Group #17948604 — subgroups

Subgroups — #17948604

Order 4352.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×#10587535	C₂	1	1	yes	maximal, characteristic
2	#14544326	C₂	1	1	yes	maximal, characteristic
3	C₄×#10587469	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587535	C₄	1	2	yes	—
5	#11763810	—	2	1	no	—
6	C₄×#63243	—	2	1	no	—
7	#11763685	—	2	1	no	—
8	#11763671	—	2	1	no	—
9	#11763670	C₄	1	2	yes	—
10	C₂×#10587469	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762584	—	2	1	no	—
13	C₄×#63214	C₄	1	1	yes	characteristic
14	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂×#63243	—	2	2	no	—
16	C₂×#63242	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	#10587544	—	4	1	no	—
21	#10587542	—	4	1	no	—
22	C₄⋊#3624	—	2	2	no	—
23	C₄×#3624	—	4	1	no	—
24	#10587535	—	2	2	no	—
25	#10587535	C₈	1	4	yes	complemented
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	#10587469	—	2	4	no	—
33	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
36	C₂×#3624	—	4	2	no	—
37	C₂×#3624	—	4	2	no	—
38	#63243	—	2	4	no	—
39	#63243	—	4	2	no	—
40	#63242	—	8	1	no	—
41	#63242	—	8	1	no	—
42	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×Dic₁₇	D₈	1	1	yes	characteristic
47	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
48	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₈×D₁₇	—	2	2	no	—
51	C₈×D₁₇	C₂×C₈	1	2	yes	—
52	C₈×D₁₇	M₁₆	1	2	yes	—
53	C₁₆×C₁₇	—	2	2	no	—
54	#63214	—	2	2	no	—
55	#7173	—	17	1	no	maximal, Sylow
56	C₂²×D₁₇	#103	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
60	C₂×Dic₁₇	#102	1	1	yes	characteristic
61	C₂×Dic₁₇	#102	1	1	yes	characteristic
62	C₂×Dic₁₇	#103	1	1	yes	characteristic
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	2	1	no	—
65	C₄×D₁₇	—	4	1	no	—
66	C₄×D₁₇	C₂²⋊C₈	1	2	yes	—
67	C₄×D₁₇	#104	1	2	yes	—
68	#3624	—	4	4	no	—
69	#3624	—	8	2	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
72	C₈×C₁₇	#97	1	1	yes	characteristic
73	#3622	—	2	1	no	—
74	#3622	C₄⋊C₈	1	2	yes	—
75	C₄×M₃₂	—	17	1	no	—
76	C₄×#108	—	17	1	no	—
77	C₂×#334	—	17	1	no	—
78	C₂²×C₁₇	#340	1	1	yes	characteristic
79	C₂×D₁₇	—	2	1	no	—
80	C₂×D₁₇	—	2	1	no	—
81	C₂×D₁₇	Q₈⋊C₈	1	2	yes	—
82	C₄×C₁₇	—	2	1	no	—
83	C₄×C₁₇	#336	1	1	yes	characteristic
84	C₄×C₁₇	#339	1	1	yes	characteristic
85	C₄×C₁₇	#365	1	1	yes	characteristic
86	C₄×C₁₇	#365	1	1	yes	characteristic
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	—	2	1	no	—
89	Dic₁₇	D₄⋊C₈	1	2	yes	—
90	C₂×M₃₂	—	34	1	no	—
91	C₂×#108	—	17	2	no	—
92	C₂×C₄⋊C₈	—	34	1	no	—
93	C₄×M₁₆	—	34	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	#346	—	34	1	no	—
96	C₄×C₁₆	—	17	1	no	—
97	C₄×C₁₆	#595	1	1	yes	characteristic
98	#341	—	34	1	no	—
99	#334	—	17	2	no	—
100	#334	—	34	1	no	—
101	C₂×C₁₇	#1279	1	1	yes	characteristic
102	C₂×C₁₇	#1383	1	1	yes	characteristic
103	C₂×C₁₇	#1385	1	1	yes	characteristic
104	D₁₇	—	2	2	no	—
105	C₂×M₁₆	—	34	2	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	68	1	no	—
109	C₂×C₄²	—	17	1	no	—
110	M₃₂	—	34	4	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	34	1	no	—
113	#108	—	17	4	no	—
114	#108	—	34	2	no	—
115	C₄⋊C₈	—	34	2	no	—
116	C₄⋊C₈	—	68	1	no	—
117	#97	—	68	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	68	1	no	—
122	C₄×C₈	C₂×#595	1	1	yes	characteristic
123	C₁₇	#7173	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	—	17	1	no	—
127	M₁₆	—	34	4	no	—
128	M₁₆	—	68	2	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	2	no	—
131	C₂×C₈	—	17	2	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	34	2	no	—
134	C₂×C₈	—	68	2	no	—
135	C₂×C₈	—	68	2	no	—
136	C₂×C₈	—	136	1	no	—
137	C₂×C₈	—	136	1	no	—
138	C₂×C₈	C₄×#595	1	1	yes	characteristic
139	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₄²	—	17	1	no	—
143	C₄²	C₄⋊#595	1	1	yes	characteristic
144	C₁₆	—	2	2	no	—
145	C₁₆	—	34	2	no	—
146	C₂³	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	2	no	—
149	C₂×C₄	—	17	2	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	34	1	no	—
153	C₂×C₄	—	34	1	no	—
154	C₂×C₄	—	68	1	no	—
155	C₂×C₄	#10587533	1	1	yes	characteristic
156	C₂×C₄	#10587534	1	1	yes	characteristic
157	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	17	2	no	—
160	C₈	—	34	1	no	—
161	C₈	—	68	4	no	—
162	C₈	—	136	2	no	—
163	C₈	C₈×#595	1	1	yes	characteristic
164	C₈	#10587532	1	1	yes	characteristic
165	C₂²	—	17	2	no	—
166	C₂²	—	34	1	no	—
167	C₂²	—	34	1	no	—
168	C₂²	#11763683	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	17	2	no	—
171	C₄	—	34	1	no	—
172	C₄	—	34	1	no	—
173	C₄	#11763674	1	1	yes	characteristic
174	C₄	#11763680	1	1	yes	characteristic
175	C₄	#11763733	1	1	yes	characteristic
176	C₄	#11763734	1	1	yes	characteristic
177	C₂	—	34	2	no	—
178	C₂	#14543619	1	1	yes	characteristic
179	C₂	#14543849	1	1	yes	characteristic
180	C₂	#14543854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587535

