Group #14528296 — subgroups

Subgroups — #14528296

Order 2176.

166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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	#11763212	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763015	C₂	1	1	yes	maximal, characteristic
3	#11763011	C₂	1	1	yes	maximal, characteristic
4	#11762886	C₂	1	1	yes	maximal, characteristic
5	#11762559	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762554	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#63230	—	2	1	no	—
10	C₂×#63219	—	2	1	no	—
11	C₂×#63218	—	2	1	no	—
12	C₄×D₆₈	C₂²	1	1	yes	characteristic
13	#10587555	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	#10587506	—	2	1	no	—
17	#10587491	—	2	1	no	—
18	#10587487	—	2	1	no	—
19	#10587478	—	2	1	no	—
20	#10587478	—	2	1	no	—
21	C₄⋊#3622	C₂²	1	1	yes	characteristic
22	C₄×#3622	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	—	2	1	no	—
24	C₂×D₆₈	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	—	2	1	no	—
26	C₂×Dic₃₄	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	1	no	—
28	C₂×C₈×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	#63230	—	4	2	no	—
33	#63227	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	D₄	1	1	yes	characteristic
39	C₂×#3622	D₄	1	1	yes	characteristic
40	#63219	—	4	2	no	—
41	#63218	—	4	2	no	—
42	C₂²×D₁₇	—	2	1	no	—
43	Q₈×C₁₇	—	4	1	no	—
44	Q₈×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	—	4	1	no	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	1	no	—
50	C₂×Dic₁₇	—	4	1	no	—
51	D₆₈	—	2	1	no	—
52	D₆₈	—	2	1	no	—
53	C₄×D₁₇	—	4	1	no	—
54	C₄×D₁₇	—	4	1	no	—
55	Dic₃₄	—	4	1	no	—
56	Dic₃₄	—	4	1	no	—
57	C₈×C₁₇	—	4	1	no	—
58	C₈×C₁₇	—	4	1	no	—
59	#3622	—	2	1	no	—
60	#3622	—	4	1	no	—
61	#3622	SD₁₆	1	2	yes	complemented
62	#1693	—	17	1	no	maximal, Sylow
63	C₂²×C₁₇	#121	1	1	yes	characteristic
64	C₂×D₁₇	—	4	1	no	—
65	C₂×D₁₇	—	4	1	no	—
66	C₂×D₁₇	—	4	1	no	—
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	—	8	1	no	—
69	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₁₇	#137	1	1	yes	characteristic
73	Dic₁₇	—	4	1	no	—
74	Dic₁₇	—	8	1	no	—
75	#500	—	17	1	no	—
76	#461	—	17	1	no	—
77	#461	—	17	1	no	—
78	#436	—	17	1	no	—
79	#334	—	17	1	no	—
80	#332	—	17	1	no	—
81	#332	D₁₇	1	1	yes	characteristic, complemented
82	C₂×C₁₇	#461	1	1	yes	characteristic
83	C₂×C₁₇	#469	1	1	yes	characteristic
84	C₂×C₁₇	#473	1	1	yes	characteristic
85	D₁₇	—	8	1	no	—
86	C₂×SD₁₆	—	34	1	no	—
87	C₂×SD₁₆	—	34	1	no	—
88	C₂×M₁₆	—	34	1	no	—
89	C₄⋊Q₈	—	17	1	no	—
90	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
91	C₄×D₄	—	17	1	no	—
92	#107	—	34	1	no	—
93	#107	—	34	1	no	—
94	C₄⋊C₈	—	17	1	no	—
95	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
96	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
97	#103	—	34	1	no	—
98	#103	—	34	1	no	—
99	C₂²⋊C₈	—	34	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₁₇	#1693	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	34	1	no	—
104	C₂×D₄	—	17	1	no	—
105	C₂²×C₄	—	34	1	no	—
106	SD₁₆	—	68	2	no	—
107	SD₁₆	—	68	2	no	—
108	M₁₆	—	68	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	17	1	no	—
117	C₄⋊C₄	—	34	1	no	—
118	C₄⋊C₄	—	34	1	no	—
119	C₂²⋊C₄	—	34	1	no	—
120	C₄²	C₂²×D₁₇	1	1	yes	characteristic
121	C₂³	—	34	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	68	1	no	—
125	Q₈	—	68	1	no	—
126	D₄	—	34	1	no	—
127	D₄	—	34	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	34	1	no	—
130	C₂×C₄	—	68	1	no	—
131	C₂×C₄	—	68	1	no	—
132	C₂×C₄	—	68	1	no	—
133	C₂×C₄	#63252	1	1	yes	characteristic
134	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
135	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	17	2	no	—
139	C₈	—	34	1	no	—
140	C₈	—	68	1	no	—
141	C₂²	—	68	1	no	—
142	C₂²	—	68	1	no	—
143	C₂²	—	68	1	no	—
144	C₂²	#10587570	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	68	1	no	—
148	C₄	—	136	1	no	—
149	C₄	#10587597	1	1	yes	characteristic
150	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
151	C₄	#10587600	1	1	yes	characteristic
152	C₄	#10587604	1	1	yes	characteristic
153	C₂	—	136	1	no	—
154	C₂	#11763013	1	1	yes	characteristic
155	C₂	#11763061	1	1	yes	characteristic
156	C₂	#11763083	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763212

