Group #16934174 — subgroups

Subgroups — #16934174

Order 4352.

542 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 199 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	#14529746	C₂	1	4	yes	maximal
2	#14528941	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587488	—	2	2	no	—
6	C₂×C₄×#3622	C₂²	1	2	yes	—
7	#11762679	C₂²	1	2	yes	—
8	#11762571	—	2	8	no	—
9	#11762530	—	2	2	no	—
10	C₂²×Q₁₆×C₁₇	—	2	2	no	—
11	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₇×#494	—	2	2	no	—
13	C₁₇×#494	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₇	—	2	2	no	—
16	C₂×Q₁₆×C₁₇	—	2	8	no	—
17	C₂×Q₁₆×C₁₇	—	4	4	no	—
18	C₂×Q₁₆×C₁₇	—	4	4	no	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	4	no	—
22	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
24	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	—	2	4	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂²×#3622	D₄	1	4	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587488	—	4	4	no	—
31	C₄×#3622	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	4	no	—
34	C₂×Q₈×C₁₇	—	4	2	no	—
35	C₂×Q₈×C₁₇	—	4	4	no	—
36	C₂×Q₈×C₁₇	—	4	4	no	—
37	C₂²×C₄×C₁₇	—	4	2	no	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	4	2	no	—
41	Q₁₆×C₁₇	—	4	16	no	—
42	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₈×C₁₇	D₈	1	4	yes	—
44	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
45	C₄⋊C₄×C₁₇	—	2	8	no	—
46	C₄⋊C₄×C₁₇	—	4	4	no	—
47	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	#41	1	2	yes	—
50	C₄⋊Dic₁₇	—	4	4	no	—
51	C₄⋊Dic₁₇	—	4	8	no	—
52	C₂×#3622	—	2	4	no	—
53	C₂×#3622	—	2	8	no	—
54	#9839	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	#127	1	1	yes	characteristic
56	Q₈×C₁₇	—	4	8	no	—
57	Q₈×C₁₇	—	8	4	no	—
58	C₂×C₄×C₁₇	—	4	4	no	—
59	C₂×C₄×C₁₇	—	8	2	no	—
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₁₇	#121	1	2	yes	—
64	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	#137	1	8	yes	—
68	C₂×Dic₁₇	—	8	2	no	—
69	C₂×Dic₁₇	—	8	4	no	—
70	C₈×C₁₇	#102	1	8	yes	—
71	#3622	—	4	4	no	—
72	C₂×#494	D₁₇	1	1	yes	characteristic
73	#1990	—	17	4	no	—
74	#1844	—	17	1	no	—
75	C₂×#322	—	17	1	no	—
76	C₂²×C₁₇	#390	1	1	yes	characteristic
77	C₂²×C₁₇	#492	1	1	yes	characteristic
78	C₂²×C₁₇	#493	1	1	yes	characteristic
79	C₂²×C₁₇	#497	1	4	yes	—
80	C₄×C₁₇	—	8	4	no	—
81	C₄×C₁₇	C₂×#102	1	2	yes	—
82	C₄×C₁₇	#441	1	4	yes	—
83	C₄×C₁₇	#470	1	4	yes	—
84	C₄×C₁₇	#486	1	2	yes	—
85	Dic₁₇	—	16	2	no	—
86	C₂²×Q₁₆	—	2	2	no	—
87	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
88	#494	—	2	2	no	—
89	#494	Dic₁₇	1	4	yes	complemented
90	C₂×#106	—	34	2	no	—
91	C₂×C₄×C₈	—	17	2	no	—
92	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	#384	—	17	2	no	—
94	#340	—	34	8	no	—
95	#322	—	34	2	no	—
96	C₂×C₁₇	#1716	1	2	yes	—
97	C₂×C₁₇	#1722	1	2	yes	—
98	C₂×C₁₇	#1957	1	1	yes	characteristic
99	C₂×C₁₇	#1971	1	2	yes	—
100	C₂²×Q₈	—	2	2	no	—
101	C₂×Q₁₆	—	2	8	no	—
102	C₂×Q₁₆	—	4	4	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂²×C₈	—	17	4	no	—
105	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
106	C₄⋊Q₈	—	2	4	no	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	34	4	no	—
111	C₂×C₄⋊C₄	—	34	2	no	—
112	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
113	#106	—	68	4	no	—
114	C₄×C₈	—	34	2	no	—
115	C₄×C₈	—	34	2	no	—
116	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
117	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
118	C₁₇	#9839	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂×Q₈	—	4	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	68	2	no	—
125	C₂²×C₄	D₄×D₁₇	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
127	Q₁₆	—	4	16	no	—
128	C₂×C₈	—	34	8	no	—
129	C₂×C₈	—	34	4	no	—
130	C₂×C₈	#63228	1	4	yes	—
131	C₂×C₈	#63229	1	4	yes	—
132	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	68	8	no	—
136	C₄⋊C₄	—	68	4	no	—
137	C₄²	#63254	1	2	yes	—
138	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
139	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	C₂³	#10587626	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	136	4	no	—
147	C₂×C₄	—	136	2	no	—
148	C₂×C₄	#10587604	1	8	yes	—
149	C₂×C₄	C₂×#63228	1	2	yes	—
150	C₂×C₄	C₂×#63229	1	2	yes	—
151	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
152	C₂×C₄	#10587622	1	1	yes	characteristic
153	C₂×C₄	#10587624	1	2	yes	—
154	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
155	C₈	—	68	4	no	—
156	C₈	#10587503	1	8	yes	—
157	C₂²	#11762712	1	1	yes	characteristic
158	C₂²	#11763181	1	1	yes	characteristic
159	C₂²	#11763185	1	1	yes	characteristic
160	C₂²	#11763202	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	272	2	no	—
163	C₄	C₂×#10587503	1	2	yes	—
164	C₄	#11762916	1	4	yes	—
165	C₄	#11763064	1	4	yes	—
166	C₄	#11763155	1	2	yes	—
167	C₂	#14528415	1	2	yes	—
168	C₂	#14528456	1	2	yes	—
169	C₂	#14529591	1	1	yes	characteristic
170	C₂	#14529656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529746

