Group #17948773 — subgroups

Subgroups — #17948773

Order 4352.

214 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14544705	C₂	1	1	yes	maximal, characteristic
2	#14544347	C₂	1	1	yes	maximal, characteristic
3	#14530585	C₂	1	1	yes	maximal, characteristic
4	#11763831	—	2	1	no	—
5	#11763755	—	2	1	no	—
6	#11763683	—	2	1	no	—
7	#11763683	C₄	1	2	yes	—
8	#11763673	C₄	1	2	yes	complemented
9	#11763671	—	2	1	no	—
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	#11762627	—	2	1	no	—
12	#11762588	—	2	1	no	—
13	#11762585	C₄	1	1	yes	characteristic, complemented
14	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#595	—	2	2	no	—
16	C₂×C₄⋊#595	—	4	1	no	—
17	C₂×C₄⋊#595	—	4	1	no	—
18	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
20	C₄²×D₁₇	D₄	1	1	yes	characteristic
21	#10587543	—	4	1	no	—
22	#10587542	—	4	1	no	—
23	C₄⋊#3624	—	2	2	no	—
24	C₂×C₁₆×C₁₇	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₂×#63214	—	2	1	no	—
29	C₄×#3622	Q₈	1	1	yes	characteristic
30	C₂²×#595	—	4	2	no	—
31	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×#3624	—	4	2	no	—
35	C₄⋊#595	—	4	2	no	—
36	C₄⋊#595	—	4	2	no	—
37	C₄⋊#595	—	8	2	no	—
38	C₄⋊#595	—	8	2	no	—
39	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₁₇	—	4	1	no	—
48	C₈×D₁₇	—	4	1	no	—
49	C₁₆×C₁₇	—	4	1	no	—
50	#63214	—	4	1	no	—
51	#7222	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	#102	1	1	yes	characteristic
53	C₂×#595	—	8	2	no	—
54	C₂×#595	—	8	2	no	—
55	C₂×#595	—	8	2	no	—
56	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
59	C₂×Dic₁₇	#102	1	1	yes	characteristic
60	C₂×Dic₁₇	#103	1	1	yes	characteristic
61	C₂×Dic₁₇	#103	1	1	yes	characteristic
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	2	1	no	—
65	C₄×D₁₇	—	2	1	no	—
66	C₄×D₁₇	—	2	2	no	—
67	#3624	—	8	2	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	—	2	1	no	—
70	#3622	—	2	1	no	—
71	#3622	—	2	1	no	—
72	#2163	—	17	1	no	—
73	#1834	—	17	1	no	—
74	#1568	—	17	1	no	—
75	C₂²×C₁₇	#340	1	1	yes	characteristic
76	C₂×D₁₇	—	2	1	no	—
77	C₂×D₁₇	—	2	1	no	—
78	C₂×D₁₇	#361	1	1	yes	characteristic
79	C₂×D₁₇	#362	1	1	yes	characteristic
80	#595	—	16	2	no	—
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	—	2	1	no	—
83	C₄×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
84	C₄×C₁₇	#344	1	1	yes	characteristic
85	Dic₁₇	—	2	1	no	—
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	#360	1	2	yes	—
88	#432	—	34	1	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	#384	—	34	1	no	—
91	C₄⋊C₁₆	—	34	1	no	—
92	C₂²⋊C₁₆	—	34	1	no	—
93	#346	—	17	1	no	—
94	#346	#595	1	1	yes	characteristic, complemented
95	#340	—	17	2	no	—
96	#340	—	34	1	no	—
97	#335	—	17	2	no	—
98	#334	—	34	1	no	—
99	C₂×C₁₇	#1286	1	1	yes	characteristic
100	C₂×C₁₇	#1386	1	1	yes	characteristic
101	C₂×C₁₇	#1387	1	1	yes	characteristic
102	D₁₇	—	2	1	no	—
103	D₁₇	—	2	1	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂×C₄⋊C₄	—	34	2	no	—
107	C₂×C₄⋊C₄	—	68	1	no	—
108	C₂×C₄⋊C₄	—	68	1	no	—
109	C₂×C₄²	—	17	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	34	1	no	—
112	C₄⋊C₈	—	34	2	no	—
113	C₂²⋊C₈	—	68	1	no	—
114	#97	—	68	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	C₂×#595	1	1	yes	characteristic
119	C₁₇	#7222	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	C₂²×C₄	—	68	2	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	68	1	no	—
127	C₂×C₈	—	68	1	no	—
128	C₂×C₈	—	68	2	no	—
129	C₂×C₈	C₄×#595	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
131	C₄⋊C₄	—	68	2	no	—
132	C₄⋊C₄	—	68	2	no	—
133	C₄⋊C₄	—	136	2	no	—
134	C₄⋊C₄	—	136	2	no	—
135	C₄²	—	17	1	no	—
136	C₄²	—	17	1	no	—
137	C₄²	—	17	1	no	—
138	C₄²	C₄⋊#595	1	1	yes	characteristic
139	C₁₆	—	4	1	no	—
140	C₁₆	—	68	1	no	—
141	C₂³	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	34	2	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	136	2	no	—
151	C₂×C₄	—	136	2	no	—
152	C₂×C₄	—	136	2	no	—
153	C₂×C₄	#10587533	1	1	yes	characteristic
154	C₂×C₄	#10587534	1	1	yes	characteristic
155	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	34	1	no	—
159	C₈	—	34	1	no	—
160	C₈	—	136	2	no	—
161	C₂²	—	17	1	no	—
162	C₂²	—	17	1	no	—
163	C₂²	—	34	1	no	—
164	C₂²	—	34	1	no	—
165	C₂²	#11763683	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	272	2	no	—
169	C₄	—	17	2	no	—
170	C₄	—	34	1	no	—
171	C₄	—	34	1	no	—
172	C₄	#11763676	1	1	yes	characteristic
173	C₄	#11763691	1	1	yes	characteristic
174	C₂	—	34	1	no	—
175	C₂	—	34	1	no	—
176	C₂	#14543638	1	1	yes	characteristic
177	C₂	#14543856	1	1	yes	characteristic
178	C₂	#14543859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544705

