Group #26022328 — subgroups

Subgroups — #26022328

Order 11392.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 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	#19152033	C₂	1	2	yes	maximal, complemented
2	#19151975	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151419	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151410	C₂	1	2	yes	maximal, complemented
5	#443×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₁₂	—	2	1	no	—
7	C₂×D₇₁₂	C₂²	1	2	yes	—
8	#15668009	—	2	1	no	—
9	#15667955	—	2	2	no	—
10	#15667953	—	2	2	no	—
11	#15667952	C₂²	1	2	yes	—
12	#135×C₈₉	C₂²	1	1	yes	characteristic
13	#131×C₈₉	—	2	1	no	—
14	#108×C₈₉	C₂²	1	1	yes	characteristic
15	#104×C₈₉	—	2	1	no	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	4	2	no	—
19	D₇₁₂	—	2	2	no	—
20	D₇₁₂	—	2	2	no	—
21	D₇₁₂	—	4	1	no	—
22	D₇₁₂	—	4	1	no	—
23	#13083549	—	2	2	no	—
24	#41×C₈₉	—	2	1	no	—
25	Q₁₆×C₈₉	D₄	1	1	yes	characteristic, complemented
26	SD₁₆×C₈₉	—	2	1	no	—
27	D₈×C₈₉	D₄	1	1	yes	characteristic, complemented
28	M₁₆×C₈₉	—	2	1	no	—
29	M₁₆×C₈₉	—	4	1	no	—
30	C₂×C₈×C₈₉	—	4	1	no	—
31	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×D₈₉	—	4	2	no	—
35	Q₈×C₈₉	—	2	1	no	—
36	D₄×C₈₉	—	2	1	no	—
37	D₄×C₈₉	—	4	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	4	1	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	D₃₅₆	—	4	2	no	—
42	D₃₅₆	—	4	2	no	—
43	D₃₅₆	—	8	4	no	—
44	C₈×C₈₉	—	4	1	no	—
45	C₈×C₈₉	D₈	1	1	yes	characteristic
46	C₈×C₈₉	D₈	1	1	yes	characteristic
47	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₈×C₈₉	#41	1	1	yes	characteristic
49	C₂²×C₈₉	—	4	1	no	—
50	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×D₈₉	—	8	4	no	—
52	C₂×D₈₉	—	8	2	no	—
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	4	1	no	—
55	C₄×C₈₉	#121	1	1	yes	characteristic
56	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	—	8	1	no	—
59	C₂×C₈₉	#459	1	1	yes	characteristic
60	D₈₉	—	16	2	no	—
61	#2213	—	89	1	no	maximal, Sylow
62	C₈₉	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	89	2	no	—
64	#493	—	89	1	no	—
65	#443	D₈₉	1	1	yes	characteristic, complemented
66	#364	—	89	1	no	—
67	#361	—	89	2	no	—
68	#135	C₂×D₈₉	1	1	yes	characteristic
69	C₂×D₈	—	89	2	no	—
70	C₂×D₈	—	178	1	no	—
71	#131	—	2	1	no	—
72	#127	—	178	1	no	—
73	SD₃₂	—	178	2	no	—
74	D₁₆	—	178	2	no	—
75	M₃₂	—	89	2	no	—
76	#108	C₂×D₈₉	1	1	yes	characteristic
77	#104	—	2	1	no	—
78	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
79	#41	—	2	1	no	—
80	C₂×D₄	—	356	2	no	—
81	C₂×D₄	—	178	2	no	—
82	Q₁₆	D₃₅₆	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	—	356	1	no	—
85	D₈	—	356	1	no	—
86	D₈	—	178	2	no	—
87	D₈	—	178	2	no	—
88	D₈	D₃₅₆	1	1	yes	characteristic, complemented
89	M₁₆	—	2	1	no	—
90	M₁₆	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
95	C₁₆	—	178	2	no	—
96	C₂³	—	356	2	no	—
97	Q₈	—	2	1	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	D₄	—	356	2	no	—
101	D₄	—	356	2	no	—
102	D₄	—	712	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
106	C₈	—	4	1	no	—
107	C₈	#13083559	1	1	yes	characteristic
108	C₈	#13083559	1	1	yes	characteristic
109	C₈	C₂×D₃₅₆	1	1	yes	characteristic
110	C₈	#13083574	1	1	yes	characteristic
111	C₂²	—	4	1	no	—
112	C₂²	—	712	4	no	—
113	C₂²	—	712	2	no	—
114	C₂²	#15668037	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	1	no	—
117	C₄	#15667986	1	1	yes	characteristic
118	C₄	C₂×#13083559	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	8	1	no	—
121	C₂	—	1424	2	no	—
122	C₂	#19151793	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152033

