Group #41012312 — subgroups

Subgroups — #41012312

Order 27008.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#27213224	C₂	1	2	yes	maximal, complemented
2	#27212897	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₂₁₁⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#27212867	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	—	2	2	no	—
7	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
9	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₂₁₁	C₄	1	1	yes	characteristic
12	C₄×D₄×C₂₁₁	C₂²	1	2	yes	—
13	#107×C₂₁₁	C₄	1	1	yes	characteristic
14	#102×C₂₁₁	C₄	1	2	yes	—
15	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	#16809946	—	4	4	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	1	no	—
19	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
20	C₂×D₄×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	—	2	2	no	—
22	D₈×C₂₁₁	—	2	2	no	—
23	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
27	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂₁₁⋊C₈	—	2	2	no	—
29	C₂₁₁⋊C₈	—	4	1	no	—
30	C₂₁₁⋊C₈	—	4	2	no	—
31	C₂³×C₂₁₁	—	2	2	no	—
32	D₄×C₂₁₁	—	2	2	no	—
33	D₄×C₂₁₁	—	2	2	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	4	2	no	—
36	C₂×C₄×C₂₁₁	—	4	2	no	—
37	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
40	C₈×C₂₁₁	—	2	1	no	—
41	C₈×C₂₁₁	M₁₆	1	2	yes	complemented
42	C₂²×C₂₁₁	—	4	2	no	—
43	C₂²×C₂₁₁	—	4	2	no	—
44	C₂²×C₂₁₁	—	4	2	no	—
45	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
46	C₄×C₂₁₁	—	2	1	no	—
47	C₄×C₂₁₁	—	4	2	no	—
48	C₄×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₂₁₁	#131	1	1	yes	characteristic
50	C₄×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₂₁₁	—	8	2	no	—
52	C₂×C₂₁₁	#436	1	1	yes	characteristic
53	C₂×C₂₁₁	#442	1	1	yes	characteristic
54	C₂×C₂₁₁	#444	1	1	yes	characteristic
55	C₂₁₁	#1594	1	1	yes	characteristic, complemented, Sylow
56	#1594	—	211	1	no	maximal, Sylow
57	C₄×D₈	D₂₁₁	1	1	yes	characteristic
58	#436	—	211	2	no	—
59	#334	—	211	1	no	—
60	D₄⋊C₈	—	211	2	no	—
61	#322	—	211	1	no	—
62	C₂×D₈	Dic₂₁₁	1	1	yes	characteristic
63	C₂×M₁₆	—	422	2	no	—
64	C₄×D₄	C₂×D₂₁₁	1	2	yes	—
65	#107	Dic₂₁₁	1	1	yes	characteristic
66	C₄⋊C₈	—	211	2	no	—
67	#102	Dic₂₁₁	1	2	yes	—
68	C₂²⋊C₈	—	422	2	no	—
69	C₄×C₈	—	211	1	no	—
70	C₄×C₈	—	211	1	no	—
71	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₂₁₁	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	844	4	no	—
76	C₂×C₈	—	422	2	no	—
77	C₂×C₈	—	422	1	no	—
78	C₂×C₈	—	211	2	no	—
79	C₂×C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₂₁₁	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
91	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
92	C₂×C₄	#16809961	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	844	1	no	—
95	C₈	—	844	2	no	—
96	C₈	—	422	2	no	—
97	C₈	#16809946	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₂₁₁	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂×#16809946	1	1	yes	characteristic
105	C₄	#20269984	1	1	yes	characteristic
106	C₄	#20270001	1	2	yes	—
107	C₂	—	8	2	no	—
108	C₂	#27213224	1	1	yes	characteristic
109	C₂	#27213258	1	1	yes	characteristic
110	C₂	#27213268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213224

C₂

yes

maximal, complemented

#27212897

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, complemented

#27212867

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₂₁₁

C₄

yes

characteristic

C₄×D₄×C₂₁₁

C₂²

yes

—

#107×C₂₁₁

C₄

yes

characteristic

#102×C₂₁₁

C₄

yes

—

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×D₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

D₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

M₁₆

yes

complemented

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×M₁₆

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1594

—

211

maximal, Sylow

C₄×D₈

D₂₁₁

yes

characteristic

#436

—

211

—

#334

—

211

—

D₄⋊C₈

—

211

—

#322

—

211

—

C₂×D₈

Dic₂₁₁

yes

characteristic

C₂×M₁₆

—

422

—

C₄×D₄

C₂×D₂₁₁

yes

—

#107

Dic₂₁₁

yes

characteristic

C₄⋊C₈

—

211

—

#102

Dic₂₁₁

yes

—

C₂²⋊C₈

—

422

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₄×C₈

C₂×D₂₁₁

yes

characteristic

C₂×D₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

844

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₂×Dic₂₁₁

yes

characteristic

C₂×C₈

C₂×Dic₂₁₁

yes

characteristic

C₄⋊C₄

C₂×Dic₂₁₁

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

D₄×D₂₁₁

yes

characteristic

C₂×C₄

#16809961

yes

characteristic

C₈

—

C₈

—

844

—

C₈

—

844

—

C₈

—

422

—

C₈

#16809946

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₂₁₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#16809946

yes

characteristic

105

C₄

#20269984

yes

characteristic

106

C₄

#20270001

yes

—

107

C₂

—

108

C₂

#27213224

yes

characteristic

109

C₂

#27213258

yes

characteristic

110

C₂

#27213268

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	C₄×D₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₄×D₄
prime radical	C₂×D₈×C₂₁₁	C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	C₄×D₈	D₂₁₁
hypocenter	C₂₁₁	#1594
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₁₁

Fitting

C₄×D₈×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₄×D₄

prime radical

C₂×D₈×C₂₁₁

C₄

cosocle

C₄²×C₂₁₁

C₂³

hypercenter

C₄×D₈

D₂₁₁

hypocenter

C₂₁₁

#1594

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #27213224	C₂	semidirect
2. #27212897	C₂	semidirect
3. D₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
4. #27212867	C₂	semidirect
41. C₈×C₂₁₁	M₁₆	semidirect
55. C₂₁₁	#1594	semidirect
97. C₈	#16809946	semidirect

H ≅ G/N

type

1. #27213224

C₂

semidirect

2. #27212897

C₂

semidirect

3. D₄⋊(C₂₁₁⋊C₈)

C₂

semidirect

4. #27212867

C₂

semidirect

41. C₈×C₂₁₁

M₁₆

semidirect

55. C₂₁₁

#1594

semidirect

97. C₈

#16809946

semidirect

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

subgroup	index
#27213224	2
#27212897	2
D₄⋊(C₂₁₁⋊C₈)	2
#27212867	2
C₄×D₈×C₂₁₁	2
#1594	211

subgroup

index

D₄⋊(C₂₁₁⋊C₈)

C₄×D₈×C₂₁₁

211

Subgroups — #41012312

Conjugacy classes of subgroups

Complements

Maximal subgroups