Group #36551746 — subgroups

Subgroups — C₄²×C₈×D₈₉ #36551746

Order 22784.

2046 conjugacy classes of subgroups; only the 1227 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 1227 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×C₄×C₈×D₈₉	C₂	1	6	yes	maximal
2	C₄³×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₈×Dic₈₉	C₂	1	6	yes	maximal, complemented
5	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	C₂²	1	4	yes	—
7	C₂×C₄²×D₈₉	C₄	1	4	yes	—
8	C₂×C₄²×D₈₉	C₂²	1	3	yes	—
9	C₂×C₄×#10623494	C₂²	1	6	yes	—
10	C₂×C₈×Dic₈₉	C₂²	1	12	yes	—
11	C₄×C₈×D₈₉	C₄	1	48	yes	direct product factor
12	C₄²×Dic₈₉	C₄	1	4	yes	—
13	C₄²×Dic₈₉	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	6	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₈₉	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₈₉	C₂³	1	1	yes	characteristic
18	C₂²×#10623494	C₂³	1	4	yes	—
19	C₂×C₈×D₈₉	C₂×C₄	1	48	yes	—
20	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₈₉	C₂×C₄	1	12	yes	—
22	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
23	C₄²×D₈₉	C₈	1	32	yes	direct product factor
24	C₄²×D₈₉	C₂×C₄	1	24	yes	—
25	C₄×#10623494	C₂×C₄	1	24	yes	complemented
26	C₈×Dic₈₉	C₂×C₄	1	48	yes	complemented
27	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
30	C₄×C₈×C₈₉	C₂×C₄	1	24	yes	complemented
31	C₂³×D₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₈₉	C₂²×C₄	1	6	yes	—
34	C₂×C₄×D₈₉	C₄²	1	24	yes	—
35	C₂×C₄×D₈₉	C₂×C₈	1	48	yes	—
36	C₂×C₄×D₈₉	C₂²×C₄	1	12	yes	—
37	C₂×#10623494	C₂²×C₄	1	24	yes	—
38	C₈×D₈₉	C₄²	1	64	yes	direct product factor
39	C₄×Dic₈₉	C₄²	1	24	yes	—
40	C₄×Dic₈₉	C₂×C₈	1	48	yes	complemented
41	C₄×Dic₈₉	C₂²×C₄	1	12	yes	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
43	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂²×C₄	1	24	yes	—
45	C₄²×C₈₉	C₂×C₈	1	16	yes	complemented
46	C₄²×C₈₉	C₂²×C₄	1	12	yes	—
47	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
48	C₂²×D₈₉	C₂×C₄²	1	6	yes	—
49	C₂²×D₈₉	C₂²×C₈	1	8	yes	—
50	C₂×C₄×C₈₉	C₂×C₄²	1	12	yes	—
51	C₂×C₄×C₈₉	C₂²×C₈	1	24	yes	—
52	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
53	C₂×Dic₈₉	C₂×C₄²	1	6	yes	—
54	C₂×Dic₈₉	C₂×C₄²	1	12	yes	—
55	C₂×Dic₈₉	C₂²×C₈	1	24	yes	—
56	C₄×D₈₉	C₄×C₈	1	96	yes	direct product factor
57	C₄×D₈₉	C₂×C₄²	1	16	yes	—
58	C₈×C₈₉	C₂×C₄²	1	16	yes	complemented
59	#10623494	C₂×C₄²	1	16	yes	complemented
60	C₂²×C₈₉	C₂²×C₄²	1	3	yes	—
61	C₂²×C₈₉	C₂³×C₈	1	4	yes	—
62	C₂×D₈₉	C₄³	1	4	yes	—
