Group #26018542 — subgroups

Subgroups — #26018542

Order 11392.

208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₄×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	#19151320	C₂	1	2	yes	maximal, complemented
4	#19151319	C₂	1	2	yes	maximal, complemented
5	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₄	1	2	yes	—
7	C₂²×#10623494	C₄	1	2	yes	—
8	#15667923	C₄	1	2	yes	—
9	C₄×#10623494	C₄	1	2	yes	—
10	C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄×#10623494	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	C₄	1	2	yes	—
15	C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	2	yes	—
17	#13083558	—	2	4	no	—
18	C₂×#10623494	—	2	4	no	—
19	C₂×#10623494	C₂×C₄	1	2	yes	—
20	C₂×#10623494	C₂×C₄	1	4	yes	—
21	C₂×#10623494	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	C₂×C₄	1	4	yes	—
26	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₂³×C₈₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₈₉	—	2	2	no	—
32	C₂×C₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₈₉	M₁₆	1	4	yes	—
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₈₉	—	2	4	no	—
41	#10623494	—	2	2	no	—
42	#10623494	—	2	4	no	—
43	#10623494	M₁₆	1	4	yes	complemented
44	C₂²×C₈₉	—	2	1	no	—
45	C₂²×C₈₉	—	2	1	no	—
46	C₂²×C₈₉	#97	1	2	yes	—
47	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₉	—	2	1	no	—
49	C₄×C₈₉	—	2	1	no	—
50	C₄×C₈₉	#97	1	2	yes	—
51	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
52	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
53	C₄×C₈₉	#131	1	2	yes	—
54	C₂×C₈₉	—	2	2	no	—
55	C₂×C₈₉	C₂×#97	1	1	yes	characteristic
56	C₂×C₈₉	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₈₉	#405	1	1	yes	characteristic
58	#1451	—	89	1	no	maximal, Sylow
59	C₈₉	#1451	1	1	yes	characteristic, complemented, Sylow
60	#431	D₈₉	1	1	yes	characteristic
61	C₄×M₁₆	—	89	1	no	—
62	C₂×C₄×C₈	—	89	1	no	—
63	#322	—	89	2	no	—
64	#322	—	89	2	no	—
65	C₂×M₁₆	—	89	2	no	—
66	C₂²×C₈	—	89	2	no	—
67	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
68	C₄⋊C₈	Dic₈₉	1	2	yes	—
69	C₂²⋊C₈	Dic₈₉	1	2	yes	—
70	#97	—	89	2	no	—
71	C₄×C₈	—	89	2	no	—
72	C₄×C₈	—	89	1	no	—
73	C₄×C₈	—	89	1	no	—
74	C₄×C₈	—	89	2	no	—
75	C₄×C₈	C₂×D₈₉	1	2	yes	—
76	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
79	M₁₆	—	178	4	no	—
80	C₂×C₈	—	89	2	no	—
81	C₂×C₈	—	89	2	no	—
82	C₂×C₈	—	89	4	no	—
83	C₂×C₈	—	178	4	no	—
84	C₂×C₈	C₂×Dic₈₉	1	4	yes	—
85	C₄²	C₄×D₈₉	1	1	yes	characteristic
86	C₄²	C₄×D₈₉	1	1	yes	characteristic
87	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
88	C₄²	C₂²×D₈₉	1	1	yes	characteristic
89	C₂³	C₄×Dic₈₉	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
95	C₂×C₄	#13083557	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	89	4	no	—
101	C₈	—	178	2	no	—
102	C₈	—	178	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#15667922	1	2	yes	—
106	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#15667922	1	2	yes	—
110	C₄	M₁₆×D₈₉	1	2	yes	—
111	C₄	C₂×#13083557	1	2	yes	—
112	C₄	#15668019	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#15667922	1	1	yes	characteristic
115	C₂	M₁₆×Dic₈₉	1	1	yes	characteristic
116	C₂	#19151554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13083558

C₂

yes

maximal, characteristic

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#19151320

C₂

yes

maximal, complemented

#19151319

C₂

yes

maximal, complemented

#431×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₄

yes

—

C₂²×#10623494

C₄

yes

—

#15667923

C₄

yes

—

C₄×#10623494

C₄

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₄

yes

—

C₂²⋊C₈×C₈₉

C₄

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

M₁₆

yes

complemented

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#97

yes

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#97

yes

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

#131

yes

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#97

yes

characteristic

C₂×C₈₉

C₄×M₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#431

D₈₉

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

Dic₈₉

yes

—

C₂²⋊C₈

Dic₈₉

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₄²

C₄×D₈₉

yes

characteristic

C₄²

C₄×D₈₉

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

C₄×Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₉

yes

characteristic

C₂×C₄

C₄×Dic₈₉

yes

characteristic

C₂×C₄

C₄×Dic₈₉

yes

characteristic

C₂×C₄

#13083557

yes

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

178

—

102

C₈

—

178

—

103

C₂²

—

104

C₂²

—

105

C₂²

#15667922

yes

—

106

C₂²

C₂×C₄×Dic₈₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#15667922

yes

—

110

C₄

M₁₆×D₈₉

yes

—

111

C₄

C₂×#13083557

yes

—

112

C₄

#15668019

yes

—

113

C₂

—

114

C₂

C₂×#15667922

yes

characteristic

115

C₂

M₁₆×Dic₈₉

yes

characteristic

116

C₂

#19151554

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂×C₄²
Fitting	#431×C₈₉	C₂
prime radical	C₂³×C₈₉	C₄²
cosocle	C₄²×C₈₉	C₂³
hypercenter	#431	D₈₉
hypocenter	C₈₉	#1451
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂×C₄²

Fitting

#431×C₈₉

C₂

prime radical

C₂³×C₈₉

C₄²

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #19151320	C₂	semidirect
4. #19151319	C₂	semidirect
43. #10623494	M₁₆	semidirect
59. C₈₉	#1451	semidirect

H ≅ G/N

type

3. #19151320

C₂

semidirect

4. #19151319

C₂

semidirect

43. #10623494

M₁₆

semidirect

59. C₈₉

#1451

semidirect

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

subgroup	index
C₄×#13083558	2
C₂×C₄×#10623494	2
#19151320	2
#19151319	2
#431×C₈₉	2
#1451	89

subgroup

index

#431×C₈₉

Subgroups — #26018542

Conjugacy classes of subgroups

Complements

Maximal subgroups