Group #36453933 — subgroups

Subgroups — #36453933

Order 22784.

558 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 127 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	#26020390	C₂	1	1	yes	maximal, characteristic, complemented
2	#26018920	C₂	1	4	yes	maximal, complemented
3	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	2	yes	—
6	#19151622	C₂²	1	2	yes	—
7	#19151363	—	2	2	no	—
8	#19151342	—	2	8	no	—
9	#19151320	—	2	2	no	—
10	C₂²×D₈×C₈₉	—	2	2	no	—
11	C₂×#127×C₈₉	C₂²	1	2	yes	—
12	#493×C₈₉	—	2	2	no	—
13	#493×C₈₉	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₄⋊#10623494	—	2	4	no	—
17	C₄⋊#10623494	D₄	1	4	yes	—
18	C₂²⋊#10623494	—	2	2	no	—
19	C₄×#10623494	—	2	2	no	—
20	C₂²×D₄×C₈₉	—	2	2	no	—
21	C₂²×D₄×C₈₉	—	2	4	no	—
22	C₂×D₈×C₈₉	—	2	8	no	—
23	C₂×D₈×C₈₉	—	4	8	no	—
24	C₂²×C₈×C₈₉	D₄	1	2	yes	—
25	#127×C₈₉	—	2	2	no	—
26	#127×C₈₉	—	2	4	no	—
27	#127×C₈₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	#13083558	—	4	4	no	—
32	C₂×#10623494	—	2	4	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂⁴×C₈₉	—	4	2	no	—
35	C₂×D₄×C₈₉	—	2	4	no	—
36	C₂×D₄×C₈₉	—	2	8	no	—
37	C₂×D₄×C₈₉	—	4	4	no	—
38	C₂×D₄×C₈₉	—	4	2	no	—
39	C₂×D₄×C₈₉	—	4	8	no	—
40	C₂×D₄×C₈₉	—	8	8	no	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	#41	1	2	yes	—
43	D₈×C₈₉	—	4	16	no	—
44	C₂×C₈×C₈₉	—	2	4	no	—
45	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
46	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₈₉	C₂×D₄	1	2	yes	—
49	C₂³×C₈₉	—	4	4	no	—
50	C₂³×C₈₉	—	8	8	no	—
51	C₂³×C₈₉	—	8	4	no	—
52	C₂³×C₈₉	#124	1	1	yes	characteristic
53	D₄×C₈₉	—	4	8	no	—
54	D₄×C₈₉	—	8	8	no	—
55	D₄×C₈₉	—	8	8	no	—
56	D₄×C₈₉	—	8	4	no	—
57	C₂×C₄×C₈₉	—	2	4	no	—
58	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	#121	1	2	yes	—
61	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
62	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
63	C₈×C₈₉	—	2	4	no	—
64	#10623494	—	4	4	no	—
65	#10623494	—	8	2	no	—
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	—	8	4	no	—
68	C₂²×C₈₉	—	8	8	no	—
69	C₂²×C₈₉	—	16	8	no	—
70	C₂²×C₈₉	#390	1	1	yes	characteristic
71	C₂²×C₈₉	#490	1	2	yes	—
72	C₄×C₈₉	—	2	2	no	—
73	C₄×C₈₉	#418	1	2	yes	—
74	C₄×C₈₉	#472	1	4	yes	—
75	C₄×C₈₉	#496	1	2	yes	—
76	#9780	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	2	2	no	—
78	C₂×C₈₉	—	16	4	no	—
79	C₂×C₈₉	#1961	1	1	yes	characteristic
80	C₂×C₈₉	#1978	1	2	yes	—
81	C₂×#493	D₈₉	1	1	yes	characteristic
82	#1837	—	89	1	no	—
83	#1545	—	89	4	no	—
84	#1452	—	89	1	no	—
85	C₈₉	#9780	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₈	—	2	2	no	—
87	C₂×#127	C₂×D₈₉	1	2	yes	—
88	#493	—	2	2	no	—
89	#493	Dic₈₉	1	4	yes	—
90	#431	—	89	2	no	—
91	#423	—	89	2	no	—
92	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
93	#341	—	178	2	no	—
94	#331	—	178	8	no	—
95	#322	—	178	2	no	—
96	C₂²×D₄	—	2	2	no	—
97	C₂²×D₄	—	2	4	no	—
98	C₂×D₈	—	2	8	no	—
99	C₂×D₈	—	4	8	no	—
100	C₂×M₁₆	—	178	2	no	—
101	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
102	#127	—	2	4	no	—
103	#127	—	2	2	no	—
104	#127	C₂×Dic₈₉	1	4	yes	—
105	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
106	C₄⋊C₈	—	89	4	no	—
107	C₄⋊C₈	—	178	4	no	—
108	C₂²⋊C₈	—	178	2	no	—
109	C₄×C₈	—	178	2	no	—
110	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
111	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
112	C₂⁴	—	4	2	no	—
113	C₂×D₄	—	2	8	no	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	4	4	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	8	no	—
118	C₂×D₄	—	8	8	no	—
119	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂²×C₄	#13083573	1	2	yes	—
121	D₈	—	4	16	no	—
122	M₁₆	—	356	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	—	178	4	no	—
126	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
127	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
128	C₄²	D₄×D₈₉	1	2	yes	—
129	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂³	—	4	4	no	—
131	C₂³	—	8	8	no	—
132	C₂³	—	8	4	no	—
133	C₂³	#15667999	1	1	yes	characteristic
134	D₄	—	4	8	no	—
135	D₄	—	8	8	no	—
136	D₄	—	8	4	no	—
137	D₄	—	8	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
140	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
141	C₂×C₄	#15667983	1	2	yes	—
142	C₂×C₄	#15668008	1	1	yes	characteristic
143	C₂×C₄	#15668038	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	356	4	no	—
146	C₈	—	712	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	8	8	no	—
149	C₂²	—	8	4	no	—
150	C₂²	—	16	8	no	—
151	C₂²	#19151501	1	1	yes	characteristic
152	C₂²	#19151962	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#19151603	1	2	yes	—
155	C₄	#19151865	1	4	yes	—
156	C₄	#19151985	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	16	4	no	—
159	C₂	#26021077	1	1	yes	characteristic
160	C₂	#26021163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020390

