Group #17953839 — subgroups

Subgroups — #17953839

Order 4352.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#14544725	C₂	1	2	yes	maximal
2	#14544340	C₂	1	2	yes	maximal
3	#14544339	C₂	1	2	yes	maximal
4	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587536	—	2	4	no	—
6	#11763813	C₂²	1	4	yes	—
7	#11763671	—	2	4	no	—
8	#11763670	—	2	4	no	—
9	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	C₈×#3622	C₄	1	1	yes	characteristic
13	C₈×#3622	C₄	1	1	yes	characteristic
14	C₈×#3622	C₄	1	1	yes	characteristic
15	C₈²×C₁₇	C₄	1	1	yes	characteristic
16	C₂×#63243	—	2	4	no	—
17	C₂×C₈×D₁₇	D₄	1	2	yes	—
18	C₂×C₈×D₁₇	Q₈	1	2	yes	—
19	C₂×C₈×D₁₇	Q₈	1	2	yes	—
20	C₄²×D₁₇	C₂³	1	1	yes	characteristic
21	C₄⋊#3624	—	2	4	no	—
22	C₄⋊#3624	—	2	4	no	—
23	#10587536	—	4	8	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	2	yes	—
28	C₈×Dic₁₇	D₄	1	2	yes	—
29	C₈×Dic₁₇	Q₈	1	2	yes	—
30	C₄×#3622	C₂×C₄	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×#3624	—	4	4	no	—
37	#63243	—	4	8	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
41	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	2	yes	—
46	C₂×#3622	C₄⋊C₄	1	2	yes	—
47	C₂×#3622	C₄⋊C₄	1	2	yes	—
48	C₈×D₁₇	—	2	4	no	—
49	C₈×D₁₇	—	2	4	no	—
50	C₈×D₁₇	SD₁₆	1	8	yes	—
51	#8822	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	#127	1	1	yes	characteristic
57	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	—	2	2	no	—
61	C₄×D₁₇	C₂×SD₁₆	1	4	yes	—
62	#3624	—	8	4	no	—
63	C₈×C₁₇	—	2	2	no	—
64	C₈×C₁₇	#106	1	4	yes	—
65	C₈×C₁₇	#108	1	4	yes	complemented
66	#3622	—	2	2	no	—
67	#3622	#106	1	4	yes	—
68	#3622	#108	1	4	yes	complemented
69	#1848	—	17	2	no	—
70	#1566	—	17	2	no	—
71	#1566	—	17	2	no	—
72	C₂×C₈²	—	17	1	no	—
73	C₂²×C₁₇	#384	1	1	yes	characteristic
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	#498	1	2	yes	—
77	C₄×C₁₇	C₂×#106	1	2	yes	—
78	C₄×C₁₇	C₂×#108	1	2	yes	—
79	C₄×C₁₇	C₂×#108	1	2	yes	—
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	#492	1	2	yes	—
83	C₂×#108	—	34	4	no	—
84	#423	—	17	4	no	—
85	C₂×C₄×C₈	—	17	1	no	—
86	C₂×C₄×C₈	—	17	1	no	—
87	C₂×C₄×C₈	—	17	1	no	—
88	#334	—	34	4	no	—
89	#334	—	34	4	no	—
90	C₈²	—	17	1	no	—
91	C₈²	—	17	1	no	—
92	C₈²	—	17	1	no	—
93	C₈²	#595	1	1	yes	characteristic
94	C₂×C₁₇	#1844	1	1	yes	characteristic
95	C₂×C₁₇	#1848	1	1	yes	characteristic
96	C₂×C₁₇	#1848	1	1	yes	characteristic
97	D₁₇	—	2	2	no	—
98	C₂×M₁₆	—	34	4	no	—
99	C₂²×C₈	—	17	2	no	—
100	C₂²×C₈	—	17	2	no	—
101	C₂²×C₈	—	17	2	no	—
102	C₂×C₄²	—	17	1	no	—
103	#108	—	68	8	no	—
104	C₄⋊C₈	—	34	4	no	—
105	C₄⋊C₈	—	34	4	no	—
106	C₄×C₈	—	17	2	no	—
107	C₄×C₈	—	17	2	no	—
108	C₄×C₈	—	17	2	no	—
109	C₄×C₈	—	17	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	C₂×#595	1	1	yes	characteristic
113	C₄×C₈	C₂×#595	1	1	yes	characteristic
114	C₄×C₈	C₂×#595	1	1	yes	characteristic
115	C₁₇	#8822	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	1	no	—
119	M₁₆	—	68	8	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	17	2	no	—
122	C₂×C₈	—	17	8	no	—
123	C₂×C₈	—	17	2	no	—
124	C₂×C₈	—	34	4	no	—
125	C₂×C₈	—	34	4	no	—
126	C₂×C₈	—	68	4	no	—
127	C₂×C₈	C₄⋊#595	1	2	yes	—
128	C₂×C₈	C₄⋊#595	1	2	yes	—
129	C₂×C₈	C₄⋊#595	1	2	yes	—
130	C₄²	—	17	1	no	—
131	C₄²	—	17	1	no	—
132	C₄²	—	17	1	no	—
133	C₄²	C₂²×#595	1	1	yes	characteristic
134	C₂³	—	17	1	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	4	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	34	2	no	—
141	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	17	4	no	—
146	C₈	—	17	4	no	—
147	C₈	—	34	2	no	—
148	C₈	—	136	4	no	—
149	C₈	#10587533	1	4	yes	—
150	C₈	#10587536	1	4	yes	complemented
151	C₂²	—	17	2	no	—
152	C₂²	—	34	1	no	—
153	C₂²	—	34	1	no	—
154	C₂²	#11763755	1	1	yes	characteristic
155	C₄	—	17	2	no	—
156	C₄	—	34	1	no	—
157	C₄	—	34	1	no	—
158	C₄	C₂×#10587533	1	2	yes	—
159	C₄	D₁₇⋊#108	1	2	yes	—
160	C₄	C₂×#10587536	1	2	yes	—
161	C₂	—	34	2	no	—
162	C₂	#14544719	1	1	yes	characteristic
163	C₂	#14544724	1	1	yes	characteristic
164	C₂	#14544725	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544725