C₂

yes

maximal, characteristic, complemented

#11763015

C₂

yes

maximal, characteristic

#11763011

C₂

yes

maximal, characteristic

#11762886

C₂

yes

maximal, characteristic

#11762559

C₂

yes

maximal, characteristic, complemented

#11762554

C₂

yes

maximal, characteristic, complemented

C₁₇×#332

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

—

—

—

C₄×D₆₈

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×D₆₈

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63230

—

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

#63219

—

#63218

—

C₂²×D₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

—

#3622

SD₁₆

yes

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

C₄⋊C₈

C₂×D₁₇

yes

characteristic

#103

—

#103

—

C₂²⋊C₈

—

100

C₄×C₈

—

101

C₁₇

#1693

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

SD₁₆

—

107

SD₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×D₁₇

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#63252

yes

characteristic

134

C₂×C₄

D₄×D₁₇

yes

characteristic

135

C₂×C₄

D₄×D₁₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#10587570

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

136

—

149

C₄

#10587597

yes

characteristic

150

C₄

SD₁₆×D₁₇

yes

characteristic

151

C₄

#10587600

yes

characteristic

152

C₄

#10587604

yes

characteristic

153

C₂

—

136

—

154

C₂

#11763013

yes

characteristic

155

C₂

#11763061

yes

characteristic

156

C₂

#11763083

yes

characteristic

	subgroup	G/H
center	C₂²	#10587570
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#332	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂×D₆₈	D₄
hypercenter	#332	D₁₇
hypocenter	C₁₇	#1693
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#10587570

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#332

C₂

socle

C₂²×C₁₇

#121

prime radical

C₂×D₆₈

D₄

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #11763212	C₂	semidirect
5. #11762559	C₂	semidirect
6. #11762554	C₂	semidirect
7. C₁₇×#332	C₂	semidirect
61. #3622	SD₁₆	semidirect
81. #332	D₁₇	semidirect
101. C₁₇	#1693	semidirect

H ≅ G/N

type

1. #11763212

C₂

semidirect

5. #11762559

C₂

semidirect

6. #11762554

C₂

semidirect

7. C₁₇×#332

C₂

semidirect

61. #3622

SD₁₆

semidirect

81. #332

D₁₇

semidirect

101. C₁₇

#1693

semidirect

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

subgroup	index
#11763212	2
#11763015	2
#11763011	2
#11762886	2
#11762559	2
#11762554	2
C₁₇×#332	2
#1693	17

subgroup

index

C₁₇×#332

Subgroups — #14528296

Conjugacy classes of subgroups

Complements

Maximal subgroups