Group #41009175 — subgroups

Subgroups — #1555×C₂₁₁ #41009175

Order 27008.

332 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 96 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×#127×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#432×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#331×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₂₁₁	—	2	2	no	—
5	C₂²×D₄×C₂₁₁	C₄	1	2	yes	—
6	#127×C₂₁₁	—	2	2	no	—
7	#127×C₂₁₁	C₄	1	2	yes	—
8	#127×C₂₁₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
11	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
12	#97×C₂₁₁	—	2	2	no	—
13	C₂⁴×C₂₁₁	—	2	2	no	—
14	C₂×D₄×C₂₁₁	—	2	2	no	—
15	C₂×D₄×C₂₁₁	—	2	4	no	—
16	C₂×D₄×C₂₁₁	—	2	4	no	—
17	C₂×D₄×C₂₁₁	—	4	4	no	—
18	C₂×D₄×C₂₁₁	—	4	8	no	—
19	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
22	C₂×C₈×C₂₁₁	—	2	4	no	—
23	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂³×C₂₁₁	—	2	4	no	—
27	C₂³×C₂₁₁	—	4	8	no	—
28	C₂³×C₂₁₁	—	4	4	no	—
29	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₂₁₁	—	4	4	no	—
31	D₄×C₂₁₁	—	4	4	no	—
32	D₄×C₂₁₁	—	4	8	no	—
33	D₄×C₂₁₁	—	4	8	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	4	no	—
36	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
40	C₈×C₂₁₁	—	4	4	no	—
41	C₂²×C₂₁₁	—	2	2	no	—
42	C₂²×C₂₁₁	—	4	4	no	—
43	C₂²×C₂₁₁	—	4	8	no	—
44	C₂²×C₂₁₁	—	8	8	no	—
45	C₂²×C₂₁₁	#100	1	2	yes	—
46	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₂₁₁	—	2	1	no	—
48	C₄×C₂₁₁	—	2	1	no	—
49	C₄×C₂₁₁	—	2	2	no	—
50	C₄×C₂₁₁	AGL(1,ℤ/8)	1	4	yes	—
51	C₂×C₂₁₁	—	2	2	no	—
52	C₂×C₂₁₁	—	8	4	no	—
53	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
54	C₂×C₂₁₁	#418	1	2	yes	—
55	C₂₁₁	#1555	1	1	yes	characteristic, direct product factor, Sylow
56	#1555	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×#127	C₂×C₂₁₁	1	1	yes	characteristic
58	#432	C₂×C₂₁₁	1	2	yes	complemented
59	#331	C₂×C₂₁₁	1	4	yes	complemented
60	C₂²×D₄	—	2	2	no	—
61	C₂²×D₄	C₄×C₂₁₁	1	2	yes	—
62	#127	—	2	2	no	—
63	#127	C₄×C₂₁₁	1	2	yes	—
64	#127	C₂²×C₂₁₁	1	2	yes	—
65	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
67	C₂²⋊C₈	—	2	2	no	—
68	#97	—	2	2	no	—
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
76	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
77	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
80	C₄²	D₄×C₂₁₁	1	2	yes	—
81	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
93	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
94	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#100×C₂₁₁	1	2	yes	—
102	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	AGL(1,ℤ/8)×C₂₁₁	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#100×C₂₁₁	1	1	yes	characteristic
110	C₂	#418×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₂₁₁

C₂

yes

maximal, characteristic

#432×C₂₁₁

C₂

yes

maximal, complemented

#331×C₂₁₁

C₂

yes

maximal, complemented

C₂²×D₄×C₂₁₁

—

C₂²×D₄×C₂₁₁

C₄

yes

—

#127×C₂₁₁

—

#127×C₂₁₁

C₄

yes

—

#127×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

C₂²

yes

complemented

C₂²⋊C₈×C₂₁₁

—

#97×C₂₁₁

—

C₂⁴×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×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

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#100

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×#100

yes

characteristic

C₂×C₂₁₁

#418

yes

—

C₂₁₁

#1555

yes

characteristic, direct product factor, Sylow

#1555

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#127

C₂×C₂₁₁

yes

characteristic

#432

C₂×C₂₁₁

yes

complemented

#331

C₂×C₂₁₁

yes

complemented

C₂²×D₄

—

C₂²×D₄

C₄×C₂₁₁

yes

—

#127

—

#127

C₄×C₂₁₁

yes

—

#127

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₄⋊C₈

C₂²×C₂₁₁

yes

complemented

C₂²⋊C₈

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂×C₈

—

C₄²

C₂×C₄×C₂₁₁

yes

characteristic

C₄²

D₄×C₂₁₁

yes

—

C₄²

C₂³×C₂₁₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₁₁

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#100×C₂₁₁

yes

—

102

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

AGL(1,ℤ/8)×C₂₁₁

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#100×C₂₁₁

yes

characteristic

110

C₂

#418×C₂₁₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₄²	C₂³×C₂₁₁
prime radical	C₂×#127×C₂₁₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₄²

C₂³×C₂₁₁

prime radical

C₂×#127×C₂₁₁

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. #432×C₂₁₁	C₂	semidirect
3. #331×C₂₁₁	C₂	semidirect
10. C₄⋊C₈×C₂₁₁	C₂²	semidirect
55. C₂₁₁	#1555	direct (N × H)
56. #1555	C₂₁₁	direct (N × H)
58. #432	C₂×C₂₁₁	semidirect
59. #331	C₂×C₂₁₁	semidirect
66. C₄⋊C₈	C₂²×C₂₁₁	semidirect

H ≅ G/N

type

2. #432×C₂₁₁

C₂

semidirect

3. #331×C₂₁₁

C₂

semidirect

10. C₄⋊C₈×C₂₁₁

C₂²

semidirect

55. C₂₁₁

#1555

direct (N × H)

56. #1555

C₂₁₁

direct (N × H)

58. #432

C₂×C₂₁₁

semidirect

59. #331

C₂×C₂₁₁

semidirect

66. C₄⋊C₈

C₂²×C₂₁₁

semidirect

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

subgroup	index
C₂×#127×C₂₁₁	2
#432×C₂₁₁	2
#331×C₂₁₁	2
#1555	211

subgroup

index

C₂×#127×C₂₁₁

#432×C₂₁₁

#331×C₂₁₁

#1555

211

Subgroups — #1555×C₂₁₁ #41009175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1555×C211 #41009175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1555×C₂₁₁ #41009175