Group #16928113 — subgroups

Subgroups — #16928113

Order 4352.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#14528944	C₂	1	2	yes	maximal
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14528941	C₂	1	1	yes	maximal, characteristic
4	#14528912	C₂	1	2	yes	maximal
5	C₂×C₁₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	4	no	—
7	C₂×#10587489	—	2	2	no	—
8	C₂×#10587488	—	2	2	no	—
9	#11762679	C₂²	1	1	yes	characteristic
10	#11762679	C₂²	1	2	yes	—
11	#11762679	C₂²	1	1	yes	characteristic
12	#11762572	—	2	4	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#322	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₇	D₄	1	4	yes	—
17	C₂²×C₈×C₁₇	D₄	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×C₄⋊Dic₁₇	—	2	1	no	—
22	C₂×C₄⋊Dic₁₇	—	2	1	no	—
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	Q₈	1	4	yes	—
29	C₄×C₈×C₁₇	Q₈	1	2	yes	—
30	#10587489	—	4	8	no	—
31	#10587489	—	4	4	no	—
32	#10587488	—	4	4	no	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
35	C₂²×Dic₁₇	—	4	2	no	—
36	C₂²×Dic₁₇	—	4	1	no	—
37	C₂²×Dic₁₇	—	4	1	no	—
38	C₂×C₈×C₁₇	—	2	4	no	—
39	C₂×C₈×C₁₇	—	2	4	no	—
40	C₂×C₈×C₁₇	C₄⋊C₄	1	8	yes	—
41	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₈×C₁₇	D₈	1	4	yes	—
43	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
44	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
47	C₄⋊Dic₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	4	no	—
50	C₄⋊Dic₁₇	—	4	4	no	—
51	C₄⋊Dic₁₇	—	4	4	no	—
52	C₄⋊Dic₁₇	—	4	4	no	—
53	C₄⋊Dic₁₇	—	4	4	no	—
54	#8820	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
63	C₂×C₄×C₁₇	#137	1	4	yes	—
64	C₂×Dic₁₇	—	8	1	no	—
65	C₂×Dic₁₇	—	8	2	no	—
66	C₂×Dic₁₇	—	8	1	no	—
67	C₂×Dic₁₇	—	8	2	no	—
68	C₂×Dic₁₇	—	8	2	no	—
69	C₂×Dic₁₇	—	8	2	no	—
70	C₂×Dic₁₇	—	8	2	no	—
71	C₈×C₁₇	—	2	8	no	—
72	C₈×C₁₇	#107	1	8	yes	complemented
73	#1845	—	17	1	no	—
74	#1845	—	17	2	no	—
75	#1844	—	17	1	no	—
76	#1834	—	17	2	no	—
77	C₂×#322	D₁₇	1	1	yes	characteristic
78	C₂²×C₁₇	#384	1	1	yes	characteristic
79	C₂²×C₁₇	#493	1	1	yes	characteristic
80	C₂²×C₁₇	#494	1	1	yes	characteristic
81	C₂²×C₁₇	#496	1	2	yes	—
82	C₂²×C₁₇	#497	1	2	yes	—
83	C₄×C₁₇	C₂×#107	1	2	yes	—
84	C₄×C₁₇	#428	1	4	yes	—
85	C₄×C₁₇	#500	1	2	yes	—
86	C₄×C₁₇	#501	1	4	yes	—
87	Dic₁₇	—	16	1	no	—
88	Dic₁₇	—	16	2	no	—
89	Dic₁₇	—	16	1	no	—
90	C₂×#107	—	34	4	no	—
91	C₂×#107	—	34	2	no	—
92	C₂×#106	—	34	2	no	—
93	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	#384	—	17	1	no	—
96	#384	—	17	2	no	—
97	#384	—	17	1	no	—
98	#340	—	34	4	no	—
99	#322	Dic₁₇	1	4	yes	complemented
