Group #19105400 — subgroups

Subgroups — C₄₁×#1620 #19105400

Order 5248.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#460	C₂	1	2	yes	maximal, complemented
4	C₄₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₄₁	—	2	2	no	—
7	C₂×D₈×C₄₁	—	2	2	no	—
8	#127×C₄₁	C₂²	1	1	yes	characteristic
9	#127×C₄₁	C₂²	1	1	yes	characteristic
10	#124×C₄₁	—	2	2	no	—
11	#123×C₄₁	—	2	2	no	—
12	C₄×Q₈×C₄₁	C₂²	1	2	yes	—
13	C₄×D₄×C₄₁	—	2	2	no	—
14	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	#102×C₄₁	—	2	2	no	—
16	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₁	D₄	1	2	yes	—
18	C₂×D₄×C₄₁	—	2	2	no	—
19	C₂×D₄×C₄₁	—	2	2	no	—
20	C₂×D₄×C₄₁	—	2	1	no	—
21	C₂×D₄×C₄₁	D₄	1	2	yes	—
22	C₂²×C₄×C₄₁	—	2	2	no	—
23	SD₁₆×C₄₁	—	4	4	no	—
24	D₈×C₄₁	—	4	4	no	—
25	C₂×C₈×C₄₁	—	2	1	no	—
26	C₂×C₈×C₄₁	—	2	2	no	—
27	C₄⋊C₄×C₄₁	—	2	2	no	—
28	C₄⋊C₄×C₄₁	D₄	1	2	yes	—
29	C₂²⋊C₄×C₄₁	—	2	4	no	—
30	C₂²⋊C₄×C₄₁	—	2	2	no	—
31	C₄²×C₄₁	—	2	2	no	—
32	C₄²×C₄₁	C₂³	1	1	yes	characteristic
33	C₂³×C₄₁	—	2	2	no	—
34	C₂³×C₄₁	—	4	1	no	—
35	Q₈×C₄₁	—	2	2	no	—
36	Q₈×C₄₁	—	2	2	no	—
37	D₄×C₄₁	—	4	2	no	—
38	D₄×C₄₁	—	4	2	no	—
39	D₄×C₄₁	—	4	2	no	—
40	D₄×C₄₁	—	4	2	no	—
41	D₄×C₄₁	—	4	2	no	—
42	D₄×C₄₁	—	4	2	no	—
43	C₂×C₄×C₄₁	—	2	2	no	—
44	C₂×C₄×C₄₁	—	2	2	no	—
45	C₂×C₄×C₄₁	—	8	2	no	—
46	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
48	C₈×C₄₁	—	2	2	no	—
49	C₈×C₄₁	—	4	2	no	—
50	C₂²×C₄₁	—	4	4	no	—
51	C₂²×C₄₁	—	4	2	no	—
52	C₂²×C₄₁	—	8	2	no	—
53	C₂²×C₄₁	—	8	1	no	—
54	C₂²×C₄₁	#120	1	1	yes	characteristic
55	C₄×C₄₁	—	2	1	no	—
56	C₄×C₄₁	—	4	2	no	—
57	C₄×C₄₁	—	4	2	no	—
58	C₄×C₄₁	#135	1	2	yes	—
59	C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
60	#1620	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₄₁	—	8	2	no	—
62	C₂×C₄₁	—	16	1	no	—
63	C₂×C₄₁	#449	1	2	yes	—
64	C₂×C₄₁	#453	1	1	yes	characteristic
65	#559	C₂×C₄₁	1	1	yes	characteristic, complemented
66	#493	C₂×C₄₁	1	1	yes	characteristic
67	#460	C₂×C₄₁	1	2	yes	complemented
68	#331	C₂×C₄₁	1	1	yes	characteristic, complemented
69	Q₈⋊C₈	C₂×C₄₁	1	2	yes	complemented
70	C₄₁	#1620	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂²×C₄₁	1	1	yes	characteristic
74	#127	C₂²×C₄₁	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂²×C₄₁	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
80	#102	—	2	2	no	—
81	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
82	C₂×Q₈	D₄×C₄₁	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	D₄×C₄₁	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	4	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	D₄×C₄₁	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂³×C₄₁	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	#120×C₄₁	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#135×C₄₁	1	2	yes	—
124	C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	C₄₁×#449	1	2	yes	—
128	C₂	C₄₁×#453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₁×#559

