Group #44400555 — subgroups

Subgroups — #44400555

Order 29056.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20300212	C₂	1	4	yes	maximal
2	C₂×#20300199	C₂	1	1	yes	maximal, characteristic
3	#28408898	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
7	#20300212	—	2	8	no	—
8	#20300199	—	2	2	no	—
9	C₂₂₇⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₂₂₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₂₇	—	2	2	no	—
12	C₄⋊Q₈×C₂₂₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₂₇	—	2	2	no	—
14	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₂₇	—	2	2	no	—
16	C₂×C₂₂₇⋊C₈	—	2	2	no	—
17	C₂×C₂₂₇⋊C₈	—	2	2	no	—
18	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₂₂₇	—	2	4	no	—
20	C₂×Q₈×C₂₂₇	—	2	2	no	—
21	C₂×Q₈×C₂₂₇	—	2	4	no	—
22	C₂×Q₈×C₂₂₇	—	2	4	no	—
23	C₂×Q₈×C₂₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₂₇	—	2	2	no	—
25	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₂₇	—	2	8	no	—
28	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
29	C₄²×C₂₂₇	D₄	1	2	yes	—
30	C₂×Dic₂₂₇	—	4	2	no	—
31	C₂×Dic₂₂₇	—	4	4	no	—
32	C₂₂₇⋊C₈	—	2	4	no	—
33	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₂₇	—	2	8	no	—
35	Q₈×C₂₂₇	—	4	4	no	—
36	C₂×C₄×C₂₂₇	—	2	4	no	—
37	C₂×C₄×C₂₂₇	—	2	4	no	—
38	C₂×C₄×C₂₂₇	—	4	2	no	—
39	C₂×C₄×C₂₂₇	—	4	2	no	—
40	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₂₇	#41	1	2	yes	—
45	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₂₇	#124	1	1	yes	characteristic
47	C₂²×C₂₂₇	#127	1	1	yes	characteristic
48	C₂²×C₂₂₇	#137	1	4	yes	—
49	C₄×C₂₂₇	—	2	4	no	—
50	C₄×C₂₂₇	—	4	4	no	—
51	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
52	C₄×C₂₂₇	#121	1	2	yes	—
53	Dic₂₂₇	—	8	2	no	—
54	C₂×C₂₂₇	#390	1	1	yes	characteristic
55	C₂×C₂₂₇	#419	1	2	yes	—
56	C₂×C₂₂₇	#491	1	2	yes	—
57	C₂×C₂₂₇	#497	1	2	yes	—
58	C₂₂₇	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	227	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₂₂₇	1	1	yes	characteristic
61	C₂×#103	—	227	4	no	—
62	C₂×#97	—	227	1	no	—
63	#382	—	227	1	no	—
64	C₂²×Q₈	C₂×D₂₂₇	1	2	yes	—
65	C₂²×C₈	—	227	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₂₂₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	227	2	no	—
70	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
71	#103	—	454	8	no	—
72	#97	—	454	2	no	—
73	#95	—	454	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₂₂₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	454	2	no	—
80	C₂²×C₄	C₂²⋊D₂₂₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
82	C₂×C₈	—	454	2	no	—
83	C₂×C₈	—	454	2	no	—
84	C₂×C₈	—	227	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	454	4	no	—
87	C₄²	C₂×Dic₂₂₇	1	2	yes	—
88	C₄²	C₂²⋊D₂₂₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	908	4	no	—
97	C₂×C₄	—	908	2	no	—
98	C₂×C₄	C₂²⋊Dic₂₂₇	1	4	yes	—
99	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
102	C₂×C₄	#16813883	1	2	yes	—
103	C₈	—	454	4	no	—
104	C₂²	C₂×C₂²⋊Dic₂₂₇	1	1	yes	characteristic
105	C₂²	#20300275	1	1	yes	characteristic
106	C₂²	#20300284	1	1	yes	characteristic
107	C₂²	#20300319	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1816	2	no	—
111	C₄	D₄×Dic₂₂₇	1	2	yes	—
112	C₄	#20300259	1	2	yes	—
113	C₂	#28408935	1	1	yes	characteristic
114	C₂	#28409041	1	2	yes	—
115	C₂	#28409400	1	2	yes	—
116	C₂	#28409424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300212