C₂

yes

maximal, characteristic, complemented

#26018920

C₂

yes

maximal, complemented

#26018544

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

—

#19151363

—

#19151342

—

#19151320

—

C₂²×D₈×C₈₉

—

C₂×#127×C₈₉

C₂²

yes

—

#493×C₈₉

—

#493×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

—

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

#127×C₈₉

—

#127×C₈₉

—

#127×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

D₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#124

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#490

yes

—

C₄×C₈₉

—

C₄×C₈₉

#418

yes

—

C₄×C₈₉

#472

yes

—

C₄×C₈₉

#496

yes

—

#9780

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1961

yes

characteristic

C₂×C₈₉

#1978

yes

—

C₂×#493

D₈₉

yes

—

—

—

C₈₉

#9780

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×#127

C₂×D₈₉

yes

—

#493

—

#493

Dic₈₉

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#341

—

178

—

#331

—

178

—

#322

—

178

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

100

C₂×M₁₆

—

178

—

101

C₂²×C₈

C₂²⋊D₈₉

yes

—

102

#127

—

103

#127

—

104

#127

C₂×Dic₈₉

yes

—

105

C₂×C₄²

C₂²×D₈₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

178

—

108

C₂²⋊C₈

—

178

—

109

C₄×C₈

—

178

—

110

C₄×C₈

C₂×Dic₈₉

yes

—

111

C₄×C₈

C₂²⋊D₈₉

yes

—

112

C₂⁴

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂²×C₄

#13083573

yes

—

121

D₈

—

122

M₁₆

—

356

—

123

C₂×C₈

—

124

C₂×C₈

—

356

—

125

C₂×C₈

—

178

—

126

C₂×C₈

C₂²⋊Dic₈₉

yes

—

127

C₄²

C₂²×Dic₈₉

yes

characteristic

128

C₄²

D₄×D₈₉

yes

—

129

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#15667999

yes

characteristic

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

140

C₂×C₄

D₄×Dic₈₉

yes

—

141

C₂×C₄

#15667983

yes

—

142

C₂×C₄

#15668008

yes

characteristic

143

C₂×C₄

#15668038

yes

—

144

C₈

—

145

C₈

—

356

—

146

C₈

—

712

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#19151501

yes

characteristic

152

C₂²

#19151962

yes

—

153

C₄

—

154

C₄

#19151603

yes

—

155

C₄

#19151865

yes

—

156

C₄

#19151985

yes

—

157

C₂

—

158

C₂

—

159

C₂

#26021077

yes

characteristic

160

C₂

#26021163

yes

—

	subgroup	G/H
center	C₂²	#19151501
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting, prime radical	C₂×#493×C₈₉	C₂
socle	C₂²×C₈₉	#390
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#493	D₈₉
hypocenter	C₈₉	#9780
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#19151501

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting, prime radical

C₂×#493×C₈₉

C₂

socle

C₂²×C₈₉

#390

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #26020390	C₂	semidirect
2. #26018920	C₂	semidirect
3. #26018544	C₂	semidirect
85. C₈₉	#9780	semidirect

H ≅ G/N

type

1. #26020390

C₂

semidirect

2. #26018920

C₂

semidirect

3. #26018544

C₂

semidirect

85. C₈₉

#9780

semidirect

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

subgroup	index
#26020390	2
#26018920	2
#26018544	2
C₂×#493×C₈₉	2
#9780	89

subgroup

index

C₂×#493×C₈₉

Subgroups — #36453933

Conjugacy classes of subgroups

Complements

Maximal subgroups