Group #17953779 — subgroups

Subgroups — #17953779

Order 4352.

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

normal subgroups only

This group has 294 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	#14544724	C₂	1	1	yes	maximal, characteristic
2	#14544721	C₂	1	1	yes	maximal, characteristic
3	#14543638	C₂	1	4	yes	maximal
4	#14527066	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587536	—	2	1	no	—
6	C₂×#10587535	—	2	1	no	—
7	C₂×#10587534	—	2	1	no	—
8	C₂×#10587533	—	2	1	no	—
9	#11763814	C₂²	1	2	yes	—
10	#11763753	C₂²	1	2	yes	—
11	#11762860	C₂²	1	2	yes	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762531	C₄	1	2	yes	complemented
14	C₈×#3622	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₁₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄⋊#595	—	2	2	no	—
17	C₂×#63243	—	2	2	no	—
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
20	C₄²×D₁₇	C₂³	1	1	yes	characteristic
21	(C₂×#595)⋊C₄	—	2	4	no	—
22	#10587544	—	2	4	no	—
23	#10587536	—	4	2	no	—
24	#10587535	—	4	2	no	—
25	#10587534	—	4	2	no	—
26	#10587533	—	4	2	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	#10587491	—	2	2	no	—
30	#10587485	—	2	2	no	—
31	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
32	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
33	C₄×#3622	C₂×C₄	1	2	yes	—
34	C₄×#3622	C₂×C₄	1	1	yes	characteristic
35	C₂²×#595	—	4	2	no	—
36	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×D₁₇	#41	1	2	yes	—
38	C₂×#3624	—	4	2	no	—
39	C₄⋊#595	—	4	2	no	—
40	C₄⋊#595	—	4	2	no	—
41	#63243	—	4	2	no	—
42	#63243	—	4	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₄×Dic₁₇	#41	1	2	yes	—
49	C₂×#3622	—	2	2	no	—
50	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
52	C₈×D₁₇	D₈	1	2	yes	—
53	C₈×D₁₇	SD₁₆	1	2	yes	—
54	C₈×D₁₇	SD₁₆	1	2	yes	—
55	C₈×D₁₇	Q₁₆	1	2	yes	—
56	#8793	—	17	1	no	maximal, Sylow
57	C₂²×D₁₇	#124	1	1	yes	characteristic
58	C₂×#595	—	8	4	no	—
59	C₂×#595	—	8	2	no	—
60	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#117	1	2	yes	—
62	C₂×Dic₁₇	#124	1	1	yes	characteristic
63	C₂×Dic₁₇	#125	1	2	yes	—
64	C₄×D₁₇	—	4	2	no	—
65	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
66	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
69	#3624	—	8	2	no	—
70	C₈×C₁₇	—	4	2	no	—
71	C₈×C₁₇	#106	1	2	yes	—
72	C₈×C₁₇	#107	1	2	yes	—
73	#3622	—	4	2	no	—
74	#3622	#106	1	2	yes	—
75	#3622	#107	1	2	yes	—
76	#1848	—	17	1	no	—
77	#1846	—	17	1	no	—
78	#1450	—	17	1	no	—
79	#1286	—	17	4	no	—
80	C₂²×C₁₇	#382	1	1	yes	characteristic
81	C₂×D₁₇	—	2	2	no	—
82	C₂×D₁₇	#486	1	1	yes	characteristic
83	C₂×D₁₇	#487	1	1	yes	characteristic
84	#595	—	16	2	no	—
85	C₄×C₁₇	—	2	2	no	—
86	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
87	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
88	Dic₁₇	—	2	2	no	—
89	Dic₁₇	#486	1	1	yes	characteristic
90	Dic₁₇	#487	1	1	yes	characteristic
91	#431	—	17	2	no	—
92	C₂×#108	—	34	1	no	—
93	C₂×#108	—	34	1	no	—
94	C₂×#107	—	34	1	no	—
95	C₂×#106	—	34	1	no	—
96	#423	—	17	2	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	#382	—	17	2	no	—
99	#322	—	17	2	no	—
100	C₈²	—	17	1	no	—
101	C₈²	#595	1	1	yes	characteristic, complemented
102	C₂×C₁₇	#1833	1	2	yes	—
103	C₂×C₁₇	#1834	1	1	yes	characteristic
104	D₁₇	—	2	1	no	—
105	D₁₇	—	2	1	no	—
106	C₂×M₁₆	—	34	2	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂×C₄⋊C₄	—	34	2	no	—
110	C₂×C₄²	—	17	1	no	—
111	#108	—	68	2	no	—
112	#108	—	68	2	no	—
113	#107	—	68	2	no	—
114	#106	—	68	2	no	—
115	C₄⋊C₈	—	34	2	no	—
116	C₄⋊C₈	—	34	4	no	—
117	C₂²⋊C₈	—	34	2	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	17	2	no	—
122	C₄×C₈	C₂×#595	1	2	yes	—
123	C₄×C₈	C₂×#595	1	1	yes	characteristic
124	#95	—	34	4	no	—
125	C₁₇	#8793	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₄	—	17	1	no	—
127	C₂²×C₄	—	17	2	no	—
128	C₂²×C₄	—	68	2	no	—
129	M₁₆	—	68	2	no	—
130	M₁₆	—	68	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	17	2	no	—
135	C₂×C₈	—	17	2	no	—
136	C₂×C₈	—	17	2	no	—
137	C₂×C₈	—	17	2	no	—
138	C₂×C₈	—	34	2	no	—
139	C₂×C₈	—	68	2	no	—
140	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
141	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
142	C₄⋊C₄	—	68	2	no	—
143	C₄⋊C₄	—	68	2	no	—
144	C₄²	—	17	1	no	—
145	C₄²	—	17	2	no	—
146	C₄²	C₂²×#595	1	1	yes	characteristic
147	C₂³	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	1	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	2	no	—
154	C₂×C₄	—	68	2	no	—
155	C₂×C₄	—	136	2	no	—
156	C₂×C₄	—	136	4	no	—
157	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
158	C₂×C₄	#10587677	1	2	yes	—
159	C₈	—	4	2	no	—
160	C₈	—	17	2	no	—
161	C₈	—	17	2	no	—
162	C₈	—	68	2	no	—
163	C₈	—	136	2	no	—
164	C₈	#10587533	1	2	yes	—
165	C₈	#10587534	1	2	yes	—
166	C₂²	—	17	1	no	—
167	C₂²	—	17	1	no	—
168	C₂²	—	34	2	no	—
169	C₂²	#11763753	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	272	2	no	—
172	C₄	—	17	1	no	—
173	C₄	—	17	1	no	—
174	C₄	—	34	2	no	—
175	C₄	C₂×#10587533	1	1	yes	characteristic
176	C₄	C₂×#10587534	1	1	yes	characteristic
177	C₂	—	34	1	no	—
178	C₂	—	34	1	no	—
179	C₂	#14544703	1	2	yes	—
180	C₂	#14544705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544724

