Group #41009158 — subgroups

Subgroups — C₂×#331×C₂₁₁ #41009158

Order 27008.

360 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 136 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×#127×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#331×C₂₁₁	C₂	1	4	yes	maximal, direct product factor
4	C₂²×D₄×C₂₁₁	—	2	2	no	—
5	C₂²×D₄×C₂₁₁	C₄	1	2	yes	—
6	C₂²×C₈×C₂₁₁	—	2	2	no	—
7	#127×C₂₁₁	—	2	2	no	—
8	#127×C₂₁₁	C₄	1	2	yes	—
9	#127×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₁₁	—	2	2	no	—
12	C₄⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
13	C₂⁴×C₂₁₁	—	2	2	no	—
14	C₂×D₄×C₂₁₁	—	2	2	no	—
15	C₂×D₄×C₂₁₁	—	2	4	no	—
16	C₂×D₄×C₂₁₁	—	2	4	no	—
17	C₂×D₄×C₂₁₁	—	4	4	no	—
18	C₂×D₄×C₂₁₁	—	4	4	no	—
19	C₂×D₄×C₂₁₁	—	4	4	no	—
20	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
23	C₂×C₈×C₂₁₁	—	2	4	no	—
24	C₂×C₈×C₂₁₁	—	4	4	no	—
25	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₁₁	D₄	1	2	yes	—
27	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂³×C₂₁₁	—	2	4	no	—
29	C₂³×C₂₁₁	—	4	4	no	—
30	C₂³×C₂₁₁	—	4	8	no	—
31	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
32	D₄×C₂₁₁	—	4	4	no	—
33	D₄×C₂₁₁	—	4	4	no	—
34	D₄×C₂₁₁	—	4	8	no	—
35	D₄×C₂₁₁	—	4	4	no	—
36	D₄×C₂₁₁	—	4	4	no	—
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₁₁	D₈	1	4	yes	—
41	C₂×C₄×C₂₁₁	SD₁₆	1	4	yes	—
42	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
44	C₈×C₂₁₁	—	4	4	no	—
45	C₂²×C₂₁₁	—	4	4	no	—
46	C₂²×C₂₁₁	—	4	8	no	—
47	C₂²×C₂₁₁	—	8	8	no	—
48	C₂²×C₂₁₁	#100	1	2	yes	—
49	C₂²×C₂₁₁	#102	1	4	yes	—
50	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₂₁₁	—	2	1	no	—
52	C₄×C₂₁₁	—	2	1	no	—
53	C₄×C₂₁₁	#102	1	4	yes	—
54	C₄×C₂₁₁	C₂×D₈	1	2	yes	—
55	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
56	C₂×C₂₁₁	—	8	4	no	—
57	C₂×C₂₁₁	#331	1	4	yes	direct product factor
58	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
59	C₂×C₂₁₁	C₂×#102	1	2	yes	—
60	C₂₁₁	C₂×#331	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×#331	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×#127	C₂×C₂₁₁	1	1	yes	characteristic
63	C₂×C₄⋊C₈	C₂×C₂₁₁	1	2	yes	complemented
64	#331	C₂×C₂₁₁	1	4	yes	direct product factor
65	C₂²×D₄	—	2	2	no	—
66	C₂²×D₄	C₄×C₂₁₁	1	2	yes	—
67	C₂²×C₈	—	2	2	no	—
68	#127	—	2	2	no	—
69	#127	C₄×C₂₁₁	1	2	yes	—
70	#127	C₂²×C₂₁₁	1	2	yes	—
71	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
74	C₂⁴	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	4	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	4	4	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	4	4	no	—
81	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
82	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
83	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	4	no	—
86	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
87	C₄²	D₄×C₂₁₁	1	2	yes	—
88	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
89	C₂³	—	2	4	no	—
90	C₂³	—	4	4	no	—
91	C₂³	—	4	8	no	—
92	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
93	D₄	—	4	4	no	—
94	D₄	—	4	4	no	—
95	D₄	—	4	8	no	—
96	D₄	—	4	4	no	—
97	D₄	—	4	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
101	C₂×C₄	D₈×C₂₁₁	1	4	yes	—
102	C₂×C₄	SD₁₆×C₂₁₁	1	4	yes	—
103	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
104	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
105	C₈	—	4	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	—	4	8	no	—
108	C₂²	—	8	8	no	—
109	C₂²	#100×C₂₁₁	1	2	yes	—
110	C₂²	#102×C₂₁₁	1	4	yes	—
111	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#102×C₂₁₁	1	4	yes	—
115	C₄	C₂×D₈×C₂₁₁	1	2	yes	—
116	C₄	C₂×SD₁₆×C₂₁₁	1	2	yes	—
117	C₂	—	8	4	no	—
118	C₂	#331×C₂₁₁	1	4	yes	direct product factor
119	C₂	C₂×#100×C₂₁₁	1	1	yes	characteristic
120	C₂	C₂×#102×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, complemented

