Group #41014367 — subgroups

Subgroups — #41014367

Order 27008.

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

normal subgroups only

This group has 260 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	C₂×#20269969	C₂	1	1	yes	maximal, characteristic
2	#27213016	C₂	1	1	yes	maximal, characteristic
3	#27212909	C₂	1	4	yes	maximal
4	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#20269969	—	2	2	no	—
6	#20269969	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	C₂×Dic₂₁₁⋊C₄	—	2	2	no	—
11	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	#97×C₂₁₁	—	2	2	no	—
14	C₂²×Dic₂₁₁	—	2	2	no	—
15	C₂²×Dic₂₁₁	—	2	2	no	—
16	C₄⋊Dic₂₁₁	—	2	2	no	—
17	C₄⋊Dic₂₁₁	—	2	4	no	—
18	C₄⋊Dic₂₁₁	—	4	4	no	—
19	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	—
20	Dic₂₁₁⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	Q₈	1	4	yes	—
26	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₄²×C₂₁₁	D₄	1	2	yes	—
28	C₂×Dic₂₁₁	—	2	4	no	—
29	C₂×Dic₂₁₁	—	4	2	no	—
30	C₂×Dic₂₁₁	—	4	2	no	—
31	C₂×Dic₂₁₁	—	4	2	no	—
32	C₂×Dic₂₁₁	—	4	4	no	—
33	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	2	no	—
36	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₂₁₁	#41	1	2	yes	—
41	C₈×C₂₁₁	—	2	4	no	—
42	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₁₁	#124	1	1	yes	characteristic
44	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₂₁₁	#137	1	2	yes	—
47	C₄×C₂₁₁	—	2	4	no	—
48	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
49	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
50	Dic₂₁₁	—	4	4	no	—
51	Dic₂₁₁	—	8	2	no	—
52	C₂×C₂₁₁	#391	1	1	yes	characteristic
53	C₂×C₂₁₁	#417	1	2	yes	—
54	C₂×C₂₁₁	#490	1	1	yes	characteristic
55	C₂×C₂₁₁	#491	1	1	yes	characteristic
56	C₂×C₂₁₁	#501	1	2	yes	—
57	C₂₁₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	211	1	no	maximal, Sylow
59	C₂×#125	—	211	1	no	—
60	C₂×#97	D₂₁₁	1	1	yes	characteristic
61	#384	—	211	1	no	—
62	#340	—	211	4	no	—
63	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
64	#125	—	422	2	no	—
65	#125	—	211	4	no	—
66	C₂×C₄⋊C₄	—	422	2	no	—
67	C₂×C₄⋊C₄	—	422	2	no	—
68	C₂×C₄⋊C₄	—	211	2	no	—
69	C₂×C₄⋊C₄	—	211	2	no	—
70	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	422	2	no	—
73	C₂²×C₄	—	422	2	no	—
74	C₂²×C₄	D₈₄₄	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₄₂₂	1	4	yes	—
79	C₄⋊C₄	—	844	4	no	—
80	C₄⋊C₄	—	422	4	no	—
81	C₄⋊C₄	—	422	8	no	—
82	C₄⋊C₄	—	422	2	no	—
83	C₄⋊C₄	—	211	4	no	—
84	C₄²	C₄×D₂₁₁	1	2	yes	—
85	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
86	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	844	2	no	—
90	C₂×C₄	—	844	2	no	—
91	C₂×C₄	—	844	2	no	—
92	C₂×C₄	—	844	4	no	—
93	C₂×C₄	—	422	4	no	—
94	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
98	C₂×C₄	#16809962	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
101	C₂²	#20269965	1	1	yes	characteristic
102	C₂²	#20269977	1	1	yes	characteristic
103	C₂²	#20270005	1	2	yes	—
104	C₂²	#20270010	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	844	4	no	—
107	C₄	—	1688	2	no	—
108	C₄	C₄×Dic₄₂₂	1	2	yes	—
109	C₄	#20269956	1	2	yes	—
110	C₂	#27213052	1	1	yes	characteristic
111	C₂	#27213148	1	2	yes	—
112	C₂	#27213511	1	1	yes	characteristic
113	C₂	#27213515	1	1	yes	characteristic
114	C₂	#27213555	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269969

C₂

yes

maximal, characteristic

#27213016

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic

#20269969

—

#20269969

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×Dic₂₁₁⋊C₄

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

Dic₂₁₁⋊C₄

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

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

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#124

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#391

yes

characteristic

C₂×C₂₁₁

#417

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂₁₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

211

maximal, Sylow

C₂×#125

—

211

—

C₂×#97

D₂₁₁

yes

characteristic

#384

—

211

—

#340

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

#125

—

422

—

#125

—

211

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#97

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₂₂

yes

—

C₄⋊C₄

—

844

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

211

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂×D₈₄₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

—

422

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

C₂×Dic₄₂₂

yes

—

C₂×C₄

#16809962

yes

—

C₈

—

100

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

101

C₂²

#20269965

yes

characteristic

102

C₂²

#20269977

yes

characteristic

103

C₂²

#20270005

yes

—

104

C₂²

#20270010

yes

—

105

C₄

—

106

C₄

—

844

—

107

C₄

—

1688

—

108

C₄

C₄×Dic₄₂₂

yes

—

109

C₄

#20269956

yes

—

110

C₂

#27213052

yes

characteristic

111

C₂

#27213148

yes

—

112

C₂

#27213511

yes

characteristic

113

C₂

#27213515

yes

characteristic

114

C₂

#27213555

yes

—

	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	#1992
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

#1992

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #20269969	C₄	semidirect
57. C₂₁₁	#1992	semidirect

H ≅ G/N

type

6. #20269969

C₄

semidirect

57. C₂₁₁

#1992

semidirect

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

subgroup	index
C₂×#20269969	2
#27213016	2
#27212909	2
C₂×#97×C₂₁₁	2
#1992	211

subgroup

index

C₂×#97×C₂₁₁

211

Subgroups — #41014367

Conjugacy classes of subgroups

Complements

Maximal subgroups