Group #16921146 — subgroups

Subgroups — #16921146

Order 4352.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14526868	C₂	1	2	yes	maximal
2	C₁₇×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587483	—	2	2	no	—
4	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
5	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
7	C₁₇×#336	—	2	2	no	—
8	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₁₇	—	2	1	no	—
10	C₂²×C₈×C₁₇	D₄	1	2	yes	—
11	C₄×Q₈×C₁₇	—	4	1	no	—
12	C₄×Q₈×C₁₇	—	4	1	no	—
13	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	—	2	1	no	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₁₇×C₄⋊C₈	—	4	1	no	—
18	C₁₇×#97	—	4	1	no	—
19	#10587483	—	2	4	no	—
20	#10587483	—	4	2	no	—
21	C₂×#63214	—	4	2	no	—
22	C₂×Q₈×C₁₇	—	4	2	no	—
23	C₂×Q₈×C₁₇	—	4	1	no	—
24	C₂²×C₄×C₁₇	—	2	1	no	—
25	C₂²×C₄×C₁₇	—	2	1	no	—
26	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	—	4	1	no	—
32	C₂×C₈×C₁₇	—	4	2	no	—
33	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
34	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
35	C₄⋊C₄×C₁₇	—	4	1	no	—
36	C₄⋊C₄×C₁₇	—	4	2	no	—
37	C₄²×C₁₇	—	2	1	no	—
38	C₄²×C₁₇	—	2	1	no	—
39	C₄²×C₁₇	—	4	1	no	—
40	C₄²×C₁₇	—	4	1	no	—
41	#63214	—	4	4	no	—
42	#7133	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
44	Q₈×C₁₇	—	4	2	no	—
45	Q₈×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	2	2	no	—
47	C₂×C₄×C₁₇	—	2	1	no	—
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	—	4	1	no	—
50	C₂×C₄×C₁₇	—	4	1	no	—
51	C₂×C₄×C₁₇	—	4	1	no	—
52	C₂×C₄×C₁₇	—	4	1	no	—
53	C₂×C₄×C₁₇	—	4	2	no	—
54	C₂×C₄×C₁₇	—	4	1	no	—
55	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#100	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#103	1	2	yes	—
58	C₂×C₄×C₁₇	#104	1	2	yes	—
59	C₈×C₁₇	—	2	2	no	—
60	C₈×C₁₇	—	2	2	no	—
61	C₈×C₁₇	—	8	1	no	—
62	#1985	D₁₇	1	1	yes	characteristic
63	#1377	—	17	2	no	—
64	C₂²×C₁₇	—	2	1	no	—
65	C₂²×C₁₇	—	2	1	no	—
66	C₂²×C₁₇	#323	1	1	yes	characteristic
67	C₂²×C₁₇	Q₈⋊C₈	1	2	yes	—
68	C₄×C₁₇	—	4	2	no	—
69	C₄×C₁₇	—	4	1	no	—
70	C₄×C₁₇	—	4	1	no	—
71	C₄×C₁₇	—	8	1	no	—
72	C₄×C₁₇	#328	1	2	yes	—
73	C₄×C₁₇	#329	1	1	yes	characteristic
74	C₄×C₁₇	#333	1	1	yes	characteristic
75	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
76	C₂×M₃₂	—	34	2	no	—
77	C₂×C₄⋊C₈	Dic₁₇	1	1	yes	characteristic
78	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
79	#336	—	2	2	no	—
80	C₂×C₁₇	—	2	2	no	—
81	C₂×C₁₇	#1272	1	1	yes	characteristic
82	C₂×C₁₇	#1320	1	1	yes	characteristic
83	C₂×C₁₇	#1321	1	1	yes	characteristic
84	C₂²×Q₈	#3622	1	1	yes	characteristic
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
87	C₄×Q₈	—	4	1	no	—
88	C₄×Q₈	—	4	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	#3622	1	1	yes	characteristic
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
93	M₃₂	—	34	4	no	—
94	M₃₂	—	68	2	no	—
95	C₂×C₁₆	—	68	2	no	—
96	C₄⋊C₈	—	4	1	no	—
97	#97	—	4	1	no	—
98	C₁₇	#7133	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
104	C₂²×C₄	#63223	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	#63230	1	2	yes	—
111	C₂×C₈	#63231	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	68	4	no	—
119	C₂³	C₂²⋊#3622	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#10587504	1	1	yes	characteristic
132	C₂×C₄	#10587505	1	1	yes	characteristic
133	C₂×C₄	#10587506	1	2	yes	—
134	C₂×C₄	#10587508	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#11762533	1	1	yes	characteristic
141	C₂²	Q₈⋊#3622	1	2	yes	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	#11762546	1	2	yes	—
147	C₄	#11762549	1	1	yes	characteristic
148	C₄	#11762558	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#14526610	1	1	yes	characteristic
151	C₂	#14526720	1	1	yes	characteristic
152	C₂	#14526722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14526868

