Group #14528234 — subgroups

Subgroups — #14528234

Order 2176.

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

normal subgroups only

This group has 160 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	#11763159	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763100	C₂	1	1	yes	maximal, characteristic
3	#11763086	C₂	1	1	yes	maximal, characteristic
4	#11762874	C₂	1	1	yes	maximal, characteristic
5	#11762561	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762557	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×C₈×D₁₇	—	2	1	no	—
10	C₄×D₆₈	C₂²	1	1	yes	characteristic
11	#10587555	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	#10587506	—	2	1	no	—
15	#10587506	—	2	1	no	—
16	#10587492	—	2	1	no	—
17	#10587491	—	2	1	no	—
18	#10587488	—	2	1	no	—
19	#10587480	—	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	#63227	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
36	C₂×#3622	—	2	1	no	—
37	C₂×#3622	D₄	1	1	yes	characteristic
38	C₂×#3622	D₄	1	1	yes	characteristic
39	C₈×D₁₇	—	4	2	no	—
40	C₂²×D₁₇	—	2	1	no	—
41	Q₈×C₁₇	—	4	1	no	—
42	Q₈×C₁₇	—	4	1	no	—
43	C₂×C₄×C₁₇	—	4	1	no	—
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
47	C₂×Dic₁₇	—	2	1	no	—
48	C₂×Dic₁₇	—	4	1	no	—
49	D₆₈	—	2	1	no	—
50	D₆₈	Q₁₆	1	2	yes	—
51	C₄×D₁₇	—	4	1	no	—
52	C₄×D₁₇	—	4	1	no	—
53	Dic₃₄	—	8	1	no	—
54	C₈×C₁₇	—	4	1	no	—
55	C₈×C₁₇	—	4	1	no	—
56	#3622	—	2	1	no	—
57	#3622	—	2	1	no	—
58	#3622	—	4	1	no	—
59	#1683	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	#121	1	1	yes	characteristic
61	C₂×D₁₇	—	2	2	no	—
62	C₂×D₁₇	—	4	1	no	—
63	C₂×D₁₇	—	4	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	8	1	no	—
66	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₁₇	#135	1	1	yes	characteristic
68	C₄×C₁₇	#135	1	1	yes	characteristic
69	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Dic₁₇	—	4	1	no	—
71	Dic₁₇	—	8	1	no	—
72	#487	—	17	1	no	—
73	#476	—	17	1	no	—
74	#474	—	17	1	no	—
75	C₈×D₄	—	17	1	no	—
76	#335	—	17	1	no	—
77	#333	—	17	1	no	—
78	#332	D₁₇	1	1	yes	characteristic, complemented
79	C₂×C₁₇	#463	1	1	yes	characteristic
80	C₂×C₁₇	#467	1	1	yes	characteristic
81	C₂×C₁₇	#471	1	1	yes	characteristic
82	D₁₇	—	4	2	no	—
83	C₂²×C₈	—	34	1	no	—
84	C₄⋊Q₈	—	17	1	no	—
85	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
86	C₄×D₄	—	17	1	no	—
87	#107	—	34	1	no	—
88	#106	—	34	1	no	—
89	C₄⋊C₈	—	17	1	no	—
90	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
92	#103	—	34	1	no	—
93	#103	—	34	1	no	—
94	#102	—	34	1	no	—
95	#102	—	34	1	no	—
96	C₂²⋊C₈	—	34	1	no	—
97	C₄×C₈	—	17	1	no	—
98	C₁₇	#1683	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	34	1	no	—
101	C₂×D₄	—	17	1	no	—
102	C₂²×C₄	—	34	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	—	68	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	17	1	no	—
112	C₄⋊C₄	—	34	1	no	—
113	C₄⋊C₄	—	34	1	no	—
114	C₂²⋊C₄	—	34	1	no	—
115	C₄²	C₂²×D₁₇	1	1	yes	characteristic
116	C₂³	—	34	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	136	1	no	—
120	D₄	—	17	2	no	—
121	D₄	—	34	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	34	1	no	—
124	C₂×C₄	—	68	1	no	—
125	C₂×C₄	—	68	1	no	—
126	C₂×C₄	—	68	1	no	—
127	C₂×C₄	#63252	1	1	yes	characteristic
128	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	34	1	no	—
133	C₈	—	34	1	no	—
134	C₈	—	68	1	no	—
135	C₂²	—	34	2	no	—
136	C₂²	—	68	1	no	—
137	C₂²	—	68	1	no	—
138	C₂²	#10587570	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	1	no	—
141	C₄	—	68	1	no	—
142	C₄	—	136	1	no	—
143	C₄	#10587597	1	1	yes	characteristic
144	C₄	#10587602	1	1	yes	characteristic
145	C₄	#10587602	1	1	yes	characteristic
146	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
147	C₂	—	68	2	no	—
148	C₂	#11763023	1	1	yes	characteristic
149	C₂	#11763049	1	1	yes	characteristic
150	C₂	#11763075	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763159

C₂

yes

maximal, characteristic, complemented

#11763100

C₂

yes

maximal, characteristic

#11763086

C₂

yes

maximal, characteristic

#11762874

C₂

yes

maximal, characteristic

#11762561

C₂

yes

maximal, characteristic, complemented

#11762557

C₂

yes

maximal, characteristic, complemented

C₁₇×#332

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₈×D₁₇

—

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

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

C₈×D₁₇

—

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

Q₁₆

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

#487

—

#476

—

#474

—

C₈×D₄

—

#335

—

#333

—

#332

D₁₇

yes

characteristic, complemented

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄×D₄

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

C₄⋊C₈

C₂×D₁₇

yes

—

—

—

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#1683

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

C₂²×D₁₇

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

136

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#63252

yes

characteristic

128

C₂×C₄

D₄×D₁₇

yes

characteristic

129

C₂×C₄

D₄×D₁₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#10587570

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

136

—

143

C₄

#10587597

yes

characteristic

144

C₄

#10587602

yes

characteristic

145

C₄

#10587602

yes

characteristic

146

C₄

Q₁₆×D₁₇

yes

characteristic

147

C₂

—

148

C₂

#11763023

yes

characteristic

149

C₂

#11763049

yes

characteristic

150

C₂

#11763075

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	#1683
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. #11763159	C₂	semidirect
5. #11762561	C₂	semidirect
6. #11762557	C₂	semidirect
7. C₁₇×#332	C₂	semidirect
78. #332	D₁₇	semidirect
98. C₁₇	#1683	semidirect

H ≅ G/N

type

1. #11763159

C₂

semidirect

5. #11762561

C₂

semidirect

6. #11762557

C₂

semidirect

7. C₁₇×#332

C₂

semidirect

78. #332

D₁₇

semidirect

98. C₁₇

#1683

semidirect

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

subgroup	index
#11763159	2
#11763100	2
#11763086	2
#11762874	2
#11762561	2
#11762557	2
C₁₇×#332	2
#1683	17

subgroup

index

C₁₇×#332

Subgroups — #14528234

Conjugacy classes of subgroups

Complements

Maximal subgroups