Group #41011742 — subgroups

Subgroups — #41011742

Order 27008.

236 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 89 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	#27213203	C₂	1	2	yes	maximal
2	#27212872	C₂	1	4	yes	maximal
3	C₂×C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₂₁₁⋊C₈)	C₂²	1	4	yes	—
6	#20269888	—	2	2	no	—
7	C₂²×Q₈×C₂₁₁	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₂₁₁	—	2	2	no	—
9	C₄×Q₈×C₂₁₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₄	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₂₁₁⋊C₈	—	2	4	no	—
15	C₂×Q₈×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	—	2	4	no	—
17	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₁₁	—	2	2	no	—
23	C₄⋊C₄×C₂₁₁	—	2	4	no	—
24	C₄²×C₂₁₁	—	2	2	no	—
25	C₄²×C₂₁₁	—	2	2	no	—
26	C₄²×C₂₁₁	D₄	1	2	yes	—
27	C₄²×C₂₁₁	D₄	1	2	yes	—
28	C₂₁₁⋊C₈	—	4	4	no	—
29	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₂₁₁	—	2	8	no	—
31	C₂×C₄×C₂₁₁	—	2	2	no	—
32	C₂×C₄×C₂₁₁	—	2	2	no	—
33	C₂×C₄×C₂₁₁	—	2	2	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	2	no	—
36	C₂×C₄×C₂₁₁	—	4	1	no	—
37	C₂×C₄×C₂₁₁	—	4	2	no	—
38	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	M₁₆	1	4	yes	—
42	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
43	C₂²×C₂₁₁	—	2	1	no	—
44	C₂²×C₂₁₁	—	2	1	no	—
45	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
47	C₄×C₂₁₁	—	2	2	no	—
48	C₄×C₂₁₁	—	2	4	no	—
49	C₄×C₂₁₁	—	2	2	no	—
50	C₄×C₂₁₁	—	2	1	no	—
51	C₄×C₂₁₁	—	4	2	no	—
52	C₄×C₂₁₁	#101	1	2	yes	—
53	C₂×C₂₁₁	—	2	2	no	—
54	C₂×C₂₁₁	#407	1	1	yes	characteristic
55	C₂×C₂₁₁	#410	1	1	yes	characteristic
56	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
57	C₂₁₁	#1467	1	1	yes	characteristic, complemented, Sylow
58	#1467	—	211	1	no	maximal, Sylow
59	C₂×C₄×Q₈	D₂₁₁	1	1	yes	characteristic
60	#432	—	211	2	no	—
61	#324	—	211	4	no	—
62	C₂²×Q₈	Dic₂₁₁	1	1	yes	characteristic
63	C₄×Q₈	—	2	2	no	—
64	C₄×Q₈	—	2	2	no	—
65	C₂×C₄⋊C₄	Dic₂₁₁	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
67	C₂×C₄²	Dic₂₁₁	1	2	yes	—
68	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
69	C₄⋊C₈	—	422	2	no	—
70	C₂²⋊C₈	—	211	4	no	—
71	#97	—	422	2	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	2	4	no	—
74	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
77	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
78	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
79	C₂×C₈	—	422	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
85	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
86	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
87	Q₈	—	2	8	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
97	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
98	C₂×C₄	#16809946	1	4	yes	—
99	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
100	C₈	—	844	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
104	C₂²	C₂×#16809946	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	4	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	#20269896	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	#27213110	1	1	yes	characteristic
113	C₂	#27213120	1	1	yes	characteristic
114	C₂	C₂×#20269896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213203

C₂

yes

maximal

#27212872

C₂

yes

maximal

C₂×C₄×Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₄⋊(C₂₁₁⋊C₈)

—

C₂²⋊(C₂₁₁⋊C₈)

C₂²

yes

—

#20269888

—

C₂²×Q₈×C₂₁₁

C₄

yes

characteristic

C₄×Q₈×C₂₁₁

—

C₄×Q₈×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₂₁₁⋊C₈

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

M₁₆

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂×M₁₆

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#101

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1467

—

211

maximal, Sylow

C₂×C₄×Q₈

D₂₁₁

yes

characteristic

#432

—

211

—

#324

—

211

—

C₂²×Q₈

Dic₂₁₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

Dic₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

C₂²⋊C₈

—

211

—

#97

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂²×Dic₂₁₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

characteristic

C₂×C₄

#16809946

yes

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

—

100

C₈

—

844

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

104

C₂²

C₂×#16809946

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#20269896

yes

—

111

C₂

—

112

C₂

#27213110

yes

characteristic

113

C₂

#27213120

yes

characteristic

114

C₂

C₂×#20269896

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₁₁
commutator, prime radical	C₂³×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄×Q₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄×Q₈	D₂₁₁
hypocenter	C₂₁₁	#1467
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₁₁

commutator, prime radical

C₂³×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄×Q₈×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄×Q₈

D₂₁₁

hypocenter

C₂₁₁

#1467

LCS

G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
57. C₂₁₁	#1467	semidirect

H ≅ G/N

type

57. C₂₁₁

#1467

semidirect

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

subgroup	index
#27213203	2
#27212872	2
C₂×C₄×Q₈×C₂₁₁	2
#1467	211

subgroup

index

#27213203

#27212872

C₂×C₄×Q₈×C₂₁₁

#1467

211

Subgroups — #41011742

Conjugacy classes of subgroups

Complements

Maximal subgroups