Group #16868374 — subgroups

Subgroups — C₁₇×#10630 #16868374

Order 4352.

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

normal subgroups only

This group has 272 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1832	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1542	C₂	1	2	yes	maximal
4	C₁₇×#1301	C₂	1	2	yes	maximal
5	C₁₇×#1274	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#533	C₂²	1	1	yes	characteristic
7	C₁₇×#431	C₂²	1	2	yes	—
8	C₁₇×#423	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₁₇×#376	C₂²	1	1	yes	characteristic
11	C₁₇×#339	—	2	2	no	—
12	C₁₇×#336	—	2	2	no	—
13	C₁₇×#333	—	2	2	no	—
14	C₁₇×#332	—	2	2	no	—
15	C₂×M₁₆×C₁₇	—	2	1	no	—
16	C₂²×C₈×C₁₇	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
18	C₁₇×#125	—	2	1	no	—
19	C₁₇×C₂²⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₁₇	—	2	2	no	—
21	C₁₇×#117	—	2	1	no	—
22	C₂×C₄²×C₁₇	D₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₁₇×C₄⋊C₈	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₁₇×C₄⋊C₈	—	2	2	no	—
28	C₁₇×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₂×Q₈×C₁₇	—	2	2	no	—
32	C₂²×C₄×C₁₇	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	#41	1	2	yes	—
35	M₁₆×C₁₇	—	4	2	no	—
36	C₂×C₈×C₁₇	—	4	2	no	—
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	—	4	1	no	—
39	C₂×C₈×C₁₇	—	4	2	no	—
40	C₂×C₈×C₁₇	—	4	2	no	—
41	C₄⋊C₄×C₁₇	—	2	2	no	—
42	C₄⋊C₄×C₁₇	—	2	2	no	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₂²⋊C₄×C₁₇	—	4	1	no	—
45	C₄²×C₁₇	—	2	2	no	—
46	C₄²×C₁₇	—	2	2	no	—
47	C₄²×C₁₇	—	4	1	no	—
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	2	yes	—
51	#10630	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₇	#123	1	1	yes	characteristic
53	Q₈×C₁₇	—	8	2	no	—
54	C₂×C₄×C₁₇	—	4	2	no	—
55	C₂×C₄×C₁₇	—	4	4	no	—
56	C₂×C₄×C₁₇	—	4	2	no	—
57	C₂×C₄×C₁₇	—	4	2	no	—
58	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#123	1	2	yes	—
60	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#126	1	2	yes	—
62	C₂×C₄×C₁₇	#135	1	2	yes	—
63	C₂×C₄×C₁₇	#137	1	2	yes	—
64	C₈×C₁₇	—	8	2	no	—
65	C₈×C₁₇	—	8	2	no	—
66	C₈×C₁₇	—	8	1	no	—
67	#1995	C₂×C₁₇	1	1	yes	characteristic
68	#1832	C₂×C₁₇	1	1	yes	characteristic
69	#1542	C₂×C₁₇	1	2	yes	—
70	#1301	C₂×C₁₇	1	2	yes	—
71	#1274	C₂×C₁₇	1	1	yes	characteristic
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	#397	1	1	yes	characteristic
74	C₂²×C₁₇	#485	1	2	yes	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	—	8	2	no	—
78	C₄×C₁₇	—	8	2	no	—
79	C₄×C₁₇	#464	1	2	yes	—
80	C₄×C₁₇	#488	1	1	yes	characteristic
81	C₄×C₁₇	#491	1	1	yes	characteristic
82	#533	C₂²×C₁₇	1	1	yes	characteristic
83	#431	C₂²×C₁₇	1	2	yes	—
84	#423	C₂²×C₁₇	1	1	yes	characteristic
85	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
86	#376	C₂²×C₁₇	1	1	yes	characteristic
87	#339	—	2	2	no	—
88	#336	—	2	2	no	—
89	#333	—	2	2	no	—
90	#332	—	2	2	no	—
91	C₂×C₁₇	—	2	2	no	—
92	C₂×C₁₇	#2074	1	1	yes	characteristic
93	C₂×C₁₇	#2080	1	2	yes	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₄⋊Q₈	D₄×C₁₇	1	2	yes	—
97	#125	—	2	1	no	—
98	C₂²⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	2	2	no	—
100	#117	—	2	1	no	—
101	C₂×C₄²	D₄×C₁₇	1	2	yes	—
102	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	#95	—	2	2	no	—
110	C₁₇	#10630	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×Q₈	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
114	C₂²×C₄	#41×C₁₇	1	2	yes	—
115	M₁₆	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	4	1	no	—
128	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
129	C₄²	#41×C₁₇	1	2	yes	—
130	C₄²	#41×C₁₇	1	1	yes	characteristic
131	C₂³	C₁₇×#123	1	1	yes	characteristic
132	Q₈	—	8	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
138	C₂×C₄	C₁₇×#123	1	2	yes	—
139	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
140	C₂×C₄	C₁₇×#126	1	2	yes	—
141	C₂×C₄	C₁₇×#135	1	2	yes	—
142	C₂×C₄	C₁₇×#137	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₁₇×#397	1	1	yes	characteristic
148	C₂²	C₁₇×#485	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	C₁₇×#464	1	2	yes	—
154	C₄	C₁₇×#488	1	1	yes	characteristic
155	C₄	C₁₇×#491	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₁₇×#2074	1	1	yes	characteristic
158	C₂	C₁₇×#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#1995