#331×C₂₁₁

C₂

yes

maximal, direct product factor

C₂²×D₄×C₂₁₁

—

C₂²×D₄×C₂₁₁

C₄

yes

—

C₂²×C₈×C₂₁₁

—

#127×C₂₁₁

—

#127×C₂₁₁

C₄

yes

—

#127×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

complemented

C₂⁴×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#100

yes

—

C₂²×C₂₁₁

#102

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#102

yes

—

C₄×C₂₁₁

C₂×D₈

yes

—

C₄×C₂₁₁

C₂×SD₁₆

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#331

yes

direct product factor

C₂×C₂₁₁

C₂×#100

yes

characteristic

C₂×C₂₁₁

C₂×#102

yes

—

C₂₁₁

C₂×#331

yes

characteristic, direct product factor, Sylow

C₂×#331

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#127

C₂×C₂₁₁

yes

characteristic

C₂×C₄⋊C₈

C₂×C₂₁₁

yes

complemented

#331

C₂×C₂₁₁

yes

direct product factor

C₂²×D₄

—

C₂²×D₄

C₄×C₂₁₁

yes

—

C₂²×C₈

—

#127

—

#127

C₄×C₂₁₁

yes

—

#127

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₁₁

yes

complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×C₄×C₂₁₁

yes

characteristic

C₄²

D₄×C₂₁₁

yes

—

C₄²

C₂³×C₂₁₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₁₁

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

characteristic

100

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

101

C₂×C₄

D₈×C₂₁₁

yes

—

102

C₂×C₄

SD₁₆×C₂₁₁

yes

—

103

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

104

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#100×C₂₁₁

yes

—

110

C₂²

#102×C₂₁₁

yes

—

111

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#102×C₂₁₁

yes

—

115

C₄

C₂×D₈×C₂₁₁

yes

—

116

C₄

C₂×SD₁₆×C₂₁₁

yes

—

117

C₂

—

118

C₂

#331×C₂₁₁

yes

direct product factor

119

C₂

C₂×#100×C₂₁₁

yes

characteristic

120

C₂

C₂×#102×C₂₁₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₄²	C₂³×C₂₁₁
prime radical	C₂×#127×C₂₁₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₁₁

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₄²

C₂³×C₂₁₁

prime radical

C₂×#127×C₂₁₁

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₂₁₁	C₂	semidirect
3. #331×C₂₁₁	C₂	direct (N × H)
12. C₄⋊C₈×C₂₁₁	C₂²	semidirect
57. C₂×C₂₁₁	#331	direct (N × H)
60. C₂₁₁	C₂×#331	direct (N × H)
61. C₂×#331	C₂₁₁	direct (N × H)
63. C₂×C₄⋊C₈	C₂×C₂₁₁	semidirect
64. #331	C₂×C₂₁₁	direct (N × H)
73. C₄⋊C₈	C₂²×C₂₁₁	semidirect
118. C₂	#331×C₂₁₁	direct (N × H)

H ≅ G/N

type

2. C₂×C₄⋊C₈×C₂₁₁

C₂

semidirect

3. #331×C₂₁₁

C₂

direct (N × H)

12. C₄⋊C₈×C₂₁₁

C₂²

semidirect

57. C₂×C₂₁₁

#331

direct (N × H)

60. C₂₁₁

C₂×#331

direct (N × H)

61. C₂×#331

C₂₁₁

direct (N × H)

63. C₂×C₄⋊C₈

C₂×C₂₁₁

semidirect

64. #331

C₂×C₂₁₁

direct (N × H)

73. C₄⋊C₈

C₂²×C₂₁₁

semidirect

118. C₂

#331×C₂₁₁

direct (N × H)

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

subgroup	index
C₂×#127×C₂₁₁	2
C₂×C₄⋊C₈×C₂₁₁	2
#331×C₂₁₁	2
C₂×#331	211

subgroup

index

C₂×#127×C₂₁₁

C₂×C₄⋊C₈×C₂₁₁

#331×C₂₁₁

C₂×#331

211

Subgroups — C₂×#331×C₂₁₁ #41009158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#331×C211 #41009158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#331×C₂₁₁ #41009158