Group #26028634 — subgroups

Subgroups — #26028634

Order 11392.

408 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19151504	C₂	1	3	yes	maximal
2	#19151496	C₂	1	3	yes	maximal
3	#19151482	C₂	1	6	yes	maximal
4	#19151466	C₂	1	2	yes	maximal
5	C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
8	C₂×#13083553	C₂²	1	6	yes	—
9	#15667919	C₂²	1	6	yes	—
10	#15667919	C₂²	1	6	yes	—
11	#15667919	C₂²	1	6	yes	—
12	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₈	—	2	6	no	—
16	C₂²×Dic₈₉	C₂³	1	2	yes	—
17	C₂²×Dic₈₉	C₂³	1	6	yes	—
18	C₄⋊Dic₈₉	—	2	6	no	—
19	#13083553	—	2	12	no	—
20	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
21	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
22	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	C₄²×C₈₉	—	2	6	no	—
24	C₄²×C₈₉	—	2	6	no	—
25	C₄²×C₈₉	Q₈	1	4	yes	complemented
26	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
27	C₂×C₄×C₈₉	—	2	3	no	—
28	C₂×C₄×C₈₉	—	2	3	no	—
29	C₂×C₄×C₈₉	—	2	3	no	—
30	C₂×C₄×C₈₉	—	2	3	no	—
31	C₂×C₄×C₈₉	C₂×Q₈	1	6	yes	—
32	C₂×C₄×C₈₉	#41	1	6	yes	—
33	C₂×C₄×C₈₉	#41	1	6	yes	—
34	C₂×Dic₈₉	—	2	12	no	—
35	C₂×Dic₈₉	—	2	6	no	—
36	C₂×Dic₈₉	—	2	6	no	—
37	Dic₁₇₈	—	4	4	no	—
38	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₈₉	C₂×#41	1	3	yes	—
40	C₂²×C₈₉	C₂×#41	1	3	yes	—
41	C₄×C₈₉	—	2	6	no	—
42	C₄×C₈₉	—	2	6	no	—
43	C₄×C₈₉	#126	1	4	yes	—
44	Dic₈₉	—	4	2	no	—
45	Dic₈₉	—	4	6	no	—
46	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
47	C₂×C₈₉	#529	1	3	yes	—
48	C₂×C₈₉	#533	1	3	yes	—
49	#2863	—	89	1	no	maximal, Sylow
50	C₈₉	#2863	1	1	yes	characteristic, complemented, Sylow
51	#391	—	89	3	no	—
52	#389	—	89	3	no	—
53	#387	—	89	6	no	—
54	#383	—	89	2	no	—
55	C₄³	D₈₉	1	1	yes	characteristic
56	C₂²×Q₈	—	89	1	no	—
57	C₂×C₄⋊C₄	—	89	6	no	—
58	C₂×C₄⋊C₄	—	89	3	no	—
59	C₂×C₄²	C₂×D₈₉	1	3	yes	—
60	C₂×C₄²	C₂×D₈₉	1	3	yes	—
61	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
62	#95	—	89	6	no	—
63	#95	—	89	6	no	—
64	#95	—	89	6	no	—
65	C₂×Q₈	—	178	6	no	—
66	C₂²×C₄	—	89	2	no	—
67	C₂²×C₄	—	89	6	no	—
68	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
69	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
70	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
71	C₄⋊C₄	—	178	12	no	—
72	C₄⋊C₄	—	178	6	no	—
73	C₄²	—	2	6	no	—
74	C₄²	—	2	6	no	—
75	C₄²	Dic₁₇₈	1	4	yes	complemented
76	C₂³	C₂³×D₈₉	1	1	yes	characteristic
77	Q₈	—	356	4	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	178	12	no	—
83	C₂×C₄	—	178	6	no	—
84	C₂×C₄	—	178	6	no	—
85	C₂×C₄	C₂×Dic₁₇₈	1	6	yes	—
86	C₂×C₄	#13083574	1	6	yes	—
87	C₂×C₄	#13083574	1	6	yes	—
88	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
89	C₂²	C₂×#13083574	1	3	yes	—
90	C₂²	C₂×#13083574	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	2	6	no	—
93	C₄	—	356	6	no	—
94	C₄	—	356	2	no	—
95	C₄	#15668006	1	4	yes	—
96	C₂	C₂×#15668006	1	1	yes	characteristic
97	C₂	#19152132	1	3	yes	—
98	C₂	#19152155	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151504

C₂

yes

maximal

#19151496

C₂

yes

maximal

#19151482

C₂

yes

maximal

#19151466

C₂

yes

maximal

C₄³×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

Q₈

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#126

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#126

yes

characteristic

C₂×C₈₉

#529

yes

—

C₂×C₈₉

#533

yes

—

#2863

—

maximal, Sylow

C₈₉

#2863

yes

characteristic, complemented, Sylow

—

—

—

—

C₄³

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

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

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

C₄²

Dic₁₇₈

yes

complemented

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

#13083574

yes

—

C₂×C₄

#13083574

yes

—

C₂²

C₂²×Dic₁₇₈

yes

characteristic

C₂²

C₂×#13083574

yes

—

C₂²

C₂×#13083574

yes

—

C₄

—

C₄

—

C₄

—

356

—

C₄

—

356

—

C₄

#15668006

yes

—

C₂

C₂×#15668006

yes

characteristic

C₂

#19152132

yes

—

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	#2863
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
25. C₄²×C₈₉	Q₈	semidirect
50. C₈₉	#2863	semidirect
75. C₄²	Dic₁₇₈	semidirect

H ≅ G/N

type

25. C₄²×C₈₉

Q₈

semidirect

50. C₈₉

#2863

semidirect

75. C₄²

Dic₁₇₈

semidirect

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

subgroup	index
#19151504	2
#19151496	2
#19151482	2
#19151466	2
C₄³×C₈₉	2
#2863	89

subgroup

index

C₄³×C₈₉

Subgroups — #26028634

Conjugacy classes of subgroups

Complements

Maximal subgroups