Group #19105450 — subgroups

Subgroups — C₄₁×#1670 #19105450

Order 5248.

156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₄₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#475	C₂	1	2	yes	maximal
3	C₈×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#335	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#332	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₄₁	—	2	1	no	—
7	C₂×Q₁₆×C₄₁	—	2	1	no	—
8	C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
9	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	#106×C₄₁	—	2	2	no	—
11	C₄⋊C₈×C₄₁	—	2	1	no	—
12	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
14	#103×C₄₁	—	2	2	no	—
15	C₄×C₈×C₄₁	—	2	1	no	—
16	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₁	—	2	2	no	—
18	C₂×Q₈×C₄₁	D₄	1	1	yes	characteristic
19	Q₁₆×C₄₁	—	4	2	no	—
20	Q₁₆×C₄₁	—	4	2	no	—
21	C₂×C₈×C₄₁	—	2	1	no	—
22	C₂×C₈×C₄₁	—	2	2	no	—
23	C₂×C₈×C₄₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₄₁	D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₄₁	—	2	1	no	—
26	C₄⋊C₄×C₄₁	—	2	2	no	—
27	C₄⋊C₄×C₄₁	—	2	2	no	—
28	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
29	C₄²×C₄₁	—	2	1	no	—
30	C₄²×C₄₁	C₂³	1	1	yes	characteristic
31	Q₈×C₄₁	—	2	1	no	—
32	Q₈×C₄₁	—	4	2	no	—
33	Q₈×C₄₁	—	4	2	no	—
34	Q₈×C₄₁	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₄₁	—	2	1	no	—
36	C₂×C₄×C₄₁	—	2	1	no	—
37	C₂×C₄×C₄₁	—	4	2	no	—
38	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₁	#41	1	1	yes	characteristic
41	C₈×C₄₁	—	2	1	no	—
42	C₈×C₄₁	—	4	2	no	—
43	C₈×C₄₁	—	4	1	no	—
44	C₈×C₄₁	SD₁₆	1	2	yes	—
45	C₂²×C₄₁	#121	1	1	yes	characteristic
46	C₄×C₄₁	—	2	2	no	—
47	C₄×C₄₁	—	2	1	no	—
48	C₄×C₄₁	—	4	1	no	—
49	C₄×C₄₁	—	8	2	no	—
50	C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₄₁	#135	1	1	yes	characteristic
53	C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	#1670	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₄₁	#460	1	1	yes	characteristic
56	C₂×C₄₁	#467	1	1	yes	characteristic
57	C₂×C₄₁	#473	1	1	yes	characteristic
58	#494	C₂×C₄₁	1	1	yes	characteristic
59	#475	C₂×C₄₁	1	2	yes	—
60	C₈×Q₈	C₂×C₄₁	1	1	yes	characteristic
61	#335	C₂×C₄₁	1	1	yes	characteristic
62	#332	C₂×C₄₁	1	2	yes	—
63	C₄₁	#1670	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×Q₁₆	—	2	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
67	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
68	#106	—	2	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
71	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
72	#103	—	2	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	D₄×C₄₁	1	1	yes	characteristic
77	Q₁₆	—	4	2	no	—
78	Q₁₆	—	4	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	D₄×C₄₁	1	1	yes	characteristic
82	C₂×C₈	D₄×C₄₁	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂³×C₄₁	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	Q₁₆×C₄₁	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
97	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
98	C₂×C₄	#41×C₄₁	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	2	no	—
101	C₈	—	4	1	no	—
102	C₈	SD₁₆×C₄₁	1	2	yes	—
103	C₂²	#121×C₄₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	8	2	no	—
108	C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
109	C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
110	C₄	#135×C₄₁	1	1	yes	characteristic
111	C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
112	C₂	C₄₁×#460	1	1	yes	characteristic
113	C₂	C₄₁×#467	1	1	yes	characteristic
114	C₂	C₄₁×#473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₁×#494

