Group #14528006 — subgroups

Subgroups — #14528006

Order 2176.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11763539	C₂	1	1	yes	maximal, characteristic
2	#11763159	C₂	1	1	yes	maximal, characteristic
3	#11763095	C₂	1	1	yes	maximal, characteristic
4	#11763020	C₂	1	1	yes	maximal, characteristic
5	#11762539	C₂	1	1	yes	maximal, characteristic
6	#11762539	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#333	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#10587630	—	2	2	no	—
10	C₂×Dic₆₈	—	2	1	no	—
11	#10587574	—	2	1	no	—
12	#10587574	—	2	1	no	—
13	#10587573	—	2	1	no	—
14	#10587573	—	2	1	no	—
15	#10587555	C₂²	1	1	yes	characteristic
16	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
17	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	#10587506	—	2	2	no	—
21	#10587488	—	2	1	no	—
22	#10587487	—	2	1	no	—
23	#10587487	—	2	1	no	—
24	C₄×#3622	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₇	D₄	1	2	yes	—
26	C₂×Dic₃₄	—	2	1	no	—
27	C₂×Dic₃₄	—	2	2	no	—
28	C₂×Dic₃₄	D₄	1	1	yes	characteristic
29	C₂×Dic₃₄	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	—	2	1	no	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₄⋊C₄×C₁₇	—	2	1	no	—
34	C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
39	#63225	—	2	2	no	—
40	#63225	—	2	2	no	—
41	#63225	—	2	1	no	—
42	#63225	—	2	1	no	—
43	C₄×Dic₁₇	—	2	2	no	—
44	C₄×Dic₁₇	—	2	1	no	—
45	C₄×Dic₁₇	—	2	1	no	—
46	C₂×#3622	—	2	1	no	—
47	Dic₆₈	—	4	2	no	—
48	Q₈×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	1	no	—
54	C₂×Dic₁₇	—	2	1	no	—
55	C₂×Dic₁₇	—	2	1	no	—
56	C₂×Dic₁₇	—	2	1	no	—
57	C₂×Dic₁₇	—	4	1	no	—
58	Dic₃₄	—	2	1	no	—
59	Dic₃₄	—	2	1	no	—
60	Dic₃₄	—	4	1	no	—
61	Dic₃₄	—	4	1	no	—
62	Dic₃₄	—	8	2	no	—
63	Dic₃₄	SD₁₆	1	2	yes	—
64	Dic₃₄	Q₁₆	1	2	yes	complemented
65	C₈×C₁₇	—	4	1	no	—
66	C₈×C₁₇	—	4	1	no	—
67	#3622	—	4	1	no	—
68	#1645	—	17	1	no	maximal, Sylow
69	C₂²×C₁₇	#120	1	1	yes	characteristic
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	—	4	2	no	—
72	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₁₇	#137	1	1	yes	characteristic
75	C₄×C₁₇	#137	1	1	yes	characteristic
76	Dic₁₇	—	2	2	no	—
77	Dic₁₇	—	2	2	no	—
78	Dic₁₇	—	4	1	no	—
79	Dic₁₇	—	4	1	no	—
80	Dic₁₇	—	8	1	no	—
81	Q₈²	—	17	1	no	—
82	#487	—	17	1	no	—
83	#475	—	17	1	no	—
84	#462	—	17	1	no	—
85	#333	D₁₇	1	1	yes	characteristic
86	Q₈⋊C₈	—	17	1	no	—
87	Q₈⋊C₈	—	17	1	no	—
88	C₂×C₁₇	#448	1	1	yes	characteristic
89	C₂×C₁₇	#451	1	1	yes	characteristic
90	C₂×C₁₇	#455	1	1	yes	characteristic
91	C₂×Q₁₆	—	34	1	no	—
92	C₄⋊Q₈	—	17	1	no	—
93	C₄⋊Q₈	—	34	1	no	—
94	C₄⋊Q₈	—	34	1	no	—
95	C₄⋊Q₈	—	34	2	no	—
96	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
97	C₄×Q₈	—	17	1	no	—
98	C₄×Q₈	—	17	1	no	—
99	C₄×Q₈	—	34	1	no	—
100	C₄×Q₈	—	34	1	no	—
101	#106	—	34	1	no	—
102	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
103	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
104	#103	—	34	1	no	—
105	#103	—	34	1	no	—
106	#103	—	34	2	no	—
107	C₄×C₈	—	17	1	no	—
108	C₁₇	#1645	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	17	1	no	—
110	C₂×Q₈	—	17	1	no	—
111	C₂×Q₈	—	34	2	no	—
112	C₂×Q₈	—	34	1	no	—
113	C₂×Q₈	D₆₈	1	2	yes	—
114	Q₁₆	—	68	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	34	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	17	1	no	—
121	C₄⋊C₄	—	17	1	no	—
122	C₄⋊C₄	—	34	2	no	—
123	C₄⋊C₄	—	34	2	no	—
124	C₄⋊C₄	—	34	1	no	—
125	C₄⋊C₄	—	34	1	no	—
126	C₄⋊C₄	—	34	1	no	—
127	C₄⋊C₄	—	34	1	no	—
128	C₄²	—	34	2	no	—
129	C₄²	—	34	1	no	—
130	C₄²	—	34	1	no	—
131	C₄²	C₂²×D₁₇	1	1	yes	characteristic
132	Q₈	—	4	2	no	—
133	Q₈	—	17	2	no	—
134	Q₈	—	17	2	no	—
135	Q₈	—	34	1	no	—
136	Q₈	—	34	1	no	—
137	Q₈	—	68	1	no	—
138	Q₈	—	68	1	no	—
139	Q₈	—	136	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	—	34	1	no	—
144	C₂×C₄	—	34	1	no	—
145	C₂×C₄	—	68	1	no	—
146	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	68	1	no	—
152	C₂²	#10587568	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	—	34	2	no	—
156	C₄	—	34	2	no	—
157	C₄	—	68	1	no	—
158	C₄	—	68	1	no	—
159	C₄	—	136	1	no	—
160	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
161	C₄	#10587601	1	1	yes	characteristic
162	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
163	C₄	#10587604	1	1	yes	characteristic
164	C₂	#11762949	1	1	yes	characteristic
165	C₂	#11762966	1	1	yes	characteristic
166	C₂	#11762989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763539

