Group #17960133 — subgroups

Subgroups — #17960133

Order 4352.

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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	#14544345	C₂	1	1	yes	maximal, characteristic
2	#14544339	C₂	1	1	yes	maximal, characteristic
3	#14543810	C₂	1	4	yes	maximal, complemented
4	D₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	#11763831	—	2	1	no	—
6	#11763813	—	2	1	no	—
7	#11763672	—	2	1	no	—
8	#11763670	—	2	1	no	—
9	#11763670	C₂²	1	2	yes	—
10	#11763670	C₂²	1	2	yes	—
11	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762585	C₄	1	1	yes	characteristic
14	#11762584	C₂²	1	2	yes	—
15	C₁₇×#346	C₄	1	1	yes	characteristic
16	C₂×#63243	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	#10587543	—	4	1	no	—
21	#10587542	—	4	1	no	—
22	C₄⋊#3624	—	2	2	no	—
23	C₄⋊#3624	—	2	2	no	—
24	C₄⋊#3624	—	4	1	no	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
29	C₂×#63214	C₂×C₄	1	2	yes	—
30	C₄×#3622	Q₈	1	1	yes	characteristic
31	C₁₆×D₁₇	—	2	4	no	—
32	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	C₂×#3624	—	4	2	no	—
37	#63243	—	8	2	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
46	C₈×D₁₇	—	2	2	no	—
47	C₈×D₁₇	—	2	2	no	—
48	C₁₆×C₁₇	—	2	2	no	—
49	#63214	M₁₆	1	4	yes	complemented
50	#12019	—	17	1	no	maximal, Sylow
51	C₂²×D₁₇	#106	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	#106	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	4	1	no	—
62	#3624	—	8	2	no	—
63	#3624	—	8	2	no	—
64	C₈×C₁₇	#108	1	2	yes	—
65	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
66	#3622	—	2	1	no	—
67	#3622	—	2	1	no	—
68	C₂×#346	—	17	1	no	—
69	#1568	—	17	1	no	—
70	#1566	—	17	1	no	—
71	#1368	—	17	4	no	—
72	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
73	C₂×D₁₇	—	2	1	no	—
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	#365	1	2	yes	—
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	#423	1	1	yes	characteristic
79	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	—	2	1	no	—
83	C₂²×C₁₆	—	17	2	no	—
84	#432	—	34	1	no	—
85	#423	—	34	1	no	—
86	C₂×C₄×C₈	—	17	1	no	—
87	#346	—	17	2	no	—
88	#346	—	17	1	no	—
89	#346	#595	1	1	yes	characteristic
90	#335	—	34	1	no	—
91	#334	—	17	2	no	—
92	#334	—	17	2	no	—
93	#334	—	34	1	no	—
94	C₂×C₁₇	#1566	1	1	yes	characteristic
95	C₂×C₁₇	C₂×#365	1	1	yes	characteristic
96	C₂×C₁₇	#2155	1	1	yes	characteristic
97	D₁₇	—	2	2	no	—
98	C₂×M₁₆	—	68	1	no	—
99	C₂²×C₈	—	17	1	no	—
100	C₂²×C₈	—	17	1	no	—
101	C₂×C₄²	—	17	1	no	—
102	C₂×C₁₆	—	17	2	no	—
103	C₂×C₁₆	—	34	4	no	—
104	C₂×C₁₆	C₂×#595	1	2	yes	—
105	C₄⋊C₈	—	34	2	no	—
106	C₄⋊C₈	—	34	2	no	—
107	C₄⋊C₈	—	68	1	no	—
108	C₂²⋊C₈	—	68	1	no	—
109	#97	—	68	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	C₂×#595	1	1	yes	characteristic
114	C₁₇	#12019	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	17	1	no	—
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	—	17	1	no	—
118	M₁₆	—	136	2	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	—	68	2	no	—
124	C₂×C₈	—	68	2	no	—
125	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
126	C₂×C₈	C₂²×#595	1	1	yes	characteristic
127	C₄²	—	17	1	no	—
128	C₄²	—	17	1	no	—
129	C₄²	—	17	1	no	—
130	C₄²	C₄⋊#595	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	—	17	4	no	—
133	C₂³	—	17	1	no	—
134	C₂×C₄	—	17	1	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	34	2	no	—
138	C₂×C₄	—	34	1	no	—
139	C₂×C₄	—	34	1	no	—
140	C₂×C₄	—	68	1	no	—
141	C₂×C₄	#10587533	1	1	yes	characteristic
142	C₂×C₄	#10587533	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
144	C₈	—	34	1	no	—
145	C₈	—	34	1	no	—
146	C₈	—	136	2	no	—
147	C₈	—	136	2	no	—
148	C₈	#10587536	1	2	yes	—
149	C₈	D₁₇⋊M₁₆	1	2	yes	—
150	C₂²	—	17	2	no	—
151	C₂²	—	34	1	no	—
152	C₂²	—	34	1	no	—
153	C₂²	C₂×#10587533	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	34	1	no	—
157	C₄	—	34	1	no	—
158	C₄	—	34	1	no	—
159	C₄	#11763814	1	1	yes	characteristic
160	C₄	C₂×#10587536	1	1	yes	characteristic
161	C₂	—	34	2	no	—
162	C₂	#14544340	1	1	yes	characteristic
163	C₂	D₁₇⋊#365	1	1	yes	characteristic
164	C₂	#14545132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544345

C₂

yes

maximal, characteristic

#14544339

C₂

yes

maximal, characteristic

#14543810

C₂

yes

maximal, complemented

D₁₇×#346

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#11763670

C₂²

yes

—

C₂×C₁₆×D₁₇

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic

#11762584

C₂²

yes

—

C₁₇×#346

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

C₄×#3622

Q₈

yes

characteristic

C₁₆×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

M₁₆

yes

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#106

yes

characteristic

C₂×Dic₁₇

C₂×SD₁₆

yes

characteristic

C₂×Dic₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

C₈×C₁₇

#108

yes

—

C₈×C₁₇

C₂×M₁₆

yes

—

—

—

—

—

—

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#365

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

#432

—

#423

—

C₂×C₄×C₈

—

—

—

yes

—

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

C₂×#595

yes

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

#97

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#595

yes

characteristic

114

C₁₇

#12019

yes

characteristic, complemented, Sylow

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

136

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₄⋊#595

yes

characteristic

126

C₂×C₈

C₂²×#595

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₄⋊#595

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#10587533

yes

characteristic

142

C₂×C₄

#10587533

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

136

—

147

C₈

—

136

—

148

C₈

#10587536

yes

—

149

C₈

D₁₇⋊M₁₆

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×#10587533

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#11763814

yes

characteristic

160

C₄

C₂×#10587536

yes

characteristic

161

C₂

—

162

C₂

#14544340

yes

characteristic

163

C₂

D₁₇⋊#365

yes

characteristic

164

C₂

#14545132

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#10587533
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂²×D₁₇	#106
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12019
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#10587533

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

C₂²×C₁₇

C₂²×D₁₇

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
3. #14543810	C₂	semidirect
49. #63214	M₁₆	semidirect
114. C₁₇	#12019	semidirect

H ≅ G/N

type

3. #14543810

C₂

semidirect

49. #63214

M₁₆

semidirect

114. C₁₇

#12019

semidirect

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

subgroup	index
#14544345	2
#14544339	2
#14543810	2
D₁₇×#346	2
#12019	17

subgroup

index

D₁₇×#346

Subgroups — #17960133

Conjugacy classes of subgroups

Complements

Maximal subgroups