63	C₂×D₈₉	C₂×C₄×C₈	1	24	yes	—
64	C₄×C₈₉	C₂×C₄×C₈	1	24	yes	complemented
65	C₄×C₈₉	C₂²×C₄²	1	4	yes	—
66	Dic₈₉	C₄³	1	4	yes	—
67	Dic₈₉	C₂×C₄×C₈	1	24	yes	complemented
68	C₂×C₄²×C₈	—	89	1	no	maximal, Sylow
69	C₂×C₈₉	C₂×C₄³	1	1	yes	characteristic
70	C₂×C₈₉	C₂²×C₄×C₈	1	6	yes	—
71	D₈₉	C₄²×C₈	1	8	yes	direct product factor
72	C₂²×C₄×C₈	—	89	6	no	—
73	C₂×C₄³	—	89	1	no	—
74	C₄²×C₈	—	89	6	no	—
75	C₄²×C₈	—	89	1	no	—
76	C₄²×C₈	D₈₉	1	1	yes	characteristic, direct product factor
77	C₈₉	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₈	—	89	4	no	—
79	C₂²×C₄²	—	89	4	no	—
80	C₂²×C₄²	—	89	3	no	—
81	C₂×C₄×C₈	—	89	12	no	—
82	C₂×C₄×C₈	—	89	6	no	—
83	C₂×C₄×C₈	—	89	48	no	—
84	C₂×C₄×C₈	C₂×D₈₉	1	6	yes	—
85	C₄³	—	89	4	no	—
86	C₄³	—	89	3	no	—
87	C₄³	C₂×D₈₉	1	1	yes	characteristic
88	C₂³×C₄	—	89	6	no	—
89	C₂³×C₄	—	89	1	no	—
90	C₂²×C₈	—	89	4	no	—
91	C₂²×C₈	—	89	48	no	—
92	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
93	C₂×C₄²	—	89	12	no	—
94	C₂×C₄²	—	89	6	no	—
95	C₂×C₄²	—	89	3	no	—
96	C₂×C₄²	—	89	32	no	—
97	C₂×C₄²	—	89	24	no	—
98	C₂×C₄²	C₄×D₈₉	1	4	yes	—
99	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
100	C₄×C₈	—	89	48	no	—
101	C₄×C₈	—	89	24	no	—
102	C₄×C₈	C₄×D₈₉	1	24	yes	direct product factor
103	C₂⁴	—	89	1	no	—
104	C₂²×C₄	—	89	1	no	—
105	C₂²×C₄	—	89	6	no	—
106	C₂²×C₄	—	89	48	no	—
107	C₂²×C₄	—	89	12	no	—
108	C₂²×C₄	—	89	24	no	—
109	C₂²×C₄	C₂×C₄×D₈₉	1	6	yes	—
110	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
111	C₂×C₈	—	89	24	no	—
112	C₂×C₈	—	89	64	no	—
113	C₂×C₈	C₂×C₄×D₈₉	1	24	yes	—
114	C₄²	—	89	48	no	—
115	C₄²	—	89	12	no	—
116	C₄²	—	89	24	no	—
117	C₄²	C₈×D₈₉	1	16	yes	direct product factor
118	C₄²	C₂×C₄×D₈₉	1	12	yes	—
119	C₂³	—	89	6	no	—
120	C₂³	—	89	8	no	—
121	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
122	C₂×C₄	—	89	12	no	—
123	C₂×C₄	—	89	6	no	—
124	C₂×C₄	—	89	24	no	—
125	C₂×C₄	—	89	16	no	—
126	C₂×C₄	—	89	96	no	—
127	C₂×C₄	C₄²×D₈₉	1	12	yes	—
128	C₂×C₄	C₂×C₈×D₈₉	1	24	yes	—
129	C₂×C₄	C₂²×C₄×D₈₉	1	6	yes	—
130	C₈	—	89	16	no	—
131	C₈	C₄²×D₈₉	1	16	yes	direct product factor
132	C₂²	—	89	24	no	—
133	C₂²	—	89	4	no	—
134	C₂²	C₂×C₄²×D₈₉	1	3	yes	—
135	C₂²	C₂²×C₈×D₈₉	1	4	yes	—
136	C₄	—	89	24	no	—
137	C₄	—	89	4	no	—
138	C₄	C₄×C₈×D₈₉	1	24	yes	direct product factor
139	C₄	C₂×C₄²×D₈₉	1	4	yes	—
140	C₂	—	89	8	no	—
141	C₂	C₄³×D₈₉	1	1	yes	characteristic
142	C₂	C₂×C₄×C₈×D₈₉	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈×D₈₉

