Group #17961639 — subgroups

Subgroups — #17961639

Order 4352.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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	#14544725	C₂	1	1	yes	maximal, characteristic
2	#14544724	C₂	1	1	yes	maximal, characteristic
3	#14543844	C₂	1	4	yes	maximal
4	D₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587536	—	2	1	no	—
6	C₂×#10587536	C₂²	1	2	yes	—
7	C₂×#10587536	C₂²	1	2	yes	—
8	C₂×#10587535	—	2	1	no	—
9	#11763814	—	2	1	no	—
10	#11763813	—	2	1	no	—
11	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762585	C₄	1	1	yes	characteristic
14	#11762584	C₄	1	2	yes	—
15	C₁₇×#346	C₄	1	1	yes	characteristic
16	C₂×#63243	—	2	2	no	—
17	C₂×#63243	—	2	2	no	—
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
20	C₄²×D₁₇	D₄	1	1	yes	characteristic
21	#10587544	—	4	2	no	—
22	C₄⋊#3624	—	4	2	no	—
23	#10587536	—	2	4	no	—
24	#10587536	—	2	4	no	—
25	#10587536	—	4	2	no	—
26	#10587535	—	4	2	no	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	C₂×#63214	C₂×C₄	1	2	yes	—
32	C₄×#3622	Q₈	1	1	yes	characteristic
33	C₁₆×D₁₇	—	2	2	no	—
34	C₁₆×D₁₇	—	2	2	no	—
35	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
36	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×#3624	—	4	2	no	—
38	C₂×#3624	—	4	2	no	—
39	#63243	—	4	2	no	—
40	#63243	—	4	2	no	—
41	#63243	—	4	2	no	—
42	#63243	—	4	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₁₇	D₈	1	2	yes	—
48	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
50	C₈×D₁₇	SD₁₆	1	2	yes	—
51	C₈×D₁₇	SD₁₆	1	2	yes	—
52	C₈×D₁₇	#41	1	2	yes	—
53	C₈×D₁₇	#41	1	2	yes	—
54	C₁₆×C₁₇	—	2	2	no	—
55	#63214	—	2	2	no	—
56	#12892	—	17	1	no	maximal, Sylow
57	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#107	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×Dic₁₇	#107	1	2	yes	—
61	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
62	C₄×D₁₇	—	4	2	no	—
63	C₄×D₁₇	#124	1	1	yes	characteristic
64	C₄×D₁₇	#124	1	1	yes	characteristic
65	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
67	#3624	—	8	2	no	—
68	#3624	—	8	2	no	—
69	C₈×C₁₇	#106	1	2	yes	—
70	C₈×C₁₇	#117	1	2	yes	—
71	#3622	#106	1	2	yes	—
72	#3622	#125	1	2	yes	—
73	C₂×#346	—	17	1	no	—
74	#1848	—	17	1	no	—
75	#1848	—	17	1	no	—
76	#1381	—	17	4	no	—
77	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
78	C₂×D₁₇	—	2	2	no	—
79	C₂×D₁₇	#487	1	1	yes	characteristic
80	C₂×D₁₇	#487	1	1	yes	characteristic
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	#382	1	1	yes	characteristic
83	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
84	Dic₁₇	—	2	2	no	—
85	Dic₁₇	#486	1	1	yes	characteristic
86	Dic₁₇	#486	1	1	yes	characteristic
87	C₂²×C₁₆	—	17	2	no	—
88	C₂×#108	—	17	2	no	—
89	C₂×#108	—	17	2	no	—
90	C₂×#108	—	34	1	no	—
91	C₂×#108	—	34	1	no	—
92	#423	—	34	1	no	—
93	#423	—	34	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	#346	—	17	2	no	—
96	#346	—	17	1	no	—
97	#346	#595	1	1	yes	characteristic
98	C₂×C₁₇	#1834	1	1	yes	characteristic
99	C₂×C₁₇	#2161	1	2	yes	—
100	D₁₇	—	2	1	no	—
101	D₁₇	—	2	1	no	—
102	C₂×M₁₆	—	34	2	no	—
103	C₂×M₁₆	—	34	2	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂×C₄²	—	17	1	no	—
107	C₂×C₁₆	—	17	2	no	—
108	C₂×C₁₆	—	34	2	no	—
109	C₂×C₁₆	—	34	2	no	—
110	C₂×C₁₆	C₂×#595	1	2	yes	—
111	#108	—	34	4	no	—
112	#108	—	34	4	no	—
113	#108	—	68	2	no	—
114	#108	—	68	2	no	—
115	C₄⋊C₈	—	68	2	no	—
116	C₄⋊C₈	—	68	2	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	C₂×#595	1	1	yes	characteristic
121	C₁₇	#12892	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	—	17	1	no	—
123	C₂²×C₄	—	17	2	no	—
124	M₁₆	—	68	2	no	—
125	M₁₆	—	68	2	no	—
126	M₁₆	—	68	2	no	—
127	M₁₆	—	68	2	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	2	no	—
131	C₂×C₈	—	17	2	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	68	2	no	—
135	C₂×C₈	—	68	2	no	—
136	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
137	C₂×C₈	C₂²×#595	1	1	yes	characteristic
138	C₄²	—	17	1	no	—
139	C₄²	—	17	2	no	—
140	C₄²	C₄⋊#595	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	34	2	no	—
143	C₂³	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	2	no	—
150	C₂×C₄	—	68	2	no	—
151	C₂×C₄	#10587534	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
153	C₈	—	17	2	no	—
154	C₈	—	17	2	no	—
155	C₈	—	136	2	no	—
156	C₈	—	136	2	no	—
157	C₈	#10587533	1	2	yes	—
158	C₈	#10587677	1	2	yes	—
159	C₂²	—	17	1	no	—
160	C₂²	—	17	1	no	—
161	C₂²	—	34	2	no	—
162	C₂²	C₂×#10587534	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	17	1	no	—
165	C₄	—	17	1	no	—
166	C₄	—	34	2	no	—
167	C₄	#11763753	1	1	yes	characteristic
168	C₄	C₂×#10587533	1	1	yes	characteristic
169	C₂	—	34	1	no	—
170	C₂	—	34	1	no	—
171	C₂	#14544705	1	1	yes	characteristic
172	C₂	#14545140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544725

