Group #14530104 — subgroups

Subgroups — #14530104

Order 2176.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×#10587485	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587506	C₂	1	2	yes	maximal
3	#11762716	C₂	1	2	yes	maximal
4	#11762572	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₇	—	2	4	no	—
8	C₂×Q₁₆×C₁₇	—	2	2	no	—
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂²×#3622	C₂²	1	1	yes	characteristic
13	#10587506	—	2	4	no	—
14	#10587502	—	2	2	no	—
15	#10587502	—	2	2	no	—
16	#10587485	—	2	2	no	—
17	C₂×Q₈×C₁₇	—	2	2	no	—
18	C₂×Q₈×C₁₇	—	2	2	no	—
19	C₂×Q₈×C₁₇	—	2	2	no	—
20	C₂×Q₈×C₁₇	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	2	no	—
22	C₂×Q₈×C₁₇	D₄	1	4	yes	—
23	C₂²×C₄×C₁₇	—	2	2	no	—
24	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₇	—	2	2	no	—
26	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
27	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
28	Q₁₆×C₁₇	—	2	8	no	—
29	C₂×C₈×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	D₄	1	2	yes	—
31	C₄⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄×Dic₁₇	—	2	1	no	—
34	C₄×Dic₁₇	—	2	1	no	—
35	C₂×#3622	—	2	1	no	—
36	C₂×#3622	—	2	2	no	—
37	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₇	—	2	4	no	—
39	Q₈×C₁₇	—	2	4	no	—
40	Q₈×C₁₇	—	4	2	no	—
41	Q₈×C₁₇	—	4	2	no	—
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	—	4	2	no	—
45	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#41	1	2	yes	—
49	C₂×Dic₁₇	—	2	1	no	—
50	C₂×Dic₁₇	—	2	1	no	—
51	C₂×Dic₁₇	—	2	1	no	—
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	C₂×Dic₁₇	—	4	2	no	—
55	C₂×Dic₁₇	—	4	2	no	—
56	C₂×Dic₁₇	Q₁₆	1	2	yes	complemented
57	C₈×C₁₇	—	2	2	no	—
58	#3622	—	4	1	no	—
59	#2056	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	#121	1	1	yes	characteristic
61	C₂²×C₁₇	#121	1	2	yes	—
62	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₇	#135	1	1	yes	characteristic
64	C₂²×C₁₇	#137	1	2	yes	—
65	C₄×C₁₇	—	4	4	no	—
66	C₄×C₁₇	#120	1	1	yes	characteristic
67	C₄×C₁₇	#123	1	2	yes	—
68	C₄×C₁₇	#124	1	1	yes	characteristic
69	Dic₁₇	—	2	2	no	—
70	Dic₁₇	—	4	1	no	—
71	Dic₁₇	—	8	2	no	—
72	C₂²×Q₁₆	D₁₇	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	17	1	no	—
74	C₂×#103	—	17	2	no	—
75	#391	—	17	2	no	—
76	#340	—	17	1	no	—
77	C₂×C₁₇	#394	1	1	yes	characteristic
78	C₂×C₁₇	#462	1	2	yes	—
79	C₂×C₁₇	#464	1	2	yes	—
80	C₂×C₁₇	#467	1	1	yes	characteristic
81	C₂×C₁₇	#470	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₁₇	1	2	yes	—
83	C₂×Q₁₆	—	2	4	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂²×C₈	—	17	1	no	—
86	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	17	2	no	—
88	C₂×C₄²	—	17	1	no	—
89	C₄⋊C₈	—	34	2	no	—
90	#103	—	34	4	no	—
91	#95	—	34	2	no	—
92	#95	—	34	2	no	—
93	C₁₇	#2056	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	C₂²⋊D₁₇	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	17	1	no	—
102	C₂²×C₄	—	17	1	no	—
103	C₂²×C₄	—	34	2	no	—
104	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
105	Q₁₆	—	2	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	34	2	no	—
108	C₂×C₈	—	34	1	no	—
109	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
110	C₄⋊C₄	—	34	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄²	—	34	1	no	—
113	C₄²	—	34	1	no	—
114	C₂³	D₄×D₁₇	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	17	2	no	—
123	C₂×C₄	—	34	1	no	—
124	C₂×C₄	—	34	1	no	—
125	C₂×C₄	—	34	2	no	—
126	C₂×C₄	—	34	1	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	—	68	2	no	—
130	C₂×C₄	#63254	1	1	yes	characteristic
131	C₂×C₄	#63254	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	68	1	no	—
136	C₂²	Q₁₆×D₁₇	1	1	yes	characteristic
137	C₂²	#10587604	1	2	yes	—
138	C₂²	#10587605	1	1	yes	characteristic
139	C₂²	#10587624	1	1	yes	characteristic
140	C₂²	#10587625	1	2	yes	—
141	C₄	—	4	4	no	—
142	C₄	—	34	2	no	—
143	C₄	—	68	1	no	—
144	C₄	—	136	2	no	—
145	C₄	#10587621	1	2	yes	—
146	C₄	#10587622	1	1	yes	characteristic
147	C₄	#10587639	1	1	yes	characteristic
148	C₂	#11762723	1	1	yes	characteristic
149	C₂	#11763017	1	2	yes	—
150	C₂	#11763033	1	2	yes	—
151	C₂	#11763047	1	1	yes	characteristic
152	C₂	#11763064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587485

C₂

yes

maximal, characteristic

C₂×#10587506

C₂

yes

maximal

#11762716

C₂

yes

maximal

#11762572

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₄

yes

characteristic

C₂²×Dic₁₇

D₄

yes

characteristic

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

Q₁₆

yes

complemented

C₈×C₁₇

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#124

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂²×Q₁₆

D₁₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#391

—

#340

—

C₂×C₁₇

#394

yes

characteristic

C₂×C₁₇

#462

yes

—

C₂×C₁₇

#464

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

C₂×D₁₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₁₇

#2056

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₁₇

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₁₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₁₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

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

—

130

C₂×C₄

#63254

yes

characteristic

131

C₂×C₄

#63254

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

Q₁₆×D₁₇

yes

characteristic

137

C₂²

#10587604

yes

—

138

C₂²

#10587605

yes

characteristic

139

C₂²

#10587624

yes

characteristic

140

C₂²

#10587625

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

136

—

145

C₄

#10587621

yes

—

146

C₄

#10587622

yes

characteristic

147

C₄

#10587639

yes

characteristic

148

C₂

#11762723

yes

characteristic

149

C₂

#11763017

yes

—

150

C₂

#11763033

yes

—

151

C₂

#11763047

yes

characteristic

152

C₂

#11763064

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂²×Q₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂²×Q₁₆	D₁₇
hypocenter	C₁₇	#2056
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂²×Q₁₆×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₂²×Q₁₆

D₁₇

hypocenter

C₁₇

#2056

LCS

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
56. C₂×Dic₁₇	Q₁₆	semidirect
93. C₁₇	#2056	semidirect

H ≅ G/N

type

56. C₂×Dic₁₇

Q₁₆

semidirect

93. C₁₇

#2056

semidirect

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

subgroup	index
C₂×#10587485	2
C₂×#10587506	2
#11762716	2
#11762572	2
C₂²×Q₁₆×C₁₇	2
#2056	17

subgroup

index

C₂²×Q₁₆×C₁₇

Subgroups — #14530104

Conjugacy classes of subgroups

Complements

Maximal subgroups