Group #26028615 — subgroups

Subgroups — #26028615

Order 11392.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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×#15668004	C₂	1	2	yes	maximal
2	#19151496	C₂	1	8	yes	maximal
3	#19151468	C₂	1	2	yes	maximal
4	#19151464	C₂	1	2	yes	maximal
5	C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668004	—	2	8	no	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	8	yes	—
9	C₂×#13083553	C₂²	1	8	yes	—
10	#15667919	C₂²	1	8	yes	—
11	C₂×C₄²×C₈₉	C₂²	1	4	yes	—
12	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₉	C₂³	1	8	yes	—
16	C₄⋊Dic₈₉	—	2	8	no	—
17	C₄⋊Dic₈₉	—	2	16	no	—
18	#13083553	—	2	16	no	—
19	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
20	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
23	C₄²×C₈₉	—	2	8	no	—
24	C₄²×C₈₉	D₄	1	4	yes	—
25	C₄²×C₈₉	Q₈	1	4	yes	—
26	C₄²×C₈₉	Q₈	1	4	yes	—
27	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₈₉	—	2	4	no	—
29	C₂×C₄×C₈₉	—	2	4	no	—
30	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₈₉	#41	1	8	yes	—
37	C₂×Dic₈₉	—	2	8	no	—
38	C₂×Dic₈₉	—	2	16	no	—
39	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
40	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₈₉	C₂×#41	1	4	yes	—
43	C₄×C₈₉	—	2	4	no	—
44	C₄×C₈₉	—	2	4	no	—
45	C₄×C₈₉	#125	1	8	yes	complemented
46	C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
47	Dic₈₉	—	4	8	no	—
48	C₂×C₈₉	C₂×#125	1	2	yes	—
49	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₈₉	#532	1	2	yes	—
51	C₂×C₈₉	#533	1	2	yes	—
52	#2859	—	89	1	no	maximal, Sylow
53	C₈₉	#2859	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	—	89	2	no	—
55	#389	—	89	8	no	—
56	#384	—	89	2	no	—
57	#382	—	89	2	no	—
58	C₄³	D₈₉	1	1	yes	characteristic
59	#125	—	178	8	no	—
60	C₂×C₄⋊C₄	—	89	4	no	—
61	C₂×C₄⋊C₄	—	89	8	no	—
62	C₂×C₄⋊C₄	—	89	8	no	—
63	C₂×C₄²	C₂×D₈₉	1	4	yes	—
64	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
67	#95	—	89	8	no	—
68	C₂²×C₄	—	89	8	no	—
69	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
70	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
71	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
73	C₄⋊C₄	—	178	16	no	—
74	C₄⋊C₄	—	178	8	no	—
75	C₄⋊C₄	—	178	16	no	—
76	C₄²	—	2	8	no	—
77	C₄²	Dic₁₇₈	1	4	yes	—
78	C₄²	Dic₁₇₈	1	4	yes	—
79	C₄²	D₃₅₆	1	4	yes	—
80	C₂³	C₂³×D₈₉	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	178	8	no	—
84	C₂×C₄	—	178	16	no	—
85	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
86	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
87	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
88	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
89	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
90	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
91	C₂×C₄	#13083574	1	8	yes	—
92	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
93	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
94	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
95	C₂²	C₂×#13083574	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	356	8	no	—
99	C₄	#15668004	1	8	yes	complemented
100	C₄	#15668012	1	4	yes	—
101	C₂	C₂×#15668004	1	2	yes	—
102	C₂	C₂×#15668012	1	1	yes	characteristic
103	C₂	#19152148	1	2	yes	—
104	C₂	#19152155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal

#19151496

C₂

yes

maximal

#19151468

C₂

yes

maximal

#19151464

C₂

yes

maximal

C₄³×C₈₉

C₂

yes

maximal, characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

Q₈

yes

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#125

yes

complemented

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

C₂×C₈₉

C₂×#125

yes

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₉

#532

yes

—

C₂×C₈₉

#533

yes

—

#2859

—

maximal, Sylow

C₈₉

#2859

yes

characteristic, complemented, Sylow

—

—

—

—

C₄³

D₈₉

yes

characteristic

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

Dic₁₇₈

yes

—

C₄²

Dic₁₇₈

yes

—

C₄²

D₃₅₆

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

#13083574

yes

—

C₂²

C₂²×D₃₅₆

yes

characteristic

C₂²

C₂²×Dic₁₇₈

yes

characteristic

C₂²

C₂²×Dic₁₇₈

yes

characteristic

C₂²

C₂×#13083574

yes

—

C₄

—

C₄

—

C₄

—

356

—

C₄

#15668004

yes

complemented

100

C₄

#15668012

yes

—

101

C₂

C₂×#15668004

yes

—

102

C₂

C₂×#15668012

yes

characteristic

103

C₂

#19152148

yes

—

104

C₂

#19152155

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₄³×C₈₉	C₂
hypercenter	C₄³	D₈₉
hypocenter	C₈₉	#2859
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₄³×C₈₉

C₂

hypercenter

C₄³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
45. C₄×C₈₉	#125	semidirect
53. C₈₉	#2859	semidirect
99. C₄	#15668004	semidirect

H ≅ G/N

type

45. C₄×C₈₉

#125

semidirect

53. C₈₉

#2859

semidirect

99. C₄

#15668004

semidirect

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

subgroup	index
C₂×#15668004	2
#19151496	2
#19151468	2
#19151464	2
C₄³×C₈₉	2
#2859	89

subgroup

index

C₄³×C₈₉

Subgroups — #26028615

Conjugacy classes of subgroups

Complements

Maximal subgroups