Group #14528351 — subgroups

Subgroups — #14528351

Order 2176.

162 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 49 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	#11763155	C₂	1	1	yes	maximal, characteristic
2	#11763020	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763006	C₂	1	1	yes	maximal, characteristic
4	#11762936	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762559	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762554	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#331	C₂	1	1	yes	maximal, characteristic
8	C₁₇×#127	C₂²	1	1	yes	characteristic
9	C₂×#63229	—	2	1	no	—
10	C₂×Dic₆₈	—	2	1	no	—
11	#10587555	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#10587503	—	2	1	no	—
16	#10587503	—	2	1	no	—
17	#10587503	—	2	1	no	—
18	#10587487	—	2	1	no	—
19	#10587486	—	2	1	no	—
20	#10587485	—	2	1	no	—
21	C₄⋊#3622	C₂²	1	1	yes	characteristic
22	C₄×#3622	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₁₇	—	2	1	no	—
24	C₂×D₄×C₁₇	—	2	1	no	—
25	C₂×D₄×C₁₇	—	2	1	no	—
26	C₂×Dic₃₄	—	2	1	no	—
27	C₂×Dic₃₄	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	—	2	1	no	—
29	C₂×C₈×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	#63229	—	4	2	no	—
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
35	#63225	—	2	1	no	—
36	C₄×Dic₁₇	—	2	1	no	—
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	D₄	1	1	yes	characteristic
39	C₂×#3622	D₄	1	1	yes	characteristic
40	Dic₆₈	—	4	2	no	—
41	C₂³×C₁₇	—	4	1	no	—
42	D₄×C₁₇	—	4	1	no	—
43	D₄×C₁₇	—	4	1	no	—
44	D₄×C₁₇	—	4	1	no	—
45	D₄×C₁₇	—	4	1	no	—
46	D₄×C₁₇	—	8	1	no	—
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	1	no	—
51	C₂×Dic₁₇	—	2	1	no	—
52	C₂×Dic₁₇	—	4	1	no	—
53	Dic₃₄	—	2	1	no	—
54	Dic₃₄	—	2	1	no	—
55	Dic₃₄	—	4	1	no	—
56	Dic₃₄	—	4	1	no	—
57	C₈×C₁₇	—	4	1	no	—
58	C₈×C₁₇	—	4	1	no	—
59	#3622	—	2	1	no	—
60	#3622	—	2	1	no	—
61	#3622	—	4	1	no	—
62	#1702	—	17	1	no	maximal, Sylow
63	C₂²×C₁₇	—	8	1	no	—
64	C₂²×C₁₇	—	8	1	no	—
65	C₂²×C₁₇	—	8	1	no	—
66	C₂²×C₁₇	#121	1	1	yes	characteristic
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	#135	1	1	yes	characteristic
69	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₁₇	#137	1	1	yes	characteristic
72	Dic₁₇	—	4	1	no	—
73	Dic₁₇	—	4	1	no	—
74	Dic₁₇	—	8	1	no	—
75	#486	—	17	1	no	—
76	#462	—	17	1	no	—
77	#460	—	17	1	no	—
78	#446	—	17	1	no	—
79	#334	—	17	1	no	—
80	#332	—	17	1	no	—
81	#331	D₁₇	1	1	yes	characteristic
82	C₂×C₁₇	—	16	1	no	—
83	C₂×C₁₇	#464	1	1	yes	characteristic
84	C₂×C₁₇	#469	1	1	yes	characteristic
85	C₂×C₁₇	#471	1	1	yes	characteristic
86	C₂×Q₁₆	—	34	1	no	—
87	C₂×SD₁₆	—	34	1	no	—
88	C₄⋊Q₈	—	17	1	no	—
89	#127	C₂×D₁₇	1	1	yes	characteristic
90	C₄×Q₈	—	17	1	no	—
91	C₄⋊C₈	—	17	1	no	—
92	C₄⋊C₈	—	34	1	no	—
93	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
95	#103	—	34	1	no	—
96	#102	—	34	1	no	—
97	#102	—	34	1	no	—
98	#102	—	34	1	no	—
99	#97	—	34	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₁₇	#1702	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	17	1	no	—
103	C₂×Q₈	—	34	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	Q₁₆	—	68	2	no	—
108	SD₁₆	—	68	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₄⋊C₄	—	17	1	no	—
115	C₄⋊C₄	—	34	1	no	—
116	C₄⋊C₄	—	34	1	no	—
117	C₄⋊C₄	—	34	1	no	—
118	C₄²	—	34	1	no	—
119	C₄²	C₂²×D₁₇	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	Q₈	—	34	1	no	—
122	Q₈	—	34	1	no	—
123	Q₈	—	68	1	no	—
124	Q₈	—	68	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	C₂×C₄	—	34	1	no	—
131	C₂×C₄	—	34	1	no	—
132	C₂×C₄	—	68	1	no	—
133	C₂×C₄	#63252	1	1	yes	characteristic
134	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
135	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	34	1	no	—
139	C₈	—	34	1	no	—
140	C₈	—	68	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	#10587570	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	68	1	no	—
147	C₄	—	68	1	no	—
148	C₄	—	136	1	no	—
149	C₄	#10587597	1	1	yes	characteristic
150	C₄	#10587598	1	1	yes	characteristic
151	C₄	#10587600	1	1	yes	characteristic
152	C₄	#10587601	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	#11763030	1	1	yes	characteristic
155	C₂	#11763061	1	1	yes	characteristic
156	C₂	#11763071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763155

