Group #14529534 — subgroups

Subgroups — #14529534

Order 2176.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#10587493	C₂	1	1	yes	maximal, characteristic
2	#11762845	C₂	1	1	yes	maximal, characteristic
3	#11762835	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587486	C₂	1	1	yes	maximal, characteristic
5	#11762570	C₂	1	1	yes	maximal, characteristic
6	#11762568	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#424	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#117	C₂²	1	1	yes	characteristic
11	#10587612	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×#63223	C₂²	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊C₈	—	2	2	no	—
16	#10587493	—	2	2	no	—
17	#10587486	—	2	2	no	—
18	#10587485	—	2	2	no	—
19	#10587479	C₄	1	2	yes	complemented
20	#10587478	C₄	1	2	yes	complemented
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	1	no	—
23	M₁₆×C₁₇	—	2	1	no	—
24	M₁₆×C₁₇	—	2	1	no	—
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	1	no	—
28	C₂×C₈×C₁₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²⋊C₄×C₁₇	—	2	1	no	—
31	C₄²×C₁₇	—	2	1	no	—
32	C₂²⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₇	—	2	1	no	—
35	C₄×Dic₁₇	—	2	1	no	—
36	C₄×Dic₁₇	Q₈	1	2	yes	—
37	#63223	—	2	1	no	—
38	#63223	—	2	1	no	—
39	C₂×#3622	—	2	1	no	—
40	C₂×#3622	C₂×C₄	1	1	yes	characteristic
41	C₂×#3622	C₂×C₄	1	1	yes	characteristic
42	C₂×#3622	D₄	1	2	yes	—
43	C₂×#3622	Q₈	1	2	yes	—
44	C₂³×C₁₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	—	4	2	no	—
55	C₈×C₁₇	—	2	2	no	—
56	C₈×C₁₇	—	4	1	no	—
57	#3622	—	4	1	no	—
58	#3622	C₄⋊C₄	1	2	yes	—
59	#3622	C₄⋊C₄	1	2	yes	—
60	#1946	—	17	1	no	maximal, Sylow
61	C₂²×C₁₇	—	2	2	no	—
62	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₇	#125	1	1	yes	characteristic
65	C₄×C₁₇	—	4	2	no	—
66	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₁₇	#121	1	1	yes	characteristic
69	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₁₇	—	4	2	no	—
71	Dic₁₇	—	4	2	no	—
72	C₂×#108	—	17	1	no	—
73	#427	—	17	1	no	—
74	#424	—	17	1	no	—
75	#424	D₁₇	1	1	yes	characteristic
76	C₂×#97	—	17	1	no	—
77	#339	—	17	1	no	—
78	#339	—	17	1	no	—
79	C₂×C₁₇	—	2	1	no	—
80	C₂×C₁₇	—	2	1	no	—
81	C₂×C₁₇	#389	1	1	yes	characteristic
82	C₂×C₁₇	#444	1	2	yes	—
83	C₂×M₁₆	—	17	1	no	—
84	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
85	C₂²×C₈	—	17	1	no	—
86	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
87	#117	—	17	1	no	—
88	#117	C₂×D₁₇	1	1	yes	characteristic
89	C₂×C₄²	—	17	1	no	—
90	#108	—	34	2	no	—
91	#107	—	17	2	no	—
92	#106	—	17	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	34	2	no	—
95	#97	—	34	2	no	—
96	C₁₇	#1946	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄	—	34	1	no	—
98	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	34	1	no	—
102	M₁₆	—	34	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	17	2	no	—
109	C₂×C₈	—	17	2	no	—
110	C₂×C₈	—	34	1	no	—
111	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
112	C₄⋊C₄	—	17	2	no	—
113	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	34	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	17	2	no	—
118	C₄²	—	34	1	no	—
119	C₄²	—	34	1	no	—
120	C₂³	#63252	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	68	2	no	—
125	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
127	C₂×C₄	#63254	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	17	2	no	—
134	C₈	—	17	2	no	—
135	C₈	—	68	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#10587575	1	1	yes	characteristic
138	C₂²	#10587582	1	1	yes	characteristic
139	C₂²	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	68	2	no	—
142	C₄	—	68	2	no	—
143	C₄	#10587573	1	1	yes	characteristic
144	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
145	C₄	#10587625	1	1	yes	characteristic
146	C₄	#10587630	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#11762704	1	1	yes	characteristic
150	C₂	#11762929	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587493

C₂

yes

maximal, characteristic

#11762845

C₂

yes

maximal, characteristic

#11762835

C₂

yes

maximal, characteristic

C₂×#10587486

C₂

yes

maximal, characteristic

#11762570

C₂

yes

maximal, characteristic

#11762568

C₂

yes

maximal, characteristic

C₁₇×#424

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

#10587612

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

—

—

—

—

C₄

yes

complemented

#10587478

C₄

yes

complemented

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

D₄

yes

—

C₂×#3622

Q₈

yes

—

C₂³×C₁₇

#41

yes

characteristic

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×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

C₄⋊C₄

yes

—

#3622

C₄⋊C₄

yes

—

#1946

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#125

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×Q₈

yes

characteristic

C₄×C₁₇

#121

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic

C₂×#97

—

#339

—

#339

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#389

yes

characteristic

C₂×C₁₇

#444

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

—

#117

C₂×D₁₇

yes

characteristic

C₂×C₄²

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₁₇

#1946

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂²⋊D₁₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₁₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

#63252

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

126

C₂×C₄

D₄×D₁₇

yes

characteristic

127

C₂×C₄

#63254

yes

characteristic

128

C₂×C₄

Q₈×D₁₇

yes

characteristic

129

C₂×C₄

Q₈×D₁₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

#10587575

yes

characteristic

138

C₂²

#10587582

yes

characteristic

139

C₂²

C₄×C₂²⋊D₁₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#10587573

yes

characteristic

144

C₄

C₄⋊C₄×D₁₇

yes

characteristic

145

C₄

#10587625

yes

characteristic

146

C₄

#10587630

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#11762704

yes

characteristic

150

C₂

#11762929

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#424	D₁₇
hypocenter	C₁₇	#1946
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#424

C₂

socle

C₂²×C₁₇

C₄×D₄

prime radical

C₂³×C₁₇

#41

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
19. #10587479	C₄	semidirect
20. #10587478	C₄	semidirect
96. C₁₇	#1946	semidirect

H ≅ G/N

type

19. #10587479

C₄

semidirect

20. #10587478

C₄

semidirect

96. C₁₇

#1946

semidirect

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

subgroup	index
C₂×#10587493	2
#11762845	2
#11762835	2
C₂×#10587486	2
#11762570	2
#11762568	2
C₁₇×#424	2
#1946	17

subgroup

index

C₁₇×#424

Subgroups — #14529534

Conjugacy classes of subgroups

Complements

Maximal subgroups