C₂

yes

maximal

C₂×#20300199

C₂

yes

maximal, characteristic

#28408898

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₂₇

C₂²

yes

—

#20300212

—

#20300199

—

C₂₂₇⋊#95

—

C₂²×Q₈×C₂₂₇

C₂²

yes

—

C₄⋊Q₈×C₂₂₇

—

C₄⋊Q₈×C₂₂₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₂₇

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₂²×Dic₂₂₇

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂×C₄

yes

—

C₄²×C₂₂₇

D₄

yes

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

—

C₂²×C₂₂₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₂₇

yes

C₂²×C₂₂₇

yes

C₂²×C₂₂₇

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

#121

yes

—

Dic₂₂₇

—

C₂×C₂₂₇

#390

yes

characteristic

C₂×C₂₂₇

#419

yes

—

C₂×C₂₂₇

#491

yes

—

C₂×C₂₂₇

#497

yes

—

C₂₂₇

#1960

yes

characteristic, complemented, Sylow

#1960

—

227

maximal, Sylow

C₂×C₄⋊Q₈

D₂₂₇

yes

characteristic

C₂×#103

—

227

—

C₂×#97

—

227

—

#382

—

227

—

C₂²×Q₈

C₂×D₂₂₇

yes

—

C₂²×C₈

—

227

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₂₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

#103

—

454

—

#97

—

454

—

#95

—

454

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₂₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

454

—

C₂²×C₄

C₂²⋊D₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

C₂×C₈

—

454

—

C₂×C₈

—

454

—

C₂×C₈

—

227

—

C₄⋊C₄

—

C₄⋊C₄

—

454

—

C₄²

C₂×Dic₂₂₇

yes

—

C₄²

C₂²⋊D₂₂₇

yes

—

C₂³

C₂×C₂²⋊D₂₂₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

908

—

C₂×C₄

—

908

—

C₂×C₄

C₂²⋊Dic₂₂₇

yes

—

C₂×C₄

C₂²×Dic₂₂₇

yes

characteristic

100

C₂×C₄

D₄×D₂₂₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

102

C₂×C₄

#16813883

yes

—

103

C₈

—

454

—

104

C₂²

C₂×C₂²⋊Dic₂₂₇

yes

characteristic

105

C₂²

#20300275

yes

characteristic

106

C₂²

#20300284

yes

characteristic

107

C₂²

#20300319

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1816

—

111

C₄

D₄×Dic₂₂₇

yes

—

112

C₄

#20300259

yes

—

113

C₂

#28408935

yes

characteristic

114

C₂

#28409041

yes

—

115

C₂

#28409400

yes

—

116

C₂

#28409424

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₂₇
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	C₂×C₄⋊Q₈×C₂₂₇	C₂
socle, prime radical	C₂³×C₂₂₇	C₂×D₄
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₂₇
hypocenter	C₂₂₇	#1960
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₂₇

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

C₂×C₄⋊Q₈×C₂₂₇

C₂

socle, prime radical

C₂³×C₂₂₇

C₂×D₄

cosocle

C₂²×C₄×C₂₂₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₂₇

hypocenter

C₂₂₇

#1960

LCS

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₂₇	C₄	semidirect
58. C₂₂₇	#1960	semidirect
67. C₄⋊Q₈	Dic₂₂₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₂₇

C₄

semidirect

58. C₂₂₇

#1960

semidirect

67. C₄⋊Q₈

Dic₂₂₇

semidirect

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

subgroup	index
C₂×#20300212	2
C₂×#20300199	2
#28408898	2
C₂×C₄⋊Q₈×C₂₂₇	2
#1960	227

subgroup

index

C₂×C₄⋊Q₈×C₂₂₇

227

Subgroups — #44400555

Conjugacy classes of subgroups

Complements

Maximal subgroups