Group #14529322 — subgroups

Subgroups — #14529322

Order 2176.

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

normal subgroups only

This group has 314 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	#11763287	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762826	C₂	1	2	yes	maximal
3	C₂×#10587507	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762565	C₂	1	2	yes	maximal, complemented
5	C₁₇×#536	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×C₄⋊Q₈	C₄	1	2	yes	complemented
9	C₁₇×#124	C₄	1	2	yes	complemented
10	C₁₇×#123	—	2	2	no	—
11	C₁₇×C₂²⋊Q₈	—	2	2	no	—
12	C₁₇×#117	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₇	—	2	4	no	—
14	#10587612	—	2	2	no	—
15	#10587612	C₂²	1	2	yes	—
16	C₂×#63223	C₂²	1	2	yes	—
17	#10587508	—	2	4	no	—
18	#10587507	—	2	2	no	—
19	#41×C₁₇	—	2	2	no	—
20	#41×C₁₇	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	2	no	—
22	C₂×Q₈×C₁₇	—	2	2	no	—
23	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₁₇	D₄	1	2	yes	—
26	C₂×D₄×C₁₇	—	2	1	no	—
27	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	—	2	1	no	—
29	C₂²×C₄×C₁₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₇	—	2	4	no	—
32	C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂²⋊C₄×C₁₇	—	2	1	no	—
34	C₂²⋊C₄×C₁₇	—	2	4	no	—
35	C₄²×C₁₇	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₁₇	—	2	2	no	—
37	C₄⋊Dic₁₇	—	2	4	no	—
38	C₄⋊Dic₁₇	—	2	2	no	—
39	C₄×Dic₁₇	—	2	4	no	—
40	C₄×Dic₁₇	D₄	1	4	yes	—
41	#63223	—	2	2	no	—
42	#63223	—	2	2	no	—
43	C₂×#3622	—	2	2	no	—
44	C₂³×C₁₇	—	2	1	no	—
45	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₇	—	2	4	no	—
47	Q₈×C₁₇	—	2	4	no	—
48	Q₈×C₁₇	—	2	2	no	—
49	D₄×C₁₇	—	2	2	no	—
50	D₄×C₁₇	—	4	1	no	—
51	D₄×C₁₇	—	4	1	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	1	no	—
56	C₂×C₄×C₁₇	—	4	2	no	—
57	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#41	1	2	yes	—
62	C₂×Dic₁₇	—	2	4	no	—
63	#3622	—	4	2	no	—
64	#1912	—	17	1	no	maximal, Sylow
65	C₂²×C₁₇	—	4	1	no	—
66	C₂²×C₁₇	—	4	2	no	—
67	C₂²×C₁₇	—	4	1	no	—
68	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₇	#120	1	1	yes	characteristic
70	C₂²×C₁₇	#123	1	1	yes	characteristic
71	C₄×C₁₇	—	2	4	no	—
72	C₄×C₁₇	—	4	2	no	—
73	C₄×C₁₇	—	4	1	no	—
74	C₄×C₁₇	C₄×D₄	1	2	yes	—
75	C₄×C₁₇	#121	1	2	yes	—
76	Dic₁₇	—	4	4	no	—
77	#536	D₁₇	1	1	yes	characteristic
78	#519	—	17	1	no	—
79	#421	—	17	2	no	—
80	C₂×#101	—	17	1	no	—
81	H(3,ℤ/4)	—	17	2	no	—
82	C₂×C₁₇	—	2	1	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	—	2	1	no	—
85	C₂×C₁₇	—	8	1	no	—
86	C₂×C₁₇	#386	1	1	yes	characteristic
87	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
89	C₂×M₁₆	—	17	2	no	—
90	C₄⋊Q₈	Dic₁₇	1	2	yes	complemented
91	#124	Dic₁₇	1	2	yes	complemented
92	#123	—	2	2	no	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	34	4	no	—
95	#117	—	17	2	no	—
96	#117	—	34	2	no	—
97	#117	C₂×D₁₇	1	1	yes	characteristic
98	#104	—	34	4	no	—
99	#101	—	34	2	no	—
100	C₁₇	#1912	1	1	yes	characteristic, complemented, Sylow
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
106	C₂×Q₈	C₂×Dic₁₇	1	2	yes	—
107	C₂×Q₈	C₂²⋊D₁₇	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₂×Dic₁₇	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
113	M₁₆	—	34	2	no	—
114	M₁₆	—	34	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	34	4	no	—
119	C₄⋊C₄	—	34	2	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	34	2	no	—
123	C₄²	—	17	4	no	—
124	C₄²	—	34	4	no	—
125	C₄²	C₂×Dic₁₇	1	2	yes	—
126	C₂³	—	2	1	no	—
127	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	2	2	no	—
131	D₄	—	2	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	34	4	no	—
140	C₂×C₄	C₂²⋊Dic₁₇	1	4	yes	—
141	C₂×C₄	D₄×D₁₇	1	2	yes	—
142	C₂×C₄	#63254	1	2	yes	—
143	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
145	C₈	—	68	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#10587621	1	1	yes	characteristic
150	C₂²	#10587623	1	1	yes	characteristic
151	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₄	—	68	4	no	—
156	C₄	D₄×Dic₁₇	1	2	yes	—
157	C₄	#10587625	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	8	1	no	—
162	C₂	#11762689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763287

