Group #16934261 — subgroups

Subgroups — #16934261

Order 4352.

422 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 151 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	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14527238	C₂	1	4	yes	maximal
3	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	#11763366	C₂²	1	2	yes	—
6	#11763214	C₄	1	2	yes	—
7	#11763211	—	2	2	no	—
8	#11763207	C₄	1	2	yes	—
9	#11762847	—	2	2	no	—
10	C₂×C₄×#3622	C₂²	1	2	yes	—
11	#11762539	—	2	8	no	—
12	C₈×#3622	—	2	2	no	—
13	C₂×C₁₇×#108	—	2	2	no	—
14	C₁₇×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	D₄	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	#10587612	—	2	2	no	—
20	#10587610	—	2	4	no	—
21	C₂²×#3622	—	2	2	no	—
22	#10587556	—	2	2	no	—
23	#10587555	C₂×C₄	1	4	yes	—
24	C₄×Dic₃₄	—	2	4	no	—
25	C₁₇×#108	—	4	4	no	—
26	C₁₇×C₄⋊C₈	—	2	4	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587489	—	2	4	no	—
31	#10587488	—	2	4	no	—
32	C₄×#3622	—	2	2	no	—
33	C₄×#3622	D₄	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	#41	1	2	yes	—
36	C₂×Dic₃₄	—	2	4	no	—
37	M₁₆×C₁₇	—	4	4	no	—
38	C₂×C₈×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
42	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
43	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×D₄	1	2	yes	—
46	C₂²⋊Dic₁₇	—	4	2	no	—
47	C₄⋊Dic₁₇	—	2	4	no	—
48	C₄⋊Dic₁₇	—	2	4	no	—
49	#63225	—	4	4	no	—
50	C₄×Dic₁₇	—	4	2	no	—
51	C₂×#3622	—	2	8	no	—
52	C₂×#3622	—	2	4	no	—
53	#9855	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	4	no	—
56	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₁₇	#121	1	2	yes	—
59	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
62	C₂×Dic₁₇	—	4	4	no	—
63	Dic₃₄	—	4	8	no	—
64	C₈×C₁₇	—	8	2	no	—
65	C₈×C₁₇	#103	1	4	yes	—
66	C₈×C₁₇	#103	1	4	yes	—
67	#3622	—	2	8	no	—
68	#3160	—	17	1	no	—
69	#1848	D₁₇	1	1	yes	characteristic
70	#1493	—	17	4	no	—
71	C₂×C₈²	—	17	1	no	—
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	#390	1	1	yes	characteristic
74	C₂²×C₁₇	#487	1	2	yes	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	C₂×#103	1	2	yes	—
77	C₄×C₁₇	#443	1	4	yes	—
78	C₄×C₁₇	#492	1	1	yes	characteristic
79	C₄×C₁₇	#494	1	1	yes	characteristic
80	Dic₁₇	—	8	4	no	—
81	#533	—	17	2	no	—
82	#500	—	17	2	no	—
83	#499	—	34	2	no	—
84	#498	—	17	2	no	—
85	C₂×#108	—	2	2	no	—
86	#427	—	34	2	no	—
87	#423	C₂×D₁₇	1	2	yes	—
88	C₂×C₄×C₈	—	17	2	no	—
89	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	Q₈⋊C₈	—	34	8	no	—
91	C₈²	—	34	2	no	—
92	C₂×C₁₇	—	2	2	no	—
93	C₂×C₁₇	#1956	1	1	yes	characteristic
94	C₂×C₁₇	#1974	1	2	yes	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂²×C₈	—	34	2	no	—
97	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
98	C₄⋊Q₈	—	17	4	no	—
99	#125	—	34	2	no	—
100	C₂²⋊Q₈	—	34	4	no	—
101	C₄×Q₈	—	34	4	no	—
102	#117	—	34	2	no	—
103	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
104	#108	—	4	4	no	—
105	#107	—	34	4	no	—
106	#106	—	34	4	no	—
107	C₄⋊C₈	—	2	4	no	—
108	C₄×C₈	—	17	4	no	—
109	C₄×C₈	—	34	2	no	—
110	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
111	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
112	C₄×C₈	D₆₈	1	2	yes	—
113	C₁₇	#9855	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	34	4	no	—
115	C₂²×C₄	#63256	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	34	8	no	—
120	C₂×C₈	—	34	4	no	—
121	C₂×C₈	#63227	1	2	yes	—
122	C₂×C₈	#63227	1	2	yes	—
123	C₂×C₈	#63229	1	4	yes	—
124	C₂×C₈	#63231	1	4	yes	—
125	C₄⋊C₄	—	34	4	no	—
126	C₄⋊C₄	—	34	4	no	—
127	C₄⋊C₄	—	68	4	no	—
128	C₂²⋊C₄	—	68	2	no	—
129	C₄²	—	68	2	no	—
130	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₄²	C₂×D₆₈	1	1	yes	characteristic
132	C₄²	D₄×D₁₇	1	2	yes	—
133	C₂³	#10587633	1	1	yes	characteristic
134	Q₈	—	68	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	68	4	no	—
137	C₂×C₄	#10587579	1	2	yes	—
138	C₂×C₄	#10587581	1	2	yes	—
139	C₂×C₄	C₂×#63227	1	1	yes	characteristic
140	C₂×C₄	C₂×#63229	1	2	yes	—
141	C₂×C₄	#10587626	1	1	yes	characteristic
142	C₂×C₄	C₂×#63231	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	34	8	no	—
145	C₈	#10587481	1	4	yes	—
146	C₈	#10587481	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#11762710	1	1	yes	characteristic
149	C₂²	#11763161	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	136	4	no	—
152	C₄	C₂×#10587481	1	2	yes	—
153	C₄	#11762925	1	4	yes	—
154	C₄	#11763181	1	1	yes	characteristic
155	C₄	#11763188	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#14529585	1	1	yes	characteristic
158	C₂	#14529670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539298

