Group #41014218 — subgroups

Subgroups — #41014218

Order 27008.

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

normal subgroups only

This group has 308 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×#20269977	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269902	C₂	1	4	yes	maximal
3	#27213012	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
6	#20269977	—	2	2	no	—
7	#20269977	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
9	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	#20269902	—	2	8	no	—
11	C₂₁₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	#97×C₂₁₁	—	2	2	no	—
15	C₂×Dic₄₂₂	—	2	2	no	—
16	C₂×Dic₄₂₂	—	2	4	no	—
17	C₂×Dic₄₂₂	—	2	4	no	—
18	C₂×Dic₄₂₂	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₁₁	—	2	2	no	—
20	C₂²×Dic₂₁₁	—	2	2	no	—
21	C₄⋊Dic₂₁₁	—	2	8	no	—
22	C₄⋊Dic₂₁₁	—	2	4	no	—
23	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
24	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	—	2	2	no	—
27	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
28	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₄²×C₂₁₁	D₄	1	2	yes	—
30	Dic₄₂₂	—	2	8	no	—
31	Dic₄₂₂	—	4	4	no	—
32	C₂×Dic₂₁₁	—	2	4	no	—
33	C₂×Dic₂₁₁	—	4	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	4	4	no	—
37	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	—	2	4	no	—
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₁₁	#41	1	2	yes	—
44	C₈×C₂₁₁	—	2	4	no	—
45	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₁₁	#124	1	1	yes	characteristic
47	C₂²×C₂₁₁	#127	1	1	yes	characteristic
48	C₂²×C₂₁₁	#137	1	4	yes	—
49	C₄×C₂₁₁	—	2	4	no	—
50	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
51	C₄×C₂₁₁	#121	1	2	yes	—
52	Dic₂₁₁	—	4	4	no	—
53	Dic₂₁₁	—	8	2	no	—
54	C₂×C₂₁₁	#390	1	1	yes	characteristic
55	C₂×C₂₁₁	#419	1	2	yes	—
56	C₂×C₂₁₁	#491	1	2	yes	—
57	C₂×C₂₁₁	#497	1	2	yes	—
58	C₂₁₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	211	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	—	211	1	no	—
61	C₂×#103	—	211	4	no	—
62	C₂×#97	D₂₁₁	1	1	yes	characteristic
63	#382	—	211	1	no	—
64	C₂²×Q₈	—	211	2	no	—
65	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
66	C₄⋊Q₈	—	422	2	no	—
67	C₄⋊Q₈	—	211	4	no	—
68	C₂×C₄⋊C₄	—	422	2	no	—
69	C₂×C₄⋊C₄	—	211	2	no	—
70	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
71	#103	—	422	8	no	—
72	#97	—	2	2	no	—
73	#95	—	422	2	no	—
74	C₂×Q₈	—	422	4	no	—
75	C₂×Q₈	—	422	4	no	—
76	C₂×Q₈	—	422	2	no	—
77	C₂×Q₈	—	211	4	no	—
78	C₂²×C₄	—	422	2	no	—
79	C₂²×C₄	—	422	2	no	—
80	C₂²×C₄	D₈₄₄	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₈₄₄	1	4	yes	—
85	C₄⋊C₄	—	422	4	no	—
86	C₄⋊C₄	—	422	8	no	—
87	C₄²	C₄×D₂₁₁	1	2	yes	—
88	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
89	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
90	Q₈	—	844	4	no	—
91	Q₈	—	422	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	844	2	no	—
94	C₂×C₄	—	844	2	no	—
95	C₂×C₄	—	844	2	no	—
96	C₂×C₄	—	844	4	no	—
97	C₂×C₄	—	422	4	no	—
98	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₈₄₄	1	2	yes	—
102	C₂×C₄	#16809962	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
105	C₂²	#20269965	1	1	yes	characteristic
106	C₂²	#20269974	1	1	yes	characteristic
107	C₂²	#20270010	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	844	4	no	—
110	C₄	—	1688	2	no	—
111	C₄	C₄×D₈₄₄	1	2	yes	—
112	C₄	#20269951	1	2	yes	—
113	C₂	#27213048	1	1	yes	characteristic
114	C₂	#27213156	1	2	yes	—
115	C₂	#27213515	1	2	yes	—
116	C₂	#27213540	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269977

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

#27213012

C₂

yes

maximal, characteristic

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

#20269977

—

#20269977

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269902

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×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

D₄

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

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×D₄

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

#121

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

#419

yes

—

C₂×C₂₁₁

#491

yes

—

C₂×C₂₁₁

#497

yes

—

C₂₁₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

C₂×#103

—

211

—

C₂×#97

D₂₁₁

yes

characteristic

#382

—

211

—

C₂²×Q₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

211

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#97

—

#95

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈₄₄

yes

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂×D₈₄₄

yes

characteristic

Q₈

—

844

—

Q₈

—

422

—

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

100

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

101

C₂×C₄

C₂×D₈₄₄

yes

—

102

C₂×C₄

#16809962

yes

—

103

C₈

—

104

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

105

C₂²

#20269965

yes

characteristic

106

C₂²

#20269974

yes

characteristic

107

C₂²

#20270010

yes

—

108

C₄

—

109

C₄

—

844

—

110

C₄

—

1688

—

111

C₄

C₄×D₈₄₄

yes

—

112

C₄

#20269951

yes

—

113

C₂

#27213048

yes

characteristic

114

C₂

#27213156

yes

—

115

C₂

#27213515

yes

—

116

C₂

#27213540

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	#1960
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

#1960

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #20269977	C₄	semidirect
58. C₂₁₁	#1960	semidirect

H ≅ G/N

type

7. #20269977

C₄

semidirect

58. C₂₁₁

#1960

semidirect

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

subgroup	index
C₂×#20269977	2
C₂×#20269902	2
#27213012	2
C₂×#97×C₂₁₁	2
#1960	211

subgroup

index

C₂×#97×C₂₁₁

211

Subgroups — #41014218

Conjugacy classes of subgroups

Complements

Maximal subgroups