Group #36441819 — subgroups

Subgroups — #36441819

Order 22784.

180 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26018198	C₂	1	2	yes	maximal
2	C₈₉×#1992	C₂	1	1	yes	maximal, characteristic
3	#19151324	—	2	2	no	—
4	C₂×#125×C₈₉	C₄	1	1	yes	characteristic
5	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
6	#384×C₈₉	C₂²	1	1	yes	characteristic
7	#340×C₈₉	—	2	2	no	—
8	C₂²⋊#10623494	—	4	2	no	—
9	C₂²×C₈×C₈₉	—	2	1	no	—
10	#125×C₈₉	—	2	1	no	—
11	#125×C₈₉	—	4	1	no	—
12	#125×C₈₉	C₈	1	2	yes	complemented
13	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
15	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
16	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
18	#97×C₈₉	—	4	1	no	—
19	C₂×#10623494	—	8	2	no	—
20	C₂²×C₄×C₈₉	—	2	1	no	—
21	C₂²×C₄×C₈₉	—	2	2	no	—
22	C₂²×C₄×C₈₉	—	4	1	no	—
23	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	—	2	2	no	—
25	C₂×C₈×C₈₉	—	4	1	no	—
26	C₂×C₈×C₈₉	—	4	1	no	—
27	C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₄⋊C₄×C₈₉	—	4	2	no	—
29	C₄⋊C₄×C₈₉	—	4	2	no	—
30	C₄⋊C₄×C₈₉	—	4	2	no	—
31	C₄⋊C₄×C₈₉	—	4	1	no	—
32	C₄⋊C₄×C₈₉	—	8	2	no	—
33	C₄²×C₈₉	—	2	1	no	—
34	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
36	C₂³×C₈₉	#99	1	1	yes	characteristic
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	—	4	2	no	—
44	C₂×C₄×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	—	8	1	no	—
46	C₂×C₄×C₈₉	—	8	1	no	—
47	C₂×C₄×C₈₉	—	8	2	no	—
48	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
50	C₈×C₈₉	—	4	2	no	—
51	#10623494	—	16	2	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	#323	1	1	yes	characteristic
55	C₂²×C₈₉	#354	1	1	yes	characteristic
56	C₂²×C₈₉	#356	1	1	yes	characteristic
57	C₄×C₈₉	—	2	1	no	—
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	—	4	1	no	—
60	C₄×C₈₉	—	4	1	no	—
61	C₄×C₈₉	—	8	2	no	—
62	C₄×C₈₉	—	8	2	no	—
63	#7358	—	89	1	no	maximal, Sylow
64	C₂×C₈₉	—	2	1	no	—
65	C₂×C₈₉	—	2	1	no	—
66	C₂×C₈₉	#1319	1	1	yes	characteristic
67	C₂×C₈₉	#1412	1	1	yes	characteristic
68	C₂×C₈₉	#1413	1	1	yes	characteristic
69	#1992	D₈₉	1	1	yes	characteristic
70	#1325	—	89	2	no	—
71	C₈₉	#7358	1	1	yes	characteristic, complemented, Sylow
72	C₂×#125	Dic₈₉	1	1	yes	characteristic
73	C₂×#97	Dic₈₉	1	1	yes	characteristic
74	#384	C₂×D₈₉	1	1	yes	characteristic
75	#340	—	2	2	no	—
76	#324	—	178	2	no	—
77	C₂²×C₈	—	2	1	no	—
78	#125	—	2	1	no	—
79	#125	—	4	1	no	—
80	#125	#10623494	1	2	yes	complemented
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
85	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
86	C₂²⋊C₈	—	356	2	no	—
87	#97	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	712	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	C₂×#10623494	1	1	yes	characteristic
104	C₄²	#13083558	1	1	yes	characteristic
105	C₂³	#15667927	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
118	C₂×C₄	#15667929	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	1424	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#19151322	1	1	yes	characteristic
124	C₂²	#19151393	1	1	yes	characteristic
125	C₂²	#19151397	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	2	no	—
131	C₄	—	8	2	no	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#26018186	1	1	yes	characteristic
135	C₂	#26018423	1	1	yes	characteristic
136	C₂	#26018425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018198

C₂

yes

maximal

C₈₉×#1992

C₂

yes

maximal, characteristic

#19151324

—

C₂×#125×C₈₉

C₄

yes

characteristic

C₂×#97×C₈₉

C₄

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

#340×C₈₉

—

C₂²⋊#10623494

—

C₂²×C₈×C₈₉

—

#125×C₈₉

—

#125×C₈₉

—

#125×C₈₉

C₈

yes

complemented

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#97×C₈₉

—

C₂×#10623494

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₂³×C₈₉

#99

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

#100

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

#7358

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

#1325

—

C₈₉

#7358

yes

characteristic, complemented, Sylow

C₂×#125

Dic₈₉

yes

characteristic

C₂×#97

Dic₈₉

yes

characteristic

#384

C₂×D₈₉

yes

characteristic

#340

—

#324

—

178

—

C₂²×C₈

—

—

—

yes

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

C₂²⋊C₈

—

356

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

712

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

C₂×#10623494

yes

characteristic

104

C₄²

#13083558

yes

characteristic

105

C₂³

#15667927

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊#10623494

yes

characteristic

118

C₂×C₄

#15667929

yes

characteristic

119

C₈

—

120

C₈

—

1424

—

121

C₂²

—

122

C₂²

—

123

C₂²

#19151322

yes

characteristic

124

C₂²

#19151393

yes

characteristic

125

C₂²

#19151397

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₂

—

133

C₂

—

134

C₂

#26018186

yes

characteristic

135

C₂

#26018423

yes

characteristic

136

C₂

#26018425

yes

characteristic

	subgroup	G/H
center	C₂²	#19151393
commutator	C₄²×C₈₉	C₂×C₈
Frattini	#384	C₂×D₈₉
Fitting	C₈₉×#1992	C₂
socle	C₂²×C₈₉	#354
prime radical	C₂³×C₈₉	#99
cosocle	#384×C₈₉	C₂²
hypercenter	#1992	D₈₉
hypocenter	C₈₉	#7358
LCS	G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1992

subgroup

G/H

center

C₂²

#19151393

commutator

C₄²×C₈₉

C₂×C₈

Frattini

#384

C₂×D₈₉

Fitting

C₈₉×#1992

C₂

C₂²×C₈₉

C₂³×C₈₉

#384×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1992

N	H ≅ G/N	type
12. #125×C₈₉	C₈	semidirect
71. C₈₉	#7358	semidirect
80. #125	#10623494	semidirect

H ≅ G/N

type

12. #125×C₈₉

C₈

semidirect

71. C₈₉

#7358

semidirect

80. #125

#10623494

semidirect

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

subgroup	index
#26018198	2
C₈₉×#1992	2
#7358	89

subgroup

index

#26018198

C₈₉×#1992

#7358

Subgroups — #36441819

Conjugacy classes of subgroups

Complements

Maximal subgroups