Group #14530207 — subgroups

Subgroups — #14530207

Order 2176.

256 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762669	C₂	1	2	yes	maximal
3	#11762572	C₂	1	2	yes	maximal, complemented
4	#11762572	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Dic₁₇	—	2	2	no	—
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
12	C₁₇×C₂²⋊C₈	—	2	2	no	—
13	#10587502	—	2	2	no	—
14	#10587489	—	2	2	no	—
15	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	—	2	1	no	—
17	C₂²×C₄×C₁₇	—	2	1	no	—
18	C₂²×C₄×C₁₇	—	2	2	no	—
19	C₂²×C₄×C₁₇	—	2	2	no	—
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₂²×Dic₁₇	—	2	2	no	—
24	C₂×C₈×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	D₄	1	2	yes	—
28	C₂²⋊Dic₁₇	—	4	4	no	—
29	C₄⋊Dic₁₇	—	2	2	no	—
30	C₄⋊Dic₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₂³×C₁₇	—	2	1	no	—
34	C₂³×C₁₇	—	2	2	no	—
35	C₂³×C₁₇	—	2	1	no	—
36	C₂³×C₁₇	—	2	2	no	—
37	C₂³×C₁₇	D₈	1	2	yes	—
38	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	4	1	no	—
42	C₂×C₄×C₁₇	—	4	1	no	—
43	C₂×C₄×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂×C₄×C₁₇	#41	1	2	yes	—
48	C₂×Dic₁₇	—	4	2	no	—
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	4	1	no	—
56	#2075	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	—	2	2	no	—
58	C₂²×C₁₇	—	2	4	no	—
59	C₂²×C₁₇	—	4	1	no	—
60	C₂²×C₁₇	—	4	1	no	—
61	C₂²×C₁₇	—	4	2	no	—
62	C₂²×C₁₇	#121	1	1	yes	characteristic
63	C₂²×C₁₇	#123	1	2	yes	—
64	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₁₇	#135	1	1	yes	characteristic
66	C₂²×C₁₇	#137	1	2	yes	—
67	C₄×C₁₇	—	4	1	no	—
68	C₄×C₁₇	#123	1	1	yes	characteristic
69	C₄×C₁₇	#124	1	1	yes	characteristic
70	C₄×C₁₇	#126	1	2	yes	—
71	Dic₁₇	—	8	2	no	—
72	Dic₁₇	—	8	2	no	—
73	C₂×#107	—	17	1	no	—
74	C₂×C₂²⋊C₈	D₁₇	1	1	yes	characteristic
75	#380	—	17	2	no	—
76	#340	—	17	1	no	—
77	#340	—	17	2	no	—
78	C₂×C₁₇	—	4	2	no	—
79	C₂×C₁₇	#397	1	1	yes	characteristic
80	C₂×C₁₇	#466	1	1	yes	characteristic
81	C₂×C₁₇	#470	1	1	yes	characteristic
82	C₂×C₁₇	#480	1	2	yes	—
83	C₂×C₁₇	#485	1	2	yes	—
84	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	17	2	no	—
88	C₂×C₄⋊C₄	—	17	2	no	—
89	C₂×C₂²⋊C₄	—	34	2	no	—
90	#107	—	34	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	34	2	no	—
93	C₁₇	#2075	1	1	yes	characteristic, complemented, Sylow
94	C₂⁴	D₆₈	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	34	2	no	—
100	C₂²×C₄	—	34	2	no	—
101	C₂²×C₄	D₆₈	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
107	C₄⋊C₄	—	34	2	no	—
108	C₄⋊C₄	—	34	2	no	—
109	C₄⋊C₄	—	34	2	no	—
110	C₄⋊C₄	—	34	2	no	—
111	C₂²⋊C₄	—	68	4	no	—
112	C₂³	—	2	2	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	1	no	—
116	C₂³	D₁₃₆	1	2	yes	—
117	C₂³	C₂×D₆₈	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	68	2	no	—
124	C₂×C₄	—	68	2	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	#63252	1	1	yes	characteristic
130	C₂×C₄	#63254	1	2	yes	—
131	C₂×C₄	#63254	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	2	no	—
140	C₂²	#10587572	1	2	yes	—
141	C₂²	C₂×D₁₃₆	1	1	yes	characteristic
142	C₂²	#10587590	1	1	yes	characteristic
143	C₂²	#10587595	1	2	yes	—
144	C₂²	#10587616	1	1	yes	characteristic
145	C₄	—	4	1	no	—
146	C₄	—	136	2	no	—
147	C₄	—	136	2	no	—
148	C₄	#10587565	1	2	yes	—
149	C₄	#10587621	1	1	yes	characteristic
150	C₄	#10587622	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#11762734	1	1	yes	characteristic
153	C₂	#11763045	1	1	yes	characteristic
154	C₂	#11763067	1	1	yes	characteristic
155	C₂	#11763123	1	2	yes	—
156	C₂	#11763151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587489

