Group #44408217 — subgroups

Subgroups — #44408217

Order 29056.

704 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 289 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	#28409799	C₂	1	2	yes	maximal, complemented
2	#28409016	C₂	1	6	yes	maximal, complemented
3	C₂×#20300207	C₂	1	6	yes	maximal, complemented
4	C₂×2₋¹⁺⁴×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	#20300295	—	2	8	no	—
6	C₂×#16813868	C₂²	1	12	yes	—
7	#20300207	C₂²	1	12	yes	—
8	#20300205	C₂²	1	4	yes	—
9	2₋¹⁺⁴×C₂₂₇	—	2	8	no	—
10	C₂×#41×C₂₂₇	C₄	1	6	yes	—
11	C₂×#41×C₂₂₇	C₂²	1	3	yes	—
12	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₂₇	C₄	1	2	yes	—
14	C₂²×Q₈×C₂₂₇	C₂²	1	3	yes	—
15	#16813868	—	2	12	no	—
16	#16813868	C₂³	1	8	yes	—
17	C₂×C₂₂₇⋊C₈	—	2	12	no	—
18	#41×C₂₂₇	—	2	24	no	—
19	#41×C₂₂₇	—	2	12	no	—
20	#41×C₂₂₇	D₄	1	8	yes	—
21	C₂×Q₈×C₂₂₇	—	2	8	no	—
22	C₂×Q₈×C₂₂₇	—	2	12	no	—
23	C₂×Q₈×C₂₂₇	C₂×C₄	1	6	yes	—
24	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Q₈×C₂₂₇	C₂³	1	3	yes	—
26	C₂×D₄×C₂₂₇	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₂₂₇	C₂×C₄	1	3	yes	—
28	C₂×D₄×C₂₂₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₂₇	C₂×C₄	1	6	yes	—
30	C₂²×C₄×C₂₂₇	C₂×C₄	1	6	yes	—
31	C₂²×C₄×C₂₂₇	C₂³	1	3	yes	—
32	C₂₂₇⋊C₈	—	2	8	no	—
33	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
34	C₂³×C₂₂₇	C₂²×C₄	1	3	yes	—
35	Q₈×C₂₂₇	—	2	12	no	—
36	Q₈×C₂₂₇	—	2	6	no	—
37	Q₈×C₂₂₇	C₂²⋊C₄	1	4	yes	—
38	D₄×C₂₂₇	—	2	12	no	—
39	D₄×C₂₂₇	—	2	2	no	—
40	D₄×C₂₂₇	C₂²⋊C₄	1	12	yes	—
41	C₂×C₄×C₂₂₇	—	2	12	no	—
42	C₂×C₄×C₂₂₇	—	2	12	no	—
43	C₂×C₄×C₂₂₇	C₂²×C₄	1	6	yes	—
44	C₂×C₄×C₂₂₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₂₂₇	C₂×D₄	1	12	yes	—
46	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
47	C₂²×C₂₂₇	—	2	2	no	—
48	C₂²×C₂₂₇	—	4	3	no	—
49	C₂²×C₂₂₇	—	4	2	no	—
50	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	6	yes	—
51	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₂₂₇	—	2	6	no	—
53	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	6	yes	—
54	C₄×C₂₂₇	C₂²×D₄	1	2	yes	—
55	C₂×C₂₂₇	—	2	1	no	—
56	C₂×C₂₂₇	—	2	6	no	—
57	C₂×C₂₂₇	—	4	2	no	—
58	C₂×C₂₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
59	C₂₂₇	#2889	1	1	yes	characteristic, complemented, Sylow
60	#2889	—	227	1	no	maximal, Sylow
61	C₂×2₋¹⁺⁴	D₂₂₇	1	1	yes	characteristic
62	#568	—	227	2	no	—
63	#413	—	227	6	no	—
64	C₂×#101	—	227	6	no	—
65	2₋¹⁺⁴	—	2	8	no	—
66	C₂×#41	Dic₂₂₇	1	6	yes	—
67	C₂×#41	C₂×D₂₂₇	1	3	yes	—
68	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
69	C₂²×Q₈	Dic₂₂₇	1	2	yes	—
70	C₂²×Q₈	C₂×D₂₂₇	1	3	yes	—
71	#131	—	454	8	no	—
72	C₂×M₁₆	—	227	12	no	—
73	#101	—	227	12	no	—
74	#100	—	227	4	no	—
75	#41	—	2	24	no	—
76	#41	—	2	12	no	—
77	#41	C₂²⋊D₂₂₇	1	8	yes	—
78	C₂×Q₈	—	2	8	no	—
79	C₂×Q₈	—	2	12	no	—
80	C₂×Q₈	C₂×Dic₂₂₇	1	6	yes	—
81	C₂×Q₈	C₂×Dic₂₂₇	1	1	yes	characteristic
82	C₂×Q₈	C₂²×D₂₂₇	1	3	yes	—
83	C₂×D₄	C₂×Dic₂₂₇	1	6	yes	—
84	C₂×D₄	C₂×Dic₂₂₇	1	3	yes	—
85	C₂×D₄	C₂²×D₂₂₇	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₂₂₇	1	6	yes	—
87	C₂²×C₄	C₂×Dic₂₂₇	1	6	yes	—
88	C₂²×C₄	C₂²×D₂₂₇	1	3	yes	—
89	M₁₆	—	454	12	no	—
90	M₁₆	—	227	8	no	—
91	C₂×C₈	—	454	12	no	—
92	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
93	C₂³	C₂²×Dic₂₂₇	1	3	yes	—
94	Q₈	—	2	6	no	—
95	Q₈	—	2	12	no	—
96	Q₈	C₂²⋊Dic₂₂₇	1	4	yes	—
97	D₄	—	2	12	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²⋊Dic₂₂₇	1	12	yes	—
100	C₂×C₄	—	2	12	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	C₂²×Dic₂₂₇	1	6	yes	—
103	C₂×C₄	C₂²×Dic₂₂₇	1	3	yes	—
104	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	12	yes	—
105	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
106	C₈	—	454	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	3	no	—
109	C₂²	—	4	2	no	—
110	C₂²	C₂×C₂²⋊Dic₂₂₇	1	6	yes	—
111	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
112	C₄	—	2	6	no	—
113	C₄	C₂×C₂²⋊Dic₂₂₇	1	6	yes	—
114	C₄	C₂²×C₂²⋊D₂₂₇	1	2	yes	—
115	C₂	—	2	1	no	—
116	C₂	—	2	6	no	—
117	C₂	—	4	2	no	—
118	C₂	C₂²×C₂²⋊Dic₂₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409799

