Group #26022422 — subgroups

Subgroups — #26022422

Order 11392.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#19151995	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151714	C₂	1	1	yes	maximal, characteristic
3	#19151422	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151418	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151409	C₂	1	1	yes	maximal, characteristic
6	#19151407	C₂	1	1	yes	maximal, characteristic
7	#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668034	C₂²	1	1	yes	characteristic
9	#15668020	—	2	1	no	—
10	#15668011	—	2	1	no	—
11	#15667952	C₂²	1	1	yes	characteristic
12	#15667944	C₂²	1	1	yes	characteristic
13	#15667943	—	2	1	no	—
14	#15667938	—	2	1	no	—
15	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	M₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	#108×C₈₉	C₂²	1	1	yes	characteristic
18	#106×C₈₉	C₂²	1	1	yes	characteristic
19	#13083574	—	2	1	no	—
20	C₂×Dic₁₇₈	—	4	1	no	—
21	Dic₃₅₆	Q₈	1	1	yes	characteristic, complemented
22	#13083563	—	2	1	no	—
23	D₇₁₂	Q₈	1	1	yes	characteristic, complemented
24	#13083557	—	4	1	no	—
25	C₂×#10623494	D₄	1	1	yes	characteristic
26	C₈×D₈₉	—	4	1	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	#13083553	—	4	1	no	—
29	C₄×Dic₈₉	D₄	1	1	yes	characteristic
30	#13083549	—	2	1	no	—
31	M₁₆×C₈₉	—	2	1	no	—
32	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₁₆×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₈₉	—	4	1	no	—
38	C₂×Dic₈₉	—	2	1	no	—
39	C₂×Dic₈₉	—	4	1	no	—
40	D₃₅₆	—	2	1	no	—
41	C₄×D₈₉	—	4	1	no	—
42	Dic₁₇₈	—	2	1	no	—
43	Dic₁₇₈	—	8	2	no	—
44	C₈×C₈₉	—	4	1	no	—
45	C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₈₉	#41	1	1	yes	characteristic
47	#10623494	SD₁₆	1	1	yes	characteristic
48	#10623494	SD₁₆	1	1	yes	characteristic
49	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₈₉	—	4	1	no	—
51	C₄×C₈₉	—	8	1	no	—
52	C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₈₉	—	2	2	no	—
55	Dic₈₉	—	4	1	no	—
56	Dic₈₉	—	8	1	no	—
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	#476	1	1	yes	characteristic
59	D₈₉	—	8	1	no	—
60	#2230	—	89	1	no	maximal, Sylow
61	C₈₉	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	89	1	no	—
63	#443	—	89	1	no	—
64	#365	—	89	1	no	—
65	#365	D₈₉	1	1	yes	characteristic, complemented
66	#364	—	89	1	no	—
67	#360	—	89	1	no	—
68	#360	—	89	1	no	—
69	#135	—	89	1	no	—
70	#131	—	178	1	no	—
71	C₄⋊Q₈	—	178	1	no	—
72	M₃₂	—	89	1	no	—
73	M₃₂	C₂×D₈₉	1	1	yes	characteristic
74	#108	C₂×D₈₉	1	1	yes	characteristic
75	#106	—	89	1	no	—
76	#106	—	178	1	no	—
77	#106	C₂×D₈₉	1	1	yes	characteristic
78	#104	—	178	1	no	—
79	C₄×C₈	—	89	1	no	—
80	#41	—	178	1	no	—
81	C₂×Q₈	—	356	1	no	—
82	Q₁₆	—	89	1	no	—
83	SD₁₆	—	178	1	no	—
84	D₈	—	89	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	356	1	no	—
87	C₂×C₈	—	89	1	no	—
88	C₂×C₈	—	356	1	no	—
89	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	356	1	no	—
92	C₄⋊C₄	—	178	1	no	—
93	C₄²	—	89	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	178	1	no	—
96	Q₈	—	178	1	no	—
97	Q₈	—	712	2	no	—
98	D₄	—	356	1	no	—
99	D₄	—	178	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	356	1	no	—
102	C₂×C₄	—	356	1	no	—
103	C₂×C₄	—	178	1	no	—
104	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	89	1	no	—
107	C₈	—	89	1	no	—
108	C₈	Q₈×D₈₉	1	1	yes	characteristic
109	C₈	#13083574	1	1	yes	characteristic
110	C₂²	—	356	1	no	—
111	C₂²	#15668039	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	356	1	no	—
114	C₄	—	178	2	no	—
115	C₄	—	712	1	no	—
116	C₄	#15667990	1	1	yes	characteristic
117	C₄	SD₁₆×D₈₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	712	1	no	—
120	C₂	#19151887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151995