C₂

yes

maximal, characteristic, complemented

C₄₁×#493

C₂

yes

maximal, characteristic

C₄₁×#460

C₂

yes

maximal, complemented

C₄₁×#331

C₂

yes

maximal, characteristic, complemented

C₄₁×Q₈⋊C₈

C₂

yes

maximal, complemented

C₂×SD₁₆×C₄₁

—

C₂×D₈×C₄₁

—

#127×C₄₁

C₂²

yes

characteristic

#127×C₄₁

C₂²

yes

characteristic

#124×C₄₁

—

#123×C₄₁

—

C₄×Q₈×C₄₁

C₂²

yes

—

C₄×D₄×C₄₁

—

C₄⋊C₈×C₄₁

C₂²

yes

—

#102×C₄₁

—

C₄×C₈×C₄₁

C₂²

yes

characteristic

C₂×Q₈×C₄₁

D₄

yes

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

D₄

yes

—

C₂²×C₄×C₄₁

—

SD₁₆×C₄₁

—

D₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₄⋊C₄×C₄₁

—

C₄⋊C₄×C₄₁

D₄

yes

—

C₂²⋊C₄×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₄²×C₄₁

—

C₄²×C₄₁

C₂³

yes

characteristic

C₂³×C₄₁

—

C₂³×C₄₁

—

Q₈×C₄₁

—

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

C₂×D₄

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

#120

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

#135

yes

—

C₄×C₄₁

AGL(1,ℤ/8)

yes

—

#1620

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

#449

yes

—

C₂×C₄₁

#453

yes

characteristic

#559

C₂×C₄₁

yes

characteristic, complemented

#493

C₂×C₄₁

yes

characteristic

#460

C₂×C₄₁

yes

complemented

#331

C₂×C₄₁

yes

characteristic, complemented

Q₈⋊C₈

C₂×C₄₁

yes

complemented

C₄₁

#1620

yes

characteristic, direct product factor, Sylow

C₂×SD₁₆

—

C₂×D₈

—

#127

C₂²×C₄₁

yes

characteristic

#127

C₂²×C₄₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂²×C₄₁

yes

—

C₄×D₄

—

C₄⋊C₈

C₂²×C₄₁

yes

—

#102

—

C₄×C₈

C₂²×C₄₁

yes

characteristic

C₂×Q₈

D₄×C₄₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₄₁

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₄₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₄₁

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

112

C₂×C₄

C₂×D₄×C₄₁

yes

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#120×C₄₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#135×C₄₁

yes

—

124

C₄

AGL(1,ℤ/8)×C₄₁

yes

—

125

C₂

—

126

C₂

—

127

C₂

C₄₁×#449

yes

—

128

C₂

C₄₁×#453

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₄₁
prime radical	C₄₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₁

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₄₁

prime radical

C₄₁×#493

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

N	H ≅ G/N	type
1. C₄₁×#559	C₂	semidirect
3. C₄₁×#460	C₂	semidirect
4. C₄₁×#331	C₂	semidirect
5. C₄₁×Q₈⋊C₈	C₂	semidirect
60. #1620	C₄₁	direct (N × H)
65. #559	C₂×C₄₁	semidirect
67. #460	C₂×C₄₁	semidirect
68. #331	C₂×C₄₁	semidirect
69. Q₈⋊C₈	C₂×C₄₁	semidirect
70. C₄₁	#1620	direct (N × H)

H ≅ G/N

type

1. C₄₁×#559

C₂

semidirect

3. C₄₁×#460

C₂

semidirect

4. C₄₁×#331

C₂

semidirect

5. C₄₁×Q₈⋊C₈

C₂

semidirect

60. #1620

C₄₁

direct (N × H)

65. #559

C₂×C₄₁

semidirect

67. #460

C₂×C₄₁

semidirect

68. #331

C₂×C₄₁

semidirect

69. Q₈⋊C₈

C₂×C₄₁

semidirect

70. C₄₁

#1620

direct (N × H)

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

subgroup	index
C₄₁×#559	2
C₄₁×#493	2
C₄₁×#460	2
C₄₁×#331	2
C₄₁×Q₈⋊C₈	2
#1620	41

subgroup

index

C₄₁×#559

C₄₁×#493

C₄₁×#460

C₄₁×#331

C₄₁×Q₈⋊C₈

#1620

Subgroups — C₄₁×#1620 #19105400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#1620 #19105400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1620 #19105400