C₂

yes

maximal, complemented

#19151975

C₂

yes

maximal, characteristic, complemented

#19151419

C₂

yes

maximal, characteristic, complemented

#19151410

C₂

yes

maximal, complemented

#443×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₇₁₂

—

C₂×D₇₁₂

C₂²

yes

—

—

—

—

C₂²

yes

—

#135×C₈₉

C₂²

yes

characteristic

#131×C₈₉

—

#108×C₈₉

C₂²

yes

characteristic

#104×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

#13083549

—

#41×C₈₉

—

Q₁₆×C₈₉

D₄

yes

characteristic, complemented

SD₁₆×C₈₉

—

D₈×C₈₉

D₄

yes

characteristic, complemented

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₈×C₈₉

D₈

yes

characteristic

C₈×C₈₉

D₈

yes

characteristic

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

#41

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#459

yes

characteristic

D₈₉

—

#2213

—

maximal, Sylow

C₈₉

#2213

yes

characteristic, complemented, Sylow

#509

—

#493

—

#443

D₈₉

yes

characteristic, complemented

#364

—

#361

—

#135

C₂×D₈₉

yes

characteristic

C₂×D₈

—

C₂×D₈

—

178

—

#131

—

#127

—

178

—

SD₃₂

—

178

—

D₁₆

—

178

—

M₃₂

—

#108

C₂×D₈₉

yes

characteristic

#104

—

C₄×C₈

C₂×D₈₉

yes

characteristic

#41

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

Q₁₆

D₃₅₆

yes

characteristic, complemented

SD₁₆

—

D₈

—

356

—

D₈

—

356

—

D₈

—

178

—

D₈

—

178

—

D₈

D₃₅₆

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄²

C₂²⋊D₈₉

yes

characteristic

C₁₆

—

178

—

C₂³

—

356

—

Q₈

—

D₄

—

D₄

—

100

D₄

—

356

—

101

D₄

—

356

—

102

D₄

—

712

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

106

C₈

—

107

C₈

#13083559

yes

characteristic

108

C₈

#13083559

yes

characteristic

109

C₈

C₂×D₃₅₆

yes

characteristic

110

C₈

#13083574

yes

characteristic

111

C₂²

—

112

C₂²

—

712

—

113

C₂²

—

712

—

114

C₂²

#15668037

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

#15667986

yes

characteristic

118

C₄

C₂×#13083559

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

1424

—

122

C₂

#19151793

yes

characteristic

	subgroup	G/H
center	C₂	#19151793
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#443×C₈₉	C₂
socle	C₂×C₈₉	#459
hypercenter	#443	D₈₉
hypocenter	C₈₉	#2213
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#19151793

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#443×C₈₉

C₂

C₂×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #19152033	C₂	semidirect
2. #19151975	C₂	semidirect
3. #19151419	C₂	semidirect
4. #19151410	C₂	semidirect
5. #443×C₈₉	C₂	semidirect
25. Q₁₆×C₈₉	D₄	semidirect
27. D₈×C₈₉	D₄	semidirect
62. C₈₉	#2213	semidirect
65. #443	D₈₉	semidirect
82. Q₁₆	D₃₅₆	semidirect
88. D₈	D₃₅₆	semidirect

H ≅ G/N

type

1. #19152033

C₂

semidirect

2. #19151975

C₂

semidirect

3. #19151419

C₂

semidirect

4. #19151410

C₂

semidirect

5. #443×C₈₉

C₂

semidirect

25. Q₁₆×C₈₉

D₄

semidirect

27. D₈×C₈₉

D₄

semidirect

62. C₈₉

#2213

semidirect

65. #443

D₈₉

semidirect

82. Q₁₆

D₃₅₆

semidirect

88. D₈

D₃₅₆

semidirect

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

subgroup	index
#19152033	2
#19151975	2
#19151419	2
#19151410	2
#443×C₈₉	2
#2213	89

subgroup

index

#443×C₈₉

Subgroups — #26022328

Conjugacy classes of subgroups

Complements

Maximal subgroups