Group #16950711 — subgroups

Subgroups — #16950711

Order 4352.

318 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 155 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×#11762585	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526875	C₂	1	4	yes	maximal
4	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587488	—	2	1	no	—
8	#11762675	—	2	1	no	—
9	#11762585	C₄	1	4	yes	complemented
10	C₂×C₁₇×#107	—	2	1	no	—
11	C₂×C₁₇×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#384	—	2	1	no	—
14	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₁₇×#107	—	2	4	no	—
21	C₁₇×#107	—	4	2	no	—
22	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₇	Q₈	1	2	yes	—
24	#10587502	—	4	2	no	—
25	#10587489	—	2	4	no	—
26	#10587488	—	4	2	no	—
27	C₂×#63214	—	2	2	no	—
28	C₂×#63214	—	2	2	no	—
29	C₂×#63214	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₇	—	4	2	no	—
31	C₂²×C₄×C₁₇	D₈	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₇	—	4	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	#41	1	2	yes	—
40	C₂×C₈×C₁₇	#41	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	4	2	no	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₄⋊C₄×C₁₇	—	8	4	no	—
44	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₇	Q₁₆	1	2	yes	—
46	C₄⋊Dic₁₇	—	4	2	no	—
47	C₄⋊Dic₁₇	—	4	2	no	—
48	#63214	—	2	4	no	—
49	#12890	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	8	2	no	—
54	C₂×C₄×C₁₇	—	8	4	no	—
55	C₂×C₄×C₁₇	#107	1	4	yes	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×Dic₁₇	—	8	2	no	—
63	C₂×Dic₁₇	—	8	2	no	—
64	C₂×Dic₁₇	—	8	2	no	—
65	C₈×C₁₇	#106	1	4	yes	—
66	C₈×C₁₇	#117	1	2	yes	—
67	C₈×C₁₇	#125	1	2	yes	—
68	C₂×#346	—	17	1	no	—
69	#1846	—	17	1	no	—
70	#1845	D₁₇	1	1	yes	characteristic
71	#1380	—	17	4	no	—
72	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
73	C₂²×C₁₇	#486	1	1	yes	characteristic
74	C₂²×C₁₇	#486	1	1	yes	characteristic
75	C₂²×C₁₇	#509	1	2	yes	—
76	C₂²×C₁₇	#510	1	2	yes	—
77	C₄×C₁₇	—	2	4	no	—
78	C₄×C₁₇	—	16	2	no	—
79	C₄×C₁₇	#382	1	1	yes	characteristic
80	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
81	C₄×C₁₇	#487	1	1	yes	characteristic
82	C₄×C₁₇	#487	1	1	yes	characteristic
83	Dic₁₇	—	16	2	no	—
84	C₂²×C₁₆	—	17	2	no	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	—	17	2	no	—
87	C₂×#107	C₂×D₁₇	1	2	yes	—
88	C₂×#106	—	34	1	no	—
89	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	#384	—	2	1	no	—
91	#382	—	34	1	no	—
92	#346	—	17	4	no	—
93	C₂×C₁₇	#1834	1	1	yes	characteristic
94	C₂×C₁₇	#2159	1	2	yes	—
95	C₂×C₁₇	#2243	1	2	yes	—
96	C₂×C₁₇	#2244	1	1	yes	characteristic
97	C₂×C₁₇	#2245	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄⋊C₄	—	34	2	no	—
103	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
104	C₂×C₁₆	—	17	4	no	—
105	C₂×C₁₆	—	34	2	no	—
106	C₂×C₁₆	—	34	2	no	—
107	#107	—	2	4	no	—
108	#107	—	4	2	no	—
109	#107	—	34	4	no	—
110	#106	—	68	2	no	—
111	C₄×C₈	Dic₃₄	1	2	yes	—
112	C₄×C₈	C₄×D₁₇	1	2	yes	—
113	#95	—	68	2	no	—
114	C₁₇	#12890	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	68	2	no	—
117	C₂²×C₄	#63228	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
119	C₂×C₈	#63225	1	2	yes	—
120	C₂×C₈	#63229	1	2	yes	—
121	C₂×C₈	#63230	1	2	yes	—
122	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
124	C₂×C₈	#63254	1	2	yes	—
125	C₂×C₈	#63256	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₄⋊C₄	—	68	2	no	—
130	C₄⋊C₄	—	68	2	no	—
131	C₄²	#63225	1	2	yes	—
132	C₄²	#63231	1	2	yes	—
133	C₁₆	—	34	4	no	—
134	C₂³	C₂×#63228	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	136	2	no	—
140	C₂×C₄	—	136	2	no	—
141	C₂×C₄	—	136	2	no	—
142	C₂×C₄	#10587478	1	4	yes	—
143	C₂×C₄	C₂×#63225	1	1	yes	characteristic
144	C₂×C₄	C₂×#63229	1	1	yes	characteristic
145	C₂×C₄	#10587622	1	1	yes	characteristic
146	C₂×C₄	C₂×#63230	1	1	yes	characteristic
147	C₂×C₄	C₂×#63231	1	1	yes	characteristic
148	C₂×C₄	#10587633	1	1	yes	characteristic
149	C₈	#10587479	1	4	yes	—
150	C₈	#10587576	1	2	yes	—
151	C₈	#10587578	1	2	yes	—
152	C₂²	C₂×#10587478	1	1	yes	characteristic
153	C₂²	#11763155	1	1	yes	characteristic
154	C₂²	#11763157	1	1	yes	characteristic
155	C₂²	#11763245	1	2	yes	—
156	C₂²	#11763250	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	272	2	no	—
160	C₄	#11762674	1	1	yes	characteristic
161	C₄	C₂×#10587479	1	1	yes	characteristic
162	C₄	#11763159	1	1	yes	characteristic
163	C₄	#11763161	1	1	yes	characteristic
164	C₂	#14528911	1	1	yes	characteristic
165	C₂	#14530570	1	2	yes	—
166	C₂	#14531023	1	2	yes	—
167	C₂	#14531025	1	1	yes	characteristic
168	C₂	#14531029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762585

