Group #41012060 — subgroups

Subgroups — #41012060

Order 27008.

246 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 85 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	#27213199	C₂	1	2	yes	maximal, complemented
2	#27212895	C₂	1	2	yes	maximal, complemented
3	#27212890	C₂	1	2	yes	maximal, complemented
4	#532×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	4	yes	—
6	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
7	C₄×C₂₁₁⋊C₈	—	2	2	no	—
8	C₂×#41×C₂₁₁	C₄	1	2	yes	—
9	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	#127×C₂₁₁	C₂²	1	1	yes	characteristic
11	#124×C₂₁₁	C₄	1	2	yes	—
12	#121×C₂₁₁	—	2	2	no	—
13	C₄×D₄×C₂₁₁	—	2	2	no	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₂₁₁⋊C₈	—	2	4	no	—
16	#41×C₂₁₁	—	4	4	no	—
17	C₂×Q₈×C₂₁₁	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₂₁₁	—	2	2	no	—
19	C₂×D₄×C₂₁₁	—	2	2	no	—
20	C₂×D₄×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	—	2	2	no	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₁₁	—	2	2	no	—
25	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
26	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₂₁₁	D₄	1	2	yes	—
28	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₂₁₁⋊C₈	—	4	4	no	—
30	C₂³×C₂₁₁	—	2	2	no	—
31	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₂₁₁	—	4	2	no	—
33	D₄×C₂₁₁	—	4	2	no	—
34	D₄×C₂₁₁	—	4	4	no	—
35	D₄×C₂₁₁	—	4	4	no	—
36	C₂×C₄×C₂₁₁	—	2	2	no	—
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	—	2	2	no	—
39	C₂×C₄×C₂₁₁	—	4	4	no	—
40	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
44	C₂²×C₂₁₁	—	2	1	no	—
45	C₂²×C₂₁₁	—	2	2	no	—
46	C₂²×C₂₁₁	—	4	2	no	—
47	C₂²×C₂₁₁	—	4	4	no	—
48	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₄×C₂₁₁	—	2	1	no	—
50	C₄×C₂₁₁	—	2	2	no	—
51	C₄×C₂₁₁	—	4	2	no	—
52	C₄×C₂₁₁	#100	1	2	yes	—
53	C₄×C₂₁₁	#135	1	4	yes	—
54	C₂×C₂₁₁	—	4	1	no	—
55	C₂×C₂₁₁	—	8	2	no	—
56	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
57	C₂×C₂₁₁	#416	1	2	yes	—
58	C₂₁₁	#1543	1	1	yes	characteristic, complemented, Sylow
59	#1543	—	211	1	no	maximal, Sylow
60	#532	D₂₁₁	1	1	yes	characteristic
61	#431	—	211	2	no	—
62	#333	—	211	2	no	—
63	#331	—	211	2	no	—
64	C₂×#41	Dic₂₁₁	1	2	yes	—
65	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
66	#127	C₂×D₂₁₁	1	1	yes	characteristic
67	#124	Dic₂₁₁	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
71	C₄⋊C₈	—	211	4	no	—
72	C₂²⋊C₈	—	422	2	no	—
73	C₄×C₈	—	422	2	no	—
74	#41	—	4	4	no	—
75	C₂×Q₈	C₂×Dic₂₁₁	1	2	yes	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	C₂×Dic₂₁₁	1	2	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
82	C₂×C₈	—	422	4	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₂²⋊C₄	—	2	4	no	—
85	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
87	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
99	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
102	C₈	—	844	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#20269894	1	2	yes	—
112	C₄	#20269997	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#20269894	1	1	yes	characteristic
116	C₂	#27213142	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213199

C₂

yes

maximal, complemented

#27212895

C₂

yes

maximal, complemented

#27212890

C₂

yes

maximal, complemented

#532×C₂₁₁

C₂

yes

maximal, characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₄×C₂₁₁⋊C₈

—

C₂×#41×C₂₁₁

C₄

yes

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

#127×C₂₁₁

C₂²

yes

characteristic

#124×C₂₁₁

C₄

yes

—

#121×C₂₁₁

—

C₄×D₄×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₂₁₁⋊C₈

—

#41×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#100

yes

—

C₄×C₂₁₁

#135

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×#100

yes

characteristic

C₂×C₂₁₁

#416

yes

—

C₂₁₁

#1543

yes

characteristic, complemented, Sylow

#1543

—

211

maximal, Sylow

#532

D₂₁₁

yes

characteristic

#431

—

211

—

#333

—

211

—

#331

—

211

—

C₂×#41

Dic₂₁₁

yes

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

#127

C₂×D₂₁₁

yes

characteristic

#124

Dic₂₁₁

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

211

—

C₂²⋊C₈

—

422

—

C₄×C₈

—

422

—

#41

—

C₂×Q₈

C₂×Dic₂₁₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₂₁₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

characteristic

100

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

—

102

C₈

—

844

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#20269894

yes

—

112

C₄

#20269997

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#20269894

yes

characteristic

116

C₂

#27213142

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting, prime radical	#532×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#532	D₂₁₁
hypocenter	C₂₁₁	#1543
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₁₁

Fitting, prime radical

#532×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #27213199	C₂	semidirect
2. #27212895	C₂	semidirect
3. #27212890	C₂	semidirect
58. C₂₁₁	#1543	semidirect

H ≅ G/N

type

1. #27213199

C₂

semidirect

2. #27212895

C₂

semidirect

3. #27212890

C₂

semidirect

58. C₂₁₁

#1543

semidirect

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

subgroup	index
#27213199	2
#27212895	2
#27212890	2
#532×C₂₁₁	2
#1543	211

subgroup

index

#532×C₂₁₁

211

Subgroups — #41012060

Conjugacy classes of subgroups

Complements

Maximal subgroups