C₂

yes

maximal, characteristic

#14544347

C₂

yes

maximal, characteristic

#14530585

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11763673

C₄

yes

complemented

#11763671

—

C₄×C₈×D₁₇

C₂²

yes

—

—

C₄

yes

characteristic, complemented

C₁₇×#346

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄⋊#595

—

C₂×C₄⋊#595

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₄×#3622

Q₈

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

—

—

—

—

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

Q₁₆

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

—

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#360

yes

—

#432

—

C₂×C₄×C₈

—

#384

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

#595

yes

characteristic, complemented

—

—

—

—

C₂×C₁₇

#1286

yes

characteristic

100

C₂×C₁₇

#1386

yes

characteristic

101

C₂×C₁₇

#1387

yes

characteristic

102

D₁₇

—

103

D₁₇

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#595

yes

characteristic

119

C₁₇

#7222

yes

characteristic, complemented, Sylow

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₄×#595

yes

characteristic

130

C₂×C₈

C₂²⋊#595

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

136

—

134

C₄⋊C₄

—

136

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₄⋊#595

yes

characteristic

139

C₁₆

—

140

C₁₆

—

141

C₂³

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

136

—

151

C₂×C₄

—

136

—

152

C₂×C₄

—

136

—

153

C₂×C₄

#10587533

yes

characteristic

154

C₂×C₄

#10587534

yes

characteristic

155

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

136

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11763683

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

272

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#11763676

yes

characteristic

173

C₄

#11763691

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#14543638

yes

characteristic

177

C₂

#14543856

yes

characteristic

178

C₂

#14543859

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#102
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#346	#595
hypocenter	C₁₇	#7222
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#346

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
8. #11763673	C₄	semidirect
13. #11762585	C₄	semidirect
14. C₁₇×#346	C₄	semidirect
94. #346	#595	semidirect
119. C₁₇	#7222	semidirect

H ≅ G/N

type

8. #11763673

C₄

semidirect

13. #11762585

C₄

semidirect

14. C₁₇×#346

C₄

semidirect

94. #346

#595

semidirect

119. C₁₇

#7222

semidirect

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

subgroup	index
#14544705	2
#14544347	2
#14530585	2
#7222	17

subgroup

index

Subgroups — #17948773

Conjugacy classes of subgroups

Complements

Maximal subgroups