Group #17960686 — subgroups

Subgroups — #17960686

Order 4352.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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	#14545007	C₂	1	1	yes	maximal, characteristic
2	#14544995	C₂	1	1	yes	maximal, characteristic
3	#14543839	C₂	1	2	yes	maximal
4	#14543835	C₂	1	2	yes	maximal
5	#14528540	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂×#10587538	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	#11763702	C₄	1	4	yes	complemented
10	#11763698	C₄	1	2	yes	complemented
11	#11763695	C₄	1	2	yes	complemented
12	#11762767	C₂²	1	2	yes	—
13	#11762767	C₂²	1	1	yes	characteristic
14	#11762566	C₄	1	2	yes	—
15	#11762566	C₄	1	1	yes	characteristic
16	C₁₇×#338	C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
18	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
19	#10587612	C₂×C₄	1	2	yes	—
20	#10587612	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂³	1	1	yes	characteristic
22	#10587538	—	2	2	no	—
23	#10587538	—	2	2	no	—
24	#10587538	C₂×C₄	1	4	yes	—
25	C₂×#63216	—	2	2	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×#63216	C₂×C₄	1	1	yes	characteristic
28	C₂×#63216	C₂×C₄	1	1	yes	characteristic
29	C₂×#63216	C₂×C₄	1	2	yes	—
30	#10587486	—	2	2	no	—
31	#10587486	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₇	—	2	2	no	—
33	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₁₇	—	2	2	no	—
36	M₁₆×C₁₇	—	2	2	no	—
37	M₁₆×C₁₇	—	2	2	no	—
38	C₂×C₈×C₁₇	—	2	2	no	—
39	C₂×C₈×C₁₇	C₄²	1	2	yes	—
40	C₂²⋊Dic₁₇	—	2	1	no	—
41	C₂²⋊Dic₁₇	—	2	2	no	—
42	C₄⋊Dic₁₇	—	2	1	no	—
43	C₄⋊Dic₁₇	—	2	2	no	—
44	C₄×Dic₁₇	—	2	2	no	—
45	C₄×Dic₁₇	C₄²	1	2	yes	—
46	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
47	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3622	#41	1	2	yes	—
49	C₂×#3622	#41	1	2	yes	—
50	#63216	—	2	1	no	—
51	#63216	—	2	1	no	—
52	#63216	—	2	4	no	—
53	#63216	—	2	2	no	—
54	#12597	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	#131	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#117	1	2	yes	—
58	C₂×C₄×C₁₇	#131	1	2	yes	—
59	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
60	C₂×Dic₁₇	—	2	2	no	—
61	C₂×Dic₁₇	—	4	2	no	—
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	—	4	2	no	—
64	#3622	#117	1	1	yes	characteristic
65	#3622	#117	1	2	yes	—
66	#3622	#117	1	1	yes	characteristic
67	#2110	—	17	1	no	—
68	#2108	—	17	1	no	—
69	#1747	—	17	1	no	—
70	#1379	—	17	2	no	—
71	#1378	—	17	2	no	—
72	C₂²×C₁₇	—	4	1	no	—
73	C₂²×C₁₇	#405	1	1	yes	characteristic
74	C₂²×C₁₇	#433	1	2	yes	—
75	C₄×C₁₇	#376	1	1	yes	characteristic
76	C₄×C₁₇	#405	1	1	yes	characteristic
77	C₄×C₁₇	#433	1	2	yes	—
78	Dic₁₇	—	4	2	no	—
79	Dic₁₇	—	8	2	no	—
80	C₂×M₃₂	—	17	1	no	—
81	C₂×M₃₂	—	17	2	no	—
82	C₂²×C₁₆	—	17	1	no	—
83	#405	—	17	1	no	—
84	#405	—	17	2	no	—
85	#347	—	17	4	no	—
86	#346	—	17	2	no	—
87	C₄×C₁₆	—	17	2	no	—
88	#338	#595	1	1	yes	characteristic
89	H(3,ℤ/4)	—	17	2	no	—
90	H(3,ℤ/4)	—	17	1	no	—
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	—	2	2	no	—
93	C₂×C₁₇	#1746	1	1	yes	characteristic
94	C₂×M₁₆	C₂×#595	1	2	yes	—
95	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
96	C₂²×C₈	—	17	1	no	—
97	#117	—	17	1	no	—
98	#117	—	17	2	no	—
99	M₃₂	—	17	4	no	—
100	M₃₂	—	34	2	no	—
101	M₃₂	—	34	2	no	—
102	C₂×C₁₆	—	17	1	no	—
103	C₂×C₁₆	—	17	2	no	—
104	C₂×C₁₆	—	17	1	no	—
105	C₂×C₁₆	—	34	2	no	—
106	C₂×C₁₆	—	34	2	no	—
107	#97	—	17	2	no	—
108	#97	—	34	2	no	—
109	C₄×C₈	—	17	2	no	—
110	C₄×C₈	—	34	2	no	—
111	C₁₇	#12597	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	17	2	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	C₄×#595	1	2	yes	—
122	C₄⋊C₄	—	34	1	no	—
123	C₄⋊C₄	—	34	2	no	—
124	C₂²⋊C₄	—	34	1	no	—
125	C₂²⋊C₄	—	34	2	no	—
126	C₄²	—	17	2	no	—
127	C₄²	—	34	2	no	—
128	C₁₆	—	34	1	no	—
129	C₁₆	—	34	1	no	—
130	C₁₆	—	34	2	no	—
131	C₁₆	—	34	4	no	—
132	C₂³	#10587680	1	1	yes	characteristic
133	C₂×C₄	—	34	2	no	—
134	C₂×C₄	—	68	2	no	—
135	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
136	C₂×C₄	#10587677	1	2	yes	—
137	C₂×C₄	#10587678	1	2	yes	—
138	C₂×C₄	#10587678	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	17	1	no	—
142	C₈	—	17	1	no	—
143	C₈	—	17	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	#11763774	1	1	yes	characteristic
146	C₂²	#11763834	1	2	yes	—
147	C₄	—	68	2	no	—
148	C₄	—	136	2	no	—
149	C₄	#11763745	1	1	yes	characteristic
150	C₄	#11763773	1	1	yes	characteristic
151	C₄	#11763833	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	#14544503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545007

