Group #26026203 — subgroups

Subgroups — #26026203

Order 11392.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151537	C₂	1	8	yes	maximal
3	#19151503	C₂	1	4	yes	maximal
4	#19151444	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	#15668012	—	2	4	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
9	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
10	#15667919	C₂²	1	4	yes	—
11	#15667919	C₂²	1	16	yes	—
12	#125×C₈₉	—	2	4	no	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	2	4	no	—
17	C₂×Dic₁₇₈	—	2	8	no	—
18	C₂²×Dic₈₉	C₂³	1	4	yes	—
19	C₂²×Dic₈₉	C₂³	1	4	yes	—
20	C₄⋊Dic₈₉	—	2	8	no	—
21	C₄×Dic₈₉	—	2	4	no	—
22	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
23	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₈₉	—	2	8	no	—
26	C₄⋊C₄×C₈₉	—	2	4	no	—
27	C₄²×C₈₉	D₄	1	4	yes	—
28	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₈₉	—	2	4	no	—
30	C₂×C₄×C₈₉	—	2	4	no	—
31	C₂×C₄×C₈₉	—	2	8	no	—
32	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₈₉	#41	1	4	yes	—
35	C₂×Dic₈₉	—	2	4	no	—
36	C₂×Dic₈₉	—	2	4	no	—
37	C₂×Dic₈₉	—	2	8	no	—
38	C₂×Dic₈₉	—	2	8	no	—
39	Dic₁₇₈	—	4	8	no	—
40	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₈₉	C₂×#41	1	2	yes	—
42	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
43	C₄×C₈₉	—	2	4	no	—
44	C₄×C₈₉	—	4	4	no	—
45	C₄×C₈₉	#124	1	4	yes	—
46	Dic₈₉	—	4	4	no	—
47	Dic₈₉	—	4	4	no	—
48	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
49	C₂×C₈₉	#536	1	2	yes	—
50	C₂×C₈₉	#541	1	4	yes	—
51	#2642	—	89	1	no	maximal, Sylow
52	C₈₉	#2642	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₄⋊Q₈	—	89	1	no	—
54	C₂×#125	D₈₉	1	1	yes	characteristic
55	#400	—	89	8	no	—
56	#391	—	89	4	no	—
57	#376	—	89	1	no	—
58	C₂²×Q₈	—	89	2	no	—
59	C₄⋊Q₈	—	178	4	no	—
60	#125	—	2	4	no	—
61	C₂×C₄⋊C₄	—	89	4	no	—
62	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
63	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
64	C₂×C₄²	—	89	2	no	—
65	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
66	#95	—	89	4	no	—
67	#95	—	89	16	no	—
68	C₂×Q₈	—	178	4	no	—
69	C₂×Q₈	—	178	8	no	—
70	C₂²×C₄	—	89	4	no	—
71	C₂²×C₄	—	89	4	no	—
72	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
73	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	178	8	no	—
78	C₄²	—	178	4	no	—
79	C₄²	C₂²⋊D₈₉	1	4	yes	—
80	C₂³	C₂³×D₈₉	1	1	yes	characteristic
81	Q₈	—	356	8	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	—	178	4	no	—
86	C₂×C₄	—	178	4	no	—
87	C₂×C₄	—	178	8	no	—
88	C₂×C₄	—	178	8	no	—
89	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
90	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
91	C₂×C₄	#13083575	1	4	yes	—
92	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
93	C₂²	C₂×#13083575	1	2	yes	—
94	C₂²	#15668059	1	4	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	4	4	no	—
97	C₄	—	356	4	no	—
98	C₄	—	356	4	no	—
99	C₄	#15668001	1	4	yes	—
100	C₂	C₂×#15668001	1	1	yes	characteristic
101	C₂	#19152169	1	2	yes	—
102	C₂	#19152211	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

#19151537

C₂

yes

maximal

#19151503

C₂

yes

maximal

#19151444

C₂

yes

maximal, characteristic

C₂×#125×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#536

yes

—

C₂×C₈₉

#541

yes

—

#2642

—

maximal, Sylow

C₈₉

#2642

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

D₈₉

yes

—

—

—

C₂²×Q₈

—

C₄⋊Q₈

—

178

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²⋊D₈₉

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

C₂×C₄

#13083575

yes

—

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

C₂²

C₂×#13083575

yes

—

C₂²

#15668059

yes

—

C₄

—

C₄

—

C₄

—

356

—

C₄

—

356

—

C₄

#15668001

yes

—

100

C₂

C₂×#15668001

yes

characteristic

101

C₂

#19152169

yes

—

102

C₂

#19152211

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#125×C₈₉	C₂
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	#2642
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#125×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
52. C₈₉	#2642	semidirect

H ≅ G/N

type

52. C₈₉

#2642

semidirect

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

subgroup	index
C₂×#15668012	2
#19151537	2
#19151503	2
#19151444	2
C₂×#125×C₈₉	2
#2642	89

subgroup

index

C₂×#125×C₈₉

Subgroups — #26026203

Conjugacy classes of subgroups

Complements

Maximal subgroups