C₂

yes

maximal

C₄³×D₈₉

C₂

yes

maximal, characteristic

C₄²×#10623494

C₂

yes

maximal, characteristic, complemented

C₄×C₈×Dic₈₉

C₂

yes

maximal, complemented

C₄²×C₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₉

C₂²

yes

—

C₂×C₄²×D₈₉

C₄

yes

—

C₂×C₄²×D₈₉

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

C₄×C₈×D₈₉

C₄

yes

direct product factor

C₄²×Dic₈₉

C₄

yes

—

C₄²×Dic₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×C₄×D₈₉

C₂×C₄

yes

—

C₂²×C₄×D₈₉

C₂³

yes

characteristic

C₂²×#10623494

C₂³

yes

—

C₂×C₈×D₈₉

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂³

yes

—

C₄²×D₈₉

C₈

yes

direct product factor

C₄²×D₈₉

C₂×C₄

yes

—

C₄×#10623494

C₂×C₄

yes

complemented

C₈×Dic₈₉

C₂×C₄

yes

complemented

C₂²×C₈×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

complemented

C₂³×D₈₉

C₂²×C₄

yes

characteristic

C₂²×Dic₈₉

C₂²×C₄

yes

characteristic

C₂²×Dic₈₉

C₂²×C₄

yes

—

C₂×C₄×D₈₉

C₄²

yes

—

C₂×C₄×D₈₉

C₂×C₈

yes

—

C₂×C₄×D₈₉

C₂²×C₄

yes

—

C₂×#10623494

C₂²×C₄

yes

—

C₈×D₈₉

C₄²

yes

direct product factor

C₄×Dic₈₉

C₄²

yes

—

C₄×Dic₈₉

C₂×C₈

yes

complemented

C₄×Dic₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

complemented

C₄²×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂²×D₈₉

C₂×C₄²

yes

—

C₂²×D₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄²

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×Dic₈₉

C₂×C₄²

yes

—

C₂×Dic₈₉

C₂×C₄²

yes

—

C₂×Dic₈₉

C₂²×C₈

yes

—

C₄×D₈₉

C₄×C₈

yes

direct product factor

C₄×D₈₉

C₂×C₄²

yes

—

C₈×C₈₉

C₂×C₄²

yes

complemented

#10623494

C₂×C₄²

yes

complemented

C₂²×C₈₉

C₂²×C₄²

yes

—

C₂²×C₈₉

C₂³×C₈

yes

—

C₂×D₈₉

C₄³

yes

—

C₂×D₈₉

C₂×C₄×C₈

yes

—

C₄×C₈₉

C₂×C₄×C₈

yes

complemented

C₄×C₈₉

C₂²×C₄²

yes

—

Dic₈₉

C₄³

yes

—

Dic₈₉

C₂×C₄×C₈

yes

complemented

C₂×C₄²×C₈

—

maximal, Sylow

C₂×C₈₉

C₂×C₄³

yes

characteristic

C₂×C₈₉

C₂²×C₄×C₈

yes

—

D₈₉

C₄²×C₈

yes

direct product factor

C₂²×C₄×C₈

—

C₂×C₄³

—

C₄²×C₈

—

C₄²×C₈

—

C₄²×C₈

D₈₉

yes

characteristic, direct product factor

C₈₉

C₂×C₄²×C₈

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₄²

—

C₂²×C₄²

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₄³

—

C₄³

—

C₄³

C₂×D₈₉

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₄×D₈₉

yes

direct product factor

103

C₂⁴

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₄×D₈₉

yes

—

110

C₂²×C₄

C₂³×D₈₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×C₄×D₈₉

yes

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₈×D₈₉

yes

direct product factor

118

C₄²

C₂×C₄×D₈₉

yes

—

119

C₂³

—

120

C₂³

—

121

C₂³

C₂²×C₄×D₈₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄²×D₈₉

yes

—

128

C₂×C₄

C₂×C₈×D₈₉

yes

—

129

C₂×C₄

C₂²×C₄×D₈₉

yes

—

130

C₈

—

131

C₈

C₄²×D₈₉

yes

direct product factor

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×C₄²×D₈₉

yes

—

135

C₂²

C₂²×C₈×D₈₉

yes

—

136

C₄

—

137

C₄

—

138

C₄

C₄×C₈×D₈₉

yes

direct product factor

139

C₄

C₂×C₄²×D₈₉

yes

—

140

C₂

—

141

C₂

C₄³×D₈₉

yes

characteristic

142

C₂

C₂×C₄×C₈×D₈₉

yes

—

	subgroup	G/H
center, hypercenter	C₄²×C₈	D₈₉
commutator, hypocenter	C₈₉	C₂×C₄²×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂³×D₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₄²×C₈