C₂

yes

maximal, characteristic

#14544724

C₂

yes

maximal, characteristic

#14543844

C₂

yes

maximal

D₁₇×#346

C₂

yes

maximal, characteristic

C₂×#10587536

—

C₂×#10587536

C₂²

yes

—

C₂×#10587536

C₂²

yes

—

C₂×#10587535

—

#11763814

—

#11763813

—

C₂×C₁₆×D₁₇

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic

#11762584

C₄

yes

—

C₁₇×#346

C₄

yes

characteristic

C₂×#63243

—

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

C₄×#3622

Q₈

yes

characteristic

C₁₆×D₁₇

—

C₁₆×D₁₇

—

C₂×C₄×D₁₇

Q₁₆

yes

—

C₂×C₄×D₁₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

#41

yes

—

C₈×D₁₇

#41

yes

—

C₁₆×C₁₇

—

#63214

—

#12892

—

maximal, Sylow

C₂²×D₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#107

yes

—

C₂×Dic₁₇

C₂×D₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

#3624

—

#3624

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#117

yes

—

#3622

#106

yes

—

#3622

#125

yes

—

—

—

—

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

C₂²×C₁₆

—

—

—

—

—

—

—

C₂×C₄×C₈

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

100

D₁₇

—

101

D₁₇

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₂×#595

yes

—

111

#108

—

112

#108

—

113

#108

—

114

#108

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂×#595

yes

characteristic

121

C₁₇

#12892

yes

characteristic, complemented, Sylow

122

C₂²×C₄

—

123

C₂²×C₄

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₄⋊#595

yes

characteristic

137

C₂×C₈

C₂²×#595

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₄²

C₄⋊#595

yes

characteristic

141

C₁₆

—

142

C₁₆

—

143

C₂³

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#10587534

yes

—

152

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

136

—

156

C₈

—

136

—

157

C₈

#10587533

yes

—

158

C₈

#10587677

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

C₂×#10587534

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#11763753

yes

characteristic

168

C₄

C₂×#10587533

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

#14544705

yes

characteristic

172

C₂

#14545140

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587534
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂²×D₁₇	C₂×Q₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12892
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#10587534

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂²×D₁₇

C₂×Q₁₆

cosocle

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
121. C₁₇	#12892	semidirect

H ≅ G/N

type

121. C₁₇

#12892

semidirect

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

subgroup	index
#14544725	2
#14544724	2
#14543844	2
D₁₇×#346	2
#12892	17

subgroup

index

D₁₇×#346

Subgroups — #17961639

Conjugacy classes of subgroups

Complements

Maximal subgroups