Group #19105878 — subgroups

Subgroups — C₄₁×#2098 #19105878

Order 5248.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#424	C₂	1	2	yes	maximal, complemented
3	C₄₁×#421	C₂	1	2	yes	maximal
4	C₄₁×#408	C₂	1	1	yes	maximal, characteristic
5	C₄₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	#117×C₄₁	C₂²	1	2	yes	—
11	#108×C₄₁	—	2	2	no	—
12	C₄⋊C₈×C₄₁	—	2	4	no	—
13	#104×C₄₁	—	2	4	no	—
14	C₂²⋊C₈×C₄₁	—	2	2	no	—
15	#41×C₄₁	—	2	2	no	—
16	#41×C₄₁	—	2	2	no	—
17	C₂×Q₈×C₄₁	Q₈	1	1	yes	characteristic
18	C₂×D₄×C₄₁	—	2	1	no	—
19	C₂×D₄×C₄₁	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₁	—	2	1	no	—
21	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₄₁	—	2	2	no	—
23	M₁₆×C₄₁	—	2	2	no	—
24	M₁₆×C₄₁	—	2	2	no	—
25	C₂×C₈×C₄₁	—	2	1	no	—
26	C₂×C₈×C₄₁	—	2	2	no	—
27	C₂×C₈×C₄₁	—	2	2	no	—
28	C₂×C₈×C₄₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₄₁	—	2	2	no	—
30	C₂²⋊C₄×C₄₁	—	2	2	no	—
31	C₄²×C₄₁	D₄	1	4	yes	—
32	C₂³×C₄₁	—	2	1	no	—
33	C₂³×C₄₁	#41	1	1	yes	characteristic
34	Q₈×C₄₁	—	2	2	no	—
35	D₄×C₄₁	—	2	2	no	—
36	D₄×C₄₁	—	4	2	no	—
37	C₂×C₄×C₄₁	—	2	1	no	—
38	C₂×C₄×C₄₁	—	2	4	no	—
39	C₂×C₄×C₄₁	—	4	1	no	—
40	C₂×C₄×C₄₁	—	4	1	no	—
41	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₄₁	#41	1	2	yes	—
45	C₈×C₄₁	—	2	2	no	—
46	C₈×C₄₁	—	4	1	no	—
47	C₈×C₄₁	—	4	2	no	—
48	C₂²×C₄₁	—	4	1	no	—
49	C₂²×C₄₁	—	4	2	no	—
50	C₂²×C₄₁	—	4	1	no	—
51	C₂²×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₄₁	#123	1	2	yes	—
53	C₄×C₄₁	—	4	4	no	—
54	C₄×C₄₁	—	4	1	no	—
55	C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
56	C₄×C₄₁	#123	1	1	yes	characteristic
57	C₄×C₄₁	#127	1	1	yes	characteristic
58	#2098	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₄₁	—	2	2	no	—
60	C₂×C₄₁	—	2	1	no	—
61	C₂×C₄₁	—	8	1	no	—
62	C₂×C₄₁	#399	1	1	yes	characteristic
63	#430	C₂×C₄₁	1	1	yes	characteristic, complemented
64	#424	C₂×C₄₁	1	2	yes	complemented
65	#421	C₂×C₄₁	1	2	yes	—
66	#408	C₂×C₄₁	1	1	yes	characteristic
67	H(3,ℤ/4)	C₂×C₄₁	1	1	yes	characteristic, complemented
68	C₄₁	#2098	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×#41	C₂²×C₄₁	1	1	yes	characteristic
70	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
71	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
73	#117	C₂²×C₄₁	1	2	yes	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	2	4	no	—
76	#104	—	2	4	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Q₈×C₄₁	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Q₈×C₄₁	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	D₄×C₄₁	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₄×C₄₁	1	4	yes	—
95	C₂³	—	2	1	no	—
96	C₂³	#41×C₄₁	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
105	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
106	C₂×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
107	C₂×C₄	#41×C₄₁	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	4	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
115	C₂²	#123×C₄₁	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
119	C₄	#123×C₄₁	1	1	yes	characteristic
120	C₄	#127×C₄₁	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	C₄₁×#399	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₁×#430