subgroup

G/H

center, hypercenter

C₄²×C₈

D₈₉

commutator, hypocenter

C₈₉

C₂×C₄²×C₈

Frattini

C₂²×C₄

C₂³×D₈₉

Fitting

C₄²×C₈×C₈₉

C₂

socle

C₂³×C₈₉

C₂³×C₄

prime radical

C₂³×D₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂⁴

LCS

G ▷ C₈₉

UCS

C₁ ◁ C₄²×C₈

N	H ≅ G/N	type
3. C₄²×#10623494	C₂	semidirect
4. C₄×C₈×Dic₈₉	C₂	semidirect
5. C₄²×C₈×C₈₉	C₂	semidirect
11. C₄×C₈×D₈₉	C₄	direct (N × H)
23. C₄²×D₈₉	C₈	direct (N × H)
25. C₄×#10623494	C₂×C₄	semidirect
26. C₈×Dic₈₉	C₂×C₄	semidirect
30. C₄×C₈×C₈₉	C₂×C₄	semidirect
38. C₈×D₈₉	C₄²	direct (N × H)
40. C₄×Dic₈₉	C₂×C₈	semidirect
45. C₄²×C₈₉	C₂×C₈	semidirect
56. C₄×D₈₉	C₄×C₈	direct (N × H)
58. C₈×C₈₉	C₂×C₄²	semidirect
59. #10623494	C₂×C₄²	semidirect
64. C₄×C₈₉	C₂×C₄×C₈	semidirect
67. Dic₈₉	C₂×C₄×C₈	semidirect
71. D₈₉	C₄²×C₈	direct (N × H)
76. C₄²×C₈	D₈₉	direct (N × H)
77. C₈₉	C₂×C₄²×C₈	semidirect
102. C₄×C₈	C₄×D₈₉	direct (N × H)
117. C₄²	C₈×D₈₉	direct (N × H)
131. C₈	C₄²×D₈₉	direct (N × H)
138. C₄	C₄×C₈×D₈₉	direct (N × H)

H ≅ G/N

type

3. C₄²×#10623494

C₂

semidirect

4. C₄×C₈×Dic₈₉

C₂

semidirect

5. C₄²×C₈×C₈₉

C₂

semidirect

11. C₄×C₈×D₈₉

C₄

direct (N × H)

23. C₄²×D₈₉

C₈

direct (N × H)

25. C₄×#10623494

C₂×C₄

semidirect

26. C₈×Dic₈₉

C₂×C₄

semidirect

30. C₄×C₈×C₈₉

C₂×C₄

semidirect

38. C₈×D₈₉

C₄²

direct (N × H)

40. C₄×Dic₈₉

C₂×C₈

semidirect

45. C₄²×C₈₉

C₂×C₈

semidirect

56. C₄×D₈₉

C₄×C₈

direct (N × H)

58. C₈×C₈₉

C₂×C₄²

semidirect

59. #10623494

C₂×C₄²

semidirect

64. C₄×C₈₉

C₂×C₄×C₈

semidirect

67. Dic₈₉

C₂×C₄×C₈

semidirect

71. D₈₉

C₄²×C₈

direct (N × H)

76. C₄²×C₈

D₈₉

direct (N × H)

77. C₈₉

C₂×C₄²×C₈

semidirect

102. C₄×C₈

C₄×D₈₉

direct (N × H)

117. C₄²

C₈×D₈₉

direct (N × H)

131. C₈

C₄²×D₈₉

direct (N × H)

138. C₄

C₄×C₈×D₈₉

direct (N × H)

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

subgroup	index
C₂×C₄×C₈×D₈₉	2
C₄³×D₈₉	2
C₄²×#10623494	2
C₄×C₈×Dic₈₉	2
C₄²×C₈×C₈₉	2
C₂×C₄²×C₈	89

subgroup

index

C₂×C₄×C₈×D₈₉

C₄³×D₈₉

C₄²×#10623494

C₄×C₈×Dic₈₉

C₄²×C₈×C₈₉

C₂×C₄²×C₈

Subgroups — C₄²×C₈×D₈₉ #36551746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×C8×D89 #36551746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₈×D₈₉ #36551746