Group #41012136 — subgroups

Subgroups — #41012136

Order 27008.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#27213170	C₂	1	2	yes	maximal
2	#27212892	C₂	1	4	yes	maximal, complemented
3	#533×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16809946	—	2	2	no	—
5	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
6	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
8	#125×C₂₁₁	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₂₁₁	—	2	2	no	—
10	C₄×Q₈×C₂₁₁	—	2	2	no	—
11	#117×C₂₁₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	#16809946	—	4	4	no	—
14	C₂×C₂₁₁⋊C₈	—	2	4	no	—
15	C₂×Q₈×C₂₁₁	—	2	2	no	—
16	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
18	C₄⋊C₄×C₂₁₁	—	2	4	no	—
19	C₄⋊C₄×C₂₁₁	—	2	2	no	—
20	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
22	C₄²×C₂₁₁	—	2	2	no	—
23	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂₁₁⋊C₈	—	4	4	no	—
27	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
28	Q₈×C₂₁₁	—	4	2	no	—
29	Q₈×C₂₁₁	—	4	2	no	—
30	C₂×C₄×C₂₁₁	—	2	2	no	—
31	C₂×C₄×C₂₁₁	—	2	2	no	—
32	C₂×C₄×C₂₁₁	—	2	2	no	—
33	C₂×C₄×C₂₁₁	—	2	4	no	—
34	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
38	C₂²×C₂₁₁	—	2	1	no	—
39	C₂²×C₂₁₁	—	2	2	no	—
40	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₂₁₁	—	2	1	no	—
42	C₄×C₂₁₁	—	2	2	no	—
43	C₄×C₂₁₁	—	4	4	no	—
44	C₄×C₂₁₁	#101	1	2	yes	—
45	C₄×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₂₁₁	#137	1	2	yes	—
47	C₂×C₂₁₁	—	4	1	no	—
48	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
49	C₂×C₂₁₁	#417	1	2	yes	—
50	C₂₁₁	#1561	1	1	yes	characteristic, complemented, Sylow
51	#1561	—	211	1	no	maximal, Sylow
52	#533	D₂₁₁	1	1	yes	characteristic
53	#423	—	211	2	no	—
54	#332	—	211	4	no	—
55	C₂×M₁₆	—	422	2	no	—
56	C₄⋊Q₈	C₂×D₂₁₁	1	2	yes	—
57	#125	Dic₂₁₁	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₂₁₁	1	2	yes	—
61	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
62	C₄⋊C₈	—	422	2	no	—
63	C₄⋊C₈	—	211	4	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
67	M₁₆	—	844	4	no	—
68	C₂×C₈	—	422	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
75	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
76	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
88	C₈	—	844	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#20269896	1	2	yes	—
96	C₄	#20270002	1	2	yes	—
97	C₄	#20270007	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#20269896	1	1	yes	characteristic
100	C₂	#27213147	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213170

C₂

yes

maximal

#27212892

C₂

yes

maximal, complemented

#533×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

—

C₄⋊(C₂₁₁⋊C₈)

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

—

#125×C₂₁₁

C₄

yes

—

C₂²⋊Q₈×C₂₁₁

—

C₄×Q₈×C₂₁₁

—

#117×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×Q₈×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#101

yes

—

C₄×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₁₁

#137

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×#101

yes

characteristic

C₂×C₂₁₁

#417

yes

—

C₂₁₁

#1561

yes

characteristic, complemented, Sylow

#1561

—

211

maximal, Sylow

#533

D₂₁₁

yes

characteristic

#423

—

211

—

#332

—

211

—

C₂×M₁₆

—

422

—

C₄⋊Q₈

C₂×D₂₁₁

yes

—

#125

Dic₂₁₁

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

C₄⋊C₈

—

211

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

M₁₆

—

844

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₁₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

C₂²⋊Dic₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

characteristic

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

—

C₈

—

844

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#20269896

yes

—

C₄

#20270002

yes

—

C₄

#20270007

yes

—

C₂

—

C₂

C₂×#20269896

yes

characteristic

100

C₂

#27213147

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#533×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#533	D₂₁₁
hypocenter	C₂₁₁	#1561
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#533×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₁₁

C₂²⋊C₄

cosocle

C₄²×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #27212892	C₂	semidirect
50. C₂₁₁	#1561	semidirect

H ≅ G/N

type

2. #27212892

C₂

semidirect

50. C₂₁₁

#1561

semidirect

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

subgroup	index
#27213170	2
#27212892	2
#533×C₂₁₁	2
#1561	211

subgroup

index

#27213170

#27212892

#533×C₂₁₁

#1561

211

Subgroups — #41012136

Conjugacy classes of subgroups

Complements

Maximal subgroups