Group #17960276 — subgroups

Subgroups — #17960276

Order 4352.

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

normal subgroups only

This group has 252 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	#14545163	C₂	1	2	yes	maximal, complemented
2	#14543806	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543735	C₂	1	2	yes	maximal, complemented
4	#14543697	C₂	1	1	yes	maximal, characteristic, complemented
5	#14527578	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	—	2	2	no	—
7	C₄⋊#63216	C₂²	1	2	yes	—
8	#11763704	—	2	2	no	—
9	C₄×#63216	C₂²	1	1	yes	characteristic
10	C₄×#63216	C₂²	1	1	yes	characteristic
11	C₄×D₁₃₆	C₄	1	1	yes	characteristic
12	#11762874	C₂²	1	2	yes	—
13	#11762536	C₄	1	2	yes	—
14	C₈×#3622	C₂²	1	1	yes	characteristic
15	C₁₇×#335	C₄	1	1	yes	characteristic
16	C₂×D₁₃₆	C₈	1	1	yes	characteristic
17	C₂×C₈×D₁₇	—	2	2	no	—
18	C₄×D₆₈	C₂×C₄	1	2	yes	—
19	C₂×#63216	—	2	2	no	—
20	C₂×#63216	—	2	1	no	—
21	C₂×#63216	D₄	1	2	yes	—
22	#10587529	—	4	4	no	—
23	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	#10587492	C₈	1	2	yes	—
26	#10587491	—	2	2	no	—
27	#10587489	C₈	1	1	yes	characteristic
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂³	1	1	yes	characteristic
30	C₂×D₆₈	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₁₇	—	2	2	no	—
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	#63227	—	2	2	no	—
37	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
38	C₂×#3622	—	2	1	no	—
39	C₂×#3622	C₂×D₄	1	1	yes	characteristic
40	C₂×#3622	#41	1	1	yes	characteristic
41	D₁₃₆	C₁₆	1	4	yes	complemented
42	C₈×D₁₇	—	4	2	no	—
43	C₈×D₁₇	—	4	2	no	—
44	#63216	—	2	2	no	—
45	#63216	—	4	1	no	—
46	#63216	—	4	2	no	—
47	#12062	—	17	1	no	maximal, Sylow
48	C₂²×D₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
52	C₂×Dic₁₇	—	2	2	no	—
53	D₆₈	—	2	2	no	—
54	D₆₈	C₂×C₁₆	1	4	yes	complemented
55	C₄×D₁₇	—	4	2	no	—
56	C₄×D₁₇	—	4	2	no	—
57	C₈×C₁₇	—	2	1	no	—
58	C₈×C₁₇	—	4	2	no	—
59	C₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
60	#3622	—	2	1	no	—
61	#3622	—	4	1	no	—
62	#3622	AGL(1,ℤ/8)	1	2	yes	—
63	D₄×C₁₆	—	17	2	no	—
64	C₈×D₈	—	17	1	no	—
65	#1367	—	17	1	no	—
66	D₄⋊C₁₆	—	17	2	no	—
67	#1312	—	17	1	no	—
68	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
69	C₂×D₁₇	—	2	4	no	—
70	C₂×D₁₇	—	4	2	no	—
71	C₂×D₁₇	—	4	2	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	#442	1	1	yes	characteristic
74	C₄×C₁₇	#444	1	1	yes	characteristic
75	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
76	C₄×C₁₇	#504	1	1	yes	characteristic
77	Dic₁₇	—	4	2	no	—
78	C₂²×C₁₆	—	34	2	no	—
79	C₄×D₈	—	17	1	no	—
80	C₈×D₄	—	17	2	no	—
81	C₄⋊C₁₆	—	17	2	no	—
82	C₂²⋊C₁₆	—	34	2	no	—
83	C₄×C₁₆	—	17	1	no	—
84	C₄×C₁₆	—	17	1	no	—
85	#335	#595	1	1	yes	characteristic
86	D₄⋊C₈	—	17	2	no	—
87	C₈²	—	17	1	no	—
88	C₂×C₁₇	#1580	1	1	yes	characteristic
89	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
90	C₂×C₁₇	#2171	1	1	yes	characteristic
91	D₁₇	—	4	4	no	—
92	C₂×D₈	—	17	1	no	—
93	C₂²×C₈	—	34	2	no	—
94	C₄×D₄	—	17	2	no	—
95	C₂×C₁₆	—	17	2	no	—
96	C₂×C₁₆	—	34	1	no	—
97	C₂×C₁₆	—	34	2	no	—
98	C₂×C₁₆	—	68	4	no	—
99	#107	—	17	1	no	—
100	C₄⋊C₈	C₂×#595	1	2	yes	—
101	#102	—	17	2	no	—
102	C₂²⋊C₈	—	34	2	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	1	no	—
105	C₄×C₈	C₂×#595	1	1	yes	characteristic
106	C₁₇	#12062	1	1	yes	characteristic, complemented, Sylow
107	C₂×D₄	—	17	2	no	—
108	C₂²×C₄	—	34	2	no	—
109	D₈	—	17	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₂×C₈	—	68	2	no	—
115	C₂×C₈	—	68	2	no	—
116	C₂×C₈	#63242	1	1	yes	characteristic
117	C₂×C₈	#63242	1	1	yes	characteristic
118	C₄⋊C₄	—	17	2	no	—
119	C₂²⋊C₄	—	34	2	no	—
120	C₄²	C₂²×#595	1	1	yes	characteristic
121	C₁₆	—	34	2	no	—
122	C₁₆	—	68	2	no	—
123	C₁₆	—	68	1	no	—
124	C₂³	—	34	2	no	—
125	D₄	—	17	4	no	—
126	D₄	—	34	2	no	—
127	C₂×C₄	—	34	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	—	68	2	no	—
130	C₂×C₄	C₂×#63242	1	1	yes	characteristic
131	C₂×C₄	D₄×#595	1	1	yes	characteristic
132	C₂×C₄	#10587680	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	4	2	no	—
135	C₈	—	17	2	no	—
136	C₈	—	34	1	no	—
137	C₈	—	68	1	no	—
138	C₈	#10587529	1	2	yes	complemented
139	C₂²	—	34	4	no	—
140	C₂²	—	68	2	no	—
141	C₂²	—	68	2	no	—
142	C₂²	#11763837	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	68	2	no	—
145	C₄	#11763850	1	1	yes	characteristic
146	C₄	#11763855	1	1	yes	characteristic
147	C₄	C₂×#10587529	1	1	yes	characteristic
148	C₄	#11763868	1	1	yes	characteristic
149	C₂	—	68	4	no	—
150	C₂	#14544382	1	1	yes	characteristic
151	C₂	#14545163	1	1	yes	characteristic
152	C₂	#14545176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545163

