Group #44399398 — subgroups

Subgroups — #44399398

Order 29056.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#28409409	C₂	1	1	yes	maximal, characteristic, complemented
2	#28409408	C₂	1	2	yes	maximal, complemented
3	#28409380	C₂	1	2	yes	maximal, complemented
4	#28408753	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₈₁₆	—	2	2	no	—
7	C₂×#16813869	—	2	4	no	—
8	#20300285	C₂²	1	2	yes	—
9	#20300208	—	2	4	no	—
10	C₄×C₂₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₂₇	—	2	2	no	—
12	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
13	#127×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×D₉₀₈	—	2	2	no	—
16	C₂×D₉₀₈	—	2	2	no	—
17	C₂×D₉₀₈	—	2	2	no	—
18	D₁₈₁₆	—	4	4	no	—
19	#16813869	—	4	8	no	—
20	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	—
21	C₂×Q₈×C₂₂₇	—	2	1	no	—
22	C₂×D₄×C₂₂₇	—	2	2	no	—
23	C₂×D₄×C₂₂₇	—	2	1	no	—
24	SD₁₆×C₂₂₇	—	4	4	no	—
25	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₂₇	—	2	2	no	—
27	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
28	C₂²×D₂₂₇	—	4	2	no	—
29	D₉₀₈	—	4	4	no	—
30	D₉₀₈	—	4	4	no	—
31	D₉₀₈	—	4	4	no	—
32	C₂₂₇⋊C₈	—	2	4	no	—
33	C₂³×C₂₂₇	—	4	1	no	—
34	Q₈×C₂₂₇	—	4	2	no	—
35	D₄×C₂₂₇	—	4	2	no	—
36	D₄×C₂₂₇	—	4	4	no	—
37	C₂×C₄×C₂₂₇	—	4	1	no	—
38	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
40	C₈×C₂₂₇	D₈	1	4	yes	complemented
41	C₂²×C₂₂₇	—	8	1	no	—
42	C₂²×C₂₂₇	—	8	2	no	—
43	C₂²×C₂₂₇	#127	1	1	yes	characteristic
44	C₂×D₂₂₇	—	8	2	no	—
45	C₂×D₂₂₇	—	8	2	no	—
46	C₂×D₂₂₇	—	8	2	no	—
47	C₄×C₂₂₇	—	8	1	no	—
48	C₄×C₂₂₇	C₂×D₈	1	2	yes	—
49	C₄×C₂₂₇	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₂₂₇	—	16	1	no	—
51	C₂×C₂₂₇	#493	1	1	yes	characteristic
52	C₂×C₂₂₇	#496	1	2	yes	—
53	D₂₂₇	—	16	2	no	—
54	C₂₂₇	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	227	1	no	maximal, Sylow
56	#493	—	227	2	no	—
57	#493	—	227	1	no	—
58	#492	D₂₂₇	1	1	yes	characteristic, complemented
59	#486	—	227	2	no	—
60	#322	—	227	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	454	4	no	—
63	C₂×D₈	—	454	2	no	—
64	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
65	#127	—	227	2	no	—
66	#127	C₂×D₂₂₇	1	1	yes	characteristic
67	#102	—	454	4	no	—
68	C₄×C₈	—	227	2	no	—
69	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	454	2	no	—
74	C₂×D₄	—	454	2	no	—
75	C₂×D₄	—	454	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	908	8	no	—
78	D₈	—	908	4	no	—
79	C₂×C₈	—	227	4	no	—
80	C₂×C₈	C₂²⋊D₂₂₇	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	908	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	908	4	no	—
89	D₄	—	908	4	no	—
90	D₄	—	908	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
94	C₈	—	454	4	no	—
95	C₈	#16813869	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	1816	2	no	—
99	C₂²	—	1816	2	no	—
100	C₂²	—	1816	2	no	—
101	C₂²	#20300284	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#16813869	1	2	yes	—
104	C₄	#20300315	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	3632	2	no	—
107	C₂	#28409408	1	1	yes	characteristic
108	C₂	#28409418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#28409409

