Group #36481590 — subgroups

Subgroups — #36481590

Order 22784.

3998 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 651 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	C₂×#19151461	C₂	1	6	yes	maximal
2	#26018523	C₂	1	8	yes	maximal, complemented
3	C₂⁴×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
5	#19151461	—	2	24	no	—
6	C₂×#15667919	C₂²	1	24	yes	—
7	C₂³×Q₈×C₈₉	—	2	24	no	—
8	C₂³×Q₈×C₈₉	C₄	1	8	yes	complemented
9	C₂³×Q₈×C₈₉	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₈₉	C₂²	1	3	yes	—
11	C₂³×Dic₈₉	C₂³	1	8	yes	—
12	C₂×C₂²⋊Dic₈₉	—	2	24	no	—
13	#15667919	—	2	48	no	—
14	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
15	C₂²×Q₈×C₈₉	—	2	128	no	—
16	C₂²×Q₈×C₈₉	—	2	96	no	—
17	C₂²×Q₈×C₈₉	—	2	48	no	—
18	C₂²×Q₈×C₈₉	C₂×C₄	1	16	yes	—
19	C₂³×C₄×C₈₉	—	2	24	no	—
20	C₂³×C₄×C₈₉	C₂×C₄	1	12	yes	—
21	C₂³×C₄×C₈₉	D₄	1	24	yes	—
22	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
23	C₂²×Dic₈₉	—	2	24	no	—
24	C₂²×Dic₈₉	—	2	32	no	—
25	C₂²⋊Dic₈₉	—	4	16	no	—
26	C₂⁴×C₈₉	—	2	8	no	—
27	C₂⁴×C₈₉	C₂²×C₄	1	2	yes	—
28	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
29	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₈₉	—	2	128	no	—
31	C₂×Q₈×C₈₉	—	2	256	no	—
32	C₂×Q₈×C₈₉	—	2	96	no	—
33	C₂²×C₄×C₈₉	—	2	48	no	—
34	C₂²×C₄×C₈₉	—	2	96	no	—
35	C₂²×C₄×C₈₉	—	2	24	no	—
36	C₂²×C₄×C₈₉	C₂²⋊C₄	1	48	yes	—
37	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
39	C₂³×C₈₉	—	2	8	no	—
40	C₂³×C₈₉	—	2	48	no	—
41	C₂³×C₈₉	C₂×C₂²⋊C₄	1	12	yes	—
42	C₂³×C₈₉	#120	1	16	yes	—
43	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
45	C₂³×C₈₉	2₋¹⁺⁴	1	8	yes	—
46	Q₈×C₈₉	—	2	128	no	—
47	C₂×C₄×C₈₉	—	2	24	no	—
48	C₂×C₄×C₈₉	—	2	48	no	—
49	C₂×C₄×C₈₉	—	2	96	no	—
50	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	24	yes	—
51	C₂×Dic₈₉	—	4	8	no	—
52	C₂×Dic₈₉	—	4	48	no	—
53	C₂²×C₈₉	—	2	24	no	—
54	C₂²×C₈₉	—	2	32	no	—
55	C₂²×C₈₉	#379	1	8	yes	—
56	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	3	yes	—
57	C₂²×C₈₉	#519	1	4	yes	—
58	C₂²×C₈₉	C₂×#120	1	4	yes	—
59	C₂²×C₈₉	#536	1	24	yes	—
60	C₄×C₈₉	—	2	24	no	—
61	Dic₈₉	—	8	8	no	—
62	#14336	—	89	1	no	maximal, Sylow
