Group #17982357 — subgroups

Subgroups — #17982357

Order 4352.

236 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14546000	C₂	1	2	yes	maximal, complemented
2	#14545842	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#63241	C₂	1	2	yes	maximal, complemented
4	#14545801	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×#3624	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587670	—	2	2	no	—
7	C₄⋊#63241	C₂²	1	2	yes	—
8	C₂²⋊#63241	—	2	2	no	—
9	C₄×#63241	C₂²	1	1	yes	characteristic
10	C₄×#63241	C₂²	1	1	yes	characteristic
11	D₄×#3624	C₂²	1	2	yes	—
12	#11763672	C₄	1	1	yes	characteristic
13	D₄⋊#3624	C₄	1	2	yes	—
14	C₈×#3624	C₂²	1	1	yes	characteristic
15	D₈×Dic₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3624	—	2	2	no	—
17	#10587670	—	4	4	no	—
18	C₂×#63241	—	2	2	no	—
19	C₂×#63241	—	2	1	no	—
20	C₂×#63241	D₄	1	2	yes	—
21	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic
22	D₄×Dic₁₇	C₂×C₄	1	2	yes	—
23	C₂²⋊#3624	—	2	2	no	—
24	C₄⋊#3624	C₂×C₄	1	2	yes	—
25	C₄×#3624	C₂×C₄	1	1	yes	characteristic
26	C₄×#3624	C₂³	1	1	yes	characteristic
27	#10587503	C₈	1	2	yes	—
28	#10587489	C₈	1	1	yes	characteristic
29	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
31	C₂²×Dic₁₇	—	2	2	no	—
32	C₂×#3624	—	2	2	no	—
33	C₂×#3624	—	2	1	no	—
34	C₂×#3624	—	4	2	no	—
35	C₂×#3624	—	4	2	no	—
36	C₂×#3624	C₂×D₄	1	1	yes	characteristic
37	C₂×#3624	#41	1	1	yes	characteristic
38	#63241	—	2	2	no	—
39	#63241	—	4	1	no	—
40	#63241	—	4	2	no	—
41	D₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂²⋊Dic₁₇	—	2	2	no	—
44	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
45	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×#3622	C₂×C₈	1	1	yes	characteristic
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	C₂²×C₈	1	1	yes	characteristic
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	C₂×Dic₁₇	—	4	2	no	—
55	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
56	C₂×Dic₁₇	#131	1	1	yes	characteristic
57	#3624	—	2	1	no	—
58	#3624	—	4	1	no	—
59	#3624	—	4	2	no	—
60	#3624	AGL(1,ℤ/8)	1	2	yes	—
61	C₈×C₁₇	M₃₂	1	2	yes	complemented
62	#3622	—	2	1	no	—
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	C₂×M₃₂	1	1	yes	characteristic
73	C₄×C₁₇	#504	1	1	yes	characteristic
74	Dic₁₇	—	2	1	no	—
75	Dic₁₇	—	4	2	no	—
76	Dic₁₇	#442	1	1	yes	characteristic
77	Dic₁₇	#444	1	1	yes	characteristic
78	C₂×M₃₂	—	34	2	no	—
79	C₄×D₈	—	17	1	no	—
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	#3633	1	1	yes	characteristic
93	C₂²×C₈	—	34	2	no	—
94	C₄×D₄	—	17	2	no	—
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	—	17	1	no	—
100	C₄⋊C₈	—	17	2	no	—
101	#102	—	17	2	no	—
102	C₂²⋊C₈	—	34	2	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₁₇	#12064	1	1	yes	characteristic, complemented, Sylow
107	C₂×D₄	C₂×#3633	1	2	yes	—
108	C₂²×C₄	—	34	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	68	2	no	—
116	C₂×C₈	—	68	2	no	—
117	C₂×C₈	C₂×#3633	1	1	yes	characteristic
118	C₄⋊C₄	—	17	2	no	—
119	C₂²⋊C₄	—	34	2	no	—
120	C₄²	—	17	1	no	—
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	—	17	1	no	—
128	C₂×C₄	—	17	1	no	—
129	C₂×C₄	—	34	2	no	—
130	C₂×C₄	—	68	2	no	—
131	C₂×C₄	—	68	2	no	—
132	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
133	C₈	—	17	2	no	—
134	C₈	—	34	1	no	—
135	C₈	—	34	1	no	—
136	C₈	—	68	2	no	—
137	C₈	—	68	1	no	—
138	C₈	#10587670	1	2	yes	complemented
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	D₄×#3633	1	1	yes	characteristic
143	C₄	—	17	1	no	—
144	C₄	—	17	1	no	—
145	C₄	—	34	1	no	—
146	C₄	—	68	2	no	—
147	C₄	C₂×#10587670	1	1	yes	characteristic
148	C₄	#11763930	1	1	yes	characteristic
149	C₂	—	8	2	no	—
150	C₂	#14545915	1	1	yes	characteristic
151	C₂	#14546000	1	1	yes	characteristic
152	C₂	#14546006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546000