C₂

yes

maximal, characteristic

#14528948

C₂

yes

maximal, characteristic

#14526875

C₂

yes

maximal

C₁₇×#1845

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

C₂×#10587488

—

#11762675

—

#11762585

C₄

yes

complemented

C₂×C₁₇×#107

—

C₂×C₁₇×#107

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#384

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

Q₁₆

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

#12890

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#117

yes

—

C₈×C₁₇

#125

yes

—

C₂×#346

—

#1846

—

#1845

D₁₇

yes

characteristic

#1380

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#510

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

C₂×#107

C₂×D₁₇

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

C₂×C₁₇

#1834

yes

characteristic

C₂×C₁₇

#2159

yes

—

C₂×C₁₇

#2243

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

#107

—

108

#107

—

109

#107

—

110

#106

—

111

C₄×C₈

Dic₃₄

yes

—

112

C₄×C₈

C₄×D₁₇

yes

—

113

#95

—

114

C₁₇

#12890

yes

characteristic, complemented, Sylow

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

#63228

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

119

C₂×C₈

#63225

yes

—

120

C₂×C₈

#63229

yes

—

121

C₂×C₈

#63230

yes

—

122

C₂×C₈

C₂×Dic₃₄

yes

characteristic

123

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

124

C₂×C₈

#63254

yes

—

125

C₂×C₈

#63256

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

#63225

yes

—

132

C₄²

#63231

yes

—

133

C₁₆

—

134

C₂³

C₂×#63228

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

136

—

140

C₂×C₄

—

136

—

141

C₂×C₄

—

136

—

142

C₂×C₄

#10587478

yes

—

143

C₂×C₄

C₂×#63225

yes

characteristic

144

C₂×C₄

C₂×#63229

yes

characteristic

145

C₂×C₄

#10587622

yes

characteristic

146

C₂×C₄

C₂×#63230

yes

characteristic

147

C₂×C₄

C₂×#63231

yes

characteristic

148

C₂×C₄

#10587633

yes

characteristic

149

C₈

#10587479

yes

—

150

C₈

#10587576

yes

—

151

C₈

#10587578

yes

—

152

C₂²

C₂×#10587478

yes

characteristic

153

C₂²

#11763155

yes

characteristic

154

C₂²

#11763157

yes

characteristic

155

C₂²

#11763245

yes

—

156

C₂²

#11763250

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

272

—

160

C₄

#11762674

yes

characteristic

161

C₄

C₂×#10587479

yes

characteristic

162

C₄

#11763159

yes

characteristic

163

C₄

#11763161

yes

characteristic

164

C₂

#14528911

yes

characteristic

165

C₂

#14530570

yes

—

166

C₂

#14531023

yes

—

167

C₂

#14531025

yes

characteristic

168

C₂

#14531029

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1845	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1845	D₁₇
hypocenter	C₁₇	#12890
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#63228

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1845

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₈

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
9. #11762585	C₄	semidirect
114. C₁₇	#12890	semidirect

H ≅ G/N

type

9. #11762585

C₄

semidirect

114. C₁₇

#12890

semidirect

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

subgroup	index
C₂×#11762585	2
#14528948	2
#14526875	2
C₁₇×#1845	2
#12890	17

subgroup

index

C₁₇×#1845

Subgroups — #16950711

Conjugacy classes of subgroups

Complements

Maximal subgroups