Group #17960376 — subgroups

Subgroups — #17960376

Order 4352.

238 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 68 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	#14544417	C₂	1	2	yes	maximal, complemented
2	#14543815	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543738	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3624	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763844	—	2	2	no	—
7	#11763695	C₂²	1	1	yes	characteristic
8	#11763672	C₂²	1	2	yes	—
9	D₄⋊#3624	—	2	2	no	—
10	C₈×#3624	C₂²	1	1	yes	characteristic
11	D₈×Dic₁₇	C₂²	1	2	yes	—
12	#11762635	C₄	1	1	yes	characteristic
13	#11762610	C₄	1	2	yes	—
14	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×D₁₆	C₄	1	1	yes	characteristic
16	C₂×#63247	—	4	2	no	—
17	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
18	D₄×Dic₁₇	—	2	2	no	—
19	C₂²⋊#3624	—	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	C₁₇×D₁₆	—	2	2	no	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
26	#10587503	—	2	2	no	—
27	#10587489	C₂×C₄	1	2	yes	—
28	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
29	C₂×#63214	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₇	—	2	2	no	—
31	C₂²×Dic₁₇	—	4	2	no	—
32	#63247	—	8	4	no	—
33	C₂×#3624	—	2	1	no	—
34	C₂×#3624	—	4	2	no	—
35	C₂×#3624	D₈	1	1	yes	characteristic
36	C₂×#3624	D₈	1	1	yes	characteristic
37	D₈×C₁₇	—	2	2	no	—
38	D₈×C₁₇	—	2	2	no	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊Dic₁₇	—	4	2	no	—
41	C₄⋊Dic₁₇	—	2	2	no	—
42	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×#3622	#41	1	1	yes	characteristic
44	#63216	—	2	2	no	—
45	C₁₆×C₁₇	M₁₆	1	2	yes	complemented
46	#63214	—	2	1	no	—
47	#12081	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	—	4	2	no	—
49	D₄×C₁₇	—	4	2	no	—
50	D₄×C₁₇	—	4	2	no	—
51	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
52	C₂×Dic₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	8	2	no	—
54	C₂×Dic₁₇	—	8	2	no	—
55	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₁₇	#135	1	1	yes	characteristic
57	#3624	—	2	2	no	—
58	#3624	—	2	1	no	—
59	#3624	—	8	2	no	—
60	#3624	D₁₆	1	2	yes	complemented
61	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₁₇	#131	1	1	yes	characteristic
63	#3622	—	2	1	no	—
64	C₄×D₁₆	—	17	1	no	—
65	#1589	—	17	2	no	—
66	#1371	—	17	1	no	—
67	#1331	—	17	2	no	—
68	C₈×C₁₆	—	17	1	no	—
69	C₂²×C₁₇	—	8	2	no	—
70	C₂²×C₁₇	—	8	2	no	—
71	C₂²×C₁₇	—	8	2	no	—
72	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
73	C₄×C₁₇	#436	1	1	yes	characteristic
74	C₄×C₁₇	#443	1	1	yes	characteristic
75	Dic₁₇	—	2	1	no	—
76	Dic₁₇	—	8	2	no	—
77	Dic₁₇	C₂×D₁₆	1	1	yes	characteristic
78	Dic₁₇	#508	1	1	yes	characteristic
79	C₂×D₁₆	#595	1	1	yes	characteristic
80	C₄×D₈	—	17	2	no	—
81	#436	—	34	2	no	—
82	#366	—	17	1	no	—
83	#357	—	17	2	no	—
84	C₄×C₁₆	—	17	1	no	—
85	C₄×C₁₆	—	17	1	no	—
86	#335	—	17	2	no	—
87	D₄⋊C₈	—	34	2	no	—
88	C₈²	—	17	1	no	—
89	C₂×C₁₇	—	16	2	no	—
90	C₂×C₁₇	#1589	1	1	yes	characteristic
91	C₂×C₁₇	C₄×D₁₆	1	1	yes	characteristic
92	C₂×C₁₇	#2178	1	1	yes	characteristic
93	C₂×D₈	C₂×#595	1	2	yes	—
94	C₂×M₁₆	—	68	2	no	—
95	C₄×D₄	—	34	2	no	—
96	D₁₆	—	2	2	no	—
97	C₂×C₁₆	—	17	1	no	—
98	C₂×C₁₆	—	34	1	no	—
99	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
100	#107	—	17	2	no	—
101	C₄⋊C₈	—	34	2	no	—
102	#102	—	34	2	no	—
103	C₂²⋊C₈	—	68	2	no	—
104	C₄×C₈	—	17	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	—	17	1	no	—
107	C₁₇	#12081	1	1	yes	characteristic, complemented, Sylow
108	C₂×D₄	—	2	2	no	—
109	C₂²×C₄	—	68	2	no	—
110	D₈	—	2	2	no	—
111	D₈	—	2	2	no	—
112	M₁₆	—	136	4	no	—
113	C₂×C₈	—	17	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	—	17	1	no	—
116	C₂×C₈	—	34	1	no	—
117	C₂×C₈	—	68	2	no	—
118	C₂×C₈	C₂²×#595	1	1	yes	characteristic
119	C₄⋊C₄	—	34	2	no	—
120	C₂²⋊C₄	—	68	2	no	—
121	C₄²	—	17	1	no	—
122	C₁₆	—	34	2	no	—
123	C₁₆	—	34	1	no	—
124	C₁₆	#63247	1	2	yes	complemented
125	C₂³	—	4	2	no	—
126	D₄	—	4	2	no	—
127	D₄	—	4	2	no	—
128	C₂×C₄	—	17	1	no	—
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	68	2	no	—
131	C₂×C₄	—	136	2	no	—
132	C₂×C₄	—	136	2	no	—
133	C₂×C₄	D₄×#595	1	1	yes	characteristic
134	C₈	—	17	2	no	—
135	C₈	—	34	1	no	—
136	C₈	—	34	2	no	—
137	C₈	—	34	1	no	—
138	C₈	—	136	2	no	—
139	C₈	#10587678	1	1	yes	characteristic
140	C₈	C₂×#63247	1	1	yes	characteristic
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	—	8	2	no	—
144	C₂²	D₈×#595	1	1	yes	characteristic
145	C₄	—	17	1	no	—
146	C₄	—	17	1	no	—
147	C₄	—	34	1	no	—
148	C₄	—	136	2	no	—
149	C₄	#11763844	1	1	yes	characteristic
150	C₄	#11763851	1	1	yes	characteristic
151	C₂	—	16	2	no	—
152	C₂	#14544417	1	1	yes	characteristic
153	C₂	D₁₆×#595	1	1	yes	characteristic
154	C₂	#14545183	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544417