C₂

yes

maximal, characteristic, complemented

#28409408

C₂

yes

maximal, complemented

#28409380

C₂

yes

maximal, complemented

#28408753

C₂

yes

maximal, characteristic, complemented

#492×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₁₈₁₆

—

C₂×#16813869

—

#20300285

C₂²

yes

—

#20300208

—

C₄×C₂₂₇⋊C₈

C₂²

yes

—

C₂×SD₁₆×C₂₂₇

—

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

#127×C₂₂₇

C₂²

yes

characteristic

C₄×C₈×C₂₂₇

C₂²

yes

characteristic

C₂×D₉₀₈

—

C₂×D₉₀₈

—

C₂×D₉₀₈

—

D₁₈₁₆

—

#16813869

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×Q₈×C₂₂₇

—

C₂×D₄×C₂₂₇

—

C₂×D₄×C₂₂₇

—

SD₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

D₄

yes

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

C₂²×D₂₂₇

—

D₉₀₈

—

D₉₀₈

—

D₉₀₈

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

—

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

D₈

yes

complemented

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

#127

yes

characteristic

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×D₈

yes

—

C₄×C₂₂₇

AGL(1,ℤ/8)

yes

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#493

yes

characteristic

C₂×C₂₂₇

#496

yes

—

D₂₂₇

—

C₂₂₇

#1721

yes

characteristic, complemented, Sylow

#1721

—

227

maximal, Sylow

#493

—

227

—

#493

—

227

—

#492

D₂₂₇

yes

characteristic, complemented

#486

—

227

—

#322

—

227

—

C₂×SD₁₆

—

C₂×D₈

—

454

—

C₂×D₈

—

454

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

#127

—

227

—

#127

C₂×D₂₂₇

yes

characteristic

#102

—

454

—

C₄×C₈

—

227

—

C₄×C₈

C₂×D₂₂₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

454

—

C₂×D₄

—

454

—

C₂×D₄

—

454

—

SD₁₆

—

D₈

—

908

—

D₈

—

908

—

C₂×C₈

—

227

—

C₂×C₈

C₂²⋊D₂₂₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₂₇

yes

characteristic

C₂³

—

C₂³

—

908

—

Q₈

—

D₄

—

D₄

—

D₄

—

908

—

D₄

—

908

—

D₄

—

908

—

C₂×C₄

—

C₂×C₄

D₄×D₂₂₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

C₈

—

454

—

C₈

#16813869

yes

complemented

C₂²

—

C₂²

—

C₂²

—

1816

—

C₂²

—

1816

—

100

C₂²

—

1816

—

101

C₂²

#20300284

yes

characteristic

102

C₄

—

103

C₄

C₂×#16813869

yes

—

104

C₄

#20300315

yes

—

105

C₂

—

106

C₂

—

3632

—

107

C₂

#28409408

yes

characteristic

108

C₂

#28409418

yes

—

	subgroup	G/H
center	C₂²	#20300284
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#492×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#127
hypercenter	#492	D₂₂₇
hypocenter	C₂₂₇	#1721
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20300284

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#492×C₂₂₇

C₂

C₂²×C₂₂₇

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #28409409	C₂	semidirect
2. #28409408	C₂	semidirect
3. #28409380	C₂	semidirect
4. #28408753	C₂	semidirect
5. #492×C₂₂₇	C₂	semidirect
40. C₈×C₂₂₇	D₈	semidirect
54. C₂₂₇	#1721	semidirect
58. #492	D₂₂₇	semidirect
95. C₈	#16813869	semidirect

H ≅ G/N

type

1. #28409409

C₂

semidirect

2. #28409408

C₂

semidirect

3. #28409380

C₂

semidirect

4. #28408753

C₂

semidirect

5. #492×C₂₂₇

C₂

semidirect

40. C₈×C₂₂₇

D₈

semidirect

54. C₂₂₇

#1721

semidirect

58. #492

D₂₂₇

semidirect

95. C₈

#16813869

semidirect

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

subgroup	index
#28409409	2
#28409408	2
#28409380	2
#28408753	2
#492×C₂₂₇	2
#1721	227

subgroup

index

#492×C₂₂₇

227

Subgroups — #44399398

Conjugacy classes of subgroups

Complements

Maximal subgroups