Group #16946179 — subgroups

Subgroups — #16946179

Order 4352.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14530603	C₂	1	2	yes	maximal, complemented
2	C₈×D₁₃₆	C₂	1	1	yes	maximal, characteristic
3	#14526738	C₂	1	2	yes	maximal, complemented
4	#14526702	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₁₇	—	2	2	no	—
7	C₄×D₁₃₆	C₄	1	1	yes	characteristic
8	C₈×D₆₈	C₂²	1	2	yes	—
9	#11762588	—	2	2	no	—
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762538	C₄	1	2	yes	—
14	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₁₇	C₂²	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×C₁₆×C₁₇	—	2	2	no	—
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587492	C₈	1	2	yes	—
24	#10587491	—	2	2	no	—
25	#10587489	C₈	1	1	yes	characteristic
26	C₂×#63214	—	2	1	no	—
27	C₂×#63214	D₄	1	2	yes	—
28	C₄⋊#3622	C₂×C₄	1	2	yes	—
29	C₁₆×D₁₇	—	4	4	no	—
30	C₂×D₆₈	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₁₇	—	2	2	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
37	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	#63227	—	2	2	no	—
39	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
40	C₂×#3622	—	2	2	no	—
41	D₁₃₆	C₁₆	1	4	yes	complemented
42	C₈×D₁₇	—	4	2	no	—
43	C₈×D₁₇	—	4	2	no	—
44	C₁₆×C₁₇	—	4	2	no	—
45	#63214	—	2	2	no	—
46	#63214	—	4	1	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	—	2	1	no	—
59	C₈×C₁₇	—	4	1	no	—
60	C₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
61	C₈×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
62	#3622	—	4	2	no	—
63	D₄×C₁₆	—	17	2	no	—
64	C₈×D₈	—	17	1	no	—
65	#1367	D₁₇	1	1	yes	characteristic, complemented
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	C₂×D₁₇	1	2	yes	—
82	C₂²⋊C₁₆	—	34	2	no	—
83	C₄×C₁₆	—	17	1	no	—
84	C₄×C₁₆	—	17	1	no	—
85	#335	—	17	1	no	—
86	D₄⋊C₈	—	17	2	no	—
87	C₈²	C₂×D₁₇	1	1	yes	characteristic
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	—	2	2	no	—
96	C₂×C₁₆	—	17	2	no	—
97	C₂×C₁₆	—	34	1	no	—
98	C₂×C₁₆	—	68	4	no	—
99	#107	—	17	1	no	—
100	C₄⋊C₈	—	17	2	no	—
101	#102	—	17	2	no	—
102	C₂²⋊C₈	—	34	2	no	—
103	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
104	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₁₇	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	1	no	—
111	C₂×C₈	—	34	2	no	—
112	C₂×C₈	—	68	2	no	—
113	C₂×C₈	—	68	2	no	—
114	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
115	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
116	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
117	C₂×C₈	#63256	1	1	yes	characteristic
118	C₄⋊C₄	—	17	2	no	—
119	C₂²⋊C₄	—	34	2	no	—
120	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
121	C₁₆	—	4	2	no	—
122	C₁₆	—	34	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	#10587579	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
132	C₂×C₄	#10587593	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	68	2	no	—
137	C₈	C₁₆×D₁₇	1	2	yes	complemented
138	C₈	#10587600	1	2	yes	—
139	C₂²	—	34	4	no	—
140	C₂²	—	68	2	no	—
141	C₂²	—	68	2	no	—
142	C₂²	#11762874	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	68	2	no	—
145	C₄	#11762920	1	1	yes	characteristic
146	C₄	#11762930	1	1	yes	characteristic
147	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
148	C₄	#11763227	1	1	yes	characteristic
149	C₂	—	68	4	no	—
150	C₂	#14527607	1	1	yes	characteristic
151	C₂	#14530603	1	1	yes	characteristic
152	C₂	#14530625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530603