C₂

yes

maximal, characteristic, complemented

#11762826

C₂

yes

maximal

C₂×#10587507

C₂

yes

maximal, characteristic, complemented

#11762565

C₂

yes

maximal, complemented

C₁₇×#536

C₂

yes

maximal, characteristic

C₂×#41×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

C₄

yes

complemented

C₁₇×#124

C₄

yes

complemented

C₁₇×#123

—

C₁₇×C₂²⋊Q₈

—

C₁₇×#117

C₂²

yes

characteristic

Q₈×Dic₁₇

—

#10587612

—

#10587612

C₂²

yes

—

C₂×#63223

C₂²

yes

—

#10587508

—

#10587507

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂×Q₈×C₁₇

D₄

yes

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

—

#63223

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

#3622

—

#1912

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

#121

yes

—

Dic₁₇

—

#536

D₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#386

yes

characteristic

C₂×#41

C₂×D₁₇

yes

characteristic

C₂²×Q₈

C₂×D₁₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₇

yes

complemented

#124

Dic₁₇

yes

complemented

#123

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₁₇

yes

characteristic

#104

—

#101

—

100

C₁₇

#1912

yes

characteristic, complemented, Sylow

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₁₇

yes

characteristic

106

C₂×Q₈

C₂×Dic₁₇

yes

—

107

C₂×Q₈

C₂²⋊D₁₇

yes

—

108

C₂×D₄

—

109

C₂×D₄

C₂×Dic₁₇

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₇

yes

—

112

C₂²×C₄

C₂²×D₁₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×Dic₁₇

yes

—

126

C₂³

—

127

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂²⋊Dic₁₇

yes

—

141

C₂×C₄

D₄×D₁₇

yes

—

142

C₂×C₄

#63254

yes

—

143

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#10587621

yes

characteristic

150

C₂²

#10587623

yes

characteristic

151

C₂²

C₂×C₂²⋊Dic₁₇

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

D₄×Dic₁₇

yes

—

157

C₄

#10587625

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#11762689

yes

characteristic

	subgroup	G/H
center	C₂	#11762689
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#536	C₂
socle	C₂×C₁₇	#386
prime radical	C₂×#41×C₁₇	C₂²
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#536	D₁₇
hypocenter	C₁₇	#1912
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#11762689

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#536

C₂

socle

C₂×C₁₇

#386

prime radical

C₂×#41×C₁₇

C₂²

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
1. #11763287	C₂	semidirect
3. C₂×#10587507	C₂	semidirect
4. #11762565	C₂	semidirect
8. C₁₇×C₄⋊Q₈	C₄	semidirect
9. C₁₇×#124	C₄	semidirect
90. C₄⋊Q₈	Dic₁₇	semidirect
91. #124	Dic₁₇	semidirect
100. C₁₇	#1912	semidirect

H ≅ G/N

type

1. #11763287

C₂

semidirect

3. C₂×#10587507

C₂

semidirect

4. #11762565

C₂

semidirect

8. C₁₇×C₄⋊Q₈

C₄

semidirect

9. C₁₇×#124

C₄

semidirect

90. C₄⋊Q₈

Dic₁₇

semidirect

91. #124

Dic₁₇

semidirect

100. C₁₇

#1912

semidirect

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

subgroup	index
#11763287	2
#11762826	2
C₂×#10587507	2
#11762565	2
C₁₇×#536	2
#1912	17

subgroup

index

C₁₇×#536

Subgroups — #14529322

Conjugacy classes of subgroups

Complements

Maximal subgroups