C₂

yes

maximal, characteristic

#14544326

C₂

yes

maximal, characteristic

C₄×#10587469

C₂

yes

maximal, characteristic

C₂×#10587535

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×#10587469

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762584

—

C₄×#63214

C₄

yes

characteristic

C₄×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×#63242

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

C₈

yes

complemented

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₄×#3622

Q₈

yes

characteristic

#10587469

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

C₈×D₁₇

M₁₆

yes

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂²⋊C₈

yes

—

C₄×D₁₇

#104

yes

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

—

C₄⋊C₈

yes

—

C₄×M₃₂

—

C₄×#108

—

C₂×#334

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

Q₈⋊C₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

D₄⋊C₈

yes

—

C₂×M₃₂

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

100

#334

—

101

C₂×C₁₇

#1279

yes

characteristic

102

C₂×C₁₇

#1383

yes

characteristic

103

C₂×C₁₇

#1385

yes

characteristic

104

D₁₇

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄²

—

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

#108

—

114

#108

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#595

yes

characteristic

123

C₁₇

#7173

yes

characteristic, complemented, Sylow

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

136

—

137

C₂×C₈

—

136

—

138

C₂×C₈

C₄×#595

yes

characteristic

139

C₂×C₈

C₂²⋊#595

yes

characteristic

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₄⋊#595

yes

characteristic

144

C₁₆

—

145

C₁₆

—

146

C₂³

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

#10587533

yes

characteristic

156

C₂×C₄

#10587534

yes

characteristic

157

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

136

—

163

C₈

C₈×#595

yes

characteristic

164

C₈

#10587532

yes

characteristic

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#11763683

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#11763674

yes

characteristic

174

C₄

#11763680

yes

characteristic

175

C₄

#11763733

yes

characteristic

176

C₄

#11763734

yes

characteristic

177

C₂

—

178

C₂

#14543619

yes

characteristic

179

C₂

#14543849

yes

characteristic

180

C₂

#14543854

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587533
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#103
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#7173
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₄

#10587533

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₄×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₄×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
25. #10587535	C₈	semidirect
123. C₁₇	#7173	semidirect

H ≅ G/N

type

25. #10587535

C₈

semidirect

123. C₁₇

#7173

semidirect

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

subgroup	index
C₄×#10587535	2
#14544326	2
C₄×#10587469	2
#7173	17

subgroup

index

Subgroups — #17948604

Conjugacy classes of subgroups

Complements

Maximal subgroups