C₂

yes

maximal, characteristic

C₁₇×#1832

C₂

yes

maximal, characteristic

C₁₇×#1542

C₂

yes

maximal

C₁₇×#1301

C₂

yes

maximal

C₁₇×#1274

C₂

yes

maximal, characteristic

C₁₇×#533

C₂²

yes

characteristic

C₁₇×#431

C₂²

yes

—

C₁₇×#423

C₂²

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#376

C₂²

yes

characteristic

C₁₇×#339

—

C₁₇×#336

—

C₁₇×#333

—

C₁₇×#332

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

—

C₁₇×C₄⋊Q₈

D₄

yes

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

C₁₇×#95

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

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

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

#10630

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

#123

yes

characteristic

Q₈×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

#124

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#1995

C₂×C₁₇

yes

characteristic

#1832

C₂×C₁₇

yes

characteristic

#1542

C₂×C₁₇

yes

—

#1301

C₂×C₁₇

yes

—

#1274

C₂×C₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

#397

yes

characteristic

C₂²×C₁₇

#485

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#464

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂²×C₁₇

yes

characteristic

#431

C₂²×C₁₇

yes

—

#423

C₂²×C₁₇

yes

characteristic

C₂×C₄⋊C₈

C₂²×C₁₇

yes

—

#376

C₂²×C₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#2074

yes

characteristic

C₂×C₁₇

#2080

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₄×C₁₇

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

D₄×C₁₇

yes

—

102

C₂×C₄²

C₂³×C₁₇

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

#95

—

110

C₁₇

#10630

yes

characteristic, direct product factor, Sylow

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₁₇

yes

characteristic

114

C₂²×C₄

#41×C₁₇

yes

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×D₄×C₁₇

yes

characteristic

129

C₄²

#41×C₁₇

yes

—

130

C₄²

#41×C₁₇

yes

characteristic

131

C₂³

C₁₇×#123

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₁₇×#121

yes

characteristic

138

C₂×C₄

C₁₇×#123

yes

—

139

C₂×C₄

C₁₇×#124

yes

characteristic

140

C₂×C₄

C₁₇×#126

yes

—

141

C₂×C₄

C₁₇×#135

yes

—

142

C₂×C₄

C₁₇×#137

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

C₁₇×#397

yes

characteristic

148

C₂²

C₁₇×#485

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₁₇×#464

yes

—

154

C₄

C₁₇×#488

yes

characteristic

155

C₄

C₁₇×#491

yes

characteristic

156

C₂

—

157

C₂

C₁₇×#2074

yes

characteristic

158

C₂

C₁₇×#2080

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₇	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#397
prime radical	C₂³×C₁₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₇

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

C₂²×C₁₇

C₂³×C₁₇

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
51. #10630	C₁₇	direct (N × H)
110. C₁₇	#10630	direct (N × H)

H ≅ G/N

type

51. #10630

C₁₇

direct (N × H)

110. C₁₇

#10630

direct (N × H)

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

subgroup	index
C₁₇×#1995	2
C₁₇×#1832	2
C₁₇×#1542	2
C₁₇×#1301	2
C₁₇×#1274	2
#10630	17

subgroup

index

C₁₇×#1995

C₁₇×#1832

C₁₇×#1542

C₁₇×#1301

C₁₇×#1274

#10630

Subgroups — C₁₇×#10630 #16868374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#10630 #16868374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10630 #16868374