C₂

yes

maximal, complemented

C₈×D₁₃₆

C₂

yes

maximal, characteristic

#14526738

C₂

yes

maximal, complemented

#14526702

C₂

yes

maximal, characteristic, complemented

C₁₇×#1367

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₁₇

—

C₄×D₁₃₆

C₄

yes

characteristic

C₈×D₆₈

C₂²

yes

—

#11762588

—

C₄×#63214

C₂²

yes

characteristic

C₄×#63214

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762538

C₄

yes

—

C₁₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×D₁₃₆

C₈

yes

characteristic

C₂×C₈×D₁₇

—

C₄×D₆₈

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587492

C₈

yes

—

#10587491

—

#10587489

C₈

yes

characteristic

C₂×#63214

—

C₂×#63214

D₄

yes

—

C₄⋊#3622

C₂×C₄

yes

—

C₁₆×D₁₇

—

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

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

C₂×C₈

yes

—

C₂×#3622

—

D₁₃₆

C₁₆

yes

complemented

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

—

—

—

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

—

C₈×C₁₇

C₂×C₁₆

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

#3622

—

D₄×C₁₆

—

C₈×D₈

—

#1367

D₁₇

yes

characteristic, complemented

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

C₂×D₁₇

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×D₁₇

yes

characteristic

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

—

101

#102

—

102

C₂²⋊C₈

—

103

C₄×C₈

C₄×D₁₇

yes

characteristic

104

C₄×C₈

C₄×D₁₇

yes

characteristic

105

C₄×C₈

C₂²×D₁₇

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

C₈×D₁₇

yes

characteristic

115

C₂×C₈

C₈×D₁₇

yes

characteristic

116

C₂×C₈

D₄×D₁₇

yes

characteristic

117

C₂×C₈

#63256

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂×C₄×D₁₇

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

#10587579

yes

characteristic

131

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

132

C₂×C₄

#10587593

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

C₁₆×D₁₇

yes

complemented

138

C₈

#10587600

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11762874

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#11762920

yes

characteristic

146

C₄

#11762930

yes

characteristic

147

C₄

C₂×C₁₆×D₁₇

yes

characteristic

148

C₄

#11763227

yes

characteristic

149

C₂

—

150

C₂

#14527607

yes

characteristic

151

C₂

#14530603

yes

characteristic

152

C₂

#14530625

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1367	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×D₁₃₆	C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1367	D₁₇
hypocenter	C₁₇	#12062
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₁₇

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1367

C₂

socle

C₂²×C₁₇

C₈×D₄

prime radical

C₂×D₁₃₆

C₈

cosocle

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
1. #14530603	C₂	semidirect
3. #14526738	C₂	semidirect
4. #14526702	C₂	semidirect
5. C₁₇×#1367	C₂	semidirect
41. D₁₃₆	C₁₆	semidirect
54. D₆₈	C₂×C₁₆	semidirect
60. C₈×C₁₇	C₂×C₁₆	semidirect
65. #1367	D₁₇	semidirect
106. C₁₇	#12062	semidirect
137. C₈	C₁₆×D₁₇	semidirect

H ≅ G/N

type

1. #14530603

C₂

semidirect

3. #14526738

C₂

semidirect

4. #14526702

C₂

semidirect

5. C₁₇×#1367

C₂

semidirect

41. D₁₃₆

C₁₆

semidirect

54. D₆₈

C₂×C₁₆

semidirect

60. C₈×C₁₇

C₂×C₁₆

semidirect

65. #1367

D₁₇

semidirect

106. C₁₇

#12062

semidirect

137. C₈

C₁₆×D₁₇

semidirect

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

subgroup	index
#14530603	2
C₈×D₁₃₆	2
#14526738	2
#14526702	2
C₁₇×#1367	2
#12062	17

subgroup

index

C₈×D₁₃₆

C₁₇×#1367

Subgroups — #16946179

Conjugacy classes of subgroups

Complements

Maximal subgroups