C₂

yes

maximal, complemented

#28409016

C₂

yes

maximal, complemented

C₂×#20300207

C₂

yes

maximal, complemented

C₂×2₋¹⁺⁴×C₂₂₇

C₂

yes

maximal, characteristic

#20300295

—

C₂×#16813868

C₂²

yes

—

#20300207

C₂²

yes

—

#20300205

C₂²

yes

—

2₋¹⁺⁴×C₂₂₇

—

C₂×#41×C₂₂₇

C₄

yes

—

C₂×#41×C₂₂₇

C₂²

yes

—

C₂×#41×C₂₂₇

C₂²

yes

characteristic

C₂²×Q₈×C₂₂₇

C₄

yes

—

C₂²×Q₈×C₂₂₇

C₂²

yes

—

#16813868

—

#16813868

C₂³

yes

—

C₂×C₂₂₇⋊C₈

—

#41×C₂₂₇

—

#41×C₂₂₇

—

#41×C₂₂₇

D₄

yes

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₂₇

C₂³

yes

—

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

C₂×C₄

yes

—

C₂×D₄×C₂₂₇

C₂³

yes

characteristic

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂²×C₄

yes

—

C₂³×C₂₂₇

C₂²×C₄

yes

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

C₂²⋊C₄

yes

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂⁴

yes

characteristic

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₂₇

C₂³×C₄

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

—

C₄×C₂₂₇

C₂²×D₄

yes

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂₂₇

#2889

yes

characteristic, complemented, Sylow

#2889

—

227

maximal, Sylow

C₂×2₋¹⁺⁴

D₂₂₇

yes

characteristic

#568

—

227

—

#413

—

227

—

C₂×#101

—

227

—

2₋¹⁺⁴

—

C₂×#41

Dic₂₂₇

yes

—

C₂×#41

C₂×D₂₂₇

yes

—

C₂×#41

C₂×D₂₂₇

yes

characteristic

C₂²×Q₈

Dic₂₂₇

yes

—

C₂²×Q₈

C₂×D₂₂₇

yes

—

#131

—

454

—

C₂×M₁₆

—

227

—

#101

—

227

—

#100

—

227

—

#41

—

#41

—

#41

C₂²⋊D₂₂₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₂₇

yes

—

C₂×Q₈

C₂×Dic₂₂₇

yes

characteristic

C₂×Q₈

C₂²×D₂₂₇

yes

—

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂×D₄

C₂×Dic₂₂₇

yes

—

C₂×D₄

C₂²×D₂₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂×Dic₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

—

M₁₆

—

454

—

M₁₆

—

227

—

C₂×C₈

—

454

—

C₂³

C₂²×Dic₂₂₇

yes

—

C₂³

C₂²×Dic₂₂₇

yes

—

Q₈

—

Q₈

—

Q₈

C₂²⋊Dic₂₂₇

yes

—

D₄

—

D₄

—

D₄

C₂²⋊Dic₂₂₇

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×Dic₂₂₇

yes

—

103

C₂×C₄

C₂²×Dic₂₂₇

yes

—

104

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

—

105

C₂×C₄

C₂³×D₂₂₇

yes

characteristic

106

C₈

—

454

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₂₂₇

yes

—

111

C₂²

C₂³×Dic₂₂₇

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊Dic₂₂₇

yes

—

114

C₄

C₂²×C₂²⋊D₂₂₇

yes

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

C₂²×C₂²⋊Dic₂₂₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting, prime radical	C₂×2₋¹⁺⁴×C₂₂₇	C₂
socle	C₂×C₂₂₇	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂×2₋¹⁺⁴	D₂₂₇
hypocenter	C₂₂₇	#2889
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₂₂₇

commutator

C₂²×C₂₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₂₇

Fitting, prime radical

C₂×2₋¹⁺⁴×C₂₂₇

C₂

socle

C₂×C₂₂₇

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₂₂₇

C₂⁴

hypercenter

C₂×2₋¹⁺⁴

D₂₂₇

hypocenter

C₂₂₇

#2889

LCS

G ▷ C₂²×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
1. #28409799	C₂	semidirect
2. #28409016	C₂	semidirect
3. C₂×#20300207	C₂	semidirect
59. C₂₂₇	#2889	semidirect

H ≅ G/N

type

1. #28409799

C₂

semidirect

2. #28409016

C₂

semidirect

3. C₂×#20300207

C₂

semidirect

59. C₂₂₇

#2889

semidirect

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

subgroup	index
#28409799	2
#28409016	2
C₂×#20300207	2
C₂×2₋¹⁺⁴×C₂₂₇	2
#2889	227

subgroup

index

C₂×2₋¹⁺⁴×C₂₂₇

227

Subgroups — #44408217

Conjugacy classes of subgroups

Complements

Maximal subgroups