Group #17948737 — subgroups

Subgroups — #17948737

Order 4352.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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	#14545118	C₂	1	1	yes	maximal, characteristic
2	C₈×#63247	C₂	1	1	yes	maximal, characteristic
3	#14530580	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	C₄	1	2	yes	—
5	#11763826	C₄	1	1	yes	characteristic
6	C₄×#63247	C₄	1	1	yes	characteristic
7	#11763730	—	2	1	no	—
8	#11763730	—	2	1	no	—
9	#11763651	—	2	1	no	—
10	C₈×#3624	—	2	1	no	—
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762627	—	2	1	no	—
13	C₂²⋊#63214	C₄	1	1	yes	characteristic
14	C₁₆×Dic₁₇	—	2	1	no	—
15	C₁₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
16	C₂×#63247	—	2	1	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
20	C₂²⋊#3624	—	2	1	no	—
21	#10587544	C₈	1	2	yes	—
22	#10587542	C₈	1	2	yes	—
23	C₄×#3624	—	2	1	no	—
24	C₄×#3624	—	2	1	no	—
25	#10587538	—	4	2	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×C₁₆×C₁₇	—	2	1	no	—
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	D₄	1	1	yes	characteristic
31	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
32	C₂×#63214	—	2	1	no	—
33	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
34	C₂²×Dic₁₇	M₁₆	1	1	yes	characteristic
35	C₂²×Dic₁₇	M₁₆	1	1	yes	characteristic
36	#63247	—	2	4	no	—
37	C₂×#3624	—	2	2	no	—
38	C₂×#3624	—	2	2	no	—
39	C₂×C₈×C₁₇	—	4	1	no	—
40	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×#3622	—	2	1	no	—
47	C₂×#3622	—	2	1	no	—
48	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	#63216	—	4	2	no	—
51	C₁₆×C₁₇	—	4	1	no	—
52	#63214	—	4	1	no	—
53	#7212	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#97	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	1	no	—
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
59	C₂×Dic₁₇	—	2	2	no	—
60	C₂×Dic₁₇	—	2	2	no	—
61	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
65	#3624	—	2	4	no	—
66	#3624	—	4	2	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	2	1	no	—
69	#3622	—	2	1	no	—
70	#3622	#100	1	1	yes	characteristic
71	#3622	#101	1	1	yes	characteristic
72	#2162	—	17	1	no	—
73	#2150	—	17	1	no	—
74	C₈×M₁₆	—	17	1	no	—
75	C₂²×C₁₇	—	2	1	no	—
76	C₂²×C₁₇	—	2	1	no	—
77	C₂²×C₁₇	—	2	1	no	—
78	C₂²×C₁₇	#336	1	1	yes	characteristic
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	#343	1	1	yes	characteristic
81	C₄×C₁₇	#344	1	1	yes	characteristic
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	#350	1	2	yes	—
85	Dic₁₇	#364	1	2	yes	—
86	C₂×M₃₂	—	17	2	no	—
87	#431	—	17	1	no	—
88	C₄×M₁₆	—	17	1	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	C₄⋊C₁₆	—	34	1	no	—
91	C₄⋊C₁₆	—	34	1	no	—
92	C₄⋊C₁₆	—	34	1	no	—
93	C₂²⋊C₁₆	—	17	1	no	—
94	C₂²⋊C₁₆	#595	1	1	yes	characteristic
95	C₄×C₁₆	—	34	1	no	—
96	#322	—	34	1	no	—
97	C₈²	—	34	1	no	—
98	C₂×C₁₇	—	4	1	no	—
99	C₂×C₁₇	#1283	1	1	yes	characteristic
100	C₂×C₁₇	#1376	1	1	yes	characteristic
101	C₂×C₁₇	#1379	1	1	yes	characteristic
102	C₂×M₁₆	—	34	1	no	—
103	C₂²×C₈	—	17	1	no	—
104	C₂²×C₈	C₂×#595	1	1	yes	characteristic
105	C₂×C₄²	—	17	1	no	—
106	M₃₂	—	68	2	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	34	2	no	—
109	C₂×C₁₆	—	34	1	no	—
110	C₄⋊C₈	—	17	2	no	—
111	C₂²⋊C₈	—	34	1	no	—
112	#97	—	17	2	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	34	1	no	—
118	C₄×C₈	—	34	1	no	—
119	C₁₇	#7212	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	—	17	1	no	—
121	C₂²×C₄	—	17	1	no	—
122	C₂²×C₄	C₄×#595	1	1	yes	characteristic
123	M₁₆	—	34	4	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	34	1	no	—
128	C₂×C₈	—	34	1	no	—
129	C₂×C₈	—	34	2	no	—
130	C₂×C₈	—	34	2	no	—
131	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
133	C₄²	—	17	1	no	—
134	C₄²	—	17	1	no	—
135	C₄²	—	17	1	no	—
136	C₄²	—	17	1	no	—
137	C₁₆	—	4	1	no	—
138	C₁₆	—	68	1	no	—
139	C₁₆	—	68	2	no	—
140	C₂³	#10587532	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	34	2	no	—
148	C₂×C₄	—	34	2	no	—
149	C₂×C₄	C₈×#595	1	1	yes	characteristic
150	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	17	1	no	—
154	C₈	—	17	1	no	—
155	C₈	—	34	4	no	—
156	C₈	—	34	1	no	—
157	C₈	—	68	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	#11763674	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	17	2	no	—
164	C₄	—	17	2	no	—
165	C₄	—	34	1	no	—
166	C₄	—	34	1	no	—
167	C₄	#11763688	1	1	yes	characteristic
168	C₄	#11763690	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#14543628	1	1	yes	characteristic
171	C₂	#14543827	1	1	yes	characteristic
172	C₂	#14543837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545118