C₂

yes

maximal, complemented

#14543806

C₂

yes

maximal, characteristic, complemented

#14543735

C₂

yes

maximal, complemented

#14543697

C₂

yes

maximal, characteristic, complemented

#14527578

C₂

yes

maximal, characteristic

C₂×#10587529

—

C₄⋊#63216

C₂²

yes

—

#11763704

—

C₄×#63216

C₂²

yes

characteristic

C₄×#63216

C₂²

yes

characteristic

C₄×D₁₃₆

C₄

yes

characteristic

#11762874

C₂²

yes

—

#11762536

C₄

yes

—

C₈×#3622

C₂²

yes

characteristic

C₁₇×#335

C₄

yes

characteristic

C₂×D₁₃₆

C₈

yes

characteristic

C₂×C₈×D₁₇

—

C₄×D₆₈

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

—

C₂×#63216

D₄

yes

—

#10587529

—

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587492

C₈

yes

—

#10587491

—

#10587489

C₈

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂³

yes

characteristic

C₂×D₆₈

C₂×C₈

yes

—

C₂×C₄×D₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

C₂×C₈

yes

—

C₂×#3622

—

C₂×#3622

C₂×D₄

yes

yes

D₁₃₆

C₁₆

yes

complemented

C₈×D₁₇

—

C₈×D₁₇

—

—

—

—

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

C₂×C₁₆

yes

complemented

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

—

—

yes

—

D₄×C₁₆

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

C₂²×C₁₆

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#595

yes

characteristic

D₄⋊C₈

—

C₈²

—

C₂×C₁₇

#1580

yes

characteristic

C₂×C₁₇

D₄×C₁₆

yes

characteristic

C₂×C₁₇

#2171

yes

characteristic

D₁₇

—

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

100

C₄⋊C₈

C₂×#595

yes

—

101

#102

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×#595

yes

characteristic

106

C₁₇

#12062

yes

characteristic, complemented, Sylow

107

C₂×D₄

—

108

C₂²×C₄

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

#63242

yes

characteristic

117

C₂×C₈

#63242

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×#595

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×#63242

yes

characteristic

131

C₂×C₄

D₄×#595

yes

characteristic

132

C₂×C₄

#10587680

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#10587529

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11763837

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#11763850

yes

characteristic

146

C₄

#11763855

yes

characteristic

147

C₄

C₂×#10587529

yes

characteristic

148

C₄

#11763868

yes

characteristic

149

C₂

—

150

C₂

#14544382

yes

characteristic

151

C₂

#14545163

yes

characteristic

152

C₂

#14545176

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×D₁₃₆	C₈
cosocle	C₄×#3622	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	#12062
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#335

C₄

socle

C₂²×C₁₇

C₈×D₄

prime radical

C₂×D₁₃₆

C₈

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #14545163	C₂	semidirect
2. #14543806	C₂	semidirect
3. #14543735	C₂	semidirect
4. #14543697	C₂	semidirect
41. D₁₃₆	C₁₆	semidirect
54. D₆₈	C₂×C₁₆	semidirect
59. C₈×C₁₇	C₂×C₁₆	semidirect
106. C₁₇	#12062	semidirect
138. C₈	#10587529	semidirect

H ≅ G/N

type

1. #14545163

C₂

semidirect

2. #14543806

C₂

semidirect

3. #14543735

C₂

semidirect

4. #14543697

C₂

semidirect

41. D₁₃₆

C₁₆

semidirect

54. D₆₈

C₂×C₁₆

semidirect

59. C₈×C₁₇

C₂×C₁₆

semidirect

106. C₁₇

#12062

semidirect

138. C₈

#10587529

semidirect

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

subgroup	index
#14545163	2
#14543806	2
#14543735	2
#14543697	2
#14527578	2
#12062	17

subgroup

index

Subgroups — #17960276

Conjugacy classes of subgroups

Complements

Maximal subgroups