Group #26028861 — subgroups

Subgroups — #26028861

Order 11392.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#19152365	C₂	1	1	yes	maximal, characteristic
2	C₂×#15668019	C₂	1	1	yes	maximal, characteristic
3	#19151646	C₂	1	6	yes	maximal, complemented
4	C₂×#15667947	C₂	1	6	yes	maximal, complemented
5	#568×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668019	—	2	4	no	—
7	#15668019	C₄	1	8	yes	—
8	C₂×#13083558	C₂²	1	6	yes	—
9	C₂×#13083558	C₂²	1	3	yes	—
10	C₂²×#10623494	C₂²	1	3	yes	—
11	#15667947	C₂²	1	4	yes	—
12	#15667947	C₂²	1	12	yes	—
13	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
14	#131×C₈₉	—	2	4	no	—
15	C₂×M₁₆×C₈₉	C₂²	1	6	yes	—
16	#13083558	—	2	6	no	—
17	#13083558	C₂×C₄	1	12	yes	—
18	#13083558	C₂³	1	4	yes	—
19	C₂×#10623494	—	2	6	no	—
20	C₂×#10623494	C₂×C₄	1	12	yes	—
21	C₂×#10623494	C₂³	1	1	yes	characteristic
22	C₂×#10623494	C₂³	1	3	yes	—
23	#41×C₈₉	C₂×C₄	1	4	yes	—
24	#41×C₈₉	D₄	1	4	yes	—
25	C₂×Q₈×C₈₉	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₈₉	Q₈	1	3	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
28	M₁₆×C₈₉	—	2	6	no	—
29	M₁₆×C₈₉	C₂³	1	4	yes	—
30	C₂×C₈×C₈₉	—	2	6	no	—
31	C₂³×C₈₉	C₂×Q₈	1	3	yes	—
32	Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
33	D₄×C₈₉	C₄⋊C₄	1	12	yes	—
34	C₂×C₄×C₈₉	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₈₉	C₂×Q₈	1	3	yes	—
37	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
38	C₈×C₈₉	—	2	4	no	—
39	#10623494	—	2	4	no	—
40	#10623494	C₂²×C₄	1	2	yes	—
41	#10623494	C₂²×C₄	1	6	yes	—
42	C₂²×C₈₉	—	4	3	no	—
43	C₂²×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₈₉	—	2	1	no	—
49	C₂×C₈₉	—	2	6	no	—
50	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	#2916	—	89	1	no	maximal, Sylow
52	C₈₉	#2916	1	1	yes	characteristic, complemented, Sylow
53	#568	—	89	1	no	—
54	#568	D₈₉	1	1	yes	characteristic
55	C₂×#131	—	89	1	no	—
56	#430	—	89	6	no	—
57	C₂×#108	—	89	6	no	—
58	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	89	8	no	—
61	#131	—	178	4	no	—
62	C₂×M₁₆	—	89	3	no	—
63	C₂×M₁₆	—	89	6	no	—
64	C₂×M₁₆	C₂×D₈₉	1	6	yes	—
65	C₂²×C₈	—	89	3	no	—
66	#108	—	89	4	no	—
67	#108	—	89	12	no	—
68	#41	C₄×D₈₉	1	4	yes	—
69	#41	C₂²⋊D₈₉	1	4	yes	—
70	C₂×Q₈	Dic₁₇₈	1	1	yes	characteristic
71	C₂×D₄	Dic₁₇₈	1	3	yes	—
72	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	89	12	no	—
75	M₁₆	—	89	4	no	—
76	M₁₆	—	178	6	no	—
77	M₁₆	C₂²×D₈₉	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	89	12	no	—
80	C₂×C₈	—	89	1	no	—
81	C₂×C₈	—	89	3	no	—
82	C₂×C₈	—	178	6	no	—
83	C₂³	C₂×Dic₁₇₈	1	3	yes	—
84	Q₈	#13083553	1	4	yes	—
85	D₄	#13083553	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₈₉	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₈₉	1	6	yes	—
88	C₂×C₄	C₂×Dic₁₇₈	1	3	yes	—
89	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	89	2	no	—
92	C₈	—	89	6	no	—
93	C₈	—	178	4	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×#13083553	1	6	yes	—
96	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
97	C₄	C₂×#13083553	1	6	yes	—
98	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×#13083553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152365

C₂

yes

maximal, characteristic

C₂×#15668019

C₂

yes

maximal, characteristic

#19151646

C₂

yes

maximal, complemented

C₂×#15667947

C₂

yes

maximal, complemented

#568×C₈₉

C₂

yes

maximal, characteristic

#15668019

—

#15668019

C₄

yes

—

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15667947

C₂²

yes

—

#15667947

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

characteristic

#131×C₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

—

#13083558

—

#13083558

C₂×C₄

yes

—

#13083558

C₂³

yes

—

C₂×#10623494

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

—

#41×C₈₉

C₂×C₄

yes

—

#41×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

Q₈

yes

characteristic

C₂×D₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

—

Q₈×C₈₉

C₄⋊C₄

yes

—

D₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₈×C₈₉

—

#10623494

—

#10623494

C₂²×C₄

yes

—

#10623494

C₂²×C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₄⋊C₄

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#568

—

#568

D₈₉

yes

—

—

—

C₂×#41

C₂×D₈₉

yes

—

—

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

#108

—

#108

—

#41

C₄×D₈₉

yes

—

#41

C₂²⋊D₈₉

yes

—

C₂×Q₈

Dic₁₇₈

yes

characteristic

C₂×D₄

Dic₁₇₈

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

C₂²×D₈₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂³

C₂×Dic₁₇₈

yes

—

Q₈

#13083553

yes

—

D₄

#13083553

yes

—

C₂×C₄

C₂×C₄×D₈₉

yes

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

178

—

C₂²

—

C₂²

C₂×#13083553

yes

—

C₂²

C₂²×Dic₁₇₈

yes

characteristic

C₄

C₂×#13083553

yes

—

C₄

C₂²×C₄×D₈₉

yes

characteristic

C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×#13083553

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#568×C₈₉	C₂
socle	C₂×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#568	D₈₉
hypocenter	C₈₉	#2916
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₈₉

commutator

C₄×C₈₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#568×C₈₉

C₂

socle

C₂×C₈₉

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₈₉

C₂²

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #19151646	C₂	semidirect
4. C₂×#15667947	C₂	semidirect
52. C₈₉	#2916	semidirect

H ≅ G/N

type

3. #19151646

C₂

semidirect

4. C₂×#15667947

C₂

semidirect

52. C₈₉

#2916

semidirect

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

subgroup	index
#19152365	2
C₂×#15668019	2
#19151646	2
C₂×#15667947	2
#568×C₈₉	2
#2916	89

subgroup

index

#568×C₈₉

Subgroups — #26028861

Conjugacy classes of subgroups

Complements

Maximal subgroups