C₂

yes

maximal, characteristic

C₈×#63247

C₂

yes

maximal, characteristic

#14530580

C₂

yes

maximal, characteristic

C₂×#10587538

C₄

yes

—

#11763826

C₄

yes

characteristic

C₄×#63247

C₄

yes

—

—

—

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762627

—

C₂²⋊#63214

C₄

yes

characteristic

C₁₆×Dic₁₇

—

C₁₇×C₂²⋊C₁₆

C₄

yes

characteristic

C₂×#63247

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²⋊#3624

—

#10587544

C₈

yes

—

#10587542

C₈

yes

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

—

C₂²×C₄×C₁₇

C₄²

yes

characteristic

C₂²×Dic₁₇

M₁₆

yes

characteristic

C₂²×Dic₁₇

M₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

#63216

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

—

yes

yes

—

—

C₈×M₁₆

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#336

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#350

yes

—

Dic₁₇

#364

yes

—

C₂×M₃₂

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

#595

yes

characteristic

C₄×C₁₆

—

#322

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1283

yes

characteristic

100

C₂×C₁₇

#1376

yes

characteristic

101

C₂×C₁₇

#1379

yes

characteristic

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂×#595

yes

characteristic

105

C₂×C₄²

—

106

M₃₂

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₄⋊C₈

—

111

C₂²⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₁₇

#7212

yes

characteristic, complemented, Sylow

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×#595

yes

characteristic

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₄⋊#595

yes

characteristic

132

C₂×C₈

C₂²⋊#595

yes

characteristic

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₁₆

—

138

C₁₆

—

139

C₁₆

—

140

C₂³

#10587532

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₈×#595

yes

characteristic

150

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#11763674

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#11763688

yes

characteristic

168

C₄

#11763690

yes

characteristic

169

C₂

—

170

C₂

#14543628

yes

characteristic

171

C₂

#14543827

yes

characteristic

172

C₂

#14543837

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂²×C₈	C₂×#595
Fitting	C₁₇×C₂²⋊C₁₆	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	#97
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²⋊C₁₆	#595
hypocenter	C₁₇	#7212
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂×C₄

(C₂×#595)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₂²×C₈

C₂×#595

Fitting

C₁₇×C₂²⋊C₁₆

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₂²⋊C₁₆

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
119. C₁₇	#7212	semidirect

H ≅ G/N

type

119. C₁₇

#7212

semidirect

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

subgroup	index
#14545118	2
C₈×#63247	2
#14530580	2
#7212	17

subgroup

index

Subgroups — #17948737

Conjugacy classes of subgroups

Complements

Maximal subgroups