63	C₂×C₈₉	—	2	8	no	—
64	C₂×C₈₉	C₂×#379	1	1	yes	characteristic
65	C₂×C₈₉	#2310	1	6	yes	—
66	C₂×C₈₉	#2614	1	8	yes	—
67	C₂⁴×Q₈	D₈₉	1	1	yes	characteristic
68	C₂×#381	—	89	6	no	—
69	#1446	—	89	8	no	—
70	C₈₉	#14336	1	1	yes	characteristic, complemented, Sylow
71	C₂³×Q₈	—	2	24	no	—
72	C₂³×Q₈	Dic₈₉	1	8	yes	complemented
73	C₂³×Q₈	C₂×D₈₉	1	4	yes	—
74	C₂⁴×C₄	C₂×D₈₉	1	3	yes	—
75	C₂²×C₂²⋊C₄	—	89	4	no	—
76	#381	—	178	24	no	—
77	C₂×#95	—	89	24	no	—
78	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
79	C₂²×Q₈	—	2	128	no	—
80	C₂²×Q₈	—	2	96	no	—
81	C₂²×Q₈	—	2	48	no	—
82	C₂²×Q₈	C₂×Dic₈₉	1	16	yes	—
83	C₂³×C₄	—	2	24	no	—
84	C₂³×C₄	—	89	8	no	—
85	C₂³×C₄	C₂×Dic₈₉	1	12	yes	—
86	C₂³×C₄	C₂²⋊D₈₉	1	24	yes	—
87	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
88	C₂×C₂²⋊C₄	—	178	24	no	—
89	#95	—	178	48	no	—
90	C₂⁴	—	2	8	no	—
91	C₂⁴	C₂²×Dic₈₉	1	2	yes	—
92	C₂⁴	C₂×C₂²⋊D₈₉	1	12	yes	—
93	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
94	C₂×Q₈	—	2	256	no	—
95	C₂×Q₈	—	2	128	no	—
96	C₂×Q₈	—	2	96	no	—
97	C₂²×C₄	—	2	96	no	—
98	C₂²×C₄	—	2	24	no	—
99	C₂²×C₄	—	2	48	no	—
100	C₂²×C₄	—	178	24	no	—
101	C₂²×C₄	—	178	32	no	—
102	C₂²×C₄	C₂²⋊Dic₈₉	1	48	yes	—
103	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₈₉	1	24	yes	—
105	C₂²⋊C₄	—	356	16	no	—
106	C₂³	—	2	48	no	—
107	C₂³	—	2	8	no	—
108	C₂³	C₂×C₂²⋊Dic₈₉	1	12	yes	—
109	C₂³	#15667981	1	16	yes	—
110	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
111	C₂³	C₂²×C₂²⋊D₈₉	1	6	yes	—
112	C₂³	#15668060	1	8	yes	—
113	Q₈	—	2	128	no	—
114	C₂×C₄	—	2	96	no	—
115	C₂×C₄	—	2	24	no	—
116	C₂×C₄	—	2	48	no	—
117	C₂×C₄	—	356	48	no	—
118	C₂×C₄	—	356	8	no	—
119	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	24	yes	—
120	C₂²	—	2	24	no	—
121	C₂²	—	2	32	no	—
122	C₂²	#19151454	1	8	yes	—
123	C₂²	C₂²×C₂²⋊Dic₈₉	1	3	yes	—
124	C₂²	#19152076	1	4	yes	—
125	C₂²	C₂×#15667981	1	4	yes	—
126	C₂²	#19152172	1	24	yes	—
127	C₄	—	2	24	no	—
128	C₄	—	712	8	no	—
129	C₂	—	2	8	no	—
130	C₂	C₂×#19151454	1	1	yes	characteristic
131	C₂	#26022838	1	6	yes	—
132	C₂	#26025971	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151461