C₂

yes

maximal, characteristic

C₄₁×#475

C₂

yes

maximal

C₈×Q₈×C₄₁

C₂

yes

maximal, characteristic

C₄₁×#335

C₂

yes

maximal, characteristic

C₄₁×#332

C₂

yes

maximal

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₄⋊Q₈×C₄₁

C₂²

yes

—

C₄×Q₈×C₄₁

C₂²

yes

characteristic

#106×C₄₁

—

C₄⋊C₈×C₄₁

—

C₄⋊C₈×C₄₁

C₂²

yes

characteristic

C₄⋊C₈×C₄₁

C₂²

yes

—

#103×C₄₁

—

C₄×C₈×C₄₁

—

C₄×C₈×C₄₁

C₂²

yes

characteristic

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

D₄

yes

characteristic

Q₁₆×C₄₁

—

Q₁₆×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

D₄

yes

characteristic

C₂×C₈×C₄₁

D₄

yes

characteristic

C₄⋊C₄×C₄₁

—

C₄⋊C₄×C₄₁

—

C₄⋊C₄×C₄₁

—

C₄⋊C₄×C₄₁

D₄

yes

characteristic

C₄²×C₄₁

—

C₄²×C₄₁

C₂³

yes

characteristic

Q₈×C₄₁

—

Q₈×C₄₁

—

Q₈×C₄₁

—

Q₈×C₄₁

Q₁₆

yes

complemented

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

#41

yes

characteristic

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

SD₁₆

yes

—

C₂²×C₄₁

#121

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₂×SD₁₆

yes

characteristic

C₄×C₄₁

C₂×Q₁₆

yes

C₄×C₄₁

yes

C₄×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

yes

C₂×C₄₁

yes

C₂×C₄₁

yes

C₂×C₄₁

yes

characteristic

#475

C₂×C₄₁

yes

—

C₈×Q₈

C₂×C₄₁

yes

characteristic

#335

C₂×C₄₁

yes

characteristic

#332

C₂×C₄₁

yes

—

C₄₁

#1670

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₄₁

yes

—

C₄×Q₈

C₂²×C₄₁

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₄₁

yes

characteristic

C₄⋊C₈

C₂²×C₄₁

yes

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×C₄₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₄₁

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₄₁

yes

characteristic

C₂×C₈

D₄×C₄₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₄₁

yes

characteristic

C₄²

—

C₄²

C₂³×C₄₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₄₁

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

C₂×C₄

#41×C₄₁

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

SD₁₆×C₄₁

yes

—

103

C₂²

#121×C₄₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₂×SD₁₆×C₄₁

yes

characteristic

109

C₄

C₂×Q₁₆×C₄₁

yes

characteristic

110

C₄

#135×C₄₁

yes

characteristic

111

C₄

AGL(1,ℤ/8)×C₄₁

yes

characteristic

112

C₂

C₄₁×#460

yes

characteristic

113

C₂

C₄₁×#467

yes

characteristic

114

C₂

C₄₁×#473

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₄₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₄₁

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₄₁

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

N	H ≅ G/N	type
34. Q₈×C₄₁	Q₁₆	semidirect
54. #1670	C₄₁	direct (N × H)
63. C₄₁	#1670	direct (N × H)
92. Q₈	Q₁₆×C₄₁	semidirect

H ≅ G/N

type

34. Q₈×C₄₁

Q₁₆

semidirect

54. #1670

C₄₁

direct (N × H)

63. C₄₁

#1670

direct (N × H)

92. Q₈

Q₁₆×C₄₁

semidirect

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

subgroup	index
C₄₁×#494	2
C₄₁×#475	2
C₈×Q₈×C₄₁	2
C₄₁×#335	2
C₄₁×#332	2
#1670	41

subgroup

index

C₄₁×#494

C₄₁×#475

C₈×Q₈×C₄₁

C₄₁×#335

C₄₁×#332

#1670

Subgroups — C₄₁×#1670 #19105450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#1670 #19105450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1670 #19105450