Group #17948574 — subgroups

Subgroups — #17948574

Order 4352.

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14545148	C₂	1	1	yes	maximal, characteristic
2	C₈×#63243	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#346	C₂	1	1	yes	maximal, characteristic
4	#11763828	C₄	1	1	yes	characteristic
5	C₄×#63243	C₄	1	1	yes	characteristic
6	C₄⋊#63216	C₄	1	2	yes	—
7	#11763704	—	2	1	no	—
8	#11763698	—	2	1	no	—
9	#11763651	—	2	1	no	—
10	C₈×#3624	—	2	1	no	—
11	C₂×C₁₆×D₁₇	—	2	1	no	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762585	C₄	1	1	yes	characteristic
14	#11762584	—	2	1	no	—
15	C₁₇×#346	C₄	1	1	yes	characteristic
16	C₂×#63243	—	2	1	no	—
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
20	#10587543	C₈	1	2	yes	—
21	#10587542	C₈	1	2	yes	—
22	C₄⋊#3624	—	2	1	no	—
23	C₄×#3624	—	2	1	no	—
24	C₄×#3624	—	2	1	no	—
25	C₂×#63216	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₁₆×D₁₇	—	2	4	no	—
33	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
36	C₂×#3624	—	2	2	no	—
37	C₂×#3624	—	2	2	no	—
38	#63243	—	2	4	no	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₄²	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
45	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	2	1	no	—
49	C₈×D₁₇	—	2	2	no	—
50	#63216	—	4	2	no	—
51	C₁₆×C₁₇	—	2	2	no	—
52	#63214	—	2	2	no	—
53	#7165	—	17	1	no	maximal, Sylow
54	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
58	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
61	C₄×D₁₇	—	2	2	no	—
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	4	1	no	—
65	#3624	—	2	4	no	—
66	#3624	—	4	2	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	#108	1	1	yes	characteristic
69	C₈×C₁₇	#108	1	1	yes	characteristic
70	#3622	—	2	1	no	—
71	#3622	#104	1	2	yes	—
72	#2163	—	17	1	no	—
73	C₂×#346	—	17	1	no	—
74	C₈×M₁₆	—	17	1	no	—
75	C₂²×C₁₇	#336	1	1	yes	characteristic
76	C₂×D₁₇	—	2	1	no	—
77	C₂×D₁₇	—	2	1	no	—
78	C₂×D₁₇	#364	1	2	yes	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#339	1	1	yes	characteristic
82	C₄×C₁₇	#341	1	1	yes	characteristic
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	—	2	1	no	—
85	Dic₁₇	—	2	1	no	—
86	C₂²×C₁₆	—	34	1	no	—
87	#431	—	17	1	no	—
88	C₄×M₁₆	—	17	1	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	C₄⋊C₁₆	—	17	2	no	—
91	C₂²⋊C₁₆	—	34	1	no	—
92	#346	—	17	1	no	—
93	#346	—	34	1	no	—
94	#346	—	34	1	no	—
95	#346	#595	1	1	yes	characteristic
96	#322	—	34	1	no	—
97	C₈²	—	34	1	no	—
98	C₂×C₁₇	#1278	1	1	yes	characteristic
99	C₂×C₁₇	#1376	1	1	yes	characteristic
100	C₂×C₁₇	#1378	1	1	yes	characteristic
101	D₁₇	—	2	2	no	—
102	C₂×M₁₆	—	34	1	no	—
103	C₂²×C₈	—	17	1	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂×C₄²	—	17	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	34	4	no	—
108	C₂×C₁₆	—	34	1	no	—
109	C₂×C₁₆	—	34	2	no	—
110	C₄⋊C₈	—	34	1	no	—
111	C₂²⋊C₈	—	17	2	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	—	34	1	no	—
117	C₄×C₈	—	34	1	no	—
118	C₄×C₈	C₂×#595	1	1	yes	characteristic
119	C₁₇	#7165	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	—	17	1	no	—
121	C₂²×C₄	—	17	1	no	—
122	C₂²×C₄	—	17	1	no	—
123	M₁₆	—	34	4	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	34	2	no	—
127	C₂×C₈	—	34	2	no	—
128	C₂×C₈	—	34	1	no	—
129	C₂×C₈	—	34	1	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	C₄×#595	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	34	2	no	—
139	C₁₆	—	68	2	no	—
140	C₂³	—	17	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	34	2	no	—
145	C₂×C₄	—	34	1	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	68	1	no	—
148	C₂×C₄	C₈×#595	1	1	yes	characteristic
149	C₂×C₄	#10587532	1	1	yes	characteristic
150	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	17	2	no	—
153	C₈	—	34	4	no	—
154	C₈	—	34	1	no	—
155	C₈	—	68	2	no	—
156	C₈	#10587535	1	1	yes	characteristic
157	C₈	#10587536	1	1	yes	characteristic
158	C₂²	—	17	2	no	—
159	C₂²	—	34	1	no	—
160	C₂²	—	34	1	no	—
161	C₂²	#11763674	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	34	1	no	—
165	C₄	—	34	1	no	—
166	C₄	—	34	1	no	—
167	C₄	#11763680	1	1	yes	characteristic
168	C₄	#11763685	1	1	yes	characteristic
169	C₂	—	34	2	no	—
170	C₂	#14543616	1	1	yes	characteristic
171	C₂	#14543827	1	1	yes	characteristic
172	C₂	#14543836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545148

C₂

yes

maximal, characteristic

C₈×#63243

C₂

yes

maximal, characteristic

D₁₇×#346

C₂

yes

maximal, characteristic

#11763828

C₄

yes

characteristic

C₄×#63243

C₄

yes

characteristic

C₄⋊#63216

C₄

yes

—

—

—

—

—

C₂×C₁₆×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic

#11762584

—

C₁₇×#346

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

#10587543

C₈

yes

—

#10587542

C₈

yes

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₄×#3622

C₂×C₄

yes

characteristic

C₁₆×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

—

—

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

#63216

—

C₁₆×C₁₇

—

#63214

—

#7165

—

maximal, Sylow

C₂²×D₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

—

#2163

—

C₂×#346

—

C₈×M₁₆

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#364

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

yes

—

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

100

C₂×C₁₇

#1378

yes

characteristic

101

D₁₇

—

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₁₆

—

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

C₂×#595

yes

characteristic

119

C₁₇

#7165

yes

characteristic, complemented, Sylow

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

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

C₄×#595

yes

characteristic

137

C₁₆

—

138

C₁₆

—

139

C₁₆

—

140

C₂³

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₈×#595

yes

characteristic

149

C₂×C₄

#10587532

yes

characteristic

150

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#10587535

yes

characteristic

157

C₈

#10587536

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#11763674

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#11763680

yes

characteristic

168

C₄

#11763685

yes

characteristic

169

C₂

—

170

C₂

#14543616

yes

characteristic

171

C₂

#14543827

yes

characteristic

172

C₂

#14543836

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₄⋊C₈
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#346	#595
hypocenter	C₁₇	#7165
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂×C₄

(C₂×#595)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#346

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₄⋊C₈

cosocle

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
119. C₁₇	#7165	semidirect

H ≅ G/N

type

119. C₁₇

#7165

semidirect

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

subgroup	index
#14545148	2
C₈×#63243	2
D₁₇×#346	2
#7165	17

subgroup

index

#14545148

C₈×#63243

D₁₇×#346

#7165

Subgroups — #17948574

Conjugacy classes of subgroups

Complements

Maximal subgroups