Group #16870270 — subgroups

Subgroups — C₁₇×#12526 #16870270

Order 4352.

296 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2248	C₂	1	2	yes	maximal
2	C₁₇×#2241	C₂	1	2	yes	maximal, complemented
3	C₁₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1706	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₇×SD₃₂	—	2	4	no	—
7	C₁₇×#500	—	2	1	no	—
8	C₁₇×#500	—	2	1	no	—
9	C₁₇×#500	C₂²	1	1	yes	characteristic
10	C₁₇×#494	C₂²	1	2	yes	—
11	C₁₇×#493	C₂²	1	1	yes	characteristic
12	C₁₇×#492	—	2	1	no	—
13	C₁₇×#486	—	2	1	no	—
14	C₁₇×#358	—	2	4	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₇	—	2	2	no	—
19	C₂×SD₁₆×C₁₇	—	4	2	no	—
20	C₂×D₈×C₁₇	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	1	no	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×#127	—	2	1	no	—
24	C₁₇×SD₃₂	—	4	8	no	—
25	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
26	C₁₇×#107	—	2	2	no	—
27	C₁₇×#107	—	4	2	no	—
28	C₁₇×#107	—	4	2	no	—
29	C₁₇×#102	—	4	2	no	—
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₇	—	4	2	no	—
34	C₂×Q₈×C₁₇	—	4	1	no	—
35	C₂×D₄×C₁₇	—	4	1	no	—
36	C₂×D₄×C₁₇	—	4	1	no	—
37	C₂×D₄×C₁₇	—	4	1	no	—
38	Q₁₆×C₁₇	—	4	4	no	—
39	Q₁₆×C₁₇	—	4	4	no	—
40	SD₁₆×C₁₇	—	8	4	no	—
41	D₈×C₁₇	—	4	4	no	—
42	C₂×C₈×C₁₇	D₈	1	2	yes	—
43	C₂×C₈×C₁₇	D₈	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₁₇	—	4	1	no	—
46	C₄⋊C₄×C₁₇	—	4	1	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	C₁₆×C₁₇	—	2	4	no	—
51	#12526	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₇	—	8	1	no	—
53	Q₈×C₁₇	—	8	2	no	—
54	Q₈×C₁₇	—	8	2	no	—
55	Q₈×C₁₇	—	8	2	no	—
56	D₄×C₁₇	—	8	2	no	—
57	D₄×C₁₇	—	8	2	no	—
58	D₄×C₁₇	—	8	2	no	—
59	C₂×C₄×C₁₇	—	8	2	no	—
60	C₂×C₄×C₁₇	—	8	1	no	—
61	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
64	C₈×C₁₇	—	2	2	no	—
65	C₈×C₁₇	SD₃₂	1	4	yes	complemented
66	C₈×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
67	#2248	C₂×C₁₇	1	2	yes	—
68	#2241	C₂×C₁₇	1	2	yes	complemented
69	#1713	C₂×C₁₇	1	1	yes	characteristic, complemented
70	#1706	C₂×C₁₇	1	1	yes	characteristic
71	#1312	C₂×C₁₇	1	1	yes	characteristic, complemented
72	C₂²×C₁₇	—	16	1	no	—
73	C₂²×C₁₇	—	16	1	no	—
74	C₂²×C₁₇	—	16	1	no	—
75	C₂²×C₁₇	#493	1	1	yes	characteristic
76	C₄×C₁₇	—	16	2	no	—
77	C₄×C₁₇	—	16	1	no	—
78	C₄×C₁₇	#496	1	2	yes	—
79	C₄×C₁₇	C₂×SD₃₂	1	2	yes	—
80	C₄×C₁₇	#510	1	2	yes	—
81	C₂×SD₃₂	—	2	4	no	—
82	#500	—	2	1	no	—
83	#500	—	2	1	no	—
84	#500	C₂²×C₁₇	1	1	yes	characteristic
85	#494	C₂²×C₁₇	1	2	yes	—
86	#493	C₂²×C₁₇	1	1	yes	characteristic
87	#492	—	2	1	no	—
88	#486	—	2	1	no	—
89	#358	—	2	4	no	—
90	C₄×C₁₆	C₂²×C₁₇	1	2	yes	—
91	C₈²	C₂²×C₁₇	1	1	yes	characteristic
92	C₂×C₁₇	—	32	1	no	—
93	C₂×C₁₇	#1721	1	1	yes	characteristic
94	C₂×C₁₇	#2248	1	1	yes	characteristic
95	C₂×C₁₇	#2251	1	1	yes	characteristic
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₂×SD₁₆	—	4	2	no	—
99	C₂×D₈	—	2	2	no	—
100	C₄⋊Q₈	—	2	1	no	—
101	C₄⋊Q₈	—	2	2	no	—
102	#127	—	2	1	no	—
103	SD₃₂	—	4	8	no	—
104	C₂×C₁₆	D₄×C₁₇	1	4	yes	—
105	#107	—	2	2	no	—
106	#107	—	4	2	no	—
107	#107	—	4	2	no	—
108	#102	—	4	2	no	—
109	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
110	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
111	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
112	C₁₇	#12526	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	1	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	4	no	—
120	SD₁₆	—	8	4	no	—
121	D₈	—	4	4	no	—
122	C₂×C₈	D₈×C₁₇	1	2	yes	—
123	C₂×C₈	D₈×C₁₇	1	2	yes	—
124	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
130	C₁₆	—	2	4	no	—
131	C₂³	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	C₁₇×SD₃₂	1	4	yes	complemented
145	C₈	C₁₇×AGL(1,ℤ/8)	1	4	yes	—
146	C₂²	—	16	1	no	—
147	C₂²	—	16	1	no	—
148	C₂²	—	16	1	no	—
149	C₂²	C₁₇×#493	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	16	1	no	—
152	C₄	C₁₇×#496	1	2	yes	—
153	C₄	C₂×C₁₇×SD₃₂	1	2	yes	—
154	C₄	C₁₇×#510	1	2	yes	—
155	C₂	—	32	1	no	—
156	C₂	C₁₇×#1721	1	1	yes	characteristic
157	C₂	C₁₇×#2248	1	1	yes	characteristic
158	C₂	C₁₇×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#2248

