Group #16955209 — subgroups

Subgroups — #16955209

Order 4352.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14531058	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14526986	C₂	1	1	yes	maximal, characteristic
4	#14526985	C₂	1	1	yes	maximal, characteristic
5	#14526979	C₂	1	1	yes	maximal, characteristic
6	#14526965	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1416	C₂	1	1	yes	maximal, characteristic
8	#11763213	—	2	1	no	—
9	Q₁₆×Dic₁₇	—	2	1	no	—
10	C₂×#10587465	C₂²	1	1	yes	characteristic
11	#11762635	C₂²	1	1	yes	characteristic
12	#11762635	C₂²	1	1	yes	characteristic
13	#11762634	—	2	1	no	—
14	#11762613	—	2	1	no	—
15	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
18	C₁₇×#366	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₇	—	2	1	no	—
20	Q₈×Dic₁₇	—	4	1	no	—
21	#10587574	—	4	1	no	—
22	C₁₇×Q₃₂	—	2	1	no	—
23	C₁₇×Q₃₂	Q₈	1	2	yes	complemented
24	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
25	C₁₇×#107	—	2	1	no	—
26	#10587506	—	4	1	no	—
27	#10587489	—	2	1	no	—
28	#10587489	—	2	1	no	—
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	D₄	1	1	yes	characteristic
31	#10587478	—	4	1	no	—
32	C₁₇×C₃₂	—	2	1	no	—
33	#10587465	—	2	1	no	—
34	C₂×Q₈×C₁₇	—	4	1	no	—
35	C₂×Dic₃₄	—	8	1	no	—
36	Q₁₆×C₁₇	—	2	2	no	—
37	Q₁₆×C₁₇	—	4	1	no	—
38	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₇	—	4	1	no	—
40	C₄⋊Dic₁₇	—	4	1	no	—
41	C₄⋊Dic₁₇	—	4	1	no	—
42	C₄⋊Dic₁₇	—	8	1	no	—
43	#63225	—	8	1	no	—
44	C₄×Dic₁₇	—	8	1	no	—
45	C₄×Dic₁₇	D₈	1	1	yes	characteristic
46	C₂×#3622	D₈	1	1	yes	characteristic
47	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₁₇	#41	1	1	yes	characteristic
49	#63214	—	2	1	no	—
50	#13704	—	17	1	no	maximal, Sylow
51	Q₈×C₁₇	—	4	2	no	—
52	Q₈×C₁₇	—	8	1	no	—
53	C₂×C₄×C₁₇	—	8	1	no	—
54	C₂×C₄×C₁₇	—	8	1	no	—
55	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₁₇	—	8	1	no	—
57	C₂×Dic₁₇	—	8	1	no	—
58	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
59	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
60	Dic₃₄	—	16	2	no	—
61	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₁₇	#137	1	1	yes	characteristic
63	#3622	—	2	1	no	—
64	#2252	—	17	1	no	—
65	C₄×Q₃₂	—	17	1	no	—
66	#1424	—	17	1	no	—
67	#1424	—	17	1	no	—
68	C₄⋊C₃₂	—	17	1	no	—
69	#1416	—	17	1	no	—
70	#1416	D₁₇	1	1	yes	characteristic
71	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
72	C₄×C₁₇	—	8	2	no	—
73	C₄×C₁₇	—	16	1	no	—
74	C₄×C₁₇	#474	1	1	yes	characteristic
75	C₄×C₁₇	#510	1	1	yes	characteristic
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	—	16	1	no	—
78	Dic₁₇	—	16	1	no	—
79	Dic₁₇	SD₆₄	1	2	yes	—
80	C₂×Q₃₂	C₂×D₁₇	1	1	yes	characteristic
81	#500	—	34	1	no	—
82	C₄×Q₁₆	—	34	1	no	—
83	C₂×C₃₂	—	17	1	no	—
84	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
85	#366	—	17	1	no	—
86	#366	—	17	1	no	—
87	#366	—	34	1	no	—
88	#366	C₂×D₁₇	1	1	yes	characteristic
89	#358	—	34	1	no	—
90	C₄×C₁₆	—	17	1	no	—
91	C₂×C₁₇	#2224	1	1	yes	characteristic
92	C₂×C₁₇	C₂×SD₆₄	1	1	yes	characteristic
93	C₂×C₁₇	#2264	1	1	yes	characteristic
94	C₂×Q₁₆	—	2	1	no	—
95	C₄⋊Q₈	—	68	1	no	—
96	C₄×Q₈	—	68	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₃₄	1	2	yes	complemented
99	C₂×C₁₆	—	17	1	no	—
100	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	34	1	no	—
103	#107	—	34	1	no	—
104	#107	—	68	1	no	—
105	#103	—	68	1	no	—
106	C₄×C₈	—	17	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	34	1	no	—
109	C₁₇	#13704	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	136	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	68	1	no	—
118	C₄⋊C₄	—	68	1	no	—
119	C₄⋊C₄	—	136	1	no	—
120	C₄⋊C₄	—	136	1	no	—
121	C₄²	—	17	1	no	—
122	C₄²	—	136	1	no	—
123	C₁₆	—	34	1	no	—
124	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
125	C₁₆	#63254	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	272	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	17	1	no	—
132	C₂×C₄	—	17	1	no	—
133	C₂×C₄	—	136	1	no	—
134	C₂×C₄	—	136	1	no	—
135	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
136	C₈	—	34	1	no	—
137	C₈	#10587564	1	1	yes	characteristic
138	C₈	#10587601	1	1	yes	characteristic
139	C₂²	D₁₆×D₁₇	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	272	1	no	—
143	C₄	—	272	1	no	—
144	C₄	—	17	2	no	—
145	C₄	—	34	1	no	—
146	C₄	#11763087	1	1	yes	characteristic
147	C₄	#11763248	1	1	yes	characteristic
148	C₂	#14530919	1	1	yes	characteristic
149	C₂	D₁₇×SD₆₄	1	1	yes	characteristic
150	C₂	#14531103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531058

