Group #26032056 — subgroups

Subgroups — #26032056

Order 11392.

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

normal subgroups only

This group has 444 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	#19152411	C₂	1	4	yes	maximal, complemented
2	#19151870	C₂	1	4	yes	maximal, complemented
3	#19151780	C₂	1	4	yes	maximal, complemented
4	#19151716	C₂	1	2	yes	maximal, complemented
5	#494×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668060	C₂²	1	4	yes	—
7	#15668044	—	2	8	no	—
8	#15668043	C₂²	1	8	yes	—
9	#15668035	—	2	4	no	—
10	#15668034	C₂²	1	2	yes	—
11	Q₁₆×D₈₉	—	2	4	no	—
12	#15668018	C₂²	1	4	yes	—
13	#15667948	C₂²	1	2	yes	—
14	#15667940	C₂²	1	4	yes	—
15	#15667931	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₈₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#13083575	—	2	8	no	—
20	#13083574	—	4	4	no	—
21	#13083574	C₂³	1	4	yes	—
22	Q₈×D₈₉	—	2	8	no	—
23	Dic₃₅₆	D₄	1	2	yes	—
24	#13083564	—	2	8	no	—
25	#13083563	D₄	1	4	yes	—
26	#13083561	—	2	8	no	—
27	D₇₁₂	D₄	1	2	yes	—
28	#13083558	C₂³	1	4	yes	—
29	#13083557	—	2	4	no	—
30	C₈×D₈₉	—	2	4	no	—
31	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
32	Q₁₆×C₈₉	—	2	4	no	—
33	Q₁₆×C₈₉	—	2	4	no	—
34	C₂×C₈×C₈₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₈₉	—	4	2	no	—
36	C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	Q₈×C₈₉	—	2	8	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₈₉	—	2	4	no	—
42	D₃₅₆	—	4	4	no	—
43	D₃₅₆	C₂×D₄	1	4	yes	—
44	C₄×D₈₉	—	2	4	no	—
45	C₄×D₈₉	—	4	8	no	—
46	Dic₁₇₈	—	4	4	no	—
47	Dic₁₇₈	C₂×D₄	1	4	yes	—
48	C₈×C₈₉	C₂×D₄	1	2	yes	—
49	C₈×C₈₉	C₂×D₄	1	2	yes	—
50	#10623494	—	2	4	no	—
51	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₈₉	—	2	4	no	—
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	4	4	no	—
55	C₄×C₈₉	C₂²×D₄	1	2	yes	—
56	Dic₈₉	—	2	4	no	—
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	D₄²	1	1	yes	characteristic
59	D₈₉	—	4	4	no	—
60	#3293	—	89	1	no	maximal, Sylow
61	C₈₉	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	89	4	no	—
63	#494	D₈₉	1	1	yes	characteristic, complemented
64	#473	—	89	4	no	—
65	#456	—	89	4	no	—
66	#443	—	89	2	no	—
67	2₋¹⁺⁴	—	89	4	no	—
68	#137	—	89	8	no	—
69	#137	—	178	8	no	—
70	#135	—	89	2	no	—
71	#135	—	178	4	no	—
72	C₂×Q₁₆	—	178	4	no	—
73	C₂×Q₁₆	C₂×D₈₉	1	4	yes	—
74	#131	—	89	4	no	—
75	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
76	#108	—	89	2	no	—
77	#104	—	89	4	no	—
78	#101	—	89	4	no	—
79	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
80	#41	—	89	4	no	—
81	#41	—	356	4	no	—
82	#41	—	178	8	no	—
83	C₂×Q₈	—	178	8	no	—
84	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	89	2	no	—
88	Q₁₆	—	178	8	no	—
89	SD₁₆	—	89	4	no	—
90	SD₁₆	—	178	8	no	—
91	D₈	—	89	2	no	—
92	M₁₆	—	89	4	no	—
93	M₁₆	—	178	4	no	—
94	C₂×C₈	—	178	4	no	—
95	C₂×C₈	C₂²×D₈₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₈₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	89	4	no	—
100	Q₈	—	356	4	no	—
101	D₄	—	89	4	no	—
102	D₄	—	356	4	no	—
103	D₄	—	178	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	356	8	no	—
107	C₂×C₄	—	178	4	no	—
108	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
109	C₈	—	178	4	no	—
110	C₈	D₄×D₈₉	1	2	yes	—
111	C₈	D₄×D₈₉	1	2	yes	—
112	C₂²	—	178	4	no	—
113	C₂²	#15668057	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	178	4	no	—
117	C₄	C₂×D₄×D₈₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	356	4	no	—
120	C₂	#19152233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152411

