Group #26028521 — subgroups

Subgroups — #26028521

Order 11392.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#19151537	C₂	1	3	yes	maximal
2	#19151531	C₂	1	6	yes	maximal
3	#19151518	C₂	1	3	yes	maximal
4	#19151466	C₂	1	2	yes	maximal
5	#384×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	3	yes	—
12	#15667919	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	4	3	no	—
17	C₂²×Dic₈₉	C₂³	1	2	yes	—
18	C₂²×Dic₈₉	C₂³	1	6	yes	—
19	C₄⋊Dic₈₉	—	4	3	no	—
20	#13083553	—	4	6	no	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
23	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
24	C₄⋊C₄×C₈₉	—	4	3	no	—
25	C₄⋊C₄×C₈₉	—	4	3	no	—
26	C₄²×C₈₉	Q₈	1	4	yes	complemented
27	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₈₉	—	2	3	no	—
29	C₂×C₄×C₈₉	—	2	3	no	—
30	C₂×C₄×C₈₉	—	2	6	no	—
31	C₂×C₄×C₈₉	—	2	3	no	—
32	C₂×C₄×C₈₉	—	2	3	no	—
33	C₂×C₄×C₈₉	C₂×Q₈	1	6	yes	—
34	C₂×Dic₈₉	—	2	6	no	—
35	C₂×Dic₈₉	—	2	6	no	—
36	C₂×Dic₈₉	—	2	12	no	—
37	Dic₁₇₈	—	4	4	no	—
38	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₈₉	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₈₉	—	2	6	no	—
42	C₄×C₈₉	—	4	1	no	—
43	C₄×C₈₉	—	4	3	no	—
44	Dic₈₉	—	4	2	no	—
45	Dic₈₉	—	4	6	no	—
46	C₂×C₈₉	#544	1	3	yes	—
47	C₂×C₈₉	#560	1	1	yes	characteristic
48	C₂×C₈₉	#563	1	3	yes	—
49	#2843	—	89	1	no	maximal, Sylow
50	C₈₉	#2843	1	1	yes	characteristic, complemented, Sylow
51	#400	—	89	3	no	—
52	#398	—	89	6	no	—
53	#395	—	89	3	no	—
54	#384	D₈₉	1	1	yes	characteristic
55	#383	—	89	2	no	—
56	C₂²×Q₈	—	89	1	no	—
57	C₂×C₄⋊C₄	—	89	6	no	—
58	C₂×C₄⋊C₄	—	89	3	no	—
59	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
60	C₂×C₄⋊C₄	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	3	no	—
65	#95	—	89	3	no	—
66	C₂×Q₈	—	356	3	no	—
67	C₂²×C₄	—	89	6	no	—
68	C₂²×C₄	—	89	2	no	—
69	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
71	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	356	6	no	—
75	C₄⋊C₄	—	356	3	no	—
76	C₄²	Dic₁₇₈	1	4	yes	complemented
77	C₂³	C₂³×D₈₉	1	1	yes	characteristic
78	Q₈	—	356	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	178	6	no	—
85	C₂×C₄	—	178	6	no	—
86	C₂×C₄	—	178	12	no	—
87	C₂×C₄	C₂×Dic₁₇₈	1	6	yes	—
88	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
89	C₂²	#15668057	1	3	yes	—
90	C₂²	#15668060	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	356	2	no	—
95	C₄	—	356	6	no	—
96	C₂	#19152229	1	3	yes	—
97	C₂	#19152335	1	1	yes	characteristic
98	C₂	#19152347	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151537

C₂

yes

maximal

#19151531

C₂

yes

maximal

#19151518

C₂

yes

maximal

#19151466

C₂

yes

maximal

#384×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

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×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

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×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×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#544

yes

—

C₂×C₈₉

#560

yes

characteristic

C₂×C₈₉

#563

yes

—

#2843

—

maximal, Sylow

C₈₉

#2843

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#383

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

Dic₁₇₈

yes

complemented

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂²

C₂²×Dic₁₇₈

yes

characteristic

C₂²

#15668057

yes

—

C₂²

#15668060

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

356

—

C₄

—

356

—

C₂

#19152229

yes

—

C₂

#19152335

yes

characteristic

C₂

#19152347

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2843
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₈₉	Q₈	semidirect
50. C₈₉	#2843	semidirect
76. C₄²	Dic₁₇₈	semidirect

H ≅ G/N

type

26. C₄²×C₈₉

Q₈

semidirect

50. C₈₉

#2843

semidirect

76. C₄²

Dic₁₇₈

semidirect

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

subgroup	index
#19151537	2
#19151531	2
#19151518	2
#19151466	2
#384×C₈₉	2
#2843	89

subgroup

index

#384×C₈₉

Subgroups — #26028521

Conjugacy classes of subgroups

Complements

Maximal subgroups