100	C₂×C₁₇	#1721	1	1	yes	characteristic
101	C₂×C₁₇	#1722	1	1	yes	characteristic
102	C₂×C₁₇	#1723	1	2	yes	—
103	C₂×C₁₇	#1845	1	1	yes	characteristic
104	C₂×C₁₇	#1847	1	2	yes	—
105	C₂²×C₈	D₆₈	1	4	yes	—
106	C₂²×C₈	D₆₈	1	2	yes	—
107	C₂×C₄⋊C₄	—	34	2	no	—
108	C₂×C₄⋊C₄	—	34	2	no	—
109	C₂×C₄⋊C₄	—	34	1	no	—
110	C₂×C₄⋊C₄	—	34	1	no	—
111	C₂×C₄⋊C₄	—	34	2	no	—
112	C₂×C₄⋊C₄	—	34	2	no	—
113	C₂×C₄⋊C₄	—	34	2	no	—
114	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
115	#107	—	68	8	no	—
116	#107	—	68	4	no	—
117	#106	—	68	4	no	—
118	C₄×C₈	Dic₃₄	1	4	yes	—
119	C₄×C₈	Dic₃₄	1	2	yes	—
120	C₄×C₈	C₂×Dic₁₇	1	4	yes	—
121	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
122	C₁₇	#8820	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	—	68	1	no	—
124	C₂²×C₄	—	68	2	no	—
125	C₂²×C₄	—	68	1	no	—
126	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₆₈	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	D₁₃₆	1	4	yes	—
131	C₂×C₈	Dic₆₈	1	4	yes	—
132	C₂×C₈	C₄⋊Dic₁₇	1	8	yes	—
133	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
134	C₄⋊C₄	—	68	2	no	—
135	C₄⋊C₄	—	68	2	no	—
136	C₄⋊C₄	—	68	4	no	—
137	C₄⋊C₄	—	68	4	no	—
138	C₄⋊C₄	—	68	4	no	—
139	C₄⋊C₄	—	68	4	no	—
140	C₄⋊C₄	—	68	4	no	—
141	C₄²	C₂×Dic₃₄	1	2	yes	—
142	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
143	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
144	C₂³	#10587560	1	1	yes	characteristic
145	C₂×C₄	—	136	2	no	—
146	C₂×C₄	—	136	1	no	—
147	C₂×C₄	—	136	2	no	—
148	C₂×C₄	—	136	1	no	—
149	C₂×C₄	—	136	2	no	—
150	C₂×C₄	—	136	2	no	—
151	C₂×C₄	—	136	2	no	—
152	C₂×C₄	#10587555	1	1	yes	characteristic
153	C₂×C₄	#10587555	1	2	yes	—
154	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
155	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
156	C₂×C₄	#10587594	1	4	yes	—
157	C₂×C₄	#10587595	1	4	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
160	C₈	—	2	8	no	—
161	C₈	#10587489	1	8	yes	complemented
162	C₂²	#11762679	1	1	yes	characteristic
163	C₂²	#11763186	1	1	yes	characteristic
164	C₂²	#11763189	1	1	yes	characteristic
165	C₂²	#11763198	1	2	yes	—
166	C₂²	#11763203	1	2	yes	—
167	C₄	—	272	1	no	—
168	C₄	—	272	2	no	—
169	C₄	—	272	1	no	—
170	C₄	C₂×#10587489	1	2	yes	—
171	C₄	#11762851	1	4	yes	—
172	C₄	#11763214	1	2	yes	—
173	C₄	#11763218	1	4	yes	—
174	C₂	#14528450	1	1	yes	characteristic
175	C₂	#14528455	1	1	yes	characteristic
176	C₂	#14528461	1	2	yes	—
177	C₂	#14528944	1	1	yes	characteristic
178	C₂	#14528952	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528944

