Group #36470789 — subgroups

Subgroups — #36470789

Order 22784.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	#26022008	C₂	1	2	yes	maximal
2	#26018334	C₂	1	4	yes	maximal
3	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	C₂²	1	4	yes	—
5	C₄⋊#13083549	—	2	4	no	—
6	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₄³×C₈₉	C₄	1	1	yes	characteristic
11	#15667952	—	2	8	no	—
12	C₂×#13083549	—	2	4	no	—
13	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	—	2	2	no	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₈₉	—	2	4	no	—
20	C₄×C₈×C₈₉	—	2	4	no	—
21	C₄×C₈×C₈₉	C₈	1	2	yes	—
22	C₄×C₈×C₈₉	C₈	1	2	yes	—
23	C₄×C₈×C₈₉	D₄	1	2	yes	—
24	C₄×C₈×C₈₉	Q₈	1	2	yes	—
25	#13083549	—	4	4	no	—
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₈₉	#41	1	4	yes	—
37	C₄²×C₈₉	—	2	8	no	—
38	C₄²×C₈₉	—	2	2	no	—
39	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₈₉	M₁₆	1	4	yes	—
43	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	8	no	—
45	C₂×C₄×C₈₉	—	2	4	no	—
46	C₂×C₄×C₈₉	—	2	4	no	—
47	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#117	1	2	yes	—
51	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	—	2	4	no	—
55	C₈×C₈₉	#124	1	2	yes	—
56	C₈×C₈₉	#125	1	2	yes	—
57	C₂²×C₈₉	—	2	2	no	—
58	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₈₉	#432	1	2	yes	—
60	C₄×C₈₉	—	2	8	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	#364	1	4	yes	—
63	C₄×C₈₉	#382	1	1	yes	characteristic
64	C₄×C₈₉	#423	1	1	yes	characteristic
65	C₄×C₈₉	#431	1	2	yes	—
66	#12883	—	89	1	no	maximal, Sylow
67	C₂×C₈₉	—	2	2	no	—
68	C₂×C₈₉	#1831	1	1	yes	characteristic
69	C₂×C₈₉	C₂×#364	1	2	yes	—
70	#2150	—	89	2	no	—
71	C₄²×C₈	D₈₉	1	1	yes	characteristic
72	#1376	—	89	4	no	—
73	C₈₉	#12883	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₃₂	—	89	4	no	—
75	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
76	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
79	C₄³	Dic₈₉	1	1	yes	characteristic
80	C₄⋊C₁₆	—	178	4	no	—
81	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
83	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
86	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
87	M₃₂	—	178	8	no	—
88	C₂×C₁₆	—	178	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	#10623494	1	2	yes	—
92	C₄×C₈	#10623494	1	2	yes	—
93	C₄×C₈	Dic₁₇₈	1	2	yes	—
94	C₄×C₈	D₃₅₆	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#13083558	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
102	C₂×C₈	C₂×#10623494	1	4	yes	—
103	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
105	C₂×C₈	#13083574	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊Dic₈₉	1	2	yes	—
109	C₄²	C₂×#10623494	1	1	yes	characteristic
110	C₄²	C₂×#10623494	1	1	yes	characteristic
111	C₄²	#13083558	1	4	yes	—
112	C₁₆	—	356	4	no	—
113	C₂³	C₂×#13083558	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄⋊#10623494	1	2	yes	—
118	C₂×C₄	C₄⋊#10623494	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
120	C₂×C₄	#15667971	1	2	yes	—
121	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
122	C₂×C₄	C₂×#13083558	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	#15668000	1	2	yes	—
126	C₈	#15668004	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
129	C₂²	#19151655	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#19151419	1	4	yes	—
133	C₄	#19151464	1	1	yes	characteristic
134	C₄	#19151622	1	1	yes	characteristic
135	C₄	#19151651	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#26020369	1	1	yes	characteristic
138	C₂	C₂×#19151419	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022008

C₂

yes

maximal

#26018334

C₂

yes

maximal

C₄²×C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₄⋊#13083549

—

C₂×C₄×C₈×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄³×C₈₉

C₄

yes

characteristic

#15667952

—

C₂×#13083549

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₈

yes

—

C₄×C₈×C₈₉

C₈

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#124

yes

—

C₈×C₈₉

#125

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#432

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#364

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12883

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1831

yes

characteristic

C₂×C₈₉

C₂×#364

yes

—

#2150

—

C₄²×C₈

D₈₉

yes

characteristic

#1376

—

C₈₉

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

Dic₈₉

yes

—

C₂×C₄×C₈

Dic₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄³

Dic₈₉

yes

characteristic

C₄⋊C₁₆

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

M₃₂

—

178

—

C₂×C₁₆

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#10623494

yes

—

C₄×C₈

#10623494

yes

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

C₂²×C₄

—

C₂²×C₄

#13083558

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄⋊Dic₈₉

yes

—

102

C₂×C₈

C₂×#10623494

yes

—

103

C₂×C₈

C₂×D₃₅₆

yes

characteristic

104

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

105

C₂×C₈

#13083574

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊Dic₈₉

yes

—

109

C₄²

C₂×#10623494

yes

characteristic

110

C₄²

C₂×#10623494

yes

characteristic

111

C₄²

#13083558

yes

—

112

C₁₆

—

356

—

113

C₂³

C₂×#13083558

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊#10623494

yes

—

118

C₂×C₄

C₄⋊#10623494

yes

—

119

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

120

C₂×C₄

#15667971

yes

—

121

C₂×C₄

C₂²×#10623494

yes

characteristic

122

C₂×C₄

C₂×#13083558

yes

—

123

C₈

—

124

C₈

—

125

C₈

#15668000

yes

—

126

C₈

#15668004

yes

—

127

C₂²

—

128

C₂²

C₂×C₄⋊#10623494

yes

characteristic

129

C₂²

#19151655

yes

—

130

C₄

—

131

C₄

—

132

C₄

#19151419

yes

—

133

C₄

#19151464

yes

characteristic

134

C₄

#19151622

yes

characteristic

135

C₄

#19151651

yes

—

136

C₂

—

137

C₂

#26020369

yes

characteristic

138

C₂

C₂×#19151419

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#12883
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₄²×C₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₄²×C₈

D₈₉

hypocenter

C₈₉

#12883

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
73. C₈₉	#12883	semidirect

H ≅ G/N

type

73. C₈₉

#12883

semidirect

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

subgroup	index
#26022008	2
#26018334	2
C₄²×C₈×C₈₉	2
#12883	89

subgroup

index

#26022008

#26018334

C₄²×C₈×C₈₉

#12883

Subgroups — #36470789

Conjugacy classes of subgroups

Complements

Maximal subgroups