C₂

yes

maximal, complemented

#14545842

C₂

yes

maximal, characteristic, complemented

D₄⋊#63241

C₂

yes

maximal, complemented

#14545801

C₂

yes

maximal, characteristic, complemented

D₈×#3624

C₂

yes

maximal, characteristic

C₂×#10587670

—

C₄⋊#63241

C₂²

yes

—

C₂²⋊#63241

—

C₄×#63241

C₂²

yes

characteristic

C₄×#63241

C₂²

yes

characteristic

D₄×#3624

C₂²

yes

—

#11763672

C₄

yes

characteristic

D₄⋊#3624

C₄

yes

—

C₈×#3624

C₂²

yes

characteristic

D₈×Dic₁₇

C₄

yes

characteristic

C₂²×#3624

—

—

—

—

D₄

yes

—

C₂×D₈×C₁₇

C₈

yes

characteristic

D₄×Dic₁₇

C₂×C₄

yes

—

C₂²⋊#3624

—

C₄⋊#3624

C₂×C₄

yes

—

C₄×#3624

C₂×C₄

yes

characteristic

C₄×#3624

C₂³

yes

characteristic

#10587503

C₈

yes

—

#10587489

C₈

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₇

C₂×C₈

yes

—

C₂²×Dic₁₇

—

—

—

—

—

C₂×D₄

yes

yes

—

—

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₈

yes

—

C₄×Dic₁₇

C₂²×C₄

yes

characteristic

C₂×#3622

C₂×C₈

yes

characteristic

#12064

—

maximal, Sylow

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄×D₄

yes

C₂×Dic₁₇

yes

—

—

—

yes

—

C₈×C₁₇

M₃₂

yes

complemented

#3622

—

#2169

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

C₂×M₃₂

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

C₂×M₃₂

—

C₄×D₈

—

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

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

100

C₄⋊C₈

—

101

#102

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₁₇

#12064

yes

characteristic, complemented, Sylow

107

C₂×D₄

C₂×#3633

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

—

117

C₂×C₈

C₂×#3633

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×#3633

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#10587670

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#3633

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#10587670

yes

characteristic

148

C₄

#11763930

yes

characteristic

149

C₂

—

150

C₂

#14545915

yes

characteristic

151

C₂

#14546000

yes

characteristic

152

C₂

#14546006

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#3633
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting, prime radical	C₂×D₈×C₁₇	C₈
socle	C₂²×C₁₇	C₈×D₄
cosocle	C₄×#3624	C₂³
hypercenter	C₂×D₈	#3633
hypocenter	C₁₇	#12064
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#3633

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3633

Fitting, prime radical

C₂×D₈×C₁₇

C₈

socle

C₂²×C₁₇

C₈×D₄

C₂³

C₂×D₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. #14546000	C₂	semidirect
2. #14545842	C₂	semidirect
3. D₄⋊#63241	C₂	semidirect
4. #14545801	C₂	semidirect
61. C₈×C₁₇	M₃₂	semidirect
106. C₁₇	#12064	semidirect
138. C₈	#10587670	semidirect

H ≅ G/N

type

1. #14546000

C₂

semidirect

2. #14545842

C₂

semidirect

3. D₄⋊#63241

C₂

semidirect

4. #14545801

C₂

semidirect

61. C₈×C₁₇

M₃₂

semidirect

106. C₁₇

#12064

semidirect

138. C₈

#10587670

semidirect

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

subgroup	index
#14546000	2
#14545842	2
D₄⋊#63241	2
#14545801	2
D₈×#3624	2
#12064	17

subgroup

index

Subgroups — #17982357

Conjugacy classes of subgroups

Complements

Maximal subgroups