Group #26020425 — subgroups

Subgroups — #26020425

Order 11392.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667947	C₂	1	2	yes	maximal
3	#19151621	C₂	1	2	yes	maximal
4	C₂×#15667922	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667948	—	2	2	no	—
10	#15667947	—	2	4	no	—
11	#15667941	—	2	4	no	—
12	#15667922	C₄	1	4	yes	—
13	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#108×C₈₉	—	2	2	no	—
16	C₂²×Dic₈₉	Q₈	1	2	yes	—
17	#13083558	—	2	2	no	—
18	#13083558	—	2	2	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₂×#10623494	C₂×C₄	1	2	yes	—
21	C₂×#10623494	D₄	1	2	yes	—
22	C₂×#10623494	Q₈	1	2	yes	—
23	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₈₉	D₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₈₉	—	2	2	no	—
27	M₁₆×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₈₉	D₄	1	2	yes	—
31	C₂×C₈×C₈₉	Q₈	1	2	yes	—
32	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₈₉	—	4	2	no	—
40	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
41	C₈×C₈₉	—	4	2	no	—
42	C₈×C₈₉	C₄⋊C₄	1	4	yes	—
43	#10623494	—	4	2	no	—
44	#10623494	C₄⋊C₄	1	4	yes	—
45	C₂²×C₈₉	—	2	2	no	—
46	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈₉	#127	1	1	yes	characteristic
52	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₈₉	—	2	4	no	—
54	C₂×C₈₉	—	2	1	no	—
55	C₂×C₈₉	—	2	1	no	—
56	C₂×C₈₉	#384	1	1	yes	characteristic
57	C₂×C₈₉	#430	1	2	yes	—
58	#1849	—	89	1	no	maximal, Sylow
59	C₈₉	#1849	1	1	yes	characteristic, complemented, Sylow
60	C₂×#108	—	89	2	no	—
61	C₂×#108	—	89	1	no	—
62	C₂×#108	D₈₉	1	1	yes	characteristic
63	#423	—	89	2	no	—
64	C₂×#97	—	89	1	no	—
65	C₂×M₁₆	—	89	2	no	—
66	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
67	C₂²×C₈	—	89	1	no	—
68	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
69	C₂×C₄²	—	89	1	no	—
70	#108	—	2	2	no	—
71	#108	—	178	4	no	—
72	#108	—	178	2	no	—
73	C₄⋊C₈	—	178	4	no	—
74	#97	—	89	4	no	—
75	C₂²×C₄	—	89	2	no	—
76	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	178	2	no	—
80	M₁₆	—	178	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	89	2	no	—
83	C₂×C₈	—	89	2	no	—
84	C₂×C₈	—	89	2	no	—
85	C₂×C₈	—	178	2	no	—
86	C₂×C₈	Dic₁₇₈	1	2	yes	—
87	C₂×C₈	C₄×D₈₉	1	2	yes	—
88	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
89	C₄²	—	89	2	no	—
90	C₄²	—	89	2	no	—
91	C₂³	Q₈×D₈₉	1	1	yes	characteristic
92	C₂×C₄	—	89	4	no	—
93	C₂×C₄	—	356	2	no	—
94	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
95	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
96	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	89	4	no	—
102	C₈	—	356	2	no	—
103	C₈	#13083553	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
106	C₂²	#15668010	1	1	yes	characteristic
107	C₂²	#15668011	1	1	yes	characteristic
108	C₄	—	178	4	no	—
109	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
110	C₄	C₂×#13083553	1	1	yes	characteristic
111	C₄	#15668008	1	1	yes	characteristic
112	C₄	#15668011	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#19151467	1	1	yes	characteristic
116	C₂	#19151645	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667948

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal

#19151621

C₂

yes

maximal

C₂×#15667922

C₂

yes

maximal, characteristic

C₂×#108×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#108×C₈₉

—

C₂²×Dic₈₉

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

Q₈

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₄⋊C₄

yes

—

#10623494

—

#10623494

C₄⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

#127

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#384

yes

characteristic

C₂×C₈₉

#430

yes

—

#1849

—

maximal, Sylow

C₈₉

#1849

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

C₂×#108

D₈₉

yes

characteristic

#423

—

C₂×#97

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

178

—

#108

—

178

—

C₄⋊C₈

—

178

—

#97

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

Dic₁₇₈

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

D₄×D₈₉

yes

characteristic

C₂×C₄

D₄×D₈₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂×C₄

Q₈×D₈₉

yes

characteristic

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

356

—

103

C₈

#13083553

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

106

C₂²

#15668010

yes

characteristic

107

C₂²

#15668011

yes

characteristic

108

C₄

—

178

—

109

C₄

C₄⋊C₄×D₈₉

yes

characteristic

110

C₄

C₂×#13083553

yes

characteristic

111

C₄

#15668008

yes

characteristic

112

C₄

#15668011

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#19151467

yes

characteristic

116

C₂

#19151645

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#108×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#108	D₈₉
hypocenter	C₈₉	#1849
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#108×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₄

prime radical

C₂³×C₈₉

C₂×Q₈

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
59. C₈₉	#1849	semidirect

H ≅ G/N

type

59. C₈₉

#1849

semidirect

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

subgroup	index
C₂×#15667948	2
C₂×#15667947	2
#19151621	2
C₂×#15667922	2
C₂×#108×C₈₉	2
#1849	89

subgroup

index

C₂×#108×C₈₉

Subgroups — #26020425

Conjugacy classes of subgroups

Complements

Maximal subgroups