C₂

yes

maximal

#14544340

C₂

yes

maximal

#14544339

C₂

yes

maximal

C₈²×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587536

—

#11763813

C₂²

yes

—

#11763671

—

#11763670

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈²×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₄²×D₁₇

C₂³

yes

—

—

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

characteristic

C₂×#3624

—

#63243

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

SD₁₆

yes

—

#8822

—

maximal, Sylow

C₂²×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#127

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×SD₁₆

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

yes

—

yes

—

yes

—

—

—

C₂×C₈²

—

C₂²×C₁₇

#384

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#498

yes

—

C₄×C₁₇

C₂×#106

yes

—

C₄×C₁₇

C₂×#108

yes

—

C₄×C₁₇

C₂×#108

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#492

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

#334

—

C₈²

—

C₈²

—

C₈²

—

C₈²

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

#108

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂×#595

yes

characteristic

113

C₄×C₈

C₂×#595

yes

characteristic

114

C₄×C₈

C₂×#595

yes

characteristic

115

C₁₇

#8822

yes

characteristic, complemented, Sylow

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₄⋊#595

yes

—

128

C₂×C₈

C₄⋊#595

yes

—

129

C₂×C₈

C₄⋊#595

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×#595

yes

characteristic

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

136

—

149

C₈

#10587533

yes

—

150

C₈

#10587536

yes

complemented

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#11763755

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂×#10587533

yes

—

159

C₄

D₁₇⋊#108

yes

—

160

C₄

C₂×#10587536

yes

—

161

C₂

—

162

C₂

#14544719

yes

characteristic

163

C₂

#14544724

yes

characteristic

164

C₂

#14544725

yes

characteristic

	subgroup	G/H
center	C₂²	#11763755
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#384
prime radical	C₂²×D₁₇	C₄⋊Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8822
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11763755

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

socle

C₂²×C₁₇

#384

prime radical

C₂²×D₁₇

C₄⋊Q₈

cosocle

C₄²×D₁₇

C₂³

C₈²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
65. C₈×C₁₇	#108	semidirect
68. #3622	#108	semidirect
115. C₁₇	#8822	semidirect
150. C₈	#10587536	semidirect

H ≅ G/N

type

65. C₈×C₁₇

#108

semidirect

68. #3622

#108

semidirect

115. C₁₇

#8822

semidirect

150. C₈

#10587536

semidirect

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

subgroup	index
#14544725	2
#14544340	2
#14544339	2
C₈²×D₁₇	2
#8822	17

subgroup

index

C₈²×D₁₇

Subgroups — #17953839

Conjugacy classes of subgroups

Complements

Maximal subgroups