Group #17949023 — subgroups

Subgroups — #17949023

Order 4352.

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

normal subgroups only

This group has 204 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	C₄×#10587536	C₂	1	1	yes	maximal, characteristic
2	#14544345	C₂	1	1	yes	maximal, characteristic
3	#14530586	C₂	1	1	yes	maximal, characteristic
4	#11763831	—	2	1	no	—
5	C₂×#10587536	—	2	1	no	—
6	C₄×#63243	—	2	1	no	—
7	#11763685	C₄	1	2	yes	—
8	#11763672	—	2	1	no	—
9	#11763670	C₄	1	2	yes	—
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊#63214	C₄	1	1	yes	characteristic
12	#11762588	—	2	1	no	—
13	#11762584	—	2	1	no	—
14	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
15	C₂×#63243	—	2	2	no	—
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₄²×D₁₇	D₄	1	1	yes	characteristic
19	#10587543	—	4	1	no	—
20	#10587542	—	4	1	no	—
21	#10587542	—	4	1	no	—
22	C₄⋊#3624	—	2	2	no	—
23	C₄×#3624	—	4	1	no	—
24	#10587536	—	2	2	no	—
25	#10587536	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	C₂×#3624	—	4	2	no	—
37	#63243	—	2	4	no	—
38	#63243	—	4	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	D₈	1	1	yes	characteristic
44	C₄×Dic₁₇	SD₁₆	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	—	2	1	no	—
48	C₈×D₁₇	—	2	1	no	—
49	C₈×D₁₇	—	4	1	no	—
50	C₁₆×C₁₇	—	4	1	no	—
51	#63214	—	4	1	no	—
52	#7303	—	17	1	no	maximal, Sylow
53	C₂²×D₁₇	#103	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
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	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	4	1	no	—
63	C₄×D₁₇	—	4	1	no	—
64	#3624	—	4	4	no	—
65	#3624	—	8	2	no	—
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	#104	1	1	yes	characteristic
68	C₈×C₁₇	#104	1	1	yes	characteristic
69	#3622	—	2	1	no	—
70	#3622	—	2	1	no	—
71	#2163	—	17	1	no	—
72	C₄×#108	—	17	1	no	—
73	#1568	—	17	1	no	—
74	C₂²×C₁₇	#340	1	1	yes	characteristic
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	—	2	1	no	—
77	C₂×D₁₇	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	#339	1	1	yes	characteristic
81	C₄×C₁₇	#344	1	1	yes	characteristic
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	—	2	1	no	—
85	#432	—	34	1	no	—
86	C₂×#108	—	34	1	no	—
87	C₄×M₁₆	—	34	1	no	—
88	C₂×C₄×C₈	—	17	1	no	—
89	C₄⋊C₁₆	—	17	1	no	—
90	C₄⋊C₁₆	#595	1	1	yes	characteristic
91	C₂²⋊C₁₆	—	34	1	no	—
92	#346	—	34	1	no	—
93	#341	—	17	2	no	—
94	#335	—	34	1	no	—
95	#334	—	17	2	no	—
96	C₂×C₁₇	#1303	1	1	yes	characteristic
97	C₂×C₁₇	#1385	1	1	yes	characteristic
98	C₂×C₁₇	#1389	1	1	yes	characteristic
99	D₁₇	—	4	1	no	—
100	C₂×M₁₆	—	34	2	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂²×C₈	—	17	1	no	—
103	C₂×C₄²	—	17	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	34	1	no	—
106	#108	—	34	2	no	—
107	#108	—	34	2	no	—
108	C₄⋊C₈	—	34	2	no	—
109	C₂²⋊C₈	—	68	1	no	—
110	#97	—	68	1	no	—
111	#97	—	68	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	68	1	no	—
116	C₄×C₈	C₂×#595	1	1	yes	characteristic
117	C₁₇	#7303	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	—	17	1	no	—
119	C₂²×C₄	—	17	1	no	—
120	C₂²×C₄	—	17	1	no	—
121	M₁₆	—	34	4	no	—
122	M₁₆	—	68	2	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	34	1	no	—
126	C₂×C₈	—	34	1	no	—
127	C₂×C₈	—	68	1	no	—
128	C₂×C₈	—	68	2	no	—
129	C₂×C₈	—	68	2	no	—
130	C₂×C₈	C₄×#595	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
132	C₄²	—	17	1	no	—
133	C₄²	—	17	1	no	—
134	C₄²	—	17	1	no	—
135	C₄²	C₄⋊#595	1	1	yes	characteristic
136	C₁₆	—	4	1	no	—
137	C₁₆	—	68	1	no	—
138	C₂³	—	17	1	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	—	34	1	no	—
144	C₂×C₄	—	68	1	no	—
145	C₂×C₄	—	68	1	no	—
146	C₂×C₄	#10587533	1	1	yes	characteristic
147	C₂×C₄	#10587534	1	1	yes	characteristic
148	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	34	1	no	—
151	C₈	—	34	1	no	—
152	C₈	—	68	4	no	—
153	C₈	—	136	2	no	—
154	C₈	#10587548	1	1	yes	characteristic
155	C₈	#10587550	1	1	yes	characteristic
156	C₂²	—	34	1	no	—
157	C₂²	—	34	1	no	—
158	C₂²	—	34	1	no	—
159	C₂²	#11763683	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	34	1	no	—
163	C₄	—	34	1	no	—
164	C₄	—	34	1	no	—
165	C₄	#11763679	1	1	yes	characteristic
166	C₄	#11763692	1	1	yes	characteristic
167	C₂	—	68	1	no	—
168	C₂	#14543681	1	1	yes	characteristic
169	C₂	#14543854	1	1	yes	characteristic
170	C₂	#14543865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587536

C₂

yes

maximal, characteristic

#14544345

C₂

yes

maximal, characteristic

#14530586

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11763672

—

#11763670

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊#63214

C₄

yes

characteristic

#11762588

—

#11762584

—

C₁₇×C₄⋊C₁₆

C₄

yes

characteristic

C₂×#63243

—

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

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

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

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

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

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#432

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

#595

yes

characteristic

C₂²⋊C₁₆

—

—

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

100

C₂×M₁₆

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

#108

—

107

#108

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×#595

yes

characteristic

117

C₁₇

#7303

yes

characteristic, complemented, Sylow

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄×#595

yes

characteristic

131

C₂×C₈

C₂²⋊#595

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₄⋊#595

yes

characteristic

136

C₁₆

—

137

C₁₆

—

138

C₂³

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#10587533

yes

characteristic

147

C₂×C₄

#10587534

yes

characteristic

148

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

136

—

154

C₈

#10587548

yes

characteristic

155

C₈

#10587550

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#11763683

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#11763679

yes

characteristic

166

C₄

#11763692

yes

characteristic

167

C₂

—

168

C₂

#14543681

yes

characteristic

169

C₂

#14543854

yes

characteristic

170

C₂

#14543865

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×C₄⋊C₁₆	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#103
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#7303
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×C₄⋊C₁₆

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₄⋊C₁₆

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
117. C₁₇	#7303	semidirect

H ≅ G/N

type

117. C₁₇

#7303

semidirect

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

subgroup	index
C₄×#10587536	2
#14544345	2
#14530586	2
#7303	17

subgroup

index

Subgroups — #17949023

Conjugacy classes of subgroups

Complements

Maximal subgroups