C₂

yes

maximal, characteristic

#14544995

C₂

yes

maximal, characteristic

#14543839

C₂

yes

maximal

#14543835

C₂

yes

maximal

#14528540

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

C₂×#10587538

C₂²

yes

characteristic

C₂²×#63216

C₂²

yes

characteristic

#11763702

C₄

yes

complemented

#11763698

C₄

yes

complemented

#11763695

C₄

yes

complemented

#11762767

C₂²

yes

—

#11762767

C₂²

yes

characteristic

#11762566

C₄

yes

—

#11762566

C₄

yes

characteristic

C₁₇×#338

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

#10587612

C₂×C₄

yes

—

#10587612

C₂×C₄

yes

characteristic

C₂²×#3622

C₂³

yes

—

—

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

#10587486

—

#10587486

C₂×C₄

yes

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

—

C₂²⋊Dic₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄²

yes

—

C₂×#3622

C₂²×C₄

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

C₂×#3622

#41

yes

—

C₂×#3622

#41

yes

—

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

yes

yes

—

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#405

yes

characteristic

C₂²×C₁₇

#433

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

C₄×C₁₆

—

yes

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1746

yes

characteristic

C₂×M₁₆

C₂×#595

yes

—

C₂×M₁₆

C₂×#595

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

M₃₂

—

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

#97

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₁₇

#12597

yes

characteristic, complemented, Sylow

112

C₂²×C₄

C₂²×#595

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄×#595

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

#10587680

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×#595

yes

characteristic

136

C₂×C₄

#10587677

yes

—

137

C₂×C₄

#10587678

yes

—

138

C₂×C₄

#10587678

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

#11763774

yes

characteristic

146

C₂²

#11763834

yes

—

147

C₄

—

148

C₄

—

136

—

149

C₄

#11763745

yes

characteristic

150

C₄

#11763773

yes

characteristic

151

C₄

#11763833

yes

—

152

C₂

—

153

C₂

—

154

C₂

#14544503

yes

characteristic

	subgroup	G/H
center	C₄	#11763745
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#338	C₄
socle	C₂×C₁₇	#1746
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	#338	#595
hypocenter	C₁₇	#12597
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#11763745

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₁₇×#338

C₄

C₂×C₁₇

C₂³×C₁₇

C₂²×#3622

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #11763702	C₄	semidirect
10. #11763698	C₄	semidirect
11. #11763695	C₄	semidirect
111. C₁₇	#12597	semidirect

H ≅ G/N

type

9. #11763702

C₄

semidirect

10. #11763698

C₄

semidirect

11. #11763695

C₄

semidirect

111. C₁₇

#12597

semidirect

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

subgroup	index
#14545007	2
#14544995	2
#14543839	2
#14543835	2
#14528540	2
#12597	17

subgroup

index

Subgroups — #17960686

Conjugacy classes of subgroups

Complements

Maximal subgroups