Group #17948315 — subgroups

Subgroups — #17948315

Order 4352.

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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	#14544570	C₂	1	2	yes	maximal
2	#14528900	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#3624	—	2	2	no	—
4	C₂×#10587542	—	2	2	no	—
5	#11763675	C₄	1	4	yes	—
6	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
7	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
8	#11762563	—	2	2	no	—
9	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂²×#3624	—	2	4	no	—
11	C₂²×C₈×C₁₇	—	2	1	no	—
12	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
16	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
17	C₂²×#3622	—	2	1	no	—
18	#10587542	—	4	2	no	—
19	#10587542	—	4	2	no	—
20	C₄⋊#3624	—	2	4	no	—
21	C₄⋊#3624	—	2	4	no	—
22	C₁₇×C₄⋊C₈	—	4	1	no	—
23	C₄⋊#3622	—	4	1	no	—
24	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₁₇	C₂²⋊C₄	1	2	yes	—
28	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
29	C₂×#3624	—	2	8	no	—
30	C₂×#3624	—	4	4	no	—
31	C₂×#3624	—	4	4	no	—
32	C₂×C₈×C₁₇	—	4	1	no	—
33	C₂×C₈×C₁₇	—	4	2	no	—
34	C₄²×C₁₇	—	2	1	no	—
35	C₄²×C₁₇	—	2	1	no	—
36	C₄×Dic₁₇	—	2	2	no	—
37	C₄×Dic₁₇	—	2	2	no	—
38	C₄×Dic₁₇	—	4	2	no	—
39	C₄×Dic₁₇	—	4	1	no	—
40	C₄×Dic₁₇	D₈	1	1	yes	characteristic
41	C₄×Dic₁₇	SD₁₆	1	2	yes	—
42	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
43	C₂×#3622	—	4	1	no	—
44	C₂×#3622	—	4	2	no	—
45	#7049	—	17	1	no	maximal, Sylow
46	C₂³×C₁₇	#95	1	1	yes	characteristic
47	C₂×C₄×C₁₇	—	2	1	no	—
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	#100	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#101	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#104	1	2	yes	—
54	C₂×C₄×C₁₇	#104	1	2	yes	—
55	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	2	no	—
58	C₂×Dic₁₇	—	2	2	no	—
59	C₂×Dic₁₇	—	2	2	no	—
60	C₂×Dic₁₇	—	2	2	no	—
61	C₂×Dic₁₇	—	2	2	no	—
62	C₂×Dic₁₇	#102	1	2	yes	—
63	C₂×Dic₁₇	#103	1	2	yes	—
64	#3624	—	4	8	no	—
65	C₈×C₁₇	—	8	1	no	—
66	#3622	—	8	1	no	—
67	#1831	—	17	1	no	—
68	#1767	—	17	2	no	—
69	C₂²×C₁₇	#338	1	1	yes	characteristic
70	C₂²×C₁₇	#339	1	2	yes	—
71	C₂²×C₁₇	#340	1	1	yes	characteristic
72	C₂²×C₁₇	#343	1	1	yes	characteristic
73	C₂²×C₁₇	#344	1	2	yes	—
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	—	4	1	no	—
77	C₄×C₁₇	#329	1	1	yes	characteristic
78	C₄×C₁₇	#329	1	1	yes	characteristic
79	C₄×C₁₇	#330	1	2	yes	—
80	Dic₁₇	—	2	2	no	—
81	Dic₁₇	—	2	2	no	—
82	Dic₁₇	—	4	2	no	—
83	C₂×C₄⋊C₈	—	17	1	no	—
84	C₂×C₄⋊C₈	—	34	2	no	—
85	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
86	C₂×#97	—	34	2	no	—
87	C₄³	—	17	1	no	—
88	#336	—	17	4	no	—
89	#336	—	34	2	no	—
90	C₂×C₁₇	#1291	1	2	yes	—
91	C₂×C₁₇	#1292	1	1	yes	characteristic
92	C₂×C₁₇	#1295	1	1	yes	characteristic
93	C₂×C₁₇	#1303	1	2	yes	—
94	C₂×C₁₇	#1307	1	1	yes	characteristic
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	34	4	no	—
97	C₂²×C₈	—	34	1	no	—
98	C₂×C₄²	—	17	1	no	—
99	C₂×C₄²	—	17	2	no	—
100	C₂×C₄²	—	17	2	no	—
101	C₂×C₄²	—	17	1	no	—
102	C₂×C₄²	C₂×#595	1	1	yes	characteristic
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	34	4	no	—
105	C₄⋊C₈	—	34	4	no	—
106	C₄⋊C₈	—	68	1	no	—
107	#97	—	68	2	no	—
108	#97	—	68	2	no	—
109	C₁₇	#7049	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₄	—	17	2	no	—
111	C₂²×C₄	—	17	2	no	—
112	C₂²×C₄	C₄×#595	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	34	8	no	—
118	C₂×C₈	—	68	4	no	—
119	C₂×C₈	—	68	4	no	—
120	C₂×C₈	—	68	2	no	—
121	C₂×C₈	—	68	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	17	2	no	—
125	C₄²	—	17	1	no	—
126	C₄²	—	17	1	no	—
127	C₄²	—	34	2	no	—
128	C₄²	—	34	2	no	—
129	C₄²	—	68	1	no	—
130	C₄²	—	68	2	no	—
131	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	17	2	no	—
137	C₂×C₄	—	17	2	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	34	2	no	—
141	C₂×C₄	—	34	2	no	—
142	C₂×C₄	—	34	2	no	—
143	C₂×C₄	#10587533	1	1	yes	characteristic
144	C₂×C₄	#10587534	1	1	yes	characteristic
145	C₂×C₄	#10587545	1	1	yes	characteristic
146	C₂×C₄	#10587548	1	2	yes	—
147	C₂×C₄	#10587550	1	2	yes	—
148	C₂×C₄	#10587553	1	1	yes	characteristic
149	C₈	—	8	1	no	—
150	C₈	—	68	8	no	—
151	C₈	—	136	1	no	—
152	C₂²	#11763678	1	1	yes	characteristic
153	C₂²	#11763679	1	2	yes	—
154	C₂²	#11763683	1	1	yes	characteristic
155	C₂²	#11763689	1	1	yes	characteristic
156	C₂²	#11763692	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	34	2	no	—
161	C₄	—	34	2	no	—
162	C₄	—	68	2	no	—
163	C₄	#11763664	1	1	yes	characteristic
164	C₄	#11763665	1	1	yes	characteristic
165	C₄	#11763666	1	2	yes	—
166	C₂	#14543650	1	2	yes	—
167	C₂	#14543654	1	1	yes	characteristic
168	C₂	#14543660	1	1	yes	characteristic
169	C₂	#14543681	1	2	yes	—
170	C₂	#14543690	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544570