C₂

yes

maximal

#14528941

C₂

yes

maximal, characteristic

C₂×#11762530

C₂

yes

maximal, characteristic

C₂×C₁₇×#494

C₂

yes

maximal, characteristic

C₂×#10587488

—

C₂×C₄×#3622

C₂²

yes

—

#11762679

C₂²

yes

—

#11762571

—

#11762530

—

C₂²×Q₁₆×C₁₇

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#494

—

C₁₇×#494

C₄

yes

complemented

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

#10587488

—

C₄×#3622

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#102

yes

—

#3622

—

C₂×#494

D₁₇

yes

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#102

yes

—

C₄×C₁₇

#441

yes

—

C₄×C₁₇

#470

yes

—

C₄×C₁₇

#486

yes

—

Dic₁₇

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

#494

—

#494

Dic₁₇

yes

complemented

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

C₂×C₁₇

#1716

yes

—

C₂×C₁₇

#1722

yes

—

C₂×C₁₇

#1957

yes

characteristic

C₂×C₁₇

#1971

yes

—

100

C₂²×Q₈

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²⋊D₁₇

yes

—

106

C₄⋊Q₈

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

C₂×Dic₁₇

yes

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

C₂²×D₁₇

yes

characteristic

113

#106

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×Dic₁₇

yes

—

117

C₄×C₈

C₂²⋊D₁₇

yes

—

118

C₁₇

#9839

yes

characteristic, complemented, Sylow

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×D₁₇

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#63228

yes

—

131

C₂×C₈

#63229

yes

—

132

C₂×C₈

C₂²⋊Dic₁₇

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

#63254

yes

—

138

C₄²

C₂²×Dic₁₇

yes

characteristic

139

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

140

C₂³

#10587626

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

136

—

147

C₂×C₄

—

136

—

148

C₂×C₄

#10587604

yes

—

149

C₂×C₄

C₂×#63228

yes

—

150

C₂×C₄

C₂×#63229

yes

—

151

C₂×C₄

D₄×Dic₁₇

yes

—

152

C₂×C₄

#10587622

yes

characteristic

153

C₂×C₄

#10587624

yes

—

154

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

155

C₈

—

156

C₈

#10587503

yes

—

157

C₂²

#11762712

yes

characteristic

158

C₂²

#11763181

yes

characteristic

159

C₂²

#11763185

yes

characteristic

160

C₂²

#11763202

yes

—

161

C₄

—

162

C₄

—

272

—

163

C₄

C₂×#10587503

yes

—

164

C₄

#11762916

yes

—

165

C₄

#11763064

yes

—

166

C₄

#11763155

yes

—

167

C₂

#14528415

yes

—

168

C₂

#14528456

yes

—

169

C₂

#14529591

yes

characteristic

170

C₂

#14529656

yes

—

	subgroup	G/H
center	C₂³	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#9839
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#494

C₂

socle, prime radical

C₂³×C₁₇

#127

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
13. C₁₇×#494	C₄	semidirect
89. #494	Dic₁₇	semidirect
118. C₁₇	#9839	semidirect

H ≅ G/N

type

13. C₁₇×#494

C₄

semidirect

89. #494

Dic₁₇

semidirect

118. C₁₇

#9839

semidirect

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

subgroup	index
#14529746	2
#14528941	2
C₂×#11762530	2
C₂×C₁₇×#494	2
#9839	17

subgroup

index

C₂×C₁₇×#494

Subgroups — #16934174

Conjugacy classes of subgroups

Complements

Maximal subgroups