C₂

yes

maximal, characteristic, complemented

#11762669

C₂

yes

maximal

#11762572

C₂

yes

maximal, complemented

#11762572

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₁₇×C₂²⋊C₈

—

#10587502

—

#10587489

—

C₂⁴×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

D₈

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#2075

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#123

yes

—

C₂²×C₁₇

C₂×D₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×C₂²⋊C₈

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#485

yes

—

C₂³×C₄

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#107

—

C₂²⋊C₈

—

#95

—

C₁₇

#2075

yes

characteristic, complemented, Sylow

C₂⁴

D₆₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

D₆₈

yes

characteristic

102

C₂²×C₄

C₂²×D₁₇

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊D₁₇

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

D₁₃₆

yes

—

117

C₂³

C₂×D₆₈

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#63252

yes

characteristic

130

C₂×C₄

#63254

yes

—

131

C₂×C₄

#63254

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#10587572

yes

—

141

C₂²

C₂×D₁₃₆

yes

characteristic

142

C₂²

#10587590

yes

characteristic

143

C₂²

#10587595

yes

—

144

C₂²

#10587616

yes

characteristic

145

C₄

—

146

C₄

—

136

—

147

C₄

—

136

—

148

C₄

#10587565

yes

—

149

C₄

#10587621

yes

characteristic

150

C₄

#10587622

yes

characteristic

151

C₂

—

152

C₂

#11762734

yes

characteristic

153

C₂

#11763045

yes

characteristic

154

C₂

#11763067

yes

characteristic

155

C₂

#11763123

yes

—

156

C₂

#11763151

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₂²⋊C₈	C₂
socle	C₂³×C₁₇	C₂×D₄
prime radical	C₂⁴×C₁₇	D₄
hypercenter	C₂×C₂²⋊C₈	D₁₇
hypocenter	C₁₇	#2075
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₂²⋊C₈

C₂

socle

C₂³×C₁₇

C₂×D₄

prime radical

C₂⁴×C₁₇

D₄

hypercenter

C₂×C₂²⋊C₈

D₁₇

hypocenter

C₁₇

#2075

LCS

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#10587489	C₂	semidirect
3. #11762572	C₂	semidirect
4. #11762572	C₂	semidirect
93. C₁₇	#2075	semidirect

H ≅ G/N

type

1. C₂×#10587489

C₂

semidirect

3. #11762572

C₂

semidirect

4. #11762572

C₂

semidirect

93. C₁₇

#2075

semidirect

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

subgroup	index
C₂×#10587489	2
#11762669	2
#11762572	2
C₂×C₁₇×C₂²⋊C₈	2
#2075	17

subgroup

index

C₂×C₁₇×C₂²⋊C₈

Subgroups — #14530207

Conjugacy classes of subgroups

Complements

Maximal subgroups