Group #26032051 — subgroups

Subgroups — #26032051

Order 11392.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152399	C₂	1	4	yes	maximal, complemented
2	#19151865	C₂	1	4	yes	maximal, complemented
3	#19151765	C₂	1	4	yes	maximal, complemented
4	#19151716	C₂	1	2	yes	maximal, complemented
5	#493×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668057	C₂²	1	4	yes	complemented
7	#15668038	C₂²	1	8	yes	complemented
8	#15668036	—	2	8	no	—
9	#15668034	C₂²	1	2	yes	—
10	#15668031	—	2	4	no	—
11	D₈×D₈₉	—	2	4	no	—
12	#15668018	C₂²	1	4	yes	complemented
13	#15667948	C₂²	1	2	yes	complemented
14	#15667940	C₂²	1	4	yes	complemented
15	#15667929	C₂²	1	4	yes	complemented
16	C₂×D₈×C₈₉	C₂²	1	4	yes	complemented
17	#127×C₈₉	C₂²	1	2	yes	complemented
18	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	#13083574	C₂³	1	4	yes	—
20	#13083573	—	2	8	no	—
21	C₂×C₂²⋊D₈₉	—	4	4	no	—
22	D₄×D₈₉	—	2	8	no	—
23	Dic₃₅₆	D₄	1	2	yes	complemented
24	#13083563	D₄	1	4	yes	complemented
25	#13083562	—	2	8	no	—
26	D₇₁₂	D₄	1	2	yes	complemented
27	#13083559	—	2	8	no	—
28	#13083558	C₂³	1	4	yes	—
29	#13083557	—	2	4	no	—
30	C₈×D₈₉	—	2	4	no	—
31	C₂×D₄×C₈₉	—	4	2	no	—
32	C₂×D₄×C₈₉	C₂³	1	4	yes	—
33	D₈×C₈₉	—	2	4	no	—
34	D₈×C₈₉	—	2	4	no	—
35	C₂×C₈×C₈₉	C₂³	1	2	yes	—
36	C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₂³×C₈₉	—	2	4	no	—
38	C₂²×D₈₉	—	4	4	no	—
39	D₄×C₈₉	—	2	8	no	—
40	D₄×C₈₉	—	8	4	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₈₉	—	2	4	no	—
44	C₂²⋊D₈₉	—	4	8	no	—
45	C₂×Dic₈₉	—	4	4	no	—
46	D₃₅₆	C₂×D₄	1	4	yes	—
47	C₄×D₈₉	—	2	4	no	—
48	Dic₁₇₈	C₂×D₄	1	4	yes	—
49	C₈×C₈₉	C₂×D₄	1	2	yes	—
50	C₈×C₈₉	C₂×D₄	1	2	yes	—
51	#10623494	—	2	4	no	—
52	C₂²×C₈₉	—	4	4	no	—
53	C₂²×C₈₉	—	8	4	no	—
54	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₈₉	—	2	4	no	—
56	C₂×D₈₉	—	16	4	no	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	C₂²×D₄	1	2	yes	—
59	Dic₈₉	—	2	4	no	—
60	C₂×C₈₉	—	2	1	no	—
61	C₂×C₈₉	—	8	4	no	—
62	C₂×C₈₉	D₄²	1	1	yes	characteristic
63	D₈₉	—	4	4	no	—
64	#3292	—	89	1	no	maximal, Sylow
65	C₈₉	#3292	1	1	yes	characteristic, complemented, Sylow
66	#576	—	89	4	no	—
67	#493	D₈₉	1	1	yes	characteristic, complemented
68	#472	—	89	4	no	—
69	#453	—	89	4	no	—
70	#443	—	89	2	no	—
71	H(5,𝔽₂)	—	89	4	no	—
72	AGL(1,ℤ/8)	—	89	8	no	—
73	AGL(1,ℤ/8)	—	178	8	no	—
74	#135	—	89	2	no	—
75	#135	—	178	4	no	—
76	C₂×D₈	—	178	4	no	—
77	C₂×D₈	C₂×D₈₉	1	4	yes	complemented
78	#131	—	89	4	no	—
79	#127	C₂×D₈₉	1	2	yes	complemented
80	#108	—	89	2	no	—
81	#104	—	89	4	no	—
82	#100	—	89	4	no	—
83	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
84	#41	—	89	4	no	—
85	#41	—	178	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	356	4	no	—
88	C₂×D₄	—	178	8	no	—
89	C₂×D₄	C₂²×D₈₉	1	4	yes	—
90	Q₁₆	—	89	2	no	—
91	SD₁₆	—	89	4	no	—
92	SD₁₆	—	178	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	89	2	no	—
96	D₈	—	178	8	no	—
97	M₁₆	—	89	4	no	—
98	M₁₆	—	178	4	no	—
99	C₂×C₈	—	178	4	no	—
100	C₂×C₈	C₂²×D₈₉	1	2	yes	—
101	C₄²	C₂²×D₈₉	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	356	4	no	—
104	Q₈	—	89	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	89	4	no	—
108	D₄	—	356	8	no	—
109	D₄	—	178	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	356	4	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
114	C₈	—	178	4	no	—
115	C₈	D₄×D₈₉	1	2	yes	—
116	C₈	D₄×D₈₉	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	1424	4	no	—
120	C₂²	—	178	4	no	—
121	C₂²	#15668057	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	178	4	no	—
124	C₄	C₂×D₄×D₈₉	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	356	4	no	—
128	C₂	#19152233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152399