C₂

yes

maximal

#26018523

C₂

yes

maximal, complemented

C₂⁴×Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×C₂²⋊Dic₈₉

C₂²

yes

—

#19151461

—

C₂×#15667919

C₂²

yes

—

C₂³×Q₈×C₈₉

—

C₂³×Q₈×C₈₉

C₄

yes

complemented

C₂³×Q₈×C₈₉

C₂²

yes

—

C₂⁴×C₄×C₈₉

C₂²

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂×C₂²⋊Dic₈₉

—

#15667919

—

C₂⁵×C₈₉

C₂³

yes

characteristic

C₂²×Q₈×C₈₉

—

128

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

C₂×C₄

yes

—

C₂³×C₄×C₈₉

—

C₂³×C₄×C₈₉

C₂×C₄

yes

—

C₂³×C₄×C₈₉

D₄

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

—

C₂⁴×C₈₉

C₂²×C₄

yes

—

C₂⁴×C₈₉

C₂×D₄

yes

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂×Q₈×C₈₉

—

128

—

C₂×Q₈×C₈₉

—

256

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

—

C₂³×C₈₉

#120

yes

—

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂³×C₈₉

C₂²×D₄

yes

—

C₂³×C₈₉

2₋¹⁺⁴

yes

—

Q₈×C₈₉

—

128

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#379

yes

—

C₂²×C₈₉

C₂²×C₂²⋊C₄

yes

—

C₂²×C₈₉

#519

yes

—

C₂²×C₈₉

C₂×#120

yes

—

C₂²×C₈₉

#536

yes

—

C₄×C₈₉

—

Dic₈₉

—

#14336

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#379

yes

characteristic

C₂×C₈₉

#2310

yes

—

C₂×C₈₉

#2614

yes

—

C₂⁴×Q₈

D₈₉

yes

characteristic

C₂×#381

—

#1446

—

C₈₉

#14336

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

C₂³×Q₈

Dic₈₉

yes

complemented

C₂³×Q₈

C₂×D₈₉

yes

—

C₂⁴×C₄

C₂×D₈₉

yes

—

C₂²×C₂²⋊C₄

—

#381

—

178

—

C₂×#95

—

C₂⁵

C₂²×D₈₉

yes

characteristic

C₂²×Q₈

—

128

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₈₉

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×Dic₈₉

yes

—

C₂³×C₄

C₂²⋊D₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂×C₂²⋊C₄

—

178

—

#95

—

178

—

C₂⁴

—

C₂⁴

C₂²×Dic₈₉

yes

—

C₂⁴

C₂×C₂²⋊D₈₉

yes

—

C₂⁴

C₂³×D₈₉

yes

characteristic

C₂×Q₈

—

256

—

C₂×Q₈

—

128

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

—

178

—

102

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

103

C₂²×C₄

C₂²×Dic₈₉

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

105

C₂²⋊C₄

—

356

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×C₂²⋊Dic₈₉

yes

—

109

C₂³

#15667981

yes

—

110

C₂³

C₂³×Dic₈₉

yes

characteristic

111

C₂³

C₂²×C₂²⋊D₈₉

yes

—

112

C₂³

#15668060

yes

—

113

Q₈

—

128

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#19151454

yes

—

123

C₂²

C₂²×C₂²⋊Dic₈₉

yes

—

124

C₂²

#19152076

yes

—

125

C₂²

C₂×#15667981

yes

—

126

C₂²

#19152172

yes

—

127

C₄

—

128

C₄

—

712

—

129

C₂

—

130

C₂

C₂×#19151454

yes

characteristic

131

C₂

#26022838

yes

—

132

C₂

#26025971

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₈₉
commutator	C₂³×C₈₉	C₂³×C₄
Fitting	C₂⁴×Q₈×C₈₉	C₂
socle, cosocle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂⁵×C₈₉	C₂³
hypercenter	C₂⁴×Q₈	D₈₉
hypocenter	C₈₉	#14336
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₈₉

commutator

C₂³×C₈₉

C₂³×C₄

Fitting

C₂⁴×Q₈×C₈₉

C₂

socle, cosocle

C₂⁴×C₈₉

C₂⁴

prime radical

C₂⁵×C₈₉

C₂³

hypercenter

C₂⁴×Q₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

N	H ≅ G/N	type
2. #26018523	C₂	semidirect
8. C₂³×Q₈×C₈₉	C₄	semidirect
70. C₈₉	#14336	semidirect
72. C₂³×Q₈	Dic₈₉	semidirect

H ≅ G/N

type

2. #26018523

C₂

semidirect

8. C₂³×Q₈×C₈₉

C₄

semidirect

70. C₈₉

#14336

semidirect

72. C₂³×Q₈

Dic₈₉

semidirect

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

subgroup	index
C₂×#19151461	2
#26018523	2
C₂⁴×Q₈×C₈₉	2
#14336	89

subgroup

index

C₂×#19151461

#26018523

C₂⁴×Q₈×C₈₉

#14336

Subgroups — #36481590

Conjugacy classes of subgroups

Complements

Maximal subgroups