Group #17948984 — subgroups

Subgroups — #17948984

Order 4352.

194 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14544346	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763671	C₂	1	1	yes	maximal, characteristic
3	#14530580	C₂	1	1	yes	maximal, characteristic
4	#11763830	—	2	1	no	—
5	C₂×C₄⋊#3624	—	2	1	no	—
6	#11763672	C₄	1	2	yes	—
7	#11763671	—	2	1	no	—
8	#11763671	—	2	1	no	—
9	#11763671	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	#11762627	—	2	1	no	—
12	C₂²⋊#63214	C₄	1	1	yes	characteristic
13	C₁₆×Dic₁₇	—	2	1	no	—
14	C₁₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
15	C₂²×#3624	—	4	1	no	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
18	C₂²×#3622	Q₈	1	1	yes	characteristic
19	C₂²⋊#3624	—	4	1	no	—
20	#10587542	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₄⋊#3624	—	2	2	no	—
23	C₄⋊#3624	—	4	1	no	—
24	C₂×C₁₆×C₁₇	—	2	1	no	—
25	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₂×#63214	—	2	1	no	—
30	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
32	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
33	C₂×#3624	—	4	2	no	—
34	C₂×#3624	—	4	2	no	—
35	C₂×#3624	—	8	2	no	—
36	C₂×C₈×C₁₇	—	4	1	no	—
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
43	C₂×#3622	—	2	2	no	—
44	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₁₆×C₁₇	—	4	1	no	—
47	#63214	—	4	1	no	—
48	#7290	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	#107	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	1	no	—
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
54	C₂×Dic₁₇	—	2	1	no	—
55	C₂×Dic₁₇	—	2	1	no	—
56	C₂×Dic₁₇	—	2	2	no	—
57	C₂×Dic₁₇	#102	1	1	yes	characteristic
58	C₂×Dic₁₇	#102	1	1	yes	characteristic
59	C₂×Dic₁₇	#103	1	1	yes	characteristic
60	C₂×Dic₁₇	#103	1	1	yes	characteristic
61	#3624	—	8	2	no	—
62	#3624	—	8	2	no	—
63	C₈×C₁₇	—	2	1	no	—
64	C₈×C₁₇	—	2	1	no	—
65	#3622	—	2	1	no	—
66	#3622	#108	1	2	yes	—
67	#2162	—	17	1	no	—
68	#1568	—	17	1	no	—
69	C₂×#334	—	17	1	no	—
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	—	2	1	no	—
72	C₂²×C₁₇	#340	1	1	yes	characteristic
73	C₂²×C₁₇	#368	1	1	yes	characteristic
74	C₂²×C₁₇	#368	1	1	yes	characteristic
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	#341	1	1	yes	characteristic
77	C₄×C₁₇	#344	1	1	yes	characteristic
78	Dic₁₇	—	2	1	no	—
79	Dic₁₇	—	2	1	no	—
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	#360	1	2	yes	—
82	#432	—	34	1	no	—
83	C₂×C₄⋊C₈	—	34	1	no	—
84	C₂×C₄×C₈	—	17	1	no	—
85	C₄⋊C₁₆	—	34	1	no	—
86	C₂²⋊C₁₆	—	17	1	no	—
87	C₂²⋊C₁₆	#595	1	1	yes	characteristic
88	C₄×C₁₆	—	34	1	no	—
89	#335	—	17	2	no	—
90	#334	—	17	2	no	—
91	#334	—	34	1	no	—
92	#334	—	34	1	no	—
93	C₂×C₁₇	—	2	1	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	#1301	1	1	yes	characteristic
96	C₂×C₁₇	#1381	1	1	yes	characteristic
97	C₂×C₁₇	#1386	1	1	yes	characteristic
98	C₂²×C₈	—	17	1	no	—
99	C₂²×C₈	—	68	1	no	—
100	C₂²×C₈	C₂×#595	1	1	yes	characteristic
101	C₂×C₄²	—	17	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	34	1	no	—
104	C₄⋊C₈	—	34	2	no	—
105	C₄⋊C₈	—	34	2	no	—
106	C₄⋊C₈	—	68	1	no	—
107	C₂²⋊C₈	—	68	1	no	—
108	#97	—	68	1	no	—
109	C₄×C₈	—	17	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₁₇	#7290	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₄	—	17	1	no	—
115	C₂²×C₄	—	17	1	no	—
116	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	34	2	no	—
121	C₂×C₈	—	68	2	no	—
122	C₂×C₈	—	68	2	no	—
123	C₂×C₈	—	136	2	no	—
124	C₂×C₈	C₄×#595	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
126	C₄²	—	17	1	no	—
127	C₄²	—	17	1	no	—
128	C₄²	—	17	1	no	—
129	C₄²	—	17	1	no	—
130	C₁₆	—	4	1	no	—
131	C₁₆	—	68	1	no	—
132	C₂³	#10587534	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	34	1	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	#10587533	1	1	yes	characteristic
143	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	17	2	no	—
147	C₈	—	34	1	no	—
148	C₈	—	136	2	no	—
149	C₈	—	136	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11763683	1	1	yes	characteristic
153	C₂²	#11763737	1	1	yes	characteristic
154	C₂²	#11763738	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	17	2	no	—
157	C₄	—	34	1	no	—
158	C₄	—	34	1	no	—
159	C₄	—	34	1	no	—
160	C₄	#11763684	1	1	yes	characteristic
161	C₄	#11763690	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#14543675	1	1	yes	characteristic
165	C₂	#14543844	1	1	yes	characteristic
166	C₂	#14543856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544346

C₂

yes

maximal, characteristic

C₂×#11763671

C₂

yes

maximal, characteristic

#14530580

C₂

yes

maximal, characteristic

#11763830

—

C₂×C₄⋊#3624

—

#11763672

C₄

yes

—

#11763671

—

#11763671

—

#11763671

C₄

yes

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762627

—

C₂²⋊#63214

C₄

yes

characteristic

C₁₆×Dic₁₇

—

C₁₇×C₂²⋊C₁₆

C₄

yes

characteristic

C₂²×#3624

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

characteristic

C₂²⋊#3624

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

D₈

yes

characteristic

C₂²×Dic₁₇

Q₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

—

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

#108

yes

—

#2162

—

#1568

—

C₂×#334

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#360

yes

—

#432

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

#595

yes

characteristic

C₄×C₁₆

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

C₂×#595

yes

characteristic

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₁₇

#7290

yes

characteristic, complemented, Sylow

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₄⋊#595

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

136

—

124

C₂×C₈

C₄×#595

yes

characteristic

125

C₂×C₈

C₂²⋊#595

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

#10587534

yes

characteristic

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

#10587533

yes

characteristic

143

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

136

—

149

C₈

—

136

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11763683

yes

characteristic

153

C₂²

#11763737

yes

characteristic

154

C₂²

#11763738

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#11763684

yes

characteristic

161

C₄

#11763690

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

#14543675

yes

characteristic

165

C₂

#14543844

yes

characteristic

166

C₂

#14543856

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₁₇×C₂²⋊C₁₆	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂³×C₁₇	#107
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²⋊C₁₆	#595
hypocenter	C₁₇	#7290
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂²×C₈

C₂×#595

Fitting

C₁₇×C₂²⋊C₁₆

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₂²⋊C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
113. C₁₇	#7290	semidirect

H ≅ G/N

type

113. C₁₇

#7290

semidirect

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

subgroup	index
#14544346	2
C₂×#11763671	2
#14530580	2
#7290	17

subgroup

index

Subgroups — #17948984

Conjugacy classes of subgroups

Complements

Maximal subgroups