Group #44399413 — subgroups

Subgroups — #44399413

Order 29056.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#28409435	C₂	1	1	yes	maximal, characteristic
2	#28409435	C₂	1	1	yes	maximal, characteristic
3	#28409428	C₂	1	1	yes	maximal, characteristic
4	#28409412	C₂	1	1	yes	maximal, characteristic
5	#28409382	C₂	1	1	yes	maximal, characteristic
6	#28408754	C₂	1	1	yes	maximal, characteristic
7	#487×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₉₀₈	—	2	2	no	—
9	#20300288	C₂²	1	1	yes	characteristic
10	#20300288	C₂²	1	1	yes	characteristic
11	#20300221	—	2	2	no	—
12	#20300221	—	2	2	no	—
13	#20300219	—	2	2	no	—
14	#20300212	—	2	2	no	—
15	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	#103×C₂₂₇	—	2	2	no	—
20	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₅₄	—	2	1	no	—
22	C₂×Dic₄₅₄	—	2	1	no	—
23	Dic₉₀₈	—	4	4	no	—
24	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₂₂₇	—	2	1	no	—
27	C₄⋊Dic₂₂₇	—	2	1	no	—
28	C₄⋊Dic₂₂₇	—	2	1	no	—
29	C₄⋊Dic₂₂₇	—	2	1	no	—
30	C₂×Q₈×C₂₂₇	—	2	1	no	—
31	C₂×Q₈×C₂₂₇	—	2	1	no	—
32	C₂×C₈×C₂₂₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₂₇	—	2	1	no	—
34	C₄⋊C₄×C₂₂₇	—	2	1	no	—
35	C₄⋊C₄×C₂₂₇	—	2	1	no	—
36	C₄⋊C₄×C₂₂₇	—	2	1	no	—
37	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
38	Dic₄₅₄	—	4	2	no	—
39	Dic₄₅₄	—	4	2	no	—
40	C₂×Dic₂₂₇	—	4	1	no	—
41	C₂×Dic₂₂₇	—	4	1	no	—
42	C₂₂₇⋊C₈	—	2	2	no	—
43	C₂₂₇⋊C₈	Q₁₆	1	4	yes	complemented
44	Q₈×C₂₂₇	—	4	2	no	—
45	Q₈×C₂₂₇	—	4	2	no	—
46	C₂×C₄×C₂₂₇	—	4	1	no	—
47	C₂×C₄×C₂₂₇	—	4	1	no	—
48	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₂₇	—	2	2	no	—
52	C₂²×C₂₂₇	#127	1	1	yes	characteristic
53	C₄×C₂₂₇	—	8	1	no	—
54	C₄×C₂₂₇	—	8	1	no	—
55	C₄×C₂₂₇	C₂×Q₁₆	1	2	yes	—
56	C₄×C₂₂₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₂₂₇	#137	1	2	yes	—
58	Dic₂₂₇	—	8	1	no	—
59	Dic₂₂₇	—	8	1	no	—
60	C₂×C₂₂₇	#494	1	1	yes	characteristic
61	C₂×C₂₂₇	#496	1	1	yes	characteristic
62	C₂×C₂₂₇	#497	1	1	yes	characteristic
63	C₂₂₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	227	1	no	maximal, Sylow
65	#500	—	227	1	no	—
66	#500	—	227	1	no	—
67	#498	—	227	1	no	—
68	#494	—	227	1	no	—
69	#487	—	227	1	no	—
70	#487	D₂₂₇	1	1	yes	characteristic
71	#322	—	227	1	no	—
72	C₂×Q₁₆	—	454	2	no	—
73	C₄⋊Q₈	—	227	1	no	—
74	C₄⋊Q₈	—	227	1	no	—
75	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
77	#107	—	454	2	no	—
78	#107	—	454	2	no	—
79	#106	—	454	2	no	—
80	#103	—	2	2	no	—
81	#103	—	454	2	no	—
82	C₄×C₈	—	227	1	no	—
83	C₄×C₈	—	227	1	no	—
84	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	454	1	no	—
88	C₂×Q₈	—	454	1	no	—
89	Q₁₆	—	908	4	no	—
90	C₂×C₈	—	227	2	no	—
91	C₂×C₈	—	227	2	no	—
92	C₂×C₈	C₂²⋊D₂₂₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	454	1	no	—
98	C₄⋊C₄	—	454	1	no	—
99	C₄⋊C₄	—	454	1	no	—
100	C₄⋊C₄	—	454	1	no	—
101	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	908	2	no	—
105	Q₈	—	908	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	908	1	no	—
109	C₂×C₄	—	908	1	no	—
110	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	454	2	no	—
115	C₈	—	227	4	no	—
116	C₂²	#20300284	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1816	1	no	—
120	C₄	—	1816	1	no	—
121	C₄	Q₁₆×D₂₂₇	1	2	yes	—
122	C₄	#20300313	1	2	yes	—
123	C₄	#20300317	1	2	yes	—
124	C₂	#28409410	1	1	yes	characteristic
125	C₂	#28409422	1	1	yes	characteristic
126	C₂	#28409423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409435

