Group #41014903 — subgroups

Subgroups — #41014903

Order 27008.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27213189	C₂	1	2	yes	maximal
2	C₂×#20269911	C₂	1	1	yes	maximal, characteristic
3	#27212914	C₂	1	2	yes	maximal
4	#27212911	C₂	1	1	yes	maximal, characteristic
5	#424×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	#20269911	—	2	2	no	—
9	#20269910	—	2	2	no	—
10	#20269908	—	2	2	no	—
11	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	#117×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₁₁	—	2	2	no	—
15	C₂²×Dic₂₁₁	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	#16809946	Q₈	1	4	yes	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₄⋊Dic₂₁₁	—	2	2	no	—
21	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₁₁	—	2	2	no	—
23	C₂×C₈×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	—	2	1	no	—
25	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₂₁₁	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
29	C₄²×C₂₁₁	—	2	1	no	—
30	C₂×Dic₂₁₁	—	4	2	no	—
31	C₂×Dic₂₁₁	—	8	2	no	—
32	C₂₁₁⋊C₈	—	2	4	no	—
33	C₂³×C₂₁₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	2	1	no	—
35	C₂×C₄×C₂₁₁	—	2	1	no	—
36	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₂₁₁	#41	1	2	yes	—
40	C₈×C₂₁₁	—	2	2	no	—
41	C₈×C₂₁₁	—	4	1	no	—
42	C₂²×C₂₁₁	—	2	1	no	—
43	C₂²×C₂₁₁	—	2	1	no	—
44	C₂²×C₂₁₁	#121	1	1	yes	characteristic
45	C₂²×C₂₁₁	#125	1	2	yes	—
46	C₄×C₂₁₁	—	4	1	no	—
47	C₄×C₂₁₁	—	4	1	no	—
48	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₁₁	#123	1	1	yes	characteristic
50	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₂₁₁	—	8	2	no	—
52	C₂×C₂₁₁	—	2	2	no	—
53	C₂×C₂₁₁	#398	1	1	yes	characteristic
54	C₂×C₂₁₁	#472	1	1	yes	characteristic
55	C₂×C₂₁₁	#473	1	1	yes	characteristic
56	C₂₁₁	#2089	1	1	yes	characteristic, complemented, Sylow
57	#2089	—	211	1	no	maximal, Sylow
58	#428	—	211	2	no	—
59	C₂×#107	—	211	1	no	—
60	#424	D₂₁₁	1	1	yes	characteristic
61	#343	—	211	2	no	—
62	#341	—	211	1	no	—
63	C₂×M₁₆	—	211	2	no	—
64	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
66	#117	C₂×D₂₁₁	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	211	2	no	—
68	#107	—	422	2	no	—
69	#107	—	422	2	no	—
70	#106	—	422	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	422	2	no	—
73	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	422	2	no	—
76	M₁₆	—	211	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	422	2	no	—
80	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
81	C₄⋊C₄	—	422	2	no	—
82	C₄⋊C₄	—	422	2	no	—
83	C₄⋊C₄	D₈₄₄	1	1	yes	characteristic
84	C₄⋊C₄	D₈₄₄	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#16809962	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	844	2	no	—
91	C₂×C₄	—	1688	2	no	—
92	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
93	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
94	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
95	C₂×C₄	#16809961	1	2	yes	—
96	C₈	—	2	2	no	—
97	C₈	—	4	1	no	—
98	C₈	—	422	4	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#20269951	1	1	yes	characteristic
102	C₂²	#20269967	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	1688	2	no	—
106	C₄	#20269957	1	2	yes	—
107	C₄	#20269958	1	1	yes	characteristic
108	C₄	#20269975	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#27213076	1	1	yes	characteristic
111	C₂	#27213416	1	1	yes	characteristic
112	C₂	#27213421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213189

C₂

yes

maximal

C₂×#20269911

C₂

yes

maximal, characteristic

#27212914

C₂

yes

maximal

#27212911

C₂

yes

maximal, characteristic

#424×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269911

—

#20269910

—

#20269908

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂²×Dic₂₁₁

—

#16809946

—

#16809946

Q₈

yes

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

D₄

yes

characteristic

C₄⋊C₄×C₂₁₁

D₄

yes

characteristic

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

#125

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#123

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#2089

—

211

maximal, Sylow

#428

—

211

—

C₂×#107

—

211

—

#424

D₂₁₁

yes

characteristic

#343

—

211

—

#341

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#117

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

#107

—

422

—

#107

—

422

—

#106

—

422

—

C₄⋊C₈

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

211

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

D₈₄₄

yes

characteristic

C₄⋊C₄

D₈₄₄

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#16809962

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

1688

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

C₂×D₈₄₄

yes

characteristic

C₂×C₄

Q₈×D₂₁₁

yes

—

C₂×C₄

#16809961

yes

—

C₈

—

C₈

—

C₈

—

422

—

C₂²

—

100

C₂²

—

101

C₂²

#20269951

yes

characteristic

102

C₂²

#20269967

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

1688

—

106

C₄

#20269957

yes

—

107

C₄

#20269958

yes

characteristic

108

C₄

#20269975

yes

characteristic

109

C₂

—

110

C₂

#27213076

yes

characteristic

111

C₂

#27213416

yes

characteristic

112

C₂

#27213421

yes

characteristic

	subgroup	G/H
center	C₂²	#20269951
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#424×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#121
prime radical	C₂³×C₂₁₁	#41
hypercenter	#424	D₂₁₁
hypocenter	C₂₁₁	#2089
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#20269951

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#424×C₂₁₁

C₂

C₂²×C₂₁₁

C₂³×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
56. C₂₁₁	#2089	semidirect

H ≅ G/N

type

56. C₂₁₁

#2089

semidirect

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

subgroup	index
#27213189	2
C₂×#20269911	2
#27212914	2
#27212911	2
#424×C₂₁₁	2
#2089	211

subgroup

index

#424×C₂₁₁

211

Subgroups — #41014903

Conjugacy classes of subgroups

Complements

Maximal subgroups