Group #36580973 — subgroups

Subgroups — #36580973

Order 22784.

984 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 255 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	C₄×#15668009	C₂	1	1	yes	maximal, characteristic
2	#26020543	C₂	1	12	yes	maximal, complemented
3	#26020395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₈₉	C₂²	1	4	yes	—
6	C₂×#15668009	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₅₆	C₂²	1	6	yes	—
8	C₂×#15667924	C₂²	1	12	yes	—
9	#19151500	C₄	1	6	yes	—
10	#19151468	C₄	1	1	yes	characteristic
11	#19151353	C₂²	1	6	yes	—
12	C₄³×C₈₉	C₂²	1	1	yes	characteristic
13	#336×C₈₉	C₂²	1	6	yes	—
14	C₂²×D₃₅₆	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₈₉	C₂³	1	4	yes	—
16	C₂²×#10623494	C₂³	1	4	yes	—
17	C₂×C₈×D₈₉	—	4	12	no	—
18	#15668009	C₈	1	8	yes	complemented
19	C₄×D₃₅₆	—	4	6	no	—
20	C₄×D₃₅₆	—	4	6	no	—
21	C₂×C₄⋊Dic₈₉	C₂×C₄	1	6	yes	—
22	C₂×#13083551	C₂×C₄	1	12	yes	—
23	#15667924	—	4	12	no	—
24	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
28	C₂³×D₈₉	C₂²×C₄	1	4	yes	—
29	C₂×D₃₅₆	—	4	6	no	—
30	C₂×D₃₅₆	—	4	6	no	—
31	C₂×D₃₅₆	C₂×C₈	1	24	yes	complemented
32	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
33	C₂×C₄×D₈₉	—	2	12	no	—
34	C₂×C₄×D₈₉	—	2	12	no	—
35	C₂×#10623494	—	2	12	no	—
36	C₈×D₈₉	—	4	16	no	—
37	C₄⋊Dic₈₉	—	4	6	no	—
38	#13083551	—	4	12	no	—
39	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	12	no	—
43	C₄²×C₈₉	—	2	6	no	—
44	C₄²×C₈₉	—	4	3	no	—
45	C₄²×C₈₉	C₂×C₈	1	4	yes	complemented
46	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
47	C₂²×D₈₉	—	2	12	no	—
48	C₂²×D₈₉	—	2	12	no	—
49	C₂²×D₈₉	C₂²×C₈	1	8	yes	—
50	C₂×C₄×C₈₉	—	2	3	no	—
51	C₂×C₄×C₈₉	—	2	3	no	—
52	C₂×C₄×C₈₉	—	2	3	no	—
53	C₂×C₄×C₈₉	—	2	6	no	—
54	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₈₉	H(5,𝔽₂)	1	6	yes	—
56	C₂×Dic₈₉	—	2	12	no	—
57	D₃₅₆	—	4	24	no	—
58	C₄×D₈₉	—	4	4	no	—
59	C₄×D₈₉	—	4	12	no	—
60	C₈×C₈₉	—	4	4	no	—
61	#10623494	—	4	4	no	—
62	C₂²×C₈₉	#518	1	3	yes	—
63	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
64	C₂²×C₈₉	#568	1	3	yes	—
65	C₂×D₈₉	—	2	24	no	—
66	C₂×D₈₉	—	4	4	no	—
67	C₂×D₈₉	—	4	12	no	—
68	C₄×C₈₉	—	2	6	no	—
69	C₄×C₈₉	—	4	3	no	—
70	C₄×C₈₉	#560	1	4	yes	—
71	Dic₈₉	—	4	4	no	—
72	#23355	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	#2380	1	1	yes	characteristic
74	C₂×C₈₉	#2972	1	3	yes	—
75	C₂×C₈₉	#2980	1	3	yes	—
76	D₈₉	—	4	8	no	—
77	C₄×#127	—	89	1	no	—
78	#1869	—	89	12	no	—
79	#1839	—	89	1	no	—
80	#1839	D₈₉	1	1	yes	characteristic, complemented
81	C₈₉	#23355	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	89	4	no	—
83	C₂×#127	—	89	1	no	—
84	C₂×C₄×D₄	—	89	6	no	—
85	C₂×C₂²⋊C₈	—	89	12	no	—
86	#390	—	89	6	no	—
87	#384	—	89	1	no	—
88	C₄³	C₂×D₈₉	1	1	yes	characteristic
89	#336	—	89	6	no	—
90	#336	C₂×D₈₉	1	6	yes	—
91	C₂²×D₄	—	89	6	no	—
92	C₂³×C₄	—	89	4	no	—
93	C₂²×C₈	—	89	4	no	—
94	C₂²×C₈	—	356	12	no	—
95	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
96	#127	—	89	8	no	—
97	C₄×D₄	—	356	6	no	—
98	C₄×D₄	—	356	6	no	—
99	C₂×C₄⋊C₄	—	89	6	no	—
100	C₂×C₂²⋊C₄	—	89	12	no	—
101	C₂×C₄²	C₄×D₈₉	1	3	yes	—
102	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
104	C₂²⋊C₈	—	356	12	no	—
105	C₂⁴	—	89	4	no	—
106	C₂×D₄	—	89	24	no	—
107	C₂×D₄	—	356	6	no	—
108	C₂×D₄	—	356	6	no	—
109	C₂²×C₄	—	89	4	no	—
110	C₂²×C₄	—	178	12	no	—
111	C₂²×C₄	—	178	12	no	—
112	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
113	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
114	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
115	C₂×C₈	—	2	12	no	—
116	C₂×C₈	—	356	16	no	—
117	C₂×C₈	—	178	12	no	—
118	C₄⋊C₄	—	356	6	no	—
119	C₂²⋊C₄	—	356	12	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	C₈×D₈₉	1	4	yes	complemented
123	C₂³	—	89	8	no	—
124	C₂³	—	178	12	no	—
125	C₂³	—	178	12	no	—
126	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
127	D₄	—	356	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	356	4	no	—
133	C₂×C₄	—	356	12	no	—
134	C₂×C₄	—	178	12	no	—
135	C₂×C₄	C₂×C₈×D₈₉	1	6	yes	—
136	C₂×C₄	#15668058	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	356	4	no	—
139	C₂²	—	356	4	no	—
140	C₂²	—	356	12	no	—
141	C₂²	—	178	24	no	—
142	C₂²	#19152072	1	3	yes	—
143	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
144	C₂²	#19152364	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	356	4	no	—
148	C₄	#19152339	1	4	yes	—
149	C₂	—	356	8	no	—
150	C₂	#26023428	1	1	yes	characteristic
151	C₂	#26029275	1	3	yes	—
152	C₂	#26029322	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15668009