C₂

yes

maximal, characteristic

#11763020

C₂

yes

maximal, characteristic, complemented

#11763006

C₂

yes

maximal, characteristic

#11762936

C₂

yes

maximal, characteristic, complemented

#11762559

C₂

yes

maximal, characteristic, complemented

#11762554

C₂

yes

maximal, characteristic, complemented

C₁₇×#331

C₂

yes

maximal, characteristic

C₁₇×#127

C₂²

yes

characteristic

C₂×#63229

—

C₂×Dic₆₈

—

#10587555

C₂²

yes

characteristic

C₄×Dic₃₄

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63229

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Dic₆₈

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×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

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

C₂×D₁₇

yes

characteristic

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

C₄⋊C₈

C₂×D₁₇

yes

—

—

—

—

—

100

C₄×C₈

—

101

C₁₇

#1702

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

Q₁₆

—

108

SD₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₁₇

yes

characteristic

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#63252

yes

characteristic

134

C₂×C₄

D₄×D₁₇

yes

characteristic

135

C₂×C₄

D₄×D₁₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#10587570

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

136

—

149

C₄

#10587597

yes

characteristic

150

C₄

#10587598

yes

characteristic

151

C₄

#10587600

yes

characteristic

152

C₄

#10587601

yes

characteristic

153

C₂

—

154

C₂

#11763030

yes

characteristic

155

C₂

#11763061

yes

characteristic

156

C₂

#11763071

yes

characteristic

	subgroup	G/H
center	C₂²	#10587570
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#331	C₂
socle	C₂²×C₁₇	#121
prime radical	C₁₇×#127	C₂²
hypercenter	#331	D₁₇
hypocenter	C₁₇	#1702
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#10587570

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#331

C₂

socle

C₂²×C₁₇

#121

prime radical

C₁₇×#127

C₂²

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #11763020	C₂	semidirect
4. #11762936	C₂	semidirect
5. #11762559	C₂	semidirect
6. #11762554	C₂	semidirect
101. C₁₇	#1702	semidirect

H ≅ G/N

type

2. #11763020

C₂

semidirect

4. #11762936

C₂

semidirect

5. #11762559

C₂

semidirect

6. #11762554

C₂

semidirect

101. C₁₇

#1702

semidirect

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

subgroup	index
#11763155	2
#11763020	2
#11763006	2
#11762936	2
#11762559	2
#11762554	2
C₁₇×#331	2
#1702	17

subgroup

index

C₁₇×#331

Subgroups — #14528351

Conjugacy classes of subgroups

Complements

Maximal subgroups