C₂

yes

maximal

#14528944

C₂

yes

maximal, characteristic

#14528941

C₂

yes

maximal, characteristic

#14528912

C₂

yes

maximal

C₂×C₁₇×#322

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#11762679

C₂²

yes

—

#11762679

C₂²

yes

characteristic

#11762572

—

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#322

C₄

yes

complemented

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587489

—

#10587489

—

#10587488

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

—

—

—

—

D₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#497

yes

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

#428

yes

—

C₄×C₁₇

#500

yes

—

C₄×C₁₇

#501

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

—

Dic₁₇

yes

complemented

100

C₂×C₁₇

#1721

yes

characteristic

101

C₂×C₁₇

#1722

yes

characteristic

102

C₂×C₁₇

#1723

yes

—

103

C₂×C₁₇

#1845

yes

characteristic

104

C₂×C₁₇

#1847

yes

—

105

C₂²×C₈

D₆₈

yes

—

106

C₂²×C₈

D₆₈

yes

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄²

C₂²×D₁₇

yes

characteristic

115

#107

—

116

#107

—

117

#106

—

118

C₄×C₈

Dic₃₄

yes

—

119

C₄×C₈

Dic₃₄

yes

—

120

C₄×C₈

C₂×Dic₁₇

yes

—

121

C₄×C₈

C₂×Dic₁₇

yes

—

122

C₁₇

#8820

yes

characteristic, complemented, Sylow

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×D₆₈

yes

characteristic

127

C₂²×C₄

C₂×D₆₈

yes

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

D₁₃₆

yes

—

131

C₂×C₈

Dic₆₈

yes

—

132

C₂×C₈

C₄⋊Dic₁₇

yes

—

133

C₂×C₈

C₄⋊Dic₁₇

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄²

C₂×Dic₃₄

yes

—

142

C₄²

C₂×Dic₃₄

yes

characteristic

143

C₄²

C₂²×Dic₁₇

yes

characteristic

144

C₂³

#10587560

yes

characteristic

145

C₂×C₄

—

136

—

146

C₂×C₄

—

136

—

147

C₂×C₄

—

136

—

148

C₂×C₄

—

136

—

149

C₂×C₄

—

136

—

150

C₂×C₄

—

136

—

151

C₂×C₄

—

136

—

152

C₂×C₄

#10587555

yes

characteristic

153

C₂×C₄

#10587555

yes

—

154

C₂×C₄

C₂×D₁₃₆

yes

—

155

C₂×C₄

C₂×Dic₆₈

yes

—

156

C₂×C₄

#10587594

yes

—

157

C₂×C₄

#10587595

yes

—

158

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

159

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

—

160

C₈

—

161

C₈

#10587489

yes

complemented

162

C₂²

#11762679

yes

characteristic

163

C₂²

#11763186

yes

characteristic

164

C₂²

#11763189

yes

characteristic

165

C₂²

#11763198

yes

—

166

C₂²

#11763203

yes

—

167

C₄

—

272

—

168

C₄

—

272

—

169

C₄

—

272

—

170

C₄

C₂×#10587489

yes

—

171

C₄

#11762851

yes

—

172

C₄

#11763214

yes

—

173

C₄

#11763218

yes

—

174

C₂

#14528450

yes

characteristic

175

C₂

#14528455

yes

characteristic

176

C₂

#14528461

yes

—

177

C₂

#14528944

yes

characteristic

178

C₂

#14528952

yes

—

	subgroup	G/H
center	C₂³	#10587560
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#322	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#322	D₁₇
hypocenter	C₁₇	#8820
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#10587560

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#322

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×#322

N	H ≅ G/N	type
15. C₁₇×#322	C₄	semidirect
72. C₈×C₁₇	#107	semidirect
99. #322	Dic₁₇	semidirect
122. C₁₇	#8820	semidirect
161. C₈	#10587489	semidirect

H ≅ G/N

type

15. C₁₇×#322

C₄

semidirect

72. C₈×C₁₇

#107

semidirect

99. #322

Dic₁₇

semidirect

122. C₁₇

#8820

semidirect

161. C₈

#10587489

semidirect

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

subgroup	index
#14528944	2
#14528941	2
#14528912	2
C₂×C₁₇×#322	2
#8820	17

subgroup

index

C₂×C₁₇×#322

Subgroups — #16928113

Conjugacy classes of subgroups

Complements

Maximal subgroups