C₂

yes

maximal, characteristic

Q₃₂×Dic₁₇

C₂

yes

maximal, characteristic

#14526986

C₂

yes

maximal, characteristic

#14526985

C₂

yes

maximal, characteristic

#14526979

C₂

yes

maximal, characteristic

#14526965

C₂

yes

maximal, characteristic

C₁₇×#1416

C₂

yes

maximal, characteristic

#11763213

—

Q₁₆×Dic₁₇

—

C₂×#10587465

C₂²

yes

characteristic

#11762635

C₂²

yes

characteristic

#11762635

C₂²

yes

characteristic

#11762634

—

#11762613

—

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₂×C₁₇×Q₃₂

C₂²

yes

characteristic

C₂×C₁₇×C₃₂

C₂²

yes

characteristic

C₁₇×#366

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

Q₈×Dic₁₇

—

#10587574

—

C₁₇×Q₃₂

—

C₁₇×Q₃₂

Q₈

yes

complemented

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₁₇×#107

—

#10587506

—

#10587489

—

#10587489

—

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

D₄

yes

characteristic

#10587478

—

C₁₇×C₃₂

—

#10587465

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₈

yes

characteristic

C₂×#3622

D₈

yes

characteristic

C₁₆×C₁₇

C₂×Q₈

yes

C₁₆×C₁₇

yes

—

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

D₁₆

yes

characteristic

C₂×Dic₁₇

D₁₆

yes

characteristic

Dic₃₄

—

C₈×C₁₇

C₂²⋊Q₈

yes

C₈×C₁₇

yes

—

—

C₄×Q₃₂

—

#1424

—

#1424

—

C₄⋊C₃₂

—

#1416

—

#1416

D₁₇

yes

characteristic

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

SD₆₄

yes

—

C₂×Q₃₂

C₂×D₁₇

yes

characteristic

#500

—

C₄×Q₁₆

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₇

yes

—

—

—

C₂×D₁₇

yes

characteristic

#358

—

C₄×C₁₆

—

C₂×C₁₇

#2224

yes

characteristic

C₂×C₁₇

C₂×SD₆₄

yes

characteristic

C₂×C₁₇

#2264

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₃₄

yes

complemented

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

101

#107

—

102

#107

—

103

#107

—

104

#107

—

105

#103

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

109

C₁₇

#13704

yes

characteristic, complemented, Sylow

110

C₂×Q₈

—

111

C₂×Q₈

—

136

—

112

Q₁₆

—

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

D₄×D₁₇

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

136

—

120

C₄⋊C₄

—

136

—

121

C₄²

—

122

C₄²

—

136

—

123

C₁₆

—

124

C₁₆

C₂×Dic₃₄

yes

characteristic

125

C₁₆

#63254

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

272

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

136

—

134

C₂×C₄

—

136

—

135

C₂×C₄

D₈×D₁₇

yes

characteristic

136

C₈

—

137

C₈

#10587564

yes

characteristic

138

C₈

#10587601

yes

characteristic

139

C₂²

D₁₆×D₁₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

272

—

143

C₄

—

272

—

144

C₄

—

145

C₄

—

146

C₄

#11763087

yes

characteristic

147

C₄

#11763248

yes

characteristic

148

C₂

#14530919

yes

characteristic

149

C₂

D₁₇×SD₆₄

yes

characteristic

150

C₂

#14531103

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₁₇
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#1416	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	#1416	D₁₇
hypocenter	C₁₇	#13704
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₁₇

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₁₇×#1416

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₁₆

D₁₇

C₁₇

G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
23. C₁₇×Q₃₂	Q₈	semidirect
98. Q₃₂	Dic₃₄	semidirect
109. C₁₇	#13704	semidirect

H ≅ G/N

type

23. C₁₇×Q₃₂

Q₈

semidirect

98. Q₃₂

Dic₃₄

semidirect

109. C₁₇

#13704

semidirect

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

subgroup	index
#14531058	2
Q₃₂×Dic₁₇	2
#14526986	2
#14526985	2
#14526979	2
#14526965	2
C₁₇×#1416	2
#13704	17

subgroup

index

Q₃₂×Dic₁₇

C₁₇×#1416

Subgroups — #16955209

Conjugacy classes of subgroups

Complements

Maximal subgroups