Group #41013532 — subgroups

Subgroups — #41013532

Order 27008.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#27213016	C₂	1	1	yes	maximal, characteristic
2	#27213012	C₂	1	1	yes	maximal, characteristic
3	#27212909	C₂	1	4	yes	maximal
4	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
6	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	C₂₁₁⋊#95	—	2	2	no	—
9	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	#97×C₂₁₁	C₄	1	4	yes	complemented
12	C₂²×Dic₂₁₁	—	2	2	no	—
13	C₂²×Dic₂₁₁	—	2	2	no	—
14	C₄⋊Dic₂₁₁	—	2	4	no	—
15	C₄⋊Dic₂₁₁	—	2	4	no	—
16	C₄⋊Dic₂₁₁	—	4	4	no	—
17	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
18	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₂₁₁	—	2	2	no	—
20	C₂×C₈×C₂₁₁	—	2	2	no	—
21	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
22	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₄²×C₂₁₁	Q₈	1	2	yes	—
24	C₂×Dic₂₁₁	—	4	2	no	—
25	C₂×Dic₂₁₁	—	4	2	no	—
26	C₂×Dic₂₁₁	—	4	4	no	—
27	C₂×Dic₂₁₁	—	4	2	no	—
28	C₂×Dic₂₁₁	—	4	2	no	—
29	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	—	2	2	no	—
31	C₂×C₄×C₂₁₁	—	2	2	no	—
32	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	#41	1	2	yes	—
36	C₂×C₄×C₂₁₁	#41	1	2	yes	—
37	C₈×C₂₁₁	—	2	4	no	—
38	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₁₁	#124	1	1	yes	characteristic
40	C₂²×C₂₁₁	#124	1	1	yes	characteristic
41	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₂₁₁	#137	1	2	yes	—
43	C₄×C₂₁₁	—	2	4	no	—
44	C₄×C₂₁₁	#117	1	2	yes	—
45	C₄×C₂₁₁	#125	1	2	yes	—
46	Dic₂₁₁	—	8	2	no	—
47	Dic₂₁₁	—	8	2	no	—
48	C₂×C₂₁₁	#382	1	1	yes	characteristic
49	C₂×C₂₁₁	#428	1	2	yes	—
50	C₂×C₂₁₁	#489	1	2	yes	—
51	C₂×C₂₁₁	#490	1	1	yes	characteristic
52	C₂×C₂₁₁	#491	1	1	yes	characteristic
53	C₂₁₁	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	211	1	no	maximal, Sylow
55	C₂×#97	D₂₁₁	1	1	yes	characteristic
56	#384	—	211	1	no	—
57	#382	—	211	1	no	—
58	#340	—	211	4	no	—
59	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
60	C₂×C₄⋊C₄	—	422	2	no	—
61	C₂×C₄⋊C₄	—	211	2	no	—
62	C₂×C₄⋊C₄	—	211	2	no	—
63	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
64	#97	Dic₂₁₁	1	4	yes	complemented
65	#95	—	422	2	no	—
66	C₂²×C₄	—	422	2	no	—
67	C₂²×C₄	—	422	2	no	—
68	C₂²×C₄	D₈₄₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	—
73	C₄⋊C₄	—	844	4	no	—
74	C₄⋊C₄	—	422	4	no	—
75	C₄⋊C₄	—	422	4	no	—
76	C₄²	C₂×Dic₂₁₁	1	2	yes	—
77	C₄²	Dic₄₂₂	1	2	yes	—
78	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	844	2	no	—
82	C₂×C₄	—	844	2	no	—
83	C₂×C₄	—	844	4	no	—
84	C₂×C₄	—	844	2	no	—
85	C₂×C₄	—	844	2	no	—
86	C₂×C₄	C₄⋊Dic₂₁₁	1	4	yes	—
87	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
89	C₂×C₄	#16809962	1	2	yes	—
90	C₂×C₄	#16809962	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
93	C₂²	#20269965	1	1	yes	characteristic
94	C₂²	#20269965	1	1	yes	characteristic
95	C₂²	#20270005	1	2	yes	—
96	C₂²	#20270010	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	1688	2	no	—
99	C₄	—	1688	2	no	—
100	C₄	#20269936	1	2	yes	—
101	C₄	#20269969	1	2	yes	—
102	C₂	#27213012	1	1	yes	characteristic
103	C₂	#27213188	1	2	yes	—
104	C₂	#27213506	1	2	yes	—
105	C₂	#27213511	1	1	yes	characteristic
106	C₂	#27213515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213016

C₂

yes

maximal, characteristic

#27213012

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

C₄

yes

complemented

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#117

yes

—

C₄×C₂₁₁

#125

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#382

yes

characteristic

C₂×C₂₁₁

#428

yes

—

C₂×C₂₁₁

#489

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1836

—

211

maximal, Sylow

C₂×#97

D₂₁₁

yes

characteristic

#384

—

211

—

#382

—

211

—

#340

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#97

Dic₂₁₁

yes

complemented

#95

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₁₁

yes

—

C₄⋊C₄

—

844

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

C₂×Dic₂₁₁

yes

—

C₄²

Dic₄₂₂

yes

—

C₂³

C₂×D₈₄₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

C₄⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

C₂×C₄

#16809962

yes

—

C₂×C₄

#16809962

yes

—

C₈

—

C₂²

C₂×C₄⋊Dic₂₁₁

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#20270010

yes

—

C₄

—

C₄

—

1688

—

C₄

—

1688

—

100

C₄

#20269936

yes

—

101

C₄

#20269969

yes

—

102

C₂

#27213012

yes

characteristic

103

C₂

#27213188

yes

—

104

C₂

#27213506

yes

—

105

C₂

#27213511

yes

characteristic

106

C₂

#27213515

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#97×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#97	D₂₁₁
hypocenter	C₂₁₁	#1836
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×#97×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×#97

D₂₁₁

hypocenter

C₂₁₁

#1836

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₂₁₁	C₄	semidirect
53. C₂₁₁	#1836	semidirect
64. #97	Dic₂₁₁	semidirect

H ≅ G/N

type

11. #97×C₂₁₁

C₄

semidirect

53. C₂₁₁

#1836

semidirect

64. #97

Dic₂₁₁

semidirect

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

subgroup	index
#27213016	2
#27213012	2
#27212909	2
C₂×#97×C₂₁₁	2
#1836	211

subgroup

index

C₂×#97×C₂₁₁

211

Subgroups — #41013532

Conjugacy classes of subgroups

Complements

Maximal subgroups