Group #26033258 — subgroups

Subgroups — #26033258

Order 11392.

348 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 141 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	#19152327	C₂	1	2	yes	maximal
2	#19151978	C₂	1	3	yes	maximal
3	#19151809	C₂	1	6	yes	maximal
4	C₄×Dic₃₅₆	C₂	1	3	yes	maximal
5	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	C₂²	1	3	yes	—
7	C₂×Dic₃₅₆	C₂²	1	6	yes	—
8	#15668012	C₂²	1	6	yes	—
9	#15668003	—	2	6	no	—
10	Q₈×Dic₈₉	—	2	2	no	—
11	C₄×Dic₁₇₈	C₂²	1	6	yes	—
12	#15667946	C₂²	1	1	yes	characteristic
13	#15667937	C₂²	1	6	yes	—
14	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
16	C₄×C₈×C₈₉	C₂²	1	3	yes	—
17	C₂×Dic₁₇₈	C₂³	1	6	yes	—
18	Dic₃₅₆	—	2	6	no	—
19	Dic₃₅₆	—	4	6	no	—
20	C₄⋊Dic₈₉	—	2	6	no	—
21	C₄⋊Dic₈₉	—	2	6	no	—
22	C₄⋊Dic₈₉	C₂³	1	2	yes	—
23	#13083553	—	2	6	no	—
24	C₄×Dic₈₉	—	2	6	no	—
25	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂³	1	3	yes	—
27	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	D₄	1	3	yes	—
29	C₄²×C₈₉	C₂³	1	3	yes	—
30	Q₈×C₈₉	Q₁₆	1	4	yes	complemented
31	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
34	C₂×Dic₈₉	—	2	2	no	—
35	C₂×Dic₈₉	—	2	6	no	—
36	Dic₁₇₈	—	2	6	no	—
37	Dic₁₇₈	—	2	6	no	—
38	C₈×C₈₉	—	2	3	no	—
39	C₈×C₈₉	#41	1	2	yes	—
40	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₈₉	—	2	3	no	—
42	C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
43	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
44	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
45	Dic₈₉	—	4	6	no	—
46	Dic₈₉	—	4	2	no	—
47	C₂×C₈₉	#550	1	1	yes	characteristic
48	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₈₉	#578	1	1	yes	characteristic
50	#3395	—	89	1	no	maximal, Sylow
51	C₈₉	#3395	1	1	yes	characteristic, complemented, Sylow
52	#557	—	89	2	no	—
53	#494	—	89	3	no	—
54	#462	—	89	6	no	—
55	C₈×Q₈	D₈₉	1	1	yes	characteristic
56	C₄×Q₁₆	—	89	3	no	—
57	C₂×Q₁₆	—	89	6	no	—
58	C₂×Q₁₆	—	89	3	no	—
59	C₄⋊Q₈	—	89	6	no	—
60	#125	—	178	6	no	—
61	C₄×Q₈	—	89	6	no	—
62	C₄×Q₈	—	178	2	no	—
63	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
64	#107	—	89	1	no	—
65	C₄⋊C₈	C₂×D₈₉	1	3	yes	—
66	#103	—	89	6	no	—
67	C₄×C₈	C₂×D₈₉	1	3	yes	—
68	C₂×Q₈	—	89	6	no	—
69	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
70	Q₁₆	—	356	6	no	—
71	Q₁₆	—	178	6	no	—
72	C₂×C₈	C₂²×D₈₉	1	3	yes	—
73	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
74	C₄⋊C₄	—	89	2	no	—
75	C₄⋊C₄	—	178	6	no	—
76	C₄⋊C₄	—	178	6	no	—
77	C₄⋊C₄	—	178	6	no	—
78	C₄⋊C₄	D₃₅₆	1	3	yes	—
79	C₄²	—	178	6	no	—
80	C₄²	C₂²×D₈₉	1	3	yes	—
81	Q₈	—	178	6	no	—
82	Q₈	—	178	6	no	—
83	Q₈	Dic₃₅₆	1	4	yes	complemented
84	C₂×C₄	—	178	2	no	—
85	C₂×C₄	—	178	6	no	—
86	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
87	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
88	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#13083575	1	2	yes	—
91	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	356	2	no	—
94	C₄	—	356	6	no	—
95	C₄	C₂×Dic₃₅₆	1	6	yes	—
96	C₄	C₂×#13083575	1	1	yes	characteristic
97	C₄	#15668058	1	1	yes	characteristic
98	C₂	#19152270	1	1	yes	characteristic
99	C₂	C₂²×Dic₃₅₆	1	1	yes	characteristic
100	C₂	#19152415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152327

C₂

yes

maximal

#19151978

C₂

yes

maximal

#19151809

C₂

yes

maximal

C₄×Dic₃₅₆

C₂

yes

maximal

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

#15668012

C₂²

yes

—

#15668003

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667946

C₂²

yes

characteristic

#15667937

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

Q₈×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#41

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

C₂×#41

yes

characteristic

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#550

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#557

—

#494

—

#462

—

C₈×Q₈

D₈₉

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

C₄×C₈

C₂×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

D₃₅₆

yes

characteristic

Q₁₆

—

356

—

Q₁₆

—

178

—

C₂×C₈

C₂²×D₈₉

yes

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

D₃₅₆

yes

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

—

Q₈

—

178

—

Q₈

—

178

—

Q₈

Dic₃₅₆

yes

complemented

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

C₈

—

C₈

#13083575

yes

—

C₂²

C₂²×D₃₅₆

yes

characteristic

C₄

—

C₄

—

356

—

C₄

—

356

—

C₄

C₂×Dic₃₅₆

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

C₂²×Dic₃₅₆

yes

characteristic

100

C₂

#19152415

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂²×D₄
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	#3395
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₈×Q₈×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂²×D₄

hypercenter

C₈×Q₈

D₈₉

hypocenter

C₈₉

#3395

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
30. Q₈×C₈₉	Q₁₆	semidirect
51. C₈₉	#3395	semidirect
83. Q₈	Dic₃₅₆	semidirect

H ≅ G/N

type

30. Q₈×C₈₉

Q₁₆

semidirect

51. C₈₉

#3395

semidirect

83. Q₈

Dic₃₅₆

semidirect

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

subgroup	index
#19152327	2
#19151978	2
#19151809	2
C₄×Dic₃₅₆	2
C₈×Q₈×C₈₉	2
#3395	89

subgroup

index

#19152327

#19151978

#19151809

C₄×Dic₃₅₆

C₈×Q₈×C₈₉

#3395

Subgroups — #26033258

Conjugacy classes of subgroups

Complements

Maximal subgroups