C₂

yes

maximal, characteristic, complemented

C₄₁×#424

C₂

yes

maximal, complemented

C₄₁×#421

C₂

yes

maximal

C₄₁×#408

C₂

yes

maximal, characteristic

C₄₁×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₄₁

C₂²

yes

characteristic

C₂×M₁₆×C₄₁

C₂²

yes

—

C₂×M₁₆×C₄₁

C₂²

yes

characteristic

C₂²×C₈×C₄₁

C₂²

yes

characteristic

#117×C₄₁

C₂²

yes

—

#108×C₄₁

—

C₄⋊C₈×C₄₁

—

#104×C₄₁

—

C₂²⋊C₈×C₄₁

—

#41×C₄₁

—

#41×C₄₁

—

C₂×Q₈×C₄₁

Q₈

yes

characteristic

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

Q₈

yes

characteristic

C₂²×C₄×C₄₁

—

C₂²×C₄×C₄₁

C₂³

yes

characteristic

M₁₆×C₄₁

—

M₁₆×C₄₁

—

M₁₆×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

D₄

yes

—

C₄⋊C₄×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₄²×C₄₁

D₄

yes

—

C₂³×C₄₁

—

C₂³×C₄₁

#41

yes

characteristic

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂×D₄

yes

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₁

#41

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₁

#123

yes

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₂²⋊Q₈

yes

—

C₄×C₄₁

yes

C₄×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

#399

yes

characteristic

#430

C₂×C₄₁

yes

characteristic, complemented

#424

C₂×C₄₁

yes

complemented

#421

C₂×C₄₁

yes

—

#408

C₂×C₄₁

yes

characteristic

H(3,ℤ/4)

C₂×C₄₁

yes

characteristic, complemented

C₄₁

#2098

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₄₁

yes

characteristic

C₂×M₁₆

C₂²×C₄₁

yes

—

C₂×M₁₆

C₂²×C₄₁

yes

characteristic

C₂²×C₈

C₂²×C₄₁

yes

characteristic

#117

C₂²×C₄₁

yes

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Q₈×C₄₁

yes

characteristic

C₂×D₄

—

C₂×D₄

Q₈×C₄₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂³×C₄₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₄₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₄₁

yes

—

C₂³

—

C₂³

#41×C₄₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₄₁

yes

—

105

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

106

C₂×C₄

C₂×Q₈×C₄₁

yes

characteristic

107

C₂×C₄

#41×C₄₁

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²⋊Q₈×C₄₁

yes

characteristic

115

C₂²

#123×C₄₁

yes

—

116

C₄

—

117

C₄

—

118

C₄

C₂²⋊Q₈×C₄₁

yes

—

119

C₄

#123×C₄₁

yes

characteristic

120

C₄

#127×C₄₁

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₄₁×#399

yes

characteristic

	subgroup	G/H
center	C₄×C₄₁	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₄₁
socle	C₂×C₄₁	#399
prime radical	C₂×#41×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

subgroup

G/H

center

C₄×C₄₁

#127

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₄₁

socle

C₂×C₄₁

#399

prime radical

C₂×#41×C₄₁

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

N	H ≅ G/N	type
1. C₄₁×#430	C₂	semidirect
2. C₄₁×#424	C₂	semidirect
5. C₄₁×H(3,ℤ/4)	C₂	semidirect
58. #2098	C₄₁	direct (N × H)
63. #430	C₂×C₄₁	semidirect
64. #424	C₂×C₄₁	semidirect
67. H(3,ℤ/4)	C₂×C₄₁	semidirect
68. C₄₁	#2098	direct (N × H)

H ≅ G/N

type

1. C₄₁×#430

C₂

semidirect

2. C₄₁×#424

C₂

semidirect

5. C₄₁×H(3,ℤ/4)

C₂

semidirect

58. #2098

C₄₁

direct (N × H)

63. #430

C₂×C₄₁

semidirect

64. #424

C₂×C₄₁

semidirect

67. H(3,ℤ/4)

C₂×C₄₁

semidirect

68. C₄₁

#2098

direct (N × H)

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

subgroup	index
C₄₁×#430	2
C₄₁×#424	2
C₄₁×#421	2
C₄₁×#408	2
C₄₁×H(3,ℤ/4)	2
#2098	41

subgroup

index

C₄₁×#430

C₄₁×#424

C₄₁×#421

C₄₁×#408

C₄₁×H(3,ℤ/4)

#2098

Subgroups — C₄₁×#2098 #19105878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#2098 #19105878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2098 #19105878