Group #17960281 — subgroups

Subgroups — #17960281

Order 4352.

236 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 86 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	#14545168	C₂	1	2	yes	maximal, complemented
2	#14543806	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#63216	C₂	1	2	yes	maximal, complemented
4	#14543697	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	—	2	2	no	—
7	C₄⋊#63216	C₂²	1	2	yes	—
8	C₂²⋊#63216	—	2	2	no	—
9	C₄×#63216	C₂²	1	1	yes	characteristic
10	C₄×#63216	C₂²	1	1	yes	characteristic
11	D₄×#3622	C₂²	1	2	yes	—
12	#11762562	C₄	1	1	yes	characteristic
13	D₄⋊#3622	C₄	1	2	yes	—
14	C₈×#3622	C₂²	1	1	yes	characteristic
15	C₄×D₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×#3622	—	2	2	no	—
19	#10587538	—	4	4	no	—
20	C₂×#63216	—	2	2	no	—
21	C₂×#63216	—	2	1	no	—
22	C₂×#63216	D₄	1	2	yes	—
23	C₁₇×#107	C₈	1	1	yes	characteristic
24	C₁₇×#102	C₈	1	2	yes	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊#3622	—	2	2	no	—
27	C₄⋊#3622	C₂×C₄	1	2	yes	—
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₁₇	—	2	2	no	—
32	D₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
36	C₂²⋊C₄×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×#3622	—	2	2	no	—
39	C₂×#3622	—	2	1	no	—
40	C₂×#3622	—	4	2	no	—
41	C₂×#3622	—	4	2	no	—
42	C₂×#3622	C₂×D₄	1	1	yes	characteristic
43	C₂×#3622	#41	1	1	yes	characteristic
44	#63216	—	2	2	no	—
45	#63216	—	4	1	no	—
46	#63216	—	4	2	no	—
47	#12064	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	—	2	2	no	—
49	D₄×C₁₇	—	2	2	no	—
50	D₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	4	2	no	—
53	C₂×C₄×C₁₇	—	4	2	no	—
54	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
57	C₈×C₁₇	—	2	1	no	—
58	C₈×C₁₇	M₃₂	1	2	yes	complemented
59	#3622	—	2	1	no	—
60	#3622	—	4	1	no	—
61	#3622	—	4	2	no	—
62	#3622	AGL(1,ℤ/8)	1	2	yes	—
63	#2169	—	17	2	no	—
64	C₈×D₈	—	17	1	no	—
65	#1367	—	17	1	no	—
66	D₄⋊C₁₆	—	17	2	no	—
67	#1312	—	17	1	no	—
68	C₂²×C₁₇	—	4	2	no	—
69	C₂²×C₁₇	—	4	2	no	—
70	C₂²×C₁₇	—	4	2	no	—
71	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	#442	1	1	yes	characteristic
75	C₄×C₁₇	#444	1	1	yes	characteristic
76	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
77	C₄×C₁₇	#504	1	1	yes	characteristic
78	C₂×M₃₂	—	34	2	no	—
79	C₄×D₈	#595	1	1	yes	characteristic
80	C₈×D₄	—	17	2	no	—
81	C₄⋊C₁₆	—	17	2	no	—
82	C₂²⋊C₁₆	—	34	2	no	—
83	C₄×C₁₆	—	17	1	no	—
84	C₄×C₁₆	—	17	1	no	—
85	#335	—	17	1	no	—
86	D₄⋊C₈	—	17	2	no	—
87	C₈²	—	17	1	no	—
88	C₂×C₁₇	—	8	2	no	—
89	C₂×C₁₇	#1580	1	1	yes	characteristic
90	C₂×C₁₇	#2169	1	1	yes	characteristic
91	C₂×C₁₇	#2171	1	1	yes	characteristic
92	C₂×D₈	#3624	1	1	yes	characteristic
93	C₂²×C₈	—	34	2	no	—
94	C₄×D₄	C₂×#595	1	2	yes	—
95	M₃₂	—	68	4	no	—
96	C₂×C₁₆	—	17	2	no	—
97	C₂×C₁₆	—	34	1	no	—
98	C₂×C₁₆	—	34	2	no	—
99	#107	#3624	1	1	yes	characteristic
100	C₄⋊C₈	—	17	2	no	—
101	#102	#3624	1	2	yes	—
102	C₂²⋊C₈	—	34	2	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	1	no	—
105	C₄×C₈	C₂×#595	1	1	yes	characteristic
106	C₁₇	#12064	1	1	yes	characteristic, complemented, Sylow
107	C₂×D₄	C₂×#3624	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	34	1	no	—
113	C₂×C₈	—	34	2	no	—
114	C₂×C₈	—	68	2	no	—
115	C₂×C₈	—	68	2	no	—
116	C₂×C₈	C₂×#3624	1	1	yes	characteristic
117	C₂×C₈	C₂×#3624	1	1	yes	characteristic
118	C₄⋊C₄	C₂×#3624	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	C₂²×#595	1	1	yes	characteristic
121	C₁₆	—	34	2	no	—
122	C₁₆	—	68	2	no	—
123	C₁₆	—	68	1	no	—
124	C₂³	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	#10587678	1	1	yes	characteristic
131	C₂×C₄	D₄×#595	1	1	yes	characteristic
132	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	17	2	no	—
135	C₈	—	34	1	no	—
136	C₈	—	68	1	no	—
137	C₈	—	68	2	no	—
138	C₈	#10587538	1	2	yes	complemented
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	D₄×#3624	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	#11763850	1	1	yes	characteristic
146	C₄	#11763856	1	1	yes	characteristic
147	C₄	C₂×#10587538	1	1	yes	characteristic
148	C₄	#11763867	1	1	yes	characteristic
149	C₂	—	8	2	no	—
150	C₂	#14544380	1	1	yes	characteristic
151	C₂	#14545168	1	1	yes	characteristic
152	C₂	#14545174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545168