C₂

yes

maximal, complemented

#19151865

C₂

yes

maximal, complemented

#19151765

C₂

yes

maximal, complemented

#19151716

C₂

yes

maximal, complemented

#493×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668057

C₂²

yes

complemented

#15668038

C₂²

yes

complemented

#15668036

—

#15668034

C₂²

yes

—

#15668031

—

D₈×D₈₉

—

#15668018

C₂²

yes

complemented

#15667948

C₂²

yes

complemented

#15667940

C₂²

yes

complemented

#15667929

C₂²

yes

complemented

C₂×D₈×C₈₉

C₂²

yes

complemented

#127×C₈₉

C₂²

yes

complemented

C₄×C₈×C₈₉

C₂²

yes

characteristic, complemented

#13083574

C₂³

yes

—

#13083573

—

C₂×C₂²⋊D₈₉

—

D₄×D₈₉

—

Dic₃₅₆

D₄

yes

complemented

#13083563

D₄

yes

complemented

#13083562

—

D₇₁₂

D₄

yes

complemented

#13083559

—

#13083558

C₂³

yes

—

#13083557

—

C₈×D₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

C₂×D₄

yes

—

C₄×D₈₉

—

Dic₁₇₈

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

D₄²

yes

characteristic

D₈₉

—

#3292

—

maximal, Sylow

C₈₉

#3292

yes

characteristic, complemented, Sylow

#576

—

#493

D₈₉

yes

characteristic, complemented

—

—

—

—

—

—

178

—

#135

—

#135

—

178

—

C₂×D₈

—

178

—

C₂×D₈

C₂×D₈₉

yes

complemented

#131

—

#127

C₂×D₈₉

yes

—

—

—

C₄×C₈

C₂×D₈₉

yes

characteristic, complemented

#41

—

#41

—

178

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

C₂²×D₈₉

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

178

—

D₈

—

D₈

—

D₈

—

D₈

—

178

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

178

—

100

C₂×C₈

C₂²×D₈₉

yes

—

101

C₄²

C₂²×D₈₉

yes

characteristic

102

C₂³

—

103

C₂³

—

356

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

356

—

109

D₄

—

178

—

110

C₂×C₄

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

178

—

113

C₂×C₄

C₂³×D₈₉

yes

characteristic

114

C₈

—

178

—

115

C₈

D₄×D₈₉

yes

—

116

C₈

D₄×D₈₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1424

—

120

C₂²

—

178

—

121

C₂²

#15668057

yes

characteristic

122

C₄

—

123

C₄

—

178

—

124

C₄

C₂×D₄×D₈₉

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

356

—

128

C₂

#19152233

yes

characteristic

	subgroup	G/H
center	C₂	#19152233
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#493×C₈₉	C₂
socle	C₂×C₈₉	D₄²
hypercenter	#493	D₈₉
hypocenter	C₈₉	#3292
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#19152233

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#493×C₈₉

C₂

socle

C₂×C₈₉

D₄²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #19152399	C₂	semidirect
2. #19151865	C₂	semidirect
3. #19151765	C₂	semidirect
4. #19151716	C₂	semidirect
5. #493×C₈₉	C₂	semidirect
6. #15668057	C₂²	semidirect
7. #15668038	C₂²	semidirect
12. #15668018	C₂²	semidirect
13. #15667948	C₂²	semidirect
14. #15667940	C₂²	semidirect
15. #15667929	C₂²	semidirect
16. C₂×D₈×C₈₉	C₂²	semidirect
17. #127×C₈₉	C₂²	semidirect
18. C₄×C₈×C₈₉	C₂²	semidirect
23. Dic₃₅₆	D₄	semidirect
24. #13083563	D₄	semidirect
26. D₇₁₂	D₄	semidirect
65. C₈₉	#3292	semidirect
67. #493	D₈₉	semidirect
77. C₂×D₈	C₂×D₈₉	semidirect
79. #127	C₂×D₈₉	semidirect
83. C₄×C₈	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #19152399

C₂

semidirect

2. #19151865

C₂

semidirect

3. #19151765

C₂

semidirect

4. #19151716

C₂

semidirect

5. #493×C₈₉

C₂

semidirect

6. #15668057

C₂²

semidirect

7. #15668038

C₂²

semidirect

12. #15668018

C₂²

semidirect

13. #15667948

C₂²

semidirect

14. #15667940

C₂²

semidirect

15. #15667929

C₂²

semidirect

16. C₂×D₈×C₈₉

C₂²

semidirect

17. #127×C₈₉

C₂²

semidirect

18. C₄×C₈×C₈₉

C₂²

semidirect

23. Dic₃₅₆

D₄

semidirect

24. #13083563

D₄

semidirect

26. D₇₁₂

D₄

semidirect

65. C₈₉

#3292

semidirect

67. #493

D₈₉

semidirect

77. C₂×D₈

C₂×D₈₉

semidirect

79. #127

C₂×D₈₉

semidirect

83. C₄×C₈

C₂×D₈₉

semidirect

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

subgroup	index
#19152399	2
#19151865	2
#19151765	2
#19151716	2
#493×C₈₉	2
#3292	89

subgroup

index

#493×C₈₉

Subgroups — #26032051

Conjugacy classes of subgroups

Complements

Maximal subgroups