Group #19105548 — subgroups

Subgroups — C₄₁×#1768 #19105548

Order 5248.

260 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 168 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×C₄⋊C₈×C₄₁	C₂	1	2	yes	maximal
2	C₂×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#376	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#336	C₂	1	2	yes	maximal
6	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₁	C₄	1	8	yes	complemented
12	#97×C₄₁	—	2	2	no	—
13	C₄×C₈×C₄₁	—	2	2	no	—
14	#95×C₄₁	C₄	1	4	yes	—
15	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₄₁	—	2	2	no	—
20	C₂×C₈×C₄₁	—	2	2	no	—
21	C₂×C₈×C₄₁	—	2	2	no	—
22	C₂×C₈×C₄₁	—	2	2	no	—
23	C₂×C₈×C₄₁	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₄₁	C₂×C₄	1	4	yes	—
25	C₄²×C₄₁	—	2	2	no	—
26	C₄²×C₄₁	C₂×C₄	1	4	yes	—
27	C₄²×C₄₁	D₄	1	2	yes	—
28	C₄²×C₄₁	Q₈	1	2	yes	—
29	C₂³×C₄₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₁	—	2	4	no	—
31	C₂×C₄×C₄₁	—	2	4	no	—
32	C₂×C₄×C₄₁	—	2	2	no	—
33	C₂×C₄×C₄₁	—	2	4	no	—
34	C₂×C₄×C₄₁	C₄²	1	4	yes	—
35	C₂×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₄₁	#41	1	2	yes	—
40	C₈×C₄₁	—	2	4	no	—
41	C₈×C₄₁	—	2	4	no	—
42	C₂²×C₄₁	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄₁	#117	1	1	yes	characteristic
45	C₂²×C₄₁	#131	1	4	yes	—
46	C₄×C₄₁	—	2	4	no	—
47	C₄×C₄₁	—	2	4	no	—
48	C₄×C₄₁	—	4	2	no	—
49	C₄×C₄₁	C₄×D₄	1	2	yes	—
50	C₄×C₄₁	C₄×Q₈	1	2	yes	—
51	#1768	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄₁	#405	1	2	yes	—
54	C₂×C₄₁	#424	1	2	yes	—
55	C₂×C₄₁	#433	1	2	yes	—
56	C₂×C₄⋊C₈	C₂×C₄₁	1	2	yes	—
57	C₂×#97	C₂×C₄₁	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
59	#376	C₂×C₄₁	1	1	yes	characteristic
60	#336	C₂×C₄₁	1	2	yes	—
61	C₄₁	#1768	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
63	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
64	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
65	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
67	C₄⋊C₈	C₄×C₄₁	1	8	yes	complemented
68	#97	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	C₄×C₄₁	1	4	yes	—
71	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
72	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
73	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
74	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	C₂×C₄×C₄₁	1	4	yes	—
80	C₂×C₈	C₂×C₄×C₄₁	1	4	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂×C₄×C₄₁	1	4	yes	—
83	C₄²	D₄×C₄₁	1	2	yes	—
84	C₄²	Q₈×C₄₁	1	2	yes	—
85	C₂³	C₂²×C₄×C₄₁	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄²×C₄₁	1	4	yes	—
91	C₂×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
92	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
93	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
95	C₂×C₄	#41×C₄₁	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	4	no	—
98	C₂²	C₂×C₄²×C₄₁	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
100	C₂²	#117×C₄₁	1	1	yes	characteristic
101	C₂²	#131×C₄₁	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₄×C₄₁	1	2	yes	—
106	C₄	C₄×Q₈×C₄₁	1	2	yes	—
107	C₂	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
108	C₂	C₄₁×#405	1	2	yes	—
109	C₂	C₄₁×#424	1	2	yes	—
110	C₂	C₄₁×#433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₄₁