C₂

yes

maximal, complemented

#19151870

C₂

yes

maximal, complemented

#19151780

C₂

yes

maximal, complemented

#19151716

C₂

yes

maximal, complemented

#494×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668060

C₂²

yes

—

#15668044

—

#15668043

C₂²

yes

—

#15668035

—

#15668034

C₂²

yes

—

Q₁₆×D₈₉

—

#15668018

C₂²

yes

—

#15667948

C₂²

yes

—

#15667940

C₂²

yes

—

#15667931

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₈₉

—

Dic₃₅₆

D₄

yes

—

#13083564

—

#13083563

D₄

yes

—

#13083561

—

D₇₁₂

D₄

yes

—

#13083558

C₂³

yes

—

#13083557

—

C₈×D₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

—

D₃₅₆

C₂×D₄

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

—

#10623494

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

D₄²

yes

characteristic

D₈₉

—

#3293

—

maximal, Sylow

C₈₉

#3293

yes

characteristic, complemented, Sylow

#578

—

#494

D₈₉

yes

characteristic, complemented

#473

—

#456

—

#443

—

2₋¹⁺⁴

—

#137

—

#137

—

178

—

#135

—

#135

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

—

#131

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₈₉

yes

characteristic

#41

—

#41

—

356

—

#41

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²×D₈₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

178

—

SD₁₆

—

SD₁₆

—

178

—

D₈

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₈₉

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

356

—

101

D₄

—

102

D₄

—

356

—

103

D₄

—

178

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

178

—

108

C₂×C₄

C₂³×D₈₉

yes

characteristic

109

C₈

—

178

—

110

C₈

D₄×D₈₉

yes

—

111

C₈

D₄×D₈₉

yes

—

112

C₂²

—

178

—

113

C₂²

#15668057

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

178

—

117

C₄

C₂×D₄×D₈₉

yes

—

118

C₂

—

119

C₂

—

356

—

120

C₂

#19152233

yes

characteristic

	subgroup	G/H
center	C₂	#19152233
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#494×C₈₉	C₂
socle	C₂×C₈₉	D₄²
hypercenter	#494	D₈₉
hypocenter	C₈₉	#3293
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#19152233

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#494×C₈₉

C₂

socle

C₂×C₈₉

D₄²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #19152411	C₂	semidirect
2. #19151870	C₂	semidirect
3. #19151780	C₂	semidirect
4. #19151716	C₂	semidirect
5. #494×C₈₉	C₂	semidirect
61. C₈₉	#3293	semidirect
63. #494	D₈₉	semidirect

H ≅ G/N

type

1. #19152411

C₂

semidirect

2. #19151870

C₂

semidirect

3. #19151780

C₂

semidirect

4. #19151716

C₂

semidirect

5. #494×C₈₉

C₂

semidirect

61. C₈₉

#3293

semidirect

63. #494

D₈₉

semidirect

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

subgroup	index
#19152411	2
#19151870	2
#19151780	2
#19151716	2
#494×C₈₉	2
#3293	89

subgroup

index

#494×C₈₉

Subgroups — #26032056

Conjugacy classes of subgroups

Complements

Maximal subgroups