C₂

yes

maximal

C₁₇×#2241

C₂

yes

maximal, complemented

C₁₇×#1713

C₂

yes

maximal, characteristic, complemented

C₁₇×#1706

C₂

yes

maximal, characteristic

C₁₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×SD₃₂

—

C₁₇×#500

—

C₁₇×#500

—

C₁₇×#500

C₂²

yes

characteristic

C₁₇×#494

C₂²

yes

—

C₁₇×#493

C₂²

yes

characteristic

C₁₇×#492

—

C₁₇×#486

—

C₁₇×#358

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×SD₁₆×C₁₇

—

C₂×D₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×#127

—

C₁₇×SD₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#107

—

C₁₇×#107

—

C₁₇×#107

—

C₁₇×#102

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

SD₁₆×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₁₆×C₁₇

—

#12526

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

SD₃₂

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

#2248

C₂×C₁₇

yes

—

#2241

C₂×C₁₇

yes

complemented

#1713

C₂×C₁₇

yes

characteristic, complemented

#1706

C₂×C₁₇

yes

characteristic

#1312

C₂×C₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#496

yes

—

C₄×C₁₇

C₂×SD₃₂

yes

—

C₄×C₁₇

#510

yes

—

C₂×SD₃₂

—

#500

—

#500

—

#500

C₂²×C₁₇

yes

characteristic

#494

C₂²×C₁₇

yes

—

#493

C₂²×C₁₇

yes

—

—

—

C₄×C₁₆

C₂²×C₁₇

yes

—

C₈²

C₂²×C₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

#127

—

103

SD₃₂

—

104

C₂×C₁₆

D₄×C₁₇

yes

—

105

#107

—

106

#107

—

107

#107

—

108

#102

—

109

C₄×C₈

D₄×C₁₇

yes

characteristic

110

C₄×C₈

D₄×C₁₇

yes

characteristic

111

C₄×C₈

C₂³×C₁₇

yes

characteristic

112

C₁₇

#12526

yes

characteristic, direct product factor, Sylow

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

119

Q₁₆

—

120

SD₁₆

—

121

D₈

—

122

C₂×C₈

D₈×C₁₇

yes

—

123

C₂×C₈

D₈×C₁₇

yes

—

124

C₂×C₈

C₂×D₄×C₁₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

C₂×D₄×C₁₇

yes

characteristic

130

C₁₆

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₁₇×#127

yes

characteristic

141

C₂×C₄

C₂×D₈×C₁₇

yes

characteristic

142

C₂×C₄

C₂×D₈×C₁₇

yes

characteristic

143

C₈

—

144

C₈

C₁₇×SD₃₂

yes

complemented

145

C₈

C₁₇×AGL(1,ℤ/8)

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₁₇×#493

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

C₁₇×#496

yes

—

153

C₄

C₂×C₁₇×SD₃₂

yes

—

154

C₄

C₁₇×#510

yes

—

155

C₂

—

156

C₂

C₁₇×#1721

yes

characteristic

157

C₂

C₁₇×#2248

yes

characteristic

158

C₂

C₁₇×#2251

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₇
prime radical	C₁₇×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₄²×C₁₇ ◁ C₈²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₇

prime radical

C₁₇×#493

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₄²×C₁₇ ◁ C₈²×C₁₇ ◁ G

N	H ≅ G/N	type
2. C₁₇×#2241	C₂	semidirect
3. C₁₇×#1713	C₂	semidirect
5. C₁₇×#1312	C₂	semidirect
51. #12526	C₁₇	direct (N × H)
65. C₈×C₁₇	SD₃₂	semidirect
68. #2241	C₂×C₁₇	semidirect
69. #1713	C₂×C₁₇	semidirect
71. #1312	C₂×C₁₇	semidirect
112. C₁₇	#12526	direct (N × H)
144. C₈	C₁₇×SD₃₂	semidirect

H ≅ G/N

type

2. C₁₇×#2241

C₂

semidirect

3. C₁₇×#1713

C₂

semidirect

5. C₁₇×#1312

C₂

semidirect

51. #12526

C₁₇

direct (N × H)

65. C₈×C₁₇

SD₃₂

semidirect

68. #2241

C₂×C₁₇

semidirect

69. #1713

C₂×C₁₇

semidirect

71. #1312

C₂×C₁₇

semidirect

112. C₁₇

#12526

direct (N × H)

144. C₈

C₁₇×SD₃₂

semidirect

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

subgroup	index
C₁₇×#2248	2
C₁₇×#2241	2
C₁₇×#1713	2
C₁₇×#1706	2
C₁₇×#1312	2
#12526	17

subgroup

index

C₁₇×#2248

C₁₇×#2241

C₁₇×#1713

C₁₇×#1706

C₁₇×#1312

#12526

Subgroups — C₁₇×#12526 #16870270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#12526 #16870270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#12526 #16870270