Group #16934263 — subgroups

Subgroups — #16934263

Order 4352.

422 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 159 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	#14528956	C₂	1	1	yes	maximal, characteristic
2	#14527239	C₂	1	4	yes	maximal
3	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	—	2	2	no	—
6	#11762833	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	Q₈⋊#3622	—	2	8	no	—
9	C₈×#3622	—	2	2	no	—
10	C₁₇×#533	C₂²	1	2	yes	—
11	C₁₇×#500	C₄	1	2	yes	—
12	C₁₇×#499	—	2	2	no	—
13	C₁₇×#498	C₄	1	2	yes	—
14	C₁₇×#427	—	2	2	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
18	C₁₇×#125	—	2	2	no	—
19	C₁₇×C₂²⋊Q₈	—	2	4	no	—
20	C₄×Q₈×C₁₇	—	2	4	no	—
21	C₁₇×#117	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×#63223	—	2	2	no	—
24	C₂²×#3622	—	2	2	no	—
25	C₁₇×#107	—	2	4	no	—
26	C₁₇×#106	—	2	4	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587490	—	4	4	no	—
31	C₄⋊#3622	—	2	4	no	—
32	C₄×#3622	—	2	2	no	—
33	C₄×#3622	D₄	1	4	yes	—
34	C₂×Q₈×C₁₇	—	2	4	no	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	#41	1	2	yes	—
37	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
40	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₁₇	—	2	4	no	—
42	C₄⋊C₄×C₁₇	—	2	4	no	—
43	C₄⋊C₄×C₁₇	—	4	4	no	—
44	C₂²⋊C₄×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	—	4	2	no	—
46	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₇	C₂×D₄	1	2	yes	—
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	#63223	—	4	4	no	—
50	C₂×#3622	—	2	8	no	—
51	C₂×#3622	—	2	4	no	—
52	C₂×#3622	—	4	2	no	—
53	#9855	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#124	1	1	yes	characteristic
55	Q₈×C₁₇	—	4	8	no	—
56	C₂×C₄×C₁₇	—	2	4	no	—
57	C₂×C₄×C₁₇	—	4	4	no	—
58	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₁₇	#121	1	2	yes	—
61	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
64	C₈×C₁₇	#103	1	4	yes	—
65	C₈×C₁₇	#103	1	4	yes	—
66	#3622	—	2	8	no	—
67	#3622	—	8	2	no	—
68	#3160	D₁₇	1	1	yes	characteristic
69	#1848	—	17	1	no	—
70	#1493	—	17	4	no	—
71	C₂×C₈²	—	17	1	no	—
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	#390	1	1	yes	characteristic
74	C₂²×C₁₇	#487	1	2	yes	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	—	8	4	no	—
77	C₄×C₁₇	C₂×#103	1	2	yes	—
78	C₄×C₁₇	#443	1	4	yes	—
79	C₄×C₁₇	#492	1	1	yes	characteristic
80	C₄×C₁₇	#494	1	1	yes	characteristic
81	#533	C₂×D₁₇	1	2	yes	—
82	#500	Dic₁₇	1	2	yes	—
83	#499	—	2	2	no	—
84	#498	Dic₁₇	1	2	yes	—
85	C₂×#108	—	34	2	no	—
86	#427	—	2	2	no	—
87	#423	—	17	2	no	—
88	C₂×C₄×C₈	—	17	2	no	—
89	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	Q₈⋊C₈	—	34	8	no	—
91	C₈²	—	34	2	no	—
92	C₂×C₁₇	—	2	2	no	—
93	C₂×C₁₇	#1956	1	1	yes	characteristic
94	C₂×C₁₇	#1974	1	2	yes	—
95	C₂×M₁₆	—	34	2	no	—
96	C₂²×C₈	—	34	2	no	—
97	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
98	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
99	#125	—	2	2	no	—
100	C₂²⋊Q₈	—	2	4	no	—
101	C₄×Q₈	—	2	4	no	—
102	#117	—	2	2	no	—
103	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
104	#108	—	68	4	no	—
105	#107	—	2	4	no	—
106	#106	—	2	4	no	—
107	C₄⋊C₈	—	34	4	no	—
108	C₄×C₈	—	17	4	no	—
109	C₄×C₈	—	34	2	no	—
110	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
113	C₁₇	#9855	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	#63254	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
117	M₁₆	—	68	4	no	—
118	C₂×C₈	—	34	8	no	—
119	C₂×C₈	—	34	4	no	—
120	C₂×C₈	—	68	2	no	—
121	C₂×C₈	#63230	1	4	yes	—
122	C₂×C₈	#63231	1	4	yes	—
123	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
124	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	D₄×D₁₇	1	2	yes	—
131	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
132	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
133	C₂³	#10587622	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
138	C₂×C₄	#10587624	1	2	yes	—
139	C₂×C₄	#10587626	1	1	yes	characteristic
140	C₂×C₄	C₂×#63230	1	2	yes	—
141	C₂×C₄	C₂×#63231	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
143	C₈	—	34	8	no	—
144	C₈	—	136	2	no	—
145	C₈	#10587506	1	4	yes	—
146	C₈	#10587506	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#11762712	1	1	yes	characteristic
149	C₂²	#11763159	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#10587506	1	2	yes	—
153	C₄	#11762926	1	4	yes	—
154	C₄	#11763183	1	1	yes	characteristic
155	C₄	#11763188	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#14529587	1	1	yes	characteristic
158	C₂	#14529673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal, characteristic

