Group #14529656 — subgroups

Subgroups — #14529656

Order 2176.

304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10587488	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587506	C₂	1	2	yes	maximal
3	C₂×#10587486	C₂	1	1	yes	maximal, characteristic
4	#11762716	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₇	—	2	2	no	—
8	C₂×Q₁₆×C₁₇	—	2	2	no	—
9	C₂×Q₁₆×C₁₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂²×#3622	C₂²	1	1	yes	characteristic
14	#10587506	—	2	4	no	—
15	#10587502	—	2	4	no	—
16	#10587488	—	2	2	no	—
17	#10587486	—	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	—	2	2	no	—
23	C₂×Q₈×C₁₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₇	—	2	2	no	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₇	—	2	2	no	—
27	C₂²×Dic₁₇	D₄	1	2	yes	—
28	Q₁₆×C₁₇	—	2	8	no	—
29	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₇	D₄	1	2	yes	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	C₄×Dic₁₇	—	2	1	no	—
35	C₄×Dic₁₇	—	2	1	no	—
36	C₂×#3622	—	2	1	no	—
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	D₄	1	2	yes	—
39	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₇	—	2	4	no	—
41	Q₈×C₁₇	—	2	4	no	—
42	Q₈×C₁₇	—	4	2	no	—
43	Q₈×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	—	2	4	no	—
45	C₂×C₄×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	—	2	4	no	—
55	C₂×Dic₁₇	—	4	2	no	—
56	C₂×Dic₁₇	—	4	4	no	—
57	C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
58	#3622	—	2	2	no	—
59	#1971	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₇	#121	1	1	yes	characteristic
62	C₂²×C₁₇	#127	1	1	yes	characteristic
63	C₂²×C₁₇	#137	1	4	yes	—
64	C₄×C₁₇	—	4	4	no	—
65	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₁₇	#121	1	1	yes	characteristic
68	C₄×C₁₇	#124	1	1	yes	characteristic
69	Dic₁₇	—	4	2	no	—
70	Dic₁₇	—	8	2	no	—
71	C₂²×Q₁₆	D₁₇	1	1	yes	characteristic
72	C₂×#106	—	17	1	no	—
73	C₂×#103	—	17	2	no	—
74	C₂×#97	—	17	1	no	—
75	#391	—	17	2	no	—
76	C₂×C₁₇	#390	1	1	yes	characteristic
77	C₂×C₁₇	#441	1	2	yes	—
78	C₂×C₁₇	#470	1	2	yes	—
79	C₂×C₁₇	#497	1	2	yes	—
80	C₂²×Q₈	C₂×D₁₇	1	2	yes	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	Dic₁₇	1	4	yes	complemented
84	C₂²×C₈	—	17	1	no	—
85	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	17	2	no	—
87	C₂×C₄²	—	17	1	no	—
88	#106	—	34	2	no	—
89	#103	—	34	4	no	—
90	#97	—	34	2	no	—
91	#95	—	34	4	no	—
92	C₁₇	#1971	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	2	2	no	—
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	C₂×Dic₁₇	1	4	yes	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	17	2	no	—
101	C₂²×C₄	—	34	2	no	—
102	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
103	Q₁₆	—	2	8	no	—
104	C₂×C₈	—	17	2	no	—
105	C₂×C₈	—	34	1	no	—
106	C₂×C₈	—	34	1	no	—
107	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
108	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
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	—	34	4	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	68	4	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
127	C₂×C₄	#63254	1	1	yes	characteristic
128	C₂×C₄	#63254	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
132	C₈	—	34	2	no	—
133	C₈	C₂²⋊Dic₁₇	1	4	yes	—
134	C₂²	#10587604	1	4	yes	—
135	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
136	C₂²	#10587624	1	1	yes	characteristic
137	C₂²	#10587626	1	1	yes	characteristic
138	C₄	—	4	4	no	—
139	C₄	—	68	2	no	—
140	C₄	—	136	2	no	—
141	C₄	D₄×Dic₁₇	1	1	yes	characteristic
142	C₄	#10587622	1	1	yes	characteristic
143	C₄	#10587624	1	1	yes	characteristic
144	C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
145	C₂	#11762712	1	1	yes	characteristic
146	C₂	#11762916	1	2	yes	—
147	C₂	#11763064	1	2	yes	—
148	C₂	#11763202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587488

C₂

yes

maximal, characteristic

C₂×#10587506

C₂

yes

maximal

C₂×#10587486

C₂

yes

maximal, characteristic

#11762716

C₂

yes

maximal

C₂²×Q₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

C₄

yes

complemented

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

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₄

yes

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

C₂²⋊C₄

yes

—

#3622

—

#1971

—

maximal, Sylow

C₂²×C₁₇

C₄×D₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂²×Q₁₆

D₁₇

yes

characteristic

C₂×#106

—

C₂×#103

—

C₂×#97

—

#391

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

#441

yes

—

C₂×C₁₇

#470

yes

—

C₂×C₁₇

#497

yes

—

C₂²×Q₈

C₂×D₁₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₁₇

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

C₁₇

#1971

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₇

yes

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₁₇

yes

characteristic

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂×Dic₁₇

yes

—

108

C₂×C₈

C₂²⋊D₁₇

yes

—

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

D₄×D₁₇

yes

characteristic

127

C₂×C₄

#63254

yes

characteristic

128

C₂×C₄

#63254

yes

characteristic

129

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

132

C₈

—

133

C₈

C₂²⋊Dic₁₇

yes

—

134

C₂²

#10587604

yes

—

135

C₂²

D₄×Dic₁₇

yes

characteristic

136

C₂²

#10587624

yes

characteristic

137

C₂²

#10587626

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

136

—

141

C₄

D₄×Dic₁₇

yes

characteristic

142

C₄

#10587622

yes

characteristic

143

C₄

#10587624

yes

characteristic

144

C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

145

C₂

#11762712

yes

characteristic

146

C₂

#11762916

yes

—

147

C₂

#11763064

yes

—

148

C₂

#11763202

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂²×Q₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂²×Q₁₆	D₁₇
hypocenter	C₁₇	#1971
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂²×Q₁₆×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂²×Q₁₆

D₁₇

hypocenter

C₁₇

#1971

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
9. C₂×Q₁₆×C₁₇	C₄	semidirect
83. C₂×Q₁₆	Dic₁₇	semidirect
92. C₁₇	#1971	semidirect

H ≅ G/N

type

9. C₂×Q₁₆×C₁₇

C₄

semidirect

83. C₂×Q₁₆

Dic₁₇

semidirect

92. C₁₇

#1971

semidirect

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

subgroup	index
C₂×#10587488	2
C₂×#10587506	2
C₂×#10587486	2
#11762716	2
C₂²×Q₁₆×C₁₇	2
#1971	17

subgroup

index

C₂²×Q₁₆×C₁₇

Subgroups — #14529656

Conjugacy classes of subgroups

Complements

Maximal subgroups