	C₂×Q₈×C₁₇	—	4	2	no	—
20	C₂²×C₄×C₁₇	—	2	2	no	—
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₇	—	2	1	no	—
25	C₂²×Dic₁₇	—	2	1	no	—
26	C₂×Dic₃₄	—	2	2	no	—
27	C₂×Dic₃₄	—	2	2	no	—
28	C₂×Dic₃₄	—	2	1	no	—
29	C₂×Dic₃₄	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₁₇	—	2	4	no	—
31	C₄⋊C₄×C₁₇	—	2	2	no	—
32	C₄⋊C₄×C₁₇	D₄	1	4	yes	—
33	C₄²×C₁₇	—	2	2	no	—
34	C₄⋊Dic₁₇	—	2	2	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	—	2	2	no	—
37	C₂×#3622	—	2	2	no	—
38	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₇	—	4	8	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
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	4	no	—
45	C₂×C₄×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	2	yes	—
52	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	—	4	1	no	—
55	C₂×Dic₁₇	—	4	1	no	—
56	C₂×Dic₁₇	—	4	1	no	—
57	C₂×Dic₁₇	—	4	1	no	—
58	C₂×Dic₁₇	—	4	1	no	—
59	Dic₃₄	—	2	4	no	—
60	Dic₃₄	—	4	2	no	—
61	#3622	—	4	2	no	—
62	#2024	—	17	1	no	maximal, Sylow
63	C₂²×C₁₇	#120	1	1	yes	characteristic
64	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂²×C₁₇	#124	1	1	yes	characteristic
66	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×C₁₇	#137	1	1	yes	characteristic
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	—	2	2	no	—
72	C₄×C₁₇	—	4	4	no	—
73	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₁₇	#124	1	2	yes	—
75	Dic₁₇	—	4	2	no	—
76	Dic₁₇	—	8	1	no	—
77	C₂×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
78	C₂×#103	—	17	2	no	—
79	#391	—	17	1	no	—
80	#340	—	17	2	no	—
81	#336	—	17	1	no	—
82	C₂×C₁₇	#393	1	1	yes	characteristic
83	C₂×C₁₇	#450	1	1	yes	characteristic
84	C₂×C₁₇	#451	1	1	yes	characteristic
85	C₂×C₁₇	#475	1	1	yes	characteristic
86	C₂×C₁₇	#477	1	1	yes	characteristic
87	C₂×C₁₇	#487	1	1	yes	characteristic
88	C₂×C₁₇	#489	1	1	yes	characteristic
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×Q₈	—	17	1	no	—
91	C₂²×C₈	—	17	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	1	no	—
94	C₂×C₄⋊C₄	—	17	1	no	—
95	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
96	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
97	#103	—	34	4	no	—
98	#95	—	34	1	no	—
99	#95	—	34	1	no	—
100	C₁₇	#2024	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	17	2	no	—
105	C₂×Q₈	—	34	1	no	—
106	C₂×Q₈	—	34	2	no	—
107	C₂×Q₈	—	34	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	34	1	no	—
110	C₂²×C₄	—	34	1	no	—
111	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	—	34	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	34	2	no	—
120	C₄⋊C₄	C₂²⋊D₁₇	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	34	4	no	—
125	Q₈	—	68	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	34	2	no	—
134	C₂×C₄	—	68	1	no	—
135	C₂×C₄	—	68	1	no	—
136	C₂×C₄	—	68	1	no	—
137	C₂×C₄	—	68	1	no	—
138	C₂×C₄	—	68	1	no	—
139	C₂×C₄	#63229	1	2	yes	—
140	C₂×C₄	#63231	1	2	yes	—
141	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
142	C₂×C₄	#63256	1	2	yes	—
143	C₂×C₄	#63256	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
145	C₈	—	68	2	no	—
146	C₂²	#10587618	1	1	yes	characteristic
147	C₂²	C₂×#63229	1	1	yes	characteristic
148	C₂²	#10587628	1	1	yes	characteristic
149	C₂²	C₂×#63231	1	1	yes	characteristic
150	C₂²	#10587632	1	1	yes	characteristic
151	C₂²	#10587633	1	1	yes	characteristic
152	C₂²	#10587639	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	68	2	no	—
157	C₄	—	136	1	no	—
158	C₄	#10587632	1	2	yes	—
159	C₄	#10587633	1	2	yes	—
160	C₂	#11762720	1	1	yes	characteristic
161	C₂	#11762960	1	1	yes	characteristic
162	C₂	#11762965	1	1	yes	characteristic
163	C₂	#11763091	1	1	yes	characteristic
164	C₂	#11763101	1	1	yes	characteristic
165	C₂	#11763161	1	1	yes	characteristic
166	C₂	#11763171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587481

C₂

yes

maximal

#11762715

C₂

yes

maximal, characteristic

#11762571

C₂

yes

maximal

#11762564

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

—

C₂²×Dic₃₄

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

—

#10587502

—

#10587502

—

#10587481

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

Q₈

yes

complemented

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×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

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#124

yes

characteristic

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

C₂²⋊Q₈

yes

—

C₄×C₁₇

#124

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×C₄⋊Q₈

D₁₇

yes

—

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

100

C₁₇

#2024

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

112

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

113

C₂²×C₄

C₂²×D₁₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂²⋊D₁₇

yes

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#63229

yes

—

140

C₂×C₄

#63231

yes

—

141

C₂×C₄

Q₈×D₁₇

yes

characteristic

142

C₂×C₄

#63256

yes

—

143

C₂×C₄

#63256

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

145

C₈

—

146

C₂²

#10587618

yes

characteristic

147

C₂²

C₂×#63229

yes

characteristic

148

C₂²

#10587628

yes

characteristic

149

C₂²

C₂×#63231

yes

characteristic

150

C₂²

#10587632

yes

characteristic

151

C₂²

#10587633

yes

characteristic

152

C₂²

#10587639

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

136

—

158

C₄

#10587632

yes

—

159

C₄

#10587633

yes

—

160

C₂

#11762720

yes

characteristic

161

C₂

#11762960

yes

characteristic

162

C₂

#11762965

yes

characteristic

163

C₂

#11763091

yes

characteristic

164

C₂

#11763101

yes

characteristic

165

C₂

#11763161

yes

characteristic

166

C₂

#11763171

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#2024
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₇

hypocenter

C₁₇

#2024

LCS

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
29. C₂×Dic₃₄	Q₈	semidirect
100. C₁₇	#2024	semidirect

H ≅ G/N

type

29. C₂×Dic₃₄

Q₈

semidirect

100. C₁₇

#2024

semidirect

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

subgroup	index
C₂×#10587481	2
#11762715	2
#11762571	2
#11762564	2
C₂×C₁₇×C₄⋊Q₈	2
#2024	17

subgroup

index

C₂×C₁₇×C₄⋊Q₈

Subgroups — #14529922

Conjugacy classes of subgroups

Complements

Maximal subgroups