Group #17960392 — subgroups

Subgroups — #17960392

Order 4352.

234 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14544416	C₂	1	2	yes	maximal, complemented
2	#14543817	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543739	C₂	1	2	yes	maximal, complemented
4	#14543695	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530630	C₂	1	1	yes	maximal, characteristic
6	#11763843	—	2	2	no	—
7	#11763695	C₂²	1	1	yes	characteristic
8	#11763672	C₂²	1	2	yes	—
9	#11763657	—	2	2	no	—
10	C₈×#3624	C₂²	1	1	yes	characteristic
11	C₂×D₂₇₂	C₄	1	1	yes	characteristic
12	#11762891	C₂²	1	2	yes	—
13	#11762609	C₄	1	2	yes	—
14	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#366	C₄	1	1	yes	characteristic
16	C₂×#63243	—	4	2	no	—
17	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
18	#10587579	—	2	2	no	—
19	#10587543	—	4	2	no	—
20	C₄⋊#3624	—	2	2	no	—
21	C₄×#3624	D₄	1	1	yes	characteristic
22	C₄×#3624	D₄	1	1	yes	characteristic
23	C₂×#63216	—	2	1	no	—
24	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₁₇×#107	C₂×C₄	1	2	yes	—
26	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂×#63214	C₂×C₄	1	1	yes	characteristic
28	#10587480	—	2	2	no	—
29	D₂₇₂	—	2	2	no	—
30	C₂×D₆₈	—	2	2	no	—
31	C₂×C₄×D₁₇	—	4	2	no	—
32	C₂×#3624	—	2	1	no	—
33	C₂×#3624	—	4	2	no	—
34	C₂×#3624	D₈	1	2	yes	—
35	#63243	—	8	4	no	—
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇	—	2	2	no	—
38	#63227	—	4	2	no	—
39	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×#3622	#41	1	1	yes	characteristic
41	D₁₃₆	—	2	2	no	—
42	D₁₃₆	—	2	2	no	—
43	#63216	—	2	2	no	—
44	C₁₆×C₁₇	M₁₆	1	2	yes	complemented
45	#63214	—	2	1	no	—
46	#12086	—	17	1	no	maximal, Sylow
47	C₂²×D₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₁₇	#135	1	1	yes	characteristic
52	D₆₈	—	4	2	no	—
53	D₆₈	—	4	2	no	—
54	C₄×D₁₇	—	8	2	no	—
55	C₄×D₁₇	—	8	2	no	—
56	#3624	—	2	2	no	—
57	#3624	—	4	1	no	—
58	#3624	—	8	2	no	—
59	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₁₇	#131	1	1	yes	characteristic
61	#3622	—	2	1	no	—
62	C₄×D₁₆	—	17	1	no	—
63	#1589	—	17	2	no	—
64	#1372	—	17	1	no	—
65	#1331	—	17	2	no	—
66	#1311	—	17	1	no	—
67	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
68	C₂×D₁₇	—	8	2	no	—
69	C₂×D₁₇	—	8	2	no	—
70	C₂×D₁₇	—	8	2	no	—
71	C₄×C₁₇	—	8	2	no	—
72	C₄×C₁₇	#436	1	1	yes	characteristic
73	C₄×C₁₇	#443	1	1	yes	characteristic
74	Dic₁₇	—	2	1	no	—
75	Dic₁₇	#509	1	2	yes	—
76	C₂×D₁₆	—	17	1	no	—
77	C₄×D₈	—	17	2	no	—
78	#436	—	34	2	no	—
79	#366	#595	1	1	yes	characteristic
80	#357	—	17	2	no	—
81	C₄×C₁₆	—	17	1	no	—
82	C₄×C₁₆	—	17	1	no	—
83	#335	—	17	2	no	—
84	D₄⋊C₈	—	34	2	no	—
85	C₈²	—	17	1	no	—
86	C₂×C₁₇	#1589	1	1	yes	characteristic
87	C₂×C₁₇	#2177	1	1	yes	characteristic
88	C₂×C₁₇	#2179	1	1	yes	characteristic
89	D₁₇	—	16	2	no	—
90	C₂×D₈	—	17	2	no	—
91	C₂×M₁₆	—	68	2	no	—
92	C₄×D₄	—	34	2	no	—
93	D₁₆	—	34	2	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	34	1	no	—
96	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
97	#107	C₂×#595	1	2	yes	—
98	C₄⋊C₈	—	34	2	no	—
99	#102	—	34	2	no	—
100	C₂²⋊C₈	—	68	2	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	17	1	no	—
104	C₁₇	#12086	1	1	yes	characteristic, complemented, Sylow
105	C₂×D₄	—	34	2	no	—
106	C₂²×C₄	—	68	2	no	—
107	D₈	—	34	2	no	—
108	D₈	—	34	2	no	—
109	M₁₆	—	136	4	no	—
110	C₂×C₈	—	17	2	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	34	1	no	—
113	C₂×C₈	—	68	2	no	—
114	C₂×C₈	C₂²×#595	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	68	2	no	—
117	C₄²	—	17	1	no	—
118	C₁₆	—	34	2	no	—
119	C₁₆	—	34	1	no	—
120	C₁₆	#63243	1	2	yes	complemented
121	C₂³	—	68	2	no	—
122	D₄	—	68	2	no	—
123	D₄	—	68	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	17	1	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	136	2	no	—
128	C₂×C₄	—	136	2	no	—
129	C₂×C₄	D₄×#595	1	1	yes	characteristic
130	C₈	—	34	2	no	—
131	C₈	—	34	1	no	—
132	C₈	—	68	1	no	—
133	C₈	—	136	2	no	—
134	C₈	C₂×#63243	1	1	yes	characteristic
135	C₈	#10587680	1	1	yes	characteristic
136	C₂²	—	136	2	no	—
137	C₂²	—	136	2	no	—
138	C₂²	—	136	2	no	—
139	C₂²	D₈×#595	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	17	2	no	—
142	C₄	—	34	1	no	—
143	C₄	#11763843	1	1	yes	characteristic
144	C₄	#11763853	1	1	yes	characteristic
145	C₂	—	272	2	no	—
146	C₂	#14544416	1	1	yes	characteristic
147	C₂	#14545182	1	1	yes	characteristic
148	C₂	#14545187	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544416