#14527239

C₂

yes

maximal

C₂×C₈×#3622

C₂

yes

maximal, characteristic

C₁₇×#3160

C₂

yes

maximal, characteristic

C₂×#10587490

—

#11762833

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

—

Q₈⋊#3622

—

C₈×#3622

—

C₁₇×#533

C₂²

yes

—

C₁₇×#500

C₄

yes

—

C₁₇×#499

—

C₁₇×#498

C₄

yes

—

C₁₇×#427

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂²×#3622

—

C₁₇×#107

—

C₁₇×#106

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×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

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

#103

yes

—

#3622

—

#3622

—

#3160

D₁₇

yes

characteristic

#1848

—

#1493

—

C₂×C₈²

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

C₄×C₁₇

#443

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂×D₁₇

yes

—

#500

Dic₁₇

yes

—

#499

—

#498

Dic₁₇

yes

—

C₂×#108

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1956

yes

characteristic

C₂×C₁₇

#1974

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₄⋊Q₈

C₂×Dic₁₇

yes

—

#125

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

C₂²×D₁₇

yes

characteristic

104

#108

—

105

#107

—

106

#106

—

107

C₄⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×Dic₁₇

yes

characteristic

111

C₄×C₈

C₂×Dic₁₇

yes

characteristic

112

C₄×C₈

C₂²⋊D₁₇

yes

—

113

C₁₇

#9855

yes

characteristic, complemented, Sylow

114

C₂×Q₈

—

115

C₂²×C₄

#63254

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

#63230

yes

—

122

C₂×C₈

#63231

yes

—

123

C₂×C₈

C₂²⋊Dic₁₇

yes

—

124

C₂×C₈

C₂²⋊Dic₁₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

D₄×D₁₇

yes

—

131

C₄²

C₂²×Dic₁₇

yes

characteristic

132

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

133

C₂³

#10587622

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

D₄×Dic₁₇

yes

—

138

C₂×C₄

#10587624

yes

—

139

C₂×C₄

#10587626

yes

characteristic

140

C₂×C₄

C₂×#63230

yes

—

141

C₂×C₄

C₂×#63231

yes

—

142

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

143

C₈

—

144

C₈

—

136

—

145

C₈

#10587506

yes

—

146

C₈

#10587506

yes

—

147

C₂²

—

148

C₂²

#11762712

yes

characteristic

149

C₂²

#11763159

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#10587506

yes

—

153

C₄

#11762926

yes

—

154

C₄

#11763183

yes

characteristic

155

C₄

#11763188

yes

characteristic

156

C₂

—

157

C₂

#14529587

yes

characteristic

158

C₂

#14529673

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#9855
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#3160

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
113. C₁₇	#9855	semidirect

H ≅ G/N

type

113. C₁₇

#9855

semidirect

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

subgroup	index
#14528956	2
#14527239	2
C₂×C₈×#3622	2
C₁₇×#3160	2
#9855	17

subgroup

index

#14528956

#14527239

C₂×C₈×#3622

C₁₇×#3160

#9855

Subgroups — #16934263

Conjugacy classes of subgroups

Complements

Maximal subgroups