C₂

yes

maximal, complemented

#14543815

C₂

yes

maximal, characteristic, complemented

#14543738

C₂

yes

maximal, complemented

C₁₆×#3624

C₂

yes

maximal, characteristic, complemented

D₁₆×Dic₁₇

C₂

yes

maximal, characteristic

#11763844

—

#11763695

C₂²

yes

characteristic

#11763672

C₂²

yes

—

D₄⋊#3624

—

C₈×#3624

C₂²

yes

characteristic

D₈×Dic₁₇

C₂²

yes

—

#11762635

C₄

yes

characteristic

#11762610

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₂×C₁₇×D₁₆

C₄

yes

characteristic

C₂×#63247

—

C₂×D₈×C₁₇

C₂×C₄

yes

—

D₄×Dic₁₇

—

C₂²⋊#3624

—

C₄⋊#3624

—

C₄×#3624

D₄

yes

characteristic

C₄×#3624

D₄

yes

characteristic

C₂×#63216

—

C₁₇×D₁₆

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

#10587503

—

#10587489

C₂×C₄

yes

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×D₄×C₁₇

—

C₂²×Dic₁₇

—

—

—

—

D₈

yes

characteristic

C₂×#3624

D₈

yes

characteristic

D₈×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₂×D₄

yes

yes

—

C₁₆×C₁₇

M₁₆

yes

—

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₈

yes

C₂×Dic₁₇

yes

—

—

—

D₁₆

yes

complemented

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

—

C₄×D₁₆

—

#1589

—

#1371

—

#1331

—

C₈×C₁₆

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₈

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂×D₁₆

yes

Dic₁₇

yes

C₂×D₁₆

yes

C₄×D₈

—

#436

—

#366

—

#357

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1589

yes

characteristic

C₂×C₁₇

C₄×D₁₆

yes

C₂×C₁₇

yes

C₂×D₈

yes

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

characteristic

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₁₇

#12081

yes

characteristic, complemented, Sylow

108

C₂×D₄

—

109

C₂²×C₄

—

110

D₈

—

111

D₈

—

112

M₁₆

—

136

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×#595

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

#63247

yes

complemented

125

C₂³

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

136

—

132

C₂×C₄

—

136

—

133

C₂×C₄

D₄×#595

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

136

—

139

C₈

#10587678

yes

characteristic

140

C₈

C₂×#63247

yes

characteristic

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

D₈×#595

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

136

—

149

C₄

#11763844

yes

characteristic

150

C₄

#11763851

yes

characteristic

151

C₂

—

152

C₂

#14544417

yes

characteristic

153

C₂

D₁₆×#595

yes

characteristic

154

C₂

#14545183

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#595
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting, prime radical	C₂×C₁₇×D₁₆	C₄
socle	C₂²×C₁₇	C₄×D₈
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂×D₁₆	#595
hypocenter	C₁₇	#12081
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×#595

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting, prime radical

C₂×C₁₇×D₁₆

C₄

socle

C₂²×C₁₇

C₄×D₈

cosocle

C₈×Dic₁₇

C₂³

C₂×D₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #14544417	C₂	semidirect
2. #14543815	C₂	semidirect
3. #14543738	C₂	semidirect
4. C₁₆×#3624	C₂	semidirect
45. C₁₆×C₁₇	M₁₆	semidirect
60. #3624	D₁₆	semidirect
107. C₁₇	#12081	semidirect
124. C₁₆	#63247	semidirect

H ≅ G/N

type

1. #14544417

C₂

semidirect

2. #14543815

C₂

semidirect

3. #14543738

C₂

semidirect

4. C₁₆×#3624

C₂

semidirect

45. C₁₆×C₁₇

M₁₆

semidirect

60. #3624

D₁₆

semidirect

107. C₁₇

#12081

semidirect

124. C₁₆

#63247

semidirect

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

subgroup	index
#14544417	2
#14543815	2
#14543738	2
C₁₆×#3624	2
D₁₆×Dic₁₇	2
#12081	17

subgroup

index

#14544417

#14543815

#14543738

C₁₆×#3624

D₁₆×Dic₁₇

#12081

Subgroups — #17960376

Conjugacy classes of subgroups

Complements

Maximal subgroups