C₂

yes

maximal

C₁₇×#1985

C₂

yes

maximal, characteristic

C₂×#10587483

—

C₂×C₄×Q₈×C₁₇

C₄

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#336

—

C₂²×Q₈×C₁₇

C₈

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₈

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×#97

—

#10587483

—

#10587483

—

C₂×#63214

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

#63214

—

#7133

—

maximal, Sylow

C₂³×C₁₇

C₂²⋊C₈

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#1985

D₁₇

yes

characteristic

#1377

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#323

yes

characteristic

C₂²×C₁₇

Q₈⋊C₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#328

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂×C₄×Q₈

Dic₁₇

yes

characteristic

C₂×M₃₂

—

C₂×C₄⋊C₈

Dic₁₇

yes

characteristic

C₂×#97

C₂×D₁₇

yes

characteristic

#336

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

yes

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#3622

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

#97

—

C₁₇

#7133

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂×#3622

yes

characteristic

104

C₂²×C₄

#63223

yes

characteristic

105

C₂²×C₄

C₂²⋊Dic₁₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#63230

yes

—

111

C₂×C₈

#63231

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₂³

C₂²⋊#3622

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#10587504

yes

characteristic

132

C₂×C₄

#10587505

yes

characteristic

133

C₂×C₄

#10587506

yes

—

134

C₂×C₄

#10587508

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

#11762533

yes

characteristic

141

C₂²

Q₈⋊#3622

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#11762546

yes

—

147

C₄

#11762549

yes

characteristic

148

C₄

#11762558

yes

characteristic

149

C₂

—

150

C₂

#14526610

yes

characteristic

151

C₂

#14526720

yes

characteristic

152

C₂

#14526722

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587505
commutator	C₂²×C₄×C₁₇	C₂×C₈
Frattini	C₂×#97	C₂×D₁₇
Fitting	C₁₇×#1985	C₂
socle	C₂²×C₁₇	#323
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₁₇×#97	C₂²
hypercenter	#1985	D₁₇
hypocenter	C₁₇	#7133
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

subgroup

G/H

center

C₂×C₄

#10587505

commutator

C₂²×C₄×C₁₇

C₂×C₈

Frattini

C₂×#97

C₂×D₁₇

Fitting

C₁₇×#1985

C₂

socle

C₂²×C₁₇

#323

prime radical

C₂³×C₁₇

C₂²⋊C₈

cosocle

C₂×C₁₇×#97

C₂²

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

N	H ≅ G/N	type
98. C₁₇	#7133	semidirect

H ≅ G/N

type

98. C₁₇

#7133

semidirect

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

subgroup	index
#14526868	2
C₁₇×#1985	2
#7133	17

subgroup

index

#14526868

C₁₇×#1985

#7133

Subgroups — #16921146

Conjugacy classes of subgroups

Complements

Maximal subgroups