C₂

yes

maximal, complemented

#14543806

C₂

yes

maximal, characteristic, complemented

D₄⋊#63216

C₂

yes

maximal, complemented

#14543697

C₂

yes

maximal, characteristic, complemented

D₈×#3622

C₂

yes

maximal, characteristic

C₂×#10587538

—

C₄⋊#63216

C₂²

yes

—

C₂²⋊#63216

—

C₄×#63216

C₂²

yes

characteristic

C₄×#63216

C₂²

yes

characteristic

D₄×#3622

C₂²

yes

—

#11762562

C₄

yes

characteristic

D₄⋊#3622

C₄

yes

—

C₈×#3622

C₂²

yes

characteristic

C₄×D₈×C₁₇

C₄

yes

characteristic

C₂×D₈×C₁₇

C₈

yes

characteristic

C₄×D₄×C₁₇

C₂×C₄

yes

—

C₂²×#3622

—

—

—

—

D₄

yes

—

C₁₇×#107

C₈

yes

characteristic

C₁₇×#102

C₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²⋊#3622

—

C₄⋊#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂³

yes

characteristic

C₂×D₄×C₁₇

C₂×C₈

yes

—

C₂²×C₄×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

—

—

—

—

C₂×D₄

yes

yes

—

—

—

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

M₃₂

yes

—

—

—

yes

—

#2169

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×M₃₂

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

C₂×M₃₂

—

C₄×D₈

#595

yes

characteristic

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×D₈

yes

C₂²×C₈

—

C₄×D₄

C₂×#595

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#3624

yes

characteristic

100

C₄⋊C₈

—

101

#102

#3624

yes

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×#595

yes

characteristic

106

C₁₇

#12064

yes

characteristic, complemented, Sylow

107

C₂×D₄

C₂×#3624

yes

—

108

C₂²×C₄

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂×#3624

yes

characteristic

117

C₂×C₈

C₂×#3624

yes

characteristic

118

C₄⋊C₄

C₂×#3624

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×#595

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10587678

yes

characteristic

131

C₂×C₄

D₄×#595

yes

characteristic

132

C₂×C₄

C₂²×#3624

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#10587538

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#3624

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#11763850

yes

characteristic

146

C₄

#11763856

yes

characteristic

147

C₄

C₂×#10587538

yes

characteristic

148

C₄

#11763867

yes

characteristic

149

C₂

—

150

C₂

#14544380

yes

characteristic

151

C₂

#14545168

yes

characteristic

152

C₂

#14545174

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₄×D₈×C₁₇	C₄
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×D₈×C₁₇	C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₄×D₈	#595
hypocenter	C₁₇	#12064
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₄²

C₂²×#595

Fitting

C₄×D₈×C₁₇

C₄

socle

C₂²×C₁₇

C₈×D₄

prime radical

C₂×D₈×C₁₇

C₈

C₂³

C₄×D₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #14545168	C₂	semidirect
2. #14543806	C₂	semidirect
3. D₄⋊#63216	C₂	semidirect
4. #14543697	C₂	semidirect
58. C₈×C₁₇	M₃₂	semidirect
106. C₁₇	#12064	semidirect
138. C₈	#10587538	semidirect

H ≅ G/N

type

1. #14545168

C₂

semidirect

2. #14543806

C₂

semidirect

3. D₄⋊#63216

C₂

semidirect

4. #14543697

C₂

semidirect

58. C₈×C₁₇

M₃₂

semidirect

106. C₁₇

#12064

semidirect

138. C₈

#10587538

semidirect

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

subgroup	index
#14545168	2
#14543806	2
D₄⋊#63216	2
#14543697	2
D₈×#3622	2
#12064	17

subgroup

index

Subgroups — #17960281

Conjugacy classes of subgroups

Complements

Maximal subgroups