C₂

yes

maximal, characteristic

#14527238

C₂

yes

maximal

C₂×C₈×#3622

C₂

yes

maximal, characteristic

C₁₇×#1848

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

—

#11763214

C₄

yes

—

#11763211

—

#11763207

C₄

yes

—

#11762847

—

C₂×C₄×#3622

C₂²

yes

—

#11762539

—

C₈×#3622

—

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

#10587612

—

#10587610

—

C₂²×#3622

—

#10587556

—

#10587555

C₂×C₄

yes

—

C₄×Dic₃₄

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

#103

yes

—

#3622

—

#3160

—

#1848

D₁₇

yes

characteristic

#1493

—

C₂×C₈²

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

C₄×C₁₇

#443

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

—

—

—

—

—

—

C₂×D₁₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1956

yes

characteristic

C₂×C₁₇

#1974

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₄⋊Q₈

—

#125

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

C₂²×D₁₇

yes

characteristic

104

#108

—

105

#107

—

106

#106

—

107

C₄⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₁₇

yes

characteristic

111

C₄×C₈

C₄×D₁₇

yes

characteristic

112

C₄×C₈

D₆₈

yes

—

113

C₁₇

#9855

yes

characteristic, complemented, Sylow

114

C₂×Q₈

—

115

C₂²×C₄

#63256

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

#63227

yes

—

122

C₂×C₈

#63227

yes

—

123

C₂×C₈

#63229

yes

—

124

C₂×C₈

#63231

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂×C₄×D₁₇

yes

characteristic

131

C₄²

C₂×D₆₈

yes

characteristic

132

C₄²

D₄×D₁₇

yes

—

133

C₂³

#10587633

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10587579

yes

—

138

C₂×C₄

#10587581

yes

—

139

C₂×C₄

C₂×#63227

yes

characteristic

140

C₂×C₄

C₂×#63229

yes

—

141

C₂×C₄

#10587626

yes

characteristic

142

C₂×C₄

C₂×#63231

yes

—

143

C₈

—

144

C₈

—

145

C₈

#10587481

yes

—

146

C₈

#10587481

yes

—

147

C₂²

—

148

C₂²

#11762710

yes

characteristic

149

C₂²

#11763161

yes

—

150

C₄

—

151

C₄

—

136

—

152

C₄

C₂×#10587481

yes

—

153

C₄

#11762925

yes

—

154

C₄

#11763181

yes

characteristic

155

C₄

#11763188

yes

characteristic

156

C₂

—

157

C₂

#14529585

yes

characteristic

158

C₂

#14529670

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#9855
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1848

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
113. C₁₇	#9855	semidirect

H ≅ G/N

type

113. C₁₇

#9855

semidirect

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

subgroup	index
#14539298	2
#14527238	2
C₂×C₈×#3622	2
C₁₇×#1848	2
#9855	17

subgroup

index

#14539298

#14527238

C₂×C₈×#3622

C₁₇×#1848

#9855

Subgroups — #16934261

Conjugacy classes of subgroups

Complements

Maximal subgroups