Group #19105302 — subgroups

Subgroups — C₂×C₄₁×#329 #19105302

Order 5248.

284 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 104 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×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#97×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₄₁×#329	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₄₁	C₄	1	1	yes	characteristic
5	C₂²×D₄×C₄₁	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₄₁	—	2	2	no	—
7	#124×C₄₁	—	2	2	no	—
8	#124×C₄₁	C₄	1	2	yes	—
9	#124×C₄₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
11	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	#97×C₄₁	—	2	2	no	—
13	#97×C₄₁	C₂²	1	4	yes	complemented
14	C₂⁴×C₄₁	—	2	1	no	—
15	C₂×Q₈×C₄₁	—	2	1	no	—
16	C₂×Q₈×C₄₁	—	4	2	no	—
17	C₂×Q₈×C₄₁	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₄₁	—	2	1	no	—
19	C₂×D₄×C₄₁	—	4	2	no	—
20	C₂×D₄×C₄₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₄₁	—	2	1	no	—
22	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₁	D₄	1	2	yes	—
24	C₂×C₈×C₄₁	—	2	8	no	—
25	C₂×C₈×C₄₁	—	2	4	no	—
26	C₂²⋊C₄×C₄₁	—	2	4	no	—
27	C₂²⋊C₄×C₄₁	—	2	4	no	—
28	C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₁	D₄	1	2	yes	—
30	C₄²×C₄₁	C₂³	1	1	yes	characteristic
31	C₂³×C₄₁	—	2	2	no	—
32	C₂³×C₄₁	—	4	2	no	—
33	C₂³×C₄₁	—	4	4	no	—
34	C₂³×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₄₁	—	4	2	no	—
36	Q₈×C₄₁	—	4	2	no	—
37	D₄×C₄₁	—	4	2	no	—
38	D₄×C₄₁	—	4	2	no	—
39	C₂×C₄×C₄₁	—	2	2	no	—
40	C₂×C₄×C₄₁	—	2	4	no	—
41	C₂×C₄×C₄₁	—	2	2	no	—
42	C₂×C₄×C₄₁	—	4	2	no	—
43	C₂×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
47	C₈×C₄₁	—	2	8	no	—
48	C₂²×C₄₁	—	4	2	no	—
49	C₂²×C₄₁	—	4	4	no	—
50	C₂²×C₄₁	—	8	4	no	—
51	C₂²×C₄₁	#100	1	2	yes	—
52	C₂²×C₄₁	#104	1	4	yes	—
53	C₂²×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₄₁	—	2	1	no	—
55	C₄×C₄₁	—	2	1	no	—
56	C₄×C₄₁	—	2	2	no	—
57	C₄×C₄₁	—	2	2	no	—
58	C₄×C₄₁	—	4	2	no	—
59	C₂×#329	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₄₁	—	8	2	no	—
61	C₂×C₄₁	#329	1	4	yes	direct product factor
62	C₂×C₄₁	C₂×#100	1	1	yes	characteristic
63	C₂×C₄₁	C₂×#104	1	2	yes	—
64	C₂×#124	C₂×C₄₁	1	1	yes	characteristic
65	C₂×#97	C₂×C₄₁	1	2	yes	complemented
66	#329	C₂×C₄₁	1	4	yes	direct product factor
67	C₄₁	C₂×#329	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×Q₈	C₄×C₄₁	1	1	yes	characteristic
69	C₂²×D₄	C₄×C₄₁	1	1	yes	characteristic
70	C₂²×C₈	—	2	2	no	—
71	#124	—	2	2	no	—
72	#124	C₄×C₄₁	1	2	yes	—
73	#124	C₂²×C₄₁	1	2	yes	—
74	C₂×C₂²⋊C₄	—	2	2	no	—
75	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
76	#97	—	2	2	no	—
77	#97	C₂²×C₄₁	1	4	yes	complemented
78	C₂⁴	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×C₄×C₄₁	1	2	yes	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	C₂×C₄×C₄₁	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
87	C₂²×C₄	D₄×C₄₁	1	2	yes	—
88	C₂×C₈	—	2	8	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂²⋊C₄	—	2	4	no	—
91	C₂²⋊C₄	—	2	4	no	—
92	C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
93	C₄²	D₄×C₄₁	1	2	yes	—
94	C₄²	C₂³×C₄₁	1	1	yes	characteristic
95	C₂³	—	2	2	no	—
96	C₂³	—	4	2	no	—
97	C₂³	—	4	4	no	—
98	C₂³	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
108	C₂×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
109	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
111	C₈	—	2	8	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	8	4	no	—
115	C₂²	#100×C₄₁	1	2	yes	—
116	C₂²	#104×C₄₁	1	4	yes	—
117	C₂²	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	2	no	—
121	C₄	—	2	2	no	—
122	C₄	—	4	2	no	—
123	C₂	—	8	2	no	—
124	C₂	C₄₁×#329	1	4	yes	direct product factor
125	C₂	C₂×#100×C₄₁	1	1	yes	characteristic
126	C₂	C₂×#104×C₄₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₄₁