C₂

yes

maximal, characteristic

#11763159

C₂

yes

maximal, characteristic

#11763095

C₂

yes

maximal, characteristic

#11763020

C₂

yes

maximal, characteristic

#11762539

C₂

yes

maximal, characteristic

#11762539

C₂

yes

maximal, characteristic

C₁₇×#333

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

#10587630

—

C₂×Dic₆₈

—

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₃₄

C₂²

yes

characteristic

C₄×Dic₃₄

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×Q₈×C₁₇

D₄

yes

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

characteristic

C₂×Dic₃₄

D₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

C₄⋊Dic₁₇

D₄

yes

—

—

—

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

SD₁₆

yes

—

Dic₃₄

Q₁₆

yes

complemented

C₈×C₁₇

—

C₈×C₁₇

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Q₈²

—

—

—

—

D₁₇

yes

characteristic

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

#106

—

102

C₄⋊C₈

C₂×D₁₇

yes

characteristic

103

C₄⋊C₈

C₂×D₁₇

yes

characteristic

104

#103

—

105

#103

—

106

#103

—

107

C₄×C₈

—

108

C₁₇

#1645

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

D₆₈

yes

—

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₁₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

136

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×D₆₈

yes

characteristic

147

C₂×C₄

D₄×D₁₇

yes

characteristic

148

C₂×C₄

D₄×D₁₇

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂²

#10587568

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

136

—

160

C₄

SD₁₆×D₁₇

yes

characteristic

161

C₄

#10587601

yes

characteristic

162

C₄

Q₁₆×D₁₇

yes

characteristic

163

C₄

#10587604

yes

characteristic

164

C₂

#11762949

yes

characteristic

165

C₂

#11762966

yes

characteristic

166

C₂

#11762989

yes

characteristic

	subgroup	G/H
center	C₂²	#10587568
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#333	C₂
socle, prime radical	C₂²×C₁₇	#120
hypercenter	#333	D₁₇
hypocenter	C₁₇	#1645
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#10587568

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#333

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
64. Dic₃₄	Q₁₆	semidirect
108. C₁₇	#1645	semidirect

H ≅ G/N

type

64. Dic₃₄

Q₁₆

semidirect

108. C₁₇

#1645

semidirect

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

subgroup	index
#11763539	2
#11763159	2
#11763095	2
#11763020	2
#11762539	2
C₁₇×#333	2
#1645	17

subgroup

index

C₁₇×#333

Subgroups — #14528006

Conjugacy classes of subgroups

Complements

Maximal subgroups