C₂

yes

maximal, characteristic

#28409435

C₂

yes

maximal, characteristic

#28409428

C₂

yes

maximal, characteristic

#28409412

C₂

yes

maximal, characteristic

#28409382

C₂

yes

maximal, characteristic

#28408754

C₂

yes

maximal, characteristic

#487×C₂₂₇

C₂

yes

maximal, characteristic

C₂×Dic₉₀₈

—

#20300288

C₂²

yes

characteristic

#20300288

C₂²

yes

—

—

—

—

C₄×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

#103×C₂₂₇

—

C₄×C₈×C₂₂₇

C₂²

yes

characteristic

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

Dic₉₀₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×C₈×C₂₂₇

D₄

yes

—

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

Q₁₆

yes

complemented

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₈×C₂₂₇

—

C₂²×C₂₂₇

#127

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×Q₁₆

yes

—

C₄×C₂₂₇

AGL(1,ℤ/8)

yes

—

C₄×C₂₂₇

#137

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂₂₇

yes

characteristic, complemented, Sylow

#1723

—

227

maximal, Sylow

#500

—

227

—

#500

—

227

—

#498

—

227

—

#494

—

227

—

#487

—

227

—

#487

D₂₂₇

yes

characteristic

#322

—

227

—

C₂×Q₁₆

—

454

—

C₄⋊Q₈

—

227

—

C₄⋊Q₈

—

227

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

#107

—

454

—

#107

—

454

—

#106

—

454

—

#103

—

#103

—

454

—

C₄×C₈

—

227

—

C₄×C₈

—

227

—

C₄×C₈

C₂×D₂₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

Q₁₆

—

908

—

C₂×C₈

—

227

—

C₂×C₈

—

227

—

C₂×C₈

C₂²⋊D₂₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

100

C₄⋊C₄

—

454

—

101

C₄²

C₂²×D₂₂₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

908

—

105

Q₈

—

908

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

908

—

109

C₂×C₄

—

908

—

110

C₂×C₄

D₄×D₂₂₇

yes

characteristic

111

C₂×C₄

D₄×D₂₂₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

113

C₈

—

114

C₈

—

454

—

115

C₈

—

227

—

116

C₂²

#20300284

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1816

—

120

C₄

—

1816

—

121

C₄

Q₁₆×D₂₂₇

yes

—

122

C₄

#20300313

yes

—

123

C₄

#20300317

yes

—

124

C₂

#28409410

yes

characteristic

125

C₂

#28409422

yes

characteristic

126

C₂

#28409423

yes

characteristic

	subgroup	G/H
center	C₂²	#20300284
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#487×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	#127
hypercenter	#487	D₂₂₇
hypocenter	C₂₂₇	#1723
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#20300284

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#487×C₂₂₇

C₂

C₂²×C₂₂₇

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
43. C₂₂₇⋊C₈	Q₁₆	semidirect
63. C₂₂₇	#1723	semidirect

H ≅ G/N

type

43. C₂₂₇⋊C₈

Q₁₆

semidirect

63. C₂₂₇

#1723

semidirect

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

subgroup	index
#28409435	2
#28409428	2
#28409412	2
#28409382	2
#28408754	2
#487×C₂₂₇	2
#1723	227

subgroup

index

#487×C₂₂₇

227

Subgroups — #44399413

Conjugacy classes of subgroups

Complements

Maximal subgroups