C₂

yes

maximal, characteristic

#14544721

C₂

yes

maximal, characteristic

#14543638

C₂

yes

maximal

#14527066

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#11763753

C₂²

yes

—

#11762860

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

C₄

yes

complemented

C₈×#3622

C₄

yes

characteristic, complemented

C₈²×C₁₇

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂³

yes

characteristic

(C₂×#595)⋊C₄

—

—

—

—

—

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587485

—

C₈×Dic₁₇

Q₈

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

#41

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×Dic₁₇

#41

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

D₈

yes

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

Q₁₆

yes

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×Dic₁₇

#124

yes

characteristic

C₂×Dic₁₇

#125

yes

—

C₄×D₁₇

—

C₄×D₁₇

C₂×D₈

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×Q₁₆

yes

characteristic

#3624

—

C₈×C₁₇

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#107

yes

—

#3622

—

#3622

#106

yes

—

#3622

#107

yes

—

—

—

—

—

C₂²×C₁₇

#382

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

—

—

—

—

—

C₂×C₄×C₈

—

#382

—

#322

—

100

C₈²

—

101

C₈²

#595

yes

characteristic, complemented

102

C₂×C₁₇

#1833

yes

—

103

C₂×C₁₇

#1834

yes

characteristic

104

D₁₇

—

105

D₁₇

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

#108

—

112

#108

—

113

#107

—

114

#106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₂²⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#595

yes

—

123

C₄×C₈

C₂×#595

yes

characteristic

124

#95

—

125

C₁₇

#8793

yes

characteristic, complemented, Sylow

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₄⋊#595

yes

characteristic

141

C₂×C₈

C₄⋊#595

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×#595

yes

characteristic

147

C₂³

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

136

—

156

C₂×C₄

—

136

—

157

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

158

C₂×C₄

#10587677

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

136

—

164

C₈

#10587533

yes

—

165

C₈

#10587534

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#11763753

yes

characteristic

170

C₄

—

171

C₄

—

272

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#10587533

yes

characteristic

176

C₄

C₂×#10587534

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#14544703

yes

—

180

C₂

#14544705

yes

characteristic

	subgroup	G/H
center	C₂²	#11763753
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#382
prime radical	C₂²×D₁₇	#124
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8793
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11763753

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄²×D₁₇

C₂³

C₈²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
13. #11762531	C₄	semidirect
14. C₈×#3622	C₄	semidirect
15. C₈²×C₁₇	C₄	semidirect
101. C₈²	#595	semidirect
125. C₁₇	#8793	semidirect

H ≅ G/N

type

13. #11762531

C₄

semidirect

14. C₈×#3622

C₄

semidirect

15. C₈²×C₁₇

C₄

semidirect

101. C₈²

#595

semidirect

125. C₁₇

#8793

semidirect

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

subgroup	index
#14544724	2
#14544721	2
#14543638	2
#14527066	2
#8793	17

subgroup

index

Subgroups — #17953779

Conjugacy classes of subgroups

Complements

Maximal subgroups