C₂

yes

maximal, complemented

#14543817

C₂

yes

maximal, characteristic, complemented

#14543739

C₂

yes

maximal, complemented

#14543695

C₂

yes

maximal, characteristic, complemented

#14530630

C₂

yes

maximal, characteristic

#11763843

—

#11763695

C₂²

yes

characteristic

#11763672

C₂²

yes

—

#11763657

—

C₈×#3624

C₂²

yes

characteristic

C₂×D₂₇₂

C₄

yes

characteristic

#11762891

C₂²

yes

—

#11762609

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₁₇×#366

C₄

yes

characteristic

C₂×#63243

—

C₂×D₁₃₆

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#3624

D₄

yes

characteristic

C₂×#63216

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#107

C₂×C₄

yes

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

#10587480

—

D₂₇₂

—

C₂×D₆₈

—

C₂×C₄×D₁₇

—

C₂×#3624

—

C₂×#3624

—

C₂×#3624

D₈

yes

—

#63243

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

#63227

—

C₄×Dic₁₇

C₂×D₄

yes

yes

D₁₃₆

—

D₁₃₆

—

#63216

—

C₁₆×C₁₇

M₁₆

yes

—

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

#135

yes

characteristic

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

#3624

—

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

—

C₄×D₁₆

—

—

—

—

—

C₂²×C₁₇

C₄×D₈

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

#509

yes

—

C₂×D₁₆

—

C₄×D₈

—

—

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×D₈

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

#102

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₁₇

#12086

yes

characteristic, complemented, Sylow

105

C₂×D₄

—

106

C₂²×C₄

—

107

D₈

—

108

D₈

—

109

M₁₆

—

136

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×#595

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

#63243

yes

complemented

121

C₂³

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

136

—

128

C₂×C₄

—

136

—

129

C₂×C₄

D₄×#595

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

136

—

134

C₈

C₂×#63243

yes

characteristic

135

C₈

#10587680

yes

characteristic

136

C₂²

—

136

—

137

C₂²

—

136

—

138

C₂²

—

136

—

139

C₂²

D₈×#595

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#11763843

yes

characteristic

144

C₄

#11763853

yes

characteristic

145

C₂

—

272

—

146

C₂

#14544416

yes

characteristic

147

C₂

#14545182

yes

characteristic

148

C₂

#14545187

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#595
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#366	C₄
socle	C₂²×C₁₇	C₄×D₈
prime radical	C₂×D₂₇₂	C₄
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#366	#595
hypocenter	C₁₇	#12086
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×#595

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#366

C₄

socle

C₂²×C₁₇

C₄×D₈

prime radical

C₂×D₂₇₂

C₄

cosocle

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
1. #14544416	C₂	semidirect
2. #14543817	C₂	semidirect
3. #14543739	C₂	semidirect
4. #14543695	C₂	semidirect
44. C₁₆×C₁₇	M₁₆	semidirect
104. C₁₇	#12086	semidirect
120. C₁₆	#63243	semidirect

H ≅ G/N

type

1. #14544416

C₂

semidirect

2. #14543817

C₂

semidirect

3. #14543739

C₂

semidirect

4. #14543695

C₂

semidirect

44. C₁₆×C₁₇

M₁₆

semidirect

104. C₁₇

#12086

semidirect

120. C₁₆

#63243

semidirect

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

subgroup	index
#14544416	2
#14543817	2
#14543739	2
#14543695	2
#14530630	2
#12086	17

subgroup

index

Subgroups — #17960392

Conjugacy classes of subgroups

Complements

Maximal subgroups