Group #26021767 — subgroups

Subgroups — #26021767

Order 11392.

222 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19151646	C₂	1	2	yes	maximal, complemented
2	#19151561	C₂	1	1	yes	maximal, characteristic
3	#19151561	C₂	1	2	yes	maximal
4	#19151356	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
7	C₂×#13083558	C₂²	1	1	yes	characteristic
8	C₂×#13083558	C₂²	1	2	yes	—
9	#15667947	—	2	4	no	—
10	#15667924	—	2	4	no	—
11	#15667924	—	2	2	no	—
12	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	#108×C₈₉	—	2	2	no	—
15	C₂³×D₈₉	Q₈	1	1	yes	characteristic
16	C₂²×Dic₈₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₈₉	—	2	2	no	—
18	C₂×C₄×D₈₉	—	2	1	no	—
19	C₂×C₄×D₈₉	—	2	2	no	—
20	C₂×C₄×D₈₉	—	2	1	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	#13083558	—	2	2	no	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	D₄	1	4	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₉	—	2	2	no	—
28	M₁₆×C₈₉	—	2	2	no	—
29	M₁₆×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	D₄	1	2	yes	—
32	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
33	C₂²×D₈₉	—	2	1	no	—
34	C₂²×D₈₉	—	2	2	no	—
35	C₂²×D₈₉	—	4	2	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#41	1	2	yes	—
39	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
40	C₂×Dic₈₉	—	2	1	no	—
41	C₂×Dic₈₉	—	2	2	no	—
42	C₄×D₈₉	—	4	1	no	—
43	C₄×D₈₉	—	4	2	no	—
44	C₄×D₈₉	—	4	1	no	—
45	C₈×C₈₉	—	2	2	no	—
46	C₈×C₈₉	—	4	2	no	—
47	#10623494	—	2	4	no	—
48	#10623494	—	4	1	no	—
49	C₂²×C₈₉	—	4	1	no	—
50	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
52	C₂×D₈₉	—	2	2	no	—
53	C₂×D₈₉	—	2	4	no	—
54	C₂×D₈₉	—	4	1	no	—
55	C₂×D₈₉	—	4	2	no	—
56	C₂×D₈₉	—	4	1	no	—
57	C₄×C₈₉	#123	1	1	yes	characteristic
58	C₄×C₈₉	#123	1	2	yes	—
59	C₄×C₈₉	#127	1	1	yes	characteristic
60	Dic₈₉	—	4	1	no	—
61	C₂×C₈₉	—	2	1	no	—
62	C₂×C₈₉	—	2	2	no	—
63	C₂×C₈₉	#399	1	1	yes	characteristic
64	D₈₉	—	4	2	no	—
65	#2091	—	89	1	no	maximal, Sylow
66	C₈₉	#2091	1	1	yes	characteristic, complemented, Sylow
67	#430	—	89	2	no	—
68	#430	D₈₉	1	1	yes	characteristic, complemented
69	#407	—	89	2	no	—
70	#407	—	89	1	no	—
71	#338	—	89	1	no	—
72	C₂³×C₄	—	89	1	no	—
73	C₂×M₁₆	—	89	2	no	—
74	C₂×M₁₆	—	89	1	no	—
75	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
76	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
77	#108	—	2	2	no	—
78	#108	—	178	4	no	—
79	C₂²⋊C₈	—	178	4	no	—
80	C₂²⋊C₈	—	178	2	no	—
81	C₂⁴	—	89	1	no	—
82	C₂²×C₄	—	89	1	no	—
83	C₂²×C₄	—	178	2	no	—
84	C₂²×C₄	—	178	1	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂²×C₄	—	178	1	no	—
87	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	178	2	no	—
92	M₁₆	—	178	2	no	—
93	M₁₆	—	178	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	89	4	no	—
96	C₂×C₈	—	178	1	no	—
97	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
98	C₂³	—	356	2	no	—
99	C₂³	—	178	1	no	—
100	C₂³	—	178	2	no	—
101	C₂³	Q₈×D₈₉	1	1	yes	characteristic
102	C₂×C₄	—	356	1	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	356	1	no	—
105	C₂×C₄	—	178	1	no	—
106	C₂×C₄	—	178	2	no	—
107	C₂×C₄	D₄×D₈₉	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
109	C₂×C₄	#13083574	1	2	yes	—
110	C₂×C₄	#13083575	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	356	1	no	—
114	C₈	—	178	4	no	—
115	C₂²	—	4	1	no	—
116	C₂²	—	356	1	no	—
117	C₂²	—	356	2	no	—
118	C₂²	—	356	1	no	—
119	C₂²	—	178	2	no	—
120	C₂²	—	178	4	no	—
121	C₂²	#15667988	1	1	yes	characteristic
122	C₂²	#15667990	1	2	yes	—
123	C₄	—	356	1	no	—
124	C₄	#15667995	1	1	yes	characteristic
125	C₄	#15667997	1	2	yes	—
126	C₄	#15668008	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	356	2	no	—
130	C₂	#19151533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151646

