Group #14527491 — subgroups

Subgroups — #14527491

Order 2176.

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

normal subgroups only

This group has 198 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	#11762861	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	#10587583	C₄	1	2	yes	—
12	#10587578	—	2	1	no	—
13	#10587576	—	2	1	no	—
14	#10587573	C₄	1	2	yes	—
15	C₄²×D₁₇	C₂²	1	1	yes	characteristic
16	#10587555	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	#10587491	—	2	1	no	—
20	#10587491	—	2	1	no	—
21	#10587486	—	2	1	no	—
22	C₈×Dic₁₇	—	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	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₁₇	—	4	1	no	—
48	Q₈×C₁₇	—	4	1	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₁₇	—	2	1	no	—
58	C₄×D₁₇	—	2	1	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	4	1	no	—
62	Dic₃₄	—	4	2	no	—
63	C₈×C₁₇	—	4	1	no	—
64	C₈×C₁₇	—	4	1	no	—
65	#3622	—	4	1	no	—
66	#3622	—	4	1	no	—
67	#1554	—	17	1	no	maximal, Sylow
68	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₁₇	—	2	1	no	—
70	C₂×D₁₇	—	2	1	no	—
71	C₂×D₁₇	—	2	1	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	#135	1	1	yes	characteristic
75	C₄×C₁₇	#135	1	1	yes	characteristic
76	C₄×C₁₇	#137	1	1	yes	characteristic
77	C₄×C₁₇	#137	1	1	yes	characteristic
78	Dic₁₇	—	2	1	no	—
79	Dic₁₇	—	2	1	no	—
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	—	4	2	no	—
82	#533	—	17	1	no	—
83	#432	—	17	1	no	—
84	#431	—	17	1	no	—
85	#333	—	17	1	no	—
86	#333	—	17	1	no	—
87	#332	—	17	1	no	—
88	#332	D₁₇	1	1	yes	characteristic, complemented
89	C₂×C₁₇	#413	1	1	yes	characteristic
90	C₂×C₁₇	#416	1	1	yes	characteristic
91	C₂×C₁₇	#419	1	1	yes	characteristic
92	D₁₇	—	4	1	no	—
93	C₄⋊Q₈	—	17	1	no	—
94	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
95	#125	—	34	1	no	—
96	C₂²⋊Q₈	—	17	2	no	—
97	C₂²⋊Q₈	—	34	1	no	—
98	C₄×Q₈	—	17	2	no	—
99	C₄×Q₈	—	34	1	no	—
100	#117	—	34	1	no	—
101	C₂×C₄²	—	17	1	no	—
102	C₄⋊C₈	—	17	1	no	—
103	C₄⋊C₈	—	17	1	no	—
104	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
106	C₂²⋊C₈	—	34	1	no	—
107	C₂²⋊C₈	—	34	1	no	—
108	#97	—	34	1	no	—
109	C₄×C₈	—	34	1	no	—
110	C₁₇	#1554	1	1	yes	characteristic, complemented, Sylow
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	17	2	no	—
113	C₂²×C₄	—	17	1	no	—
114	C₂²×C₄	—	17	1	no	—
115	C₂²×C₄	—	17	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	34	1	no	—
119	C₂×C₈	—	34	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	34	2	no	—
122	C₄⋊C₄	—	34	2	no	—
123	C₄⋊C₄	—	34	1	no	—
124	C₄⋊C₄	—	34	1	no	—
125	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
126	C₂²⋊C₄	—	34	2	no	—
127	C₄²	—	17	1	no	—
128	C₄²	—	17	1	no	—
129	C₄²	—	17	1	no	—
130	C₄²	—	34	2	no	—
131	C₄²	C₂²×D₁₇	1	1	yes	characteristic
132	C₂³	—	17	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	68	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	34	1	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	—	34	1	no	—
144	C₂×C₄	—	34	2	no	—
145	C₂×C₄	—	68	1	no	—
146	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	68	1	no	—
152	C₈	—	68	1	no	—
153	C₂²	—	34	1	no	—
154	C₂²	—	34	1	no	—
155	C₂²	—	34	1	no	—
156	C₂²	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	34	1	no	—
160	C₄	—	34	1	no	—
161	C₄	—	34	1	no	—
162	C₄	—	68	2	no	—
163	C₄	#10587598	1	1	yes	characteristic
164	C₄	#10587601	1	1	yes	characteristic
165	C₄	#10587602	1	1	yes	characteristic
166	C₄	#10587604	1	1	yes	characteristic
167	C₂	—	68	1	no	—
168	C₂	#11762792	1	1	yes	characteristic
169	C₂	#11762802	1	1	yes	characteristic
170	C₂	#11762816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763364

C₂

yes

maximal, characteristic

#11762865

C₂

yes

maximal, characteristic

#11762861

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

—

#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

—

—

—

C₈×Dic₁₇

—

C₄⋊#3622

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

characteristic

C₂×#3622

—

C₂×#3622

—

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

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₁₇

yes

characteristic

105

C₄⋊C₈

C₂×D₁₇

yes

characteristic

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

C₁₇

#1554

yes

characteristic, complemented, Sylow

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

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

C₄×D₁₇

yes

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₁₇

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

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

—

146

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

147

C₂×C₄

D₄×D₁₇

yes

characteristic

148

C₂×C₄

D₄×D₁₇

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂²⋊C₄×D₁₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#10587598

yes

characteristic

164

C₄

#10587601

yes

characteristic

165

C₄

#10587602

yes

characteristic

166

C₄

#10587604

yes

characteristic

167

C₂

—

168

C₂

#11762792

yes

characteristic

169

C₂

#11762802

yes

characteristic

170

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	#1554
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
88. #332	D₁₇	semidirect
110. C₁₇	#1554	semidirect

H ≅ G/N

type

4. #11762557

C₂

semidirect

5. #11762557

C₂

semidirect

6. #11762555

C₂

semidirect

7. C₁₇×#332

C₂

semidirect

88. #332

D₁₇

semidirect

110. C₁₇

#1554

semidirect

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

subgroup	index
#11763364	2
#11762865	2
#11762861	2
#11762557	2
#11762555	2
C₁₇×#332	2
#1554	17

subgroup

index

C₁₇×#332

Subgroups — #14527491

Conjugacy classes of subgroups

Complements

Maximal subgroups