Group #26022510 — subgroups

Subgroups — #26022510

Order 11392.

184 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 53 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×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667956	C₂	1	2	yes	maximal, complemented
3	#19151981	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	C₂²	1	2	yes	—
7	C₂×#13083562	—	2	2	no	—
8	#15667999	—	2	2	no	—
9	Dic₇₁₂	—	2	2	no	—
10	#15667956	—	2	4	no	—
11	C₂×#13083549	C₂²	1	1	yes	characteristic
12	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₈₉	C₂²	1	2	yes	—
14	D₁₆×C₈₉	—	2	2	no	—
15	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	2	2	no	—
17	Dic₃₅₆	—	2	2	no	—
18	Dic₃₅₆	—	2	2	no	—
19	#13083562	—	4	4	no	—
20	C₂×#10623494	D₄	1	1	yes	characteristic
21	C₂²⋊Dic₈₉	—	4	4	no	—
22	C₄×Dic₈₉	D₄	1	1	yes	characteristic
23	#13083549	D₄	1	2	yes	—
24	C₂×D₄×C₈₉	—	2	2	no	—
25	D₈×C₈₉	—	2	2	no	—
26	D₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₁₆×C₈₉	D₄	1	2	yes	—
29	C₂³×C₈₉	—	4	2	no	—
30	D₄×C₈₉	—	4	2	no	—
31	D₄×C₈₉	—	4	2	no	—
32	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₈₉	—	4	2	no	—
34	C₂×Dic₈₉	D₈	1	2	yes	—
35	Dic₁₇₈	—	4	2	no	—
36	Dic₁₇₈	—	4	2	no	—
37	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	#10623494	D₈	1	2	yes	—
40	C₂²×C₈₉	—	8	2	no	—
41	C₂²×C₈₉	—	8	4	no	—
42	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
43	C₄×C₈₉	#127	1	1	yes	characteristic
44	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
45	Dic₈₉	—	2	2	no	—
46	Dic₈₉	—	8	2	no	—
47	C₂×C₈₉	—	16	2	no	—
48	C₂×C₈₉	#493	1	1	yes	characteristic
49	C₂×C₈₉	#508	1	2	yes	—
50	#2249	—	89	1	no	maximal, Sylow
51	C₈₉	#2249	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₃₂	—	89	1	no	—
53	C₂×SD₃₂	—	89	2	no	—
54	C₂×D₁₆	D₈₉	1	1	yes	characteristic
55	#495	—	89	2	no	—
56	C₄×C₁₆	—	89	1	no	—
57	C₂×Q₁₆	—	89	2	no	—
58	C₂×SD₁₆	—	178	2	no	—
59	C₂×D₈	C₂×D₈₉	1	2	yes	—
60	#124	—	178	2	no	—
61	Q₃₂	—	178	2	no	—
62	SD₃₂	—	178	4	no	—
63	D₁₆	—	2	2	no	—
64	C₂×C₁₆	—	89	1	no	—
65	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
66	C₄×C₈	—	89	1	no	—
67	C₂×Q₈	—	178	2	no	—
68	C₂×D₄	—	2	2	no	—
69	Q₁₆	—	178	2	no	—
70	Q₁₆	—	178	2	no	—
71	SD₁₆	—	356	4	no	—
72	D₈	—	2	2	no	—
73	D₈	—	2	2	no	—
74	C₂×C₈	—	89	1	no	—
75	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
76	C₂²⋊C₄	—	356	4	no	—
77	C₄²	—	89	1	no	—
78	C₁₆	—	89	2	no	—
79	C₁₆	C₂²⋊D₈₉	1	2	yes	—
80	C₂³	—	4	2	no	—
81	Q₈	—	356	2	no	—
82	Q₈	—	356	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	C₂×C₄	—	89	2	no	—
86	C₂×C₄	—	356	2	no	—
87	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
88	C₈	—	89	2	no	—
89	C₈	D₄×D₈₉	1	1	yes	characteristic
90	C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
91	C₂²	—	8	2	no	—
92	C₂²	—	8	4	no	—
93	C₂²	D₈×D₈₉	1	1	yes	characteristic
94	C₄	—	178	2	no	—
95	C₄	—	712	2	no	—
96	C₄	#15668008	1	1	yes	characteristic
97	C₄	D₈×D₈₉	1	1	yes	characteristic
98	C₂	—	16	2	no	—
99	C₂	#19151973	1	1	yes	characteristic
100	C₂	#19152027	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₁₂

C₂

yes

maximal, characteristic, complemented

C₂×#15667956

C₂

yes

maximal, complemented

#19151981

C₂

yes

maximal, complemented

C₁₆×Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

C₂×#13083562

—

#15667999

—

Dic₇₁₂

—

#15667956

—

C₂×#13083549

C₂²

yes

characteristic

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×D₈×C₈₉

C₂²

yes

—

D₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083562

—

C₂×#10623494

D₄

yes

characteristic

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

D₄

yes

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

D₄

yes

—

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

C₂×D₄

yes

characteristic

#10623494

D₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

#127

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#493

yes

characteristic

C₂×C₈₉

#508

yes

—

#2249

—

maximal, Sylow

C₈₉

#2249

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

D₈₉

yes

characteristic

#495

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

178

—

C₂×D₈

C₂×D₈₉

yes

—

#124

—

178

—

Q₃₂

—

178

—

SD₃₂

—

178

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

Q₁₆

—

178

—

Q₁₆

—

178

—

SD₁₆

—

356

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₂²⋊C₄

—

356

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊D₈₉

yes

—

C₂³

—

Q₈

—

356

—

Q₈

—

356

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

D₄×D₈₉

yes

characteristic

C₈

—

C₈

D₄×D₈₉

yes

characteristic

C₈

C₂×C₂²⋊D₈₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

D₈×D₈₉

yes

characteristic

C₄

—

178

—

C₄

—

712

—

C₄

#15668008

yes

characteristic

C₄

D₈×D₈₉

yes

characteristic

C₂

—

C₂

#19151973

yes

characteristic

100

C₂

#19152027

yes

—

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting, prime radical	C₂×D₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
hypercenter	C₂×D₁₆	D₈₉
hypocenter	C₈₉	#2249
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting, prime radical

C₂×D₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

hypercenter

C₂×D₁₆

D₈₉

hypocenter

C₈₉

#2249

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. C₂×Dic₇₁₂	C₂	semidirect
2. C₂×#15667956	C₂	semidirect
3. #19151981	C₂	semidirect
4. C₁₆×Dic₈₉	C₂	semidirect
51. C₈₉	#2249	semidirect

H ≅ G/N

type

1. C₂×Dic₇₁₂

C₂

semidirect

2. C₂×#15667956

C₂

semidirect

3. #19151981

C₂

semidirect

4. C₁₆×Dic₈₉

C₂

semidirect

51. C₈₉

#2249

semidirect

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

subgroup	index
C₂×Dic₇₁₂	2
C₂×#15667956	2
#19151981	2
C₁₆×Dic₈₉	2
C₂×D₁₆×C₈₉	2
#2249	89

subgroup

index

C₂×Dic₇₁₂

C₂×#15667956

#19151981

C₁₆×Dic₈₉

C₂×D₁₆×C₈₉

#2249

Subgroups — #26022510

Conjugacy classes of subgroups

Complements

Maximal subgroups