C₂

yes

maximal, complemented

#19151561

C₂

yes

maximal, characteristic

#19151561

C₂

yes

maximal

#19151356

C₂

yes

maximal, characteristic, complemented

#430×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₈₉

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

#15667947

—

#15667924

—

#15667924

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#108×C₈₉

—

C₂³×D₈₉

Q₈

yes

characteristic

C₂²×Dic₈₉

Q₈

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#123

yes

characteristic

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#127

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#399

yes

characteristic

D₈₉

—

#2091

—

maximal, Sylow

C₈₉

#2091

yes

characteristic, complemented, Sylow

#430

—

#430

D₈₉

yes

characteristic, complemented

#407

—

#407

—

#338

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

#108

—

#108

—

178

—

C₂²⋊C₈

—

178

—

C₂²⋊C₈

—

178

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₂³

—

356

—

C₂³

—

178

—

100

C₂³

—

178

—

101

C₂³

Q₈×D₈₉

yes

characteristic

102

C₂×C₄

—

356

—

103

C₂×C₄

—

356

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

D₄×D₈₉

yes

—

108

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

109

C₂×C₄

#13083574

yes

—

110

C₂×C₄

#13083575

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

356

—

114

C₈

—

178

—

115

C₂²

—

116

C₂²

—

356

—

117

C₂²

—

356

—

118

C₂²

—

356

—

119

C₂²

—

178

—

120

C₂²

—

178

—

121

C₂²

#15667988

yes

characteristic

122

C₂²

#15667990

yes

—

123

C₄

—

356

—

124

C₄

#15667995

yes

characteristic

125

C₄

#15667997

yes

—

126

C₄

#15668008

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

356

—

130

C₂

#19151533

yes

characteristic

	subgroup	G/H
center	C₄	#15668008
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#430×C₈₉	C₂
socle	C₂×C₈₉	#399
prime radical	C₂³×D₈₉	Q₈
hypercenter	#430	D₈₉
hypocenter	C₈₉	#2091
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#15668008

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#430×C₈₉

C₂

socle

C₂×C₈₉

#399

prime radical

C₂³×D₈₉

Q₈

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #19151646	C₂	semidirect
4. #19151356	C₂	semidirect
5. #430×C₈₉	C₂	semidirect
66. C₈₉	#2091	semidirect
68. #430	D₈₉	semidirect

H ≅ G/N

type

1. #19151646

C₂

semidirect

4. #19151356

C₂

semidirect

5. #430×C₈₉

C₂

semidirect

66. C₈₉

#2091

semidirect

68. #430

D₈₉

semidirect

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

subgroup	index
#19151646	2
#19151561	2
#19151356	2
#430×C₈₉	2
#2091	89

subgroup

index

#430×C₈₉

Subgroups — #26021767

Conjugacy classes of subgroups

Complements

Maximal subgroups