C₂

yes

maximal

C₂×#97×C₄₁

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₄₁

C₂

yes

maximal, characteristic

C₄₁×#376

C₂

yes

maximal, characteristic

C₄₁×#336

C₂

yes

maximal

C₂²×C₈×C₄₁

C₂²

yes

—

C₂²×C₈×C₄₁

C₂²

yes

—

C₂×C₄²×C₄₁

C₂²

yes

characteristic

C₂×C₄²×C₄₁

C₂²

yes

characteristic

C₂×C₄²×C₄₁

C₂²

yes

characteristic

C₄⋊C₈×C₄₁

C₄

yes

complemented

#97×C₄₁

—

C₄×C₈×C₄₁

—

#95×C₄₁

C₄

yes

—

C₂²×C₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

C₂×C₄

yes

—

C₂×C₈×C₄₁

C₂×C₄

yes

—

C₄²×C₄₁

—

C₄²×C₄₁

C₂×C₄

yes

—

C₄²×C₄₁

D₄

yes

—

C₄²×C₄₁

Q₈

yes

—

C₂³×C₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₄²

yes

—

C₂×C₄×C₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₄₁

C₂²×C₄

yes

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₁

#41

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₂²×C₄₁

C₂×C₄²

yes

characteristic

C₂²×C₄₁

C₂×C₄⋊C₄

yes

C₂²×C₄₁

yes

C₂²×C₄₁

yes

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₄×D₄

yes

—

C₄×C₄₁

C₄×Q₈

yes

—

#1768

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄₁

#405

yes

—

C₂×C₄₁

#424

yes

—

C₂×C₄₁

#433

yes

—

C₂×C₄⋊C₈

C₂×C₄₁

yes

—

C₂×#97

C₂×C₄₁

yes

characteristic

C₂×C₄×C₈

C₂×C₄₁

yes

characteristic

#376

C₂×C₄₁

yes

characteristic

#336

C₂×C₄₁

yes

—

C₄₁

#1768

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₄₁

yes

—

C₂²×C₈

C₂²×C₄₁

yes

—

C₂×C₄²

C₂²×C₄₁

yes

characteristic

C₂×C₄²

C₂²×C₄₁

yes

characteristic

C₂×C₄²

C₂²×C₄₁

yes

characteristic

C₄⋊C₈

C₄×C₄₁

yes

complemented

#97

—

C₄×C₈

—

#95

C₄×C₄₁

yes

—

C₂²×C₄

C₂×C₄×C₄₁

yes

—

C₂²×C₄

C₂×C₄×C₄₁

yes

—

C₂²×C₄

C₂×C₄×C₄₁

yes

—

C₂²×C₄

C₂³×C₄₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₄₁

yes

—

C₂×C₈

C₂×C₄×C₄₁

yes

—

C₄²

—

C₄²

C₂×C₄×C₄₁

yes

—

C₄²

D₄×C₄₁

yes

—

C₄²

Q₈×C₄₁

yes

—

C₂³

C₂²×C₄×C₄₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₄₁

yes

—

C₂×C₄

C₄⋊C₄×C₄₁

yes

—

C₂×C₄

C₂²×C₄×C₄₁

yes

—

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

C₂×C₄

C₂×Q₈×C₄₁

yes

characteristic

C₂×C₄

#41×C₄₁

yes

—

C₈

—

C₈

—

C₂²

C₂×C₄²×C₄₁

yes

characteristic

C₂²

C₂×C₄⋊C₄×C₄₁

yes

characteristic

100

C₂²

#117×C₄₁

yes

characteristic

101

C₂²

#131×C₄₁

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₄×C₄₁

yes

—

106

C₄

C₄×Q₈×C₄₁

yes

—

107

C₂

C₄×C₄⋊C₄×C₄₁

yes

characteristic

108

C₂

C₄₁×#405

yes

—

109

C₂

C₄₁×#424

yes

—

110

C₂

C₄₁×#433

yes

—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂³
commutator	C₂²	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₄₁

C₂³

commutator

C₂²

C₂×C₄²×C₄₁

Frattini, cosocle

C₂²×C₄

C₂³×C₄₁

socle, prime radical

C₂³×C₄₁

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₄₁ ◁ G

N	H ≅ G/N	type
11. C₄⋊C₈×C₄₁	C₄	semidirect
51. #1768	C₄₁	direct (N × H)
61. C₄₁	#1768	direct (N × H)
67. C₄⋊C₈	C₄×C₄₁	semidirect

H ≅ G/N

type

11. C₄⋊C₈×C₄₁

C₄

semidirect

51. #1768

C₄₁

direct (N × H)

61. C₄₁

#1768

direct (N × H)

67. C₄⋊C₈

C₄×C₄₁

semidirect

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

subgroup	index
C₂×C₄⋊C₈×C₄₁	2
C₂×#97×C₄₁	2
C₂×C₄×C₈×C₄₁	2
C₄₁×#376	2
C₄₁×#336	2
#1768	41

subgroup

index

C₂×C₄⋊C₈×C₄₁

C₂×#97×C₄₁

C₂×C₄×C₈×C₄₁

C₄₁×#376

C₄₁×#336

#1768

Subgroups — C₄₁×#1768 #19105548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#1768 #19105548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1768 #19105548