Group #14529739 — subgroups

Subgroups — #14529739

Order 2176.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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×#10587556	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762572	C₂	1	2	yes	maximal
4	#11762572	C₂	1	2	yes	maximal
5	C₂×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×#63225	—	2	1	no	—
12	C₂×#63225	—	2	1	no	—
13	C₂×C₄×Dic₁₇	—	2	1	no	—
14	#10587556	—	2	2	no	—
15	#10587556	C₄	1	4	yes	complemented
16	C₁₇×C₄⋊C₈	—	2	2	no	—
17	#10587502	—	2	1	no	—
18	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₇	—	2	2	no	—
22	C₂²×Dic₁₇	—	2	1	no	—
23	C₂²×Dic₁₇	—	2	1	no	—
24	C₂×C₈×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	4	no	—
26	C₄²×C₁₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇	D₄	1	2	yes	—
28	C₄⋊Dic₁₇	—	2	1	no	—
29	C₄⋊Dic₁₇	—	2	1	no	—
30	C₄⋊Dic₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₁₇	Q₈	1	4	yes	—
34	#63225	—	2	4	no	—
35	#63225	—	2	4	no	—
36	C₄×Dic₁₇	—	4	2	no	—
37	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂×Dic₁₇	—	2	2	no	—
48	C₂×Dic₁₇	—	2	2	no	—
49	C₂×Dic₁₇	—	2	4	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	1	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	C₂×Dic₁₇	—	4	1	no	—
55	C₈×C₁₇	—	4	2	no	—
56	#1989	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
59	C₂²×C₁₇	#135	1	2	yes	—
60	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₇	#137	1	1	yes	characteristic
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	—	2	2	no	—
64	C₄×C₁₇	C₄×Q₈	1	2	yes	—
65	C₄×C₁₇	#124	1	1	yes	characteristic
66	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₇	—	4	2	no	—
68	Dic₁₇	—	4	2	no	—
69	Dic₁₇	—	4	4	no	—
70	C₂×#125	—	17	1	no	—
71	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	17	1	no	—
73	#340	—	17	2	no	—
74	#340	—	17	2	no	—
75	C₂×C₁₇	#391	1	1	yes	characteristic
76	C₂×C₁₇	#416	1	1	yes	characteristic
77	C₂×C₁₇	#417	1	1	yes	characteristic
78	C₂×C₁₇	#478	1	2	yes	—
79	C₂×C₁₇	#479	1	2	yes	—
80	C₂²×C₈	C₂×D₁₇	1	2	yes	—
81	#125	—	17	4	no	—
82	#125	—	34	2	no	—
83	C₂×C₄⋊C₄	—	17	1	no	—
84	C₂×C₄⋊C₄	—	17	1	no	—
85	C₂×C₄⋊C₄	—	17	2	no	—
86	C₂×C₄⋊C₄	—	34	1	no	—
87	C₂×C₄⋊C₄	—	34	1	no	—
88	C₂×C₄²	—	34	1	no	—
89	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#95	—	34	1	no	—
92	C₁₇	#1989	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	34	2	no	—
94	C₂²×C₄	—	34	1	no	—
95	C₂²×C₄	—	34	1	no	—
96	C₂²×C₄	D₆₈	1	1	yes	characteristic
97	C₂²×C₄	D₆₈	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	17	2	no	—
102	C₄⋊C₄	—	17	2	no	—
103	C₄⋊C₄	—	17	4	no	—
104	C₄⋊C₄	—	34	2	no	—
105	C₄⋊C₄	—	34	1	no	—
106	C₄⋊C₄	—	34	1	no	—
107	C₄⋊C₄	—	34	4	no	—
108	C₄⋊C₄	—	34	4	no	—
109	C₄²	—	68	2	no	—
110	C₄²	C₄×D₁₇	1	2	yes	—
111	C₄²	C₂²⋊D₁₇	1	2	yes	—
112	C₂³	C₂×D₆₈	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	34	4	no	—
117	C₂×C₄	—	34	2	no	—
118	C₂×C₄	—	34	2	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	—	68	2	no	—
121	C₂×C₄	—	68	2	no	—
122	C₂×C₄	—	68	1	no	—
123	C₂×C₄	—	68	1	no	—
124	C₂×C₄	#63227	1	2	yes	—
125	C₂×C₄	#63227	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
127	C₂×C₄	#63254	1	2	yes	—
128	C₂×C₄	Q₈×D₁₇	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₂²	#10587583	1	2	yes	—
132	C₂²	#10587590	1	2	yes	—
133	C₂²	#10587594	1	1	yes	characteristic
134	C₂²	#10587595	1	1	yes	characteristic
135	C₂²	C₂×#63227	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	68	2	no	—
139	C₄	—	68	2	no	—
140	C₄	—	68	4	no	—
141	C₄	#10587573	1	2	yes	—
142	C₄	#10587622	1	1	yes	characteristic
143	C₄	#10587630	1	1	yes	characteristic
144	C₂	#11762714	1	1	yes	characteristic
145	C₂	#11762806	1	1	yes	characteristic
146	C₂	#11762811	1	1	yes	characteristic
147	C₂	#11763110	1	2	yes	—
148	C₂	#11763117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587556

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₇

C₂

yes

maximal, characteristic

#11762572

C₂

yes

maximal

#11762572

C₂

yes

maximal

C₂×C₁₇×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

—

C₂×#63225

—

C₂×C₄×Dic₁₇

—

#10587556

—

#10587556

C₄

yes

complemented

C₁₇×C₄⋊C₈

—

#10587502

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

Q₈

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#1989

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₂²×C₁₇

#135

yes

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₈

yes

—

C₄×C₁₇

#124

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×#125

—

C₂×C₄⋊C₈

D₁₇

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

#340

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#479

yes

—

C₂²×C₈

C₂×D₁₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

#95

—

C₁₇

#1989

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

C₄×D₁₇

yes

—

111

C₄²

C₂²⋊D₁₇

yes

—

112

C₂³

C₂×D₆₈

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#63227

yes

—

125

C₂×C₄

#63227

yes

—

126

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

127

C₂×C₄

#63254

yes

—

128

C₂×C₄

Q₈×D₁₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

130

C₈

—

131

C₂²

#10587583

yes

—

132

C₂²

#10587590

yes

—

133

C₂²

#10587594

yes

characteristic

134

C₂²

#10587595

yes

characteristic

135

C₂²

C₂×#63227

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#10587573

yes

—

142

C₄

#10587622

yes

characteristic

143

C₄

#10587630

yes

characteristic

144

C₂

#11762714

yes

characteristic

145

C₂

#11762806

yes

characteristic

146

C₂

#11762811

yes

characteristic

147

C₂

#11763110

yes

—

148

C₂

#11763117

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#1989
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₇

hypocenter

C₁₇

#1989

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. #10587556	C₄	semidirect
92. C₁₇	#1989	semidirect

H ≅ G/N

type

15. #10587556

C₄

semidirect

92. C₁₇

#1989

semidirect

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

subgroup	index
C₂×#10587556	2
C₄×C₄⋊Dic₁₇	2
#11762572	2
C₂×C₁₇×C₄⋊C₈	2
#1989	17

subgroup

index

C₂×#10587556

C₄×C₄⋊Dic₁₇

#11762572

C₂×C₁₇×C₄⋊C₈

#1989

Subgroups — #14529739

Conjugacy classes of subgroups

Complements

Maximal subgroups