Group #16868379 — subgroups

Subgroups — C₁₇×#10635 #16868379

Order 4352.

294 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₁₇×#1960	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1838	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1546	C₂	1	2	yes	maximal
4	C₁₇×#1286	C₂	1	2	yes	maximal
5	C₁₇×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₁₇×#431	C₂²	1	2	yes	—
8	C₁₇×#423	C₂²	1	2	yes	—
9	C₂×C₁₇×#103	—	2	2	no	—
10	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
11	C₁₇×#382	C₂²	1	1	yes	characteristic
12	C₁₇×#341	—	2	2	no	—
13	C₁₇×#332	—	2	4	no	—
14	C₂²×Q₈×C₁₇	—	2	1	no	—
15	C₂×M₁₆×C₁₇	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₁₇×C₄⋊Q₈	—	2	2	no	—
18	C₁₇×C₄⋊Q₈	—	2	1	no	—
19	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	—	2	2	no	—
24	C₁₇×C₄⋊C₈	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×#103	—	4	4	no	—
27	C₁₇×C₂²⋊C₈	—	2	2	no	—
28	C₁₇×#97	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₂×Q₈×C₁₇	—	2	2	no	—
32	C₂×Q₈×C₁₇	—	4	1	no	—
33	C₂×Q₈×C₁₇	—	4	2	no	—
34	C₂×Q₈×C₁₇	—	4	2	no	—
35	C₂²×C₄×C₁₇	—	4	1	no	—
36	C₂²×C₄×C₁₇	—	4	1	no	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	#41	1	2	yes	—
39	M₁₆×C₁₇	—	4	2	no	—
40	M₁₆×C₁₇	—	4	2	no	—
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	—	4	2	no	—
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊C₄×C₁₇	—	4	2	no	—
49	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	1	yes	characteristic
51	C₄²×C₁₇	#41	1	2	yes	—
52	#10635	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₁₇	#124	1	1	yes	characteristic
54	Q₈×C₁₇	—	4	4	no	—
55	Q₈×C₁₇	—	8	2	no	—
56	C₂×C₄×C₁₇	—	4	2	no	—
57	C₂×C₄×C₁₇	—	4	2	no	—
58	C₂×C₄×C₁₇	—	8	1	no	—
59	C₂×C₄×C₁₇	—	8	1	no	—
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	—	8	1	no	—
62	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#123	1	2	yes	—
64	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#126	1	2	yes	—
66	C₂×C₄×C₁₇	#135	1	2	yes	—
67	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₈×C₁₇	—	4	2	no	—
69	C₈×C₁₇	—	8	2	no	—
70	C₈×C₁₇	—	8	2	no	—
71	#1960	C₂×C₁₇	1	1	yes	characteristic
72	#1838	C₂×C₁₇	1	1	yes	characteristic
73	#1546	C₂×C₁₇	1	2	yes	—
74	#1286	C₂×C₁₇	1	2	yes	—
75	#1281	C₂×C₁₇	1	1	yes	characteristic
76	C₂²×C₁₇	—	2	2	no	—
77	C₂²×C₁₇	#397	1	1	yes	characteristic
78	C₂²×C₁₇	#488	1	1	yes	characteristic
79	C₂²×C₁₇	#490	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	—	8	2	no	—
83	C₄×C₁₇	—	16	1	no	—
84	C₄×C₁₇	#463	1	2	yes	—
85	C₄×C₁₇	#482	1	2	yes	—
86	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
87	#431	C₂²×C₁₇	1	2	yes	—
88	#423	C₂²×C₁₇	1	2	yes	—
89	C₂×#103	—	2	2	no	—
90	C₂×#97	C₂²×C₁₇	1	1	yes	characteristic
91	#382	C₂²×C₁₇	1	1	yes	characteristic
92	#341	—	2	2	no	—
93	#332	—	2	4	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	2	1	no	—
96	C₂×C₁₇	#2073	1	1	yes	characteristic
97	C₂×C₁₇	#2082	1	2	yes	—
98	C₂²×Q₈	—	2	1	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂²×C₈	D₄×C₁₇	1	2	yes	—
101	C₄⋊Q₈	—	2	2	no	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	D₄×C₁₇	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	#103	—	4	4	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	#95	—	2	2	no	—
115	C₁₇	#10635	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
123	C₂²×C₄	#41×C₁₇	1	2	yes	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
135	C₄²	#41×C₁₇	1	2	yes	—
136	C₄²	#41×C₁₇	1	1	yes	characteristic
137	C₂³	C₁₇×#124	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
147	C₂×C₄	C₁₇×#123	1	2	yes	—
148	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
149	C₂×C₄	C₁₇×#126	1	2	yes	—
150	C₂×C₄	C₁₇×#135	1	2	yes	—
151	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	C₁₇×#397	1	1	yes	characteristic
157	C₂²	C₁₇×#488	1	1	yes	characteristic
158	C₂²	C₁₇×#490	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	C₁₇×#463	1	2	yes	—
164	C₄	C₁₇×#482	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	C₁₇×#2073	1	1	yes	characteristic
168	C₂	C₁₇×#2082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#1960