C₂

yes

maximal

#14528900

C₂

yes

maximal, characteristic

C₂×C₄⋊#3624

—

C₂×#10587542

—

#11763675

C₄

yes

—

C₂×C₄⋊#3622

C₄

yes

characteristic

C₄²×Dic₁₇

C₂²

yes

characteristic

#11762563

—

C₂×C₁₇×C₄⋊C₈

C₄

yes

characteristic

C₂²×#3624

—

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂×C₄

yes

—

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

—

C₂×C₄×Dic₁₇

Q₈

yes

characteristic

C₂²×#3622

—

—

—

—

—

C₁₇×C₄⋊C₈

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₄²

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

C₂²⋊C₄

yes

—

C₂²×Dic₁₇

C₄⋊C₄

yes

—

C₂×#3624

—

C₂×#3624

—

C₂×#3624

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

—

C₄×Dic₁₇

Q₁₆

yes

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

#102

yes

—

C₂×Dic₁₇

#103

yes

—

#3624

—

C₈×C₁₇

—

#3622

—

#1831

—

#1767

—

C₂²×C₁₇

#338

yes

characteristic

C₂²×C₁₇

#339

yes

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#595

yes

characteristic

C₂×#97

—

C₄³

—

#336

—

#336

—

C₂×C₁₇

#1291

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#1307

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×#595

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#97

—

108

#97

—

109

C₁₇

#7049

yes

characteristic, complemented, Sylow

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×#595

yes

characteristic

113

C₂²×C₄

C₄⋊#595

yes

characteristic

114

C₂²×C₄

C₂²⋊#595

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₂³

(C₂×#595)⋊C₄

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#10587533

yes

characteristic

144

C₂×C₄

#10587534

yes

characteristic

145

C₂×C₄

#10587545

yes

characteristic

146

C₂×C₄

#10587548

yes

—

147

C₂×C₄

#10587550

yes

—

148

C₂×C₄

#10587553

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

136

—

152

C₂²

#11763678

yes

characteristic

153

C₂²

#11763679

yes

—

154

C₂²

#11763683

yes

characteristic

155

C₂²

#11763689

yes

characteristic

156

C₂²

#11763692

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11763664

yes

characteristic

164

C₄

#11763665

yes

characteristic

165

C₄

#11763666

yes

—

166

C₂

#14543650

yes

—

167

C₂

#14543654

yes

characteristic

168

C₂

#14543660

yes

characteristic

169

C₂

#14543681

yes

—

170

C₂

#14543690

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#595)⋊C₄
commutator	C₂²×C₄×C₁₇	C₄²
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₇	#95
cosocle	C₄²×Dic₁₇	C₂²
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	#7049
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#595)⋊C₄

commutator

C₂²×C₄×C₁₇

C₄²

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₁₇×C₄⋊C₈

C₄

socle, prime radical

C₂³×C₁₇

#95

cosocle

C₄²×Dic₁₇

C₂²

C₂×C₄⋊C₈

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
109. C₁₇	#7049	semidirect

H ≅ G/N

type

109. C₁₇

#7049

semidirect

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

subgroup	index
#14544570	2
#14528900	2
#7049	17

subgroup

index

#14544570

#14528900

#7049

Subgroups — #17948315

Conjugacy classes of subgroups

Complements

Maximal subgroups