C₂

yes

maximal, characteristic

#26020543

C₂

yes

maximal, complemented

#26020395

C₂

yes

maximal, characteristic, complemented

C₈₉×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₉

C₂²

yes

—

C₂×#15668009

C₄

yes

characteristic

C₂×C₄×D₃₅₆

C₂²

yes

—

C₂×#15667924

C₂²

yes

—

#19151500

C₄

yes

—

#19151468

C₄

yes

characteristic

#19151353

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

C₂×C₄

yes

—

C₂²×C₄×D₈₉

C₂³

yes

—

C₂²×#10623494

C₂³

yes

—

C₂×C₈×D₈₉

—

#15668009

C₈

yes

complemented

C₄×D₃₅₆

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083551

C₂×C₄

yes

—

#15667924

—

C₂²×C₈×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

—

C₂³×D₈₉

C₂²×C₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₈

yes

complemented

C₂²×Dic₈₉

C₂²×C₄

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

complemented

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

H(5,𝔽₂)

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#518

yes

—

C₂²×C₈₉

C₂³×C₈

yes

characteristic

C₂²×C₈₉

#568

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#560

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2980

yes

—

D₈₉

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#23355

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₈₉

yes

characteristic

#336

—

#336

C₂×D₈₉

yes

—

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

356

—

C₂²×C₈

C₂²×D₈₉

yes

—

#127

—

C₄×D₄

—

356

—

C₄×D₄

—

356

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₄×D₈₉

yes

—

102

C₂×C₄²

C₄×D₈₉

yes

characteristic

103

C₂×C₄²

C₂²×D₈₉

yes

—

104

C₂²⋊C₈

—

356

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

356

—

108

C₂×D₄

—

356

—

109

C₂²×C₄

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

—

178

—

112

C₂²×C₄

C₂×C₄×D₈₉

yes

—

113

C₂²×C₄

C₂×C₄×D₈₉

yes

—

114

C₂²×C₄

C₂³×D₈₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

356

—

117

C₂×C₈

—

178

—

118

C₄⋊C₄

—

356

—

119

C₂²⋊C₄

—

356

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₈×D₈₉

yes

complemented

123

C₂³

—

124

C₂³

—

178

—

125

C₂³

—

178

—

126

C₂³

C₂²×C₄×D₈₉

yes

characteristic

127

D₄

—

356

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

178

—

135

C₂×C₄

C₂×C₈×D₈₉

yes

—

136

C₂×C₄

#15668058

yes

—

137

C₈

—

138

C₈

—

356

—

139

C₂²

—

356

—

140

C₂²

—

356

—

141

C₂²

—

178

—

142

C₂²

#19152072

yes

—

143

C₂²

C₂²×C₈×D₈₉

yes

characteristic

144

C₂²

#19152364

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

356

—

148

C₄

#19152339

yes

—

149

C₂

—

356

—

150

C₂

#26023428

yes

characteristic

151

C₂

#26029275

yes

—

152

C₂

#26029322

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#1839	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂×#15668009	C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1839	D₈₉
hypocenter	C₈₉	#23355
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₈₉

commutator

C₂²×C₈₉

C₂³×C₈

Fitting

C₈₉×#1839

C₂

socle

C₂³×C₈₉

C₂³×C₄

prime radical

C₂×#15668009

C₄

cosocle

C₂²×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #26020543	C₂	semidirect
3. #26020395	C₂	semidirect
4. C₈₉×#1839	C₂	semidirect
18. #15668009	C₈	semidirect
31. C₂×D₃₅₆	C₂×C₈	semidirect
45. C₄²×C₈₉	C₂×C₈	semidirect
80. #1839	D₈₉	semidirect
81. C₈₉	#23355	semidirect
122. C₄²	C₈×D₈₉	semidirect

H ≅ G/N

type

2. #26020543

C₂

semidirect

3. #26020395

C₂

semidirect

4. C₈₉×#1839

C₂

semidirect

18. #15668009

C₈

semidirect

31. C₂×D₃₅₆

C₂×C₈

semidirect

45. C₄²×C₈₉

C₂×C₈

semidirect

80. #1839

D₈₉

semidirect

81. C₈₉

#23355

semidirect

122. C₄²

C₈×D₈₉

semidirect

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

subgroup	index
C₄×#15668009	2
#26020543	2
#26020395	2
C₈₉×#1839	2
#23355	89

subgroup

index

C₈₉×#1839

Subgroups — #36580973

Conjugacy classes of subgroups

Complements

Maximal subgroups