C₂

yes

maximal, characteristic, complemented

#19151714

C₂

yes

maximal, characteristic

#19151422

C₂

yes

maximal, characteristic, complemented

#19151418

C₂

yes

maximal, characteristic, complemented

#19151409

C₂

yes

maximal, characteristic

#19151407

C₂

yes

maximal, characteristic

#365×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668034

C₂²

yes

—

—

C₂²

yes

characteristic

#15667944

C₂²

yes

characteristic

#15667943

—

#15667938

—

C₈×Dic₈₉

C₂²

yes

characteristic

M₃₂×C₈₉

C₂²

yes

characteristic

#108×C₈₉

C₂²

yes

characteristic

#106×C₈₉

C₂²

yes

characteristic

#13083574

—

C₂×Dic₁₇₈

—

Dic₃₅₆

Q₈

yes

characteristic, complemented

#13083563

—

D₇₁₂

Q₈

yes

characteristic, complemented

#13083557

—

C₂×#10623494

D₄

yes

characteristic

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₈

yes

C₈×C₈₉

yes

SD₁₆

yes

characteristic

#10623494

SD₁₆

yes

characteristic

C₂²×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#476

yes

characteristic

D₈₉

—

#2230

—

maximal, Sylow

C₈₉

#2230

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

—

—

—

—

—

178

—

C₄⋊Q₈

—

178

—

M₃₂

—

M₃₂

C₂×D₈₉

yes

characteristic

#108

C₂×D₈₉

yes

characteristic

#106

—

#106

—

178

—

#106

C₂×D₈₉

yes

characteristic

#104

—

178

—

C₄×C₈

—

#41

—

178

—

C₂×Q₈

—

356

—

Q₁₆

—

SD₁₆

—

178

—

D₈

—

M₁₆

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄²

—

C₁₆

—

C₁₆

—

178

—

Q₈

—

178

—

Q₈

—

712

—

D₄

—

356

—

D₄

—

178

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

—

178

—

104

C₂×C₄

D₄×D₈₉

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

Q₈×D₈₉

yes

characteristic

109

C₈

#13083574

yes

characteristic

110

C₂²

—

356

—

111

C₂²

#15668039

yes

characteristic

112

C₄

—

113

C₄

—

356

—

114

C₄

—

178

—

115

C₄

—

712

—

116

C₄

#15667990

yes

characteristic

117

C₄

SD₁₆×D₈₉

yes

characteristic

118

C₂

—

119

C₂

—

712

—

120

C₂

#19151887

yes

characteristic

	subgroup	G/H
center	C₂	#19151887
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#365×C₈₉	C₂
socle	C₂×C₈₉	#476
prime radical	#15668034	C₂²
hypercenter	#365	D₈₉
hypocenter	C₈₉	#2230
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#19151887

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#365×C₈₉

C₂

C₂×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #19151995	C₂	semidirect
3. #19151422	C₂	semidirect
4. #19151418	C₂	semidirect
7. #365×C₈₉	C₂	semidirect
21. Dic₃₅₆	Q₈	semidirect
23. D₇₁₂	Q₈	semidirect
61. C₈₉	#2230	semidirect
65. #365	D₈₉	semidirect

H ≅ G/N

type

1. #19151995

C₂

semidirect

3. #19151422

C₂

semidirect

4. #19151418

C₂

semidirect

7. #365×C₈₉

C₂

semidirect

21. Dic₃₅₆

Q₈

semidirect

23. D₇₁₂

Q₈

semidirect

61. C₈₉

#2230

semidirect

65. #365

D₈₉

semidirect

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

subgroup	index
#19151995	2
#19151714	2
#19151422	2
#19151418	2
#19151409	2
#19151407	2
#365×C₈₉	2
#2230	89

subgroup

index

#365×C₈₉

Subgroups — #26022422

Conjugacy classes of subgroups

Complements

Maximal subgroups