C₂

yes

maximal, characteristic

C₂×#97×C₄₁

C₂

yes

maximal, complemented

C₄₁×#329

C₂

yes

maximal, direct product factor

C₂²×Q₈×C₄₁

C₄

yes

characteristic

C₂²×D₄×C₄₁

C₄

yes

characteristic

C₂²×C₈×C₄₁

—

#124×C₄₁

—

#124×C₄₁

C₄

yes

—

#124×C₄₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₁

—

C₂×C₄²×C₄₁

C₂²

yes

characteristic

#97×C₄₁

—

#97×C₄₁

C₂²

yes

complemented

C₂⁴×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

C₂×C₄

yes

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

—

C₂²×C₄×C₄₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₁

D₄

yes

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₄²×C₄₁

C₂×C₄

yes

characteristic

C₄²×C₄₁

D₄

yes

—

C₄²×C₄₁

C₂³

yes

characteristic

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

C₂²⋊C₄

yes

characteristic

Q₈×C₄₁

—

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₄₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₁

C₂×D₄

yes

—

C₈×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

#100

yes

—

C₂²×C₄₁

#104

yes

—

C₂²×C₄₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₂×#329

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

—

C₂×C₄₁

#329

yes

direct product factor

C₂×C₄₁

C₂×#100

yes

characteristic

C₂×C₄₁

C₂×#104

yes

—

C₂×#124

C₂×C₄₁

yes

characteristic

C₂×#97

C₂×C₄₁

yes

complemented

#329

C₂×C₄₁

yes

direct product factor

C₄₁

C₂×#329

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₄×C₄₁

yes

characteristic

C₂²×D₄

C₄×C₄₁

yes

characteristic

C₂²×C₈

—

#124

—

#124

C₄×C₄₁

yes

—

#124

C₂²×C₄₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₄₁

yes

characteristic

#97

—

#97

C₂²×C₄₁

yes

complemented

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₄₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₄₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₄₁

yes

characteristic

C₂²×C₄

D₄×C₄₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₄₁

yes

characteristic

C₄²

D₄×C₄₁

yes

—

C₄²

C₂³×C₄₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₄₁

yes

characteristic

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊C₄×C₄₁

yes

—

108

C₂×C₄

C₂²⋊C₄×C₄₁

yes

characteristic

109

C₂×C₄

C₂²×C₄×C₄₁

yes

characteristic

110

C₂×C₄

C₂×D₄×C₄₁

yes

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#100×C₄₁

yes

—

116

C₂²

#104×C₄₁

yes

—

117

C₂²

C₂×C₂²⋊C₄×C₄₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₂

—

124

C₂

C₄₁×#329

yes

direct product factor

125

C₂

C₂×#100×C₄₁

yes

characteristic

126

C₂

C₂×#104×C₄₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₄²	C₂³×C₄₁
prime radical	C₂²×D₄×C₄₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₁

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₄₁

Frattini, cosocle

C₄²

C₂³×C₄₁

prime radical

C₂²×D₄×C₄₁

C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

N	H ≅ G/N	type
2. C₂×#97×C₄₁	C₂	semidirect
3. C₄₁×#329	C₂	direct (N × H)
13. #97×C₄₁	C₂²	semidirect
59. C₂×#329	C₄₁	direct (N × H)
61. C₂×C₄₁	#329	direct (N × H)
65. C₂×#97	C₂×C₄₁	semidirect
66. #329	C₂×C₄₁	direct (N × H)
67. C₄₁	C₂×#329	direct (N × H)
77. #97	C₂²×C₄₁	semidirect
124. C₂	C₄₁×#329	direct (N × H)

H ≅ G/N

type

2. C₂×#97×C₄₁

C₂

semidirect

3. C₄₁×#329

C₂

direct (N × H)

13. #97×C₄₁

C₂²

semidirect

59. C₂×#329

C₄₁

direct (N × H)

61. C₂×C₄₁

#329

direct (N × H)

65. C₂×#97

C₂×C₄₁

semidirect

66. #329

C₂×C₄₁

direct (N × H)

67. C₄₁

C₂×#329

direct (N × H)

77. #97

C₂²×C₄₁

semidirect

124. C₂

C₄₁×#329

direct (N × H)

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

subgroup	index
C₂×#124×C₄₁	2
C₂×#97×C₄₁	2
C₄₁×#329	2
C₂×#329	41

subgroup

index

C₂×#124×C₄₁

C₂×#97×C₄₁

C₄₁×#329

C₂×#329

Subgroups — C₂×C₄₁×#329 #19105302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C41×#329 #19105302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#329 #19105302