C₂

yes

maximal, characteristic

C₁₇×#1838

C₂

yes

maximal, characteristic

C₁₇×#1546

C₂

yes

maximal

C₁₇×#1286

C₂

yes

maximal

C₁₇×#1281

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₁₇×#431

C₂²

yes

—

C₁₇×#423

C₂²

yes

—

C₂×C₁₇×#103

—

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

characteristic

C₁₇×#341

—

C₁₇×#332

—

C₂²×Q₈×C₁₇

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#103

—

C₁₇×C₂²⋊C₈

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

C₁₇×#95

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

#10635

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

#124

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#123

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#1960

C₂×C₁₇

yes

characteristic

#1838

C₂×C₁₇

yes

characteristic

#1546

C₂×C₁₇

yes

—

#1286

C₂×C₁₇

yes

—

#1281

C₂×C₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#463

yes

—

C₄×C₁₇

#482

yes

—

C₂×C₄⋊Q₈

C₂²×C₁₇

yes

characteristic

#431

C₂²×C₁₇

yes

—

#423

C₂²×C₁₇

yes

—

C₂×#103

—

C₂×#97

C₂²×C₁₇

yes

characteristic

#382

C₂²×C₁₇

yes

characteristic

#341

—

#332

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#2073

yes

characteristic

C₂×C₁₇

#2082

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

100

C₂²×C₈

D₄×C₁₇

yes

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

D₄×C₁₇

yes

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂³×C₁₇

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

C₂²⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

#95

—

115

C₁₇

#10635

yes

characteristic, direct product factor, Sylow

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×D₄×C₁₇

yes

characteristic

123

C₂²×C₄

#41×C₁₇

yes

—

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₄×C₁₇

yes

characteristic

135

C₄²

#41×C₁₇

yes

—

136

C₄²

#41×C₁₇

yes

characteristic

137

C₂³

C₁₇×#124

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₁₇×#121

yes

characteristic

147

C₂×C₄

C₁₇×#123

yes

—

148

C₂×C₄

C₁₇×#123

yes

characteristic

149

C₂×C₄

C₁₇×#126

yes

—

150

C₂×C₄

C₁₇×#135

yes

—

151

C₂×C₄

C₁₇×AGL(1,ℤ/8)

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

C₁₇×#397

yes

characteristic

157

C₂²

C₁₇×#488

yes

characteristic

158

C₂²

C₁₇×#490

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₁₇×#463

yes

—

164

C₄

C₁₇×#482

yes

—

165

C₂

—

166

C₂

—

167

C₂

C₁₇×#2073

yes

characteristic

168

C₂

C₁₇×#2082

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂³×C₁₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

prime radical

C₂³×C₁₇

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
52. #10635	C₁₇	direct (N × H)
115. C₁₇	#10635	direct (N × H)

H ≅ G/N

type

52. #10635

C₁₇

direct (N × H)

115. C₁₇

#10635

direct (N × H)

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

subgroup	index
C₁₇×#1960	2
C₁₇×#1838	2
C₁₇×#1546	2
C₁₇×#1286	2
C₁₇×#1281	2
#10635	17

subgroup

index

C₁₇×#1960

C₁₇×#1838

C₁₇×#1546

C₁₇×#1286

C₁₇×#1281

#10635

Subgroups — C₁₇×#10635 #16868379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#10635 #16868379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10635 #16868379