Group #44398815 — subgroups

Subgroups — #44398815

Order 29056.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#28409748	C₂	1	1	yes	maximal, characteristic, complemented
2	#28409224	C₂	1	2	yes	maximal, complemented
3	#28408781	C₂	1	1	yes	maximal, characteristic, complemented
4	#28408761	C₂	1	2	yes	maximal, complemented
5	#492×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813869	—	2	2	no	—
7	#20300277	—	2	2	no	—
8	#20300263	—	2	2	no	—
9	C₄×D₉₀₈	C₂²	1	2	yes	—
10	#20300246	—	2	2	no	—
11	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
12	#20300209	—	2	2	no	—
13	C₂×SD₁₆×C₂₂₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	#127×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×D₉₀₈	—	2	2	no	—
18	C₂×D₉₀₈	D₄	1	2	yes	—
19	C₂×C₄×D₂₂₇	—	2	2	no	—
20	#16813869	—	4	4	no	—
21	C₂×C₂₂₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
23	(C₂×D₂₂₇)⋊C₄	—	2	4	no	—
24	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
25	C₄×Dic₂₂₇	—	2	2	no	—
26	C₂×Q₈×C₂₂₇	D₄	1	2	yes	—
27	C₂×D₄×C₂₂₇	—	2	2	no	—
28	C₂×D₄×C₂₂₇	—	2	1	no	—
29	SD₁₆×C₂₂₇	—	4	4	no	—
30	C₂×C₈×C₂₂₇	—	2	1	no	—
31	C₄⋊C₄×C₂₂₇	—	2	2	no	—
32	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
33	C₂²×D₂₂₇	—	2	2	no	—
34	D₉₀₈	—	2	2	no	—
35	D₉₀₈	—	2	2	no	—
36	D₉₀₈	—	8	2	no	—
37	C₂×Dic₂₂₇	—	2	2	no	—
38	C₄×D₂₂₇	—	4	2	no	—
39	C₄×D₂₂₇	—	4	2	no	—
40	C₂₂₇⋊C₈	—	4	2	no	—
41	C₂³×C₂₂₇	—	4	1	no	—
42	Q₈×C₂₂₇	—	4	2	no	—
43	D₄×C₂₂₇	—	4	2	no	—
44	D₄×C₂₂₇	—	4	2	no	—
45	D₄×C₂₂₇	—	4	2	no	—
46	C₂×C₄×C₂₂₇	—	2	2	no	—
47	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
49	C₈×C₂₂₇	—	2	2	no	—
50	C₂²×C₂₂₇	—	8	2	no	—
51	C₂²×C₂₂₇	—	8	1	no	—
52	C₂²×C₂₂₇	#120	1	1	yes	characteristic
53	C₂×D₂₂₇	—	4	2	no	—
54	C₂×D₂₂₇	—	4	2	no	—
55	C₂×D₂₂₇	—	4	2	no	—
56	C₄×C₂₂₇	—	2	1	no	—
57	C₄×C₂₂₇	—	4	2	no	—
58	C₄×C₂₂₇	#135	1	2	yes	—
59	C₄×C₂₂₇	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₂₂₇	—	4	2	no	—
61	C₂×C₂₂₇	—	16	1	no	—
62	C₂×C₂₂₇	#449	1	2	yes	—
63	C₂×C₂₂₇	#453	1	1	yes	characteristic
64	D₂₂₇	—	8	2	no	—
65	C₂₂₇	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	227	1	no	maximal, Sylow
67	#555	—	227	1	no	—
68	#492	D₂₂₇	1	1	yes	characteristic, complemented
69	#459	—	227	2	no	—
70	#333	—	227	1	no	—
71	D₄⋊C₈	—	227	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	454	2	no	—
74	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
75	#127	C₂×D₂₂₇	1	1	yes	characteristic
76	#124	—	454	2	no	—
77	#121	—	454	2	no	—
78	C₄×D₄	—	227	2	no	—
79	#117	—	454	2	no	—
80	C₄⋊C₈	—	227	2	no	—
81	#102	—	454	2	no	—
82	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₂₂₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	454	2	no	—
87	C₂×D₄	—	227	2	no	—
88	C₂²×C₄	—	454	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	908	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	454	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	227	2	no	—
95	C₂²⋊C₄	—	454	4	no	—
96	C₂²⋊C₄	—	454	2	no	—
97	C₄²	—	454	2	no	—
98	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	454	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	1816	2	no	—
106	D₄	—	454	2	no	—
107	D₄	—	454	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	908	2	no	—
110	C₂×C₄	—	908	2	no	—
111	C₂×C₄	—	454	2	no	—
112	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	908	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	908	2	no	—
119	C₂²	—	908	2	no	—
120	C₂²	—	908	2	no	—
121	C₂²	#20300256	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	908	2	no	—
125	C₄	#20300309	1	2	yes	—
126	C₄	#20300315	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1816	2	no	—
129	C₂	#28409176	1	2	yes	—
130	C₂	#28409199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409748

C₂

yes

maximal, characteristic, complemented

#28409224

C₂

yes

maximal, complemented

#28408781

C₂

yes

maximal, characteristic, complemented

#28408761

C₂

yes

maximal, complemented

#492×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813869

—

#20300277

—

#20300263

—

C₄×D₉₀₈

C₂²

yes

—

#20300246

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

—

#20300209

—

C₂×SD₁₆×C₂₂₇

—

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

#127×C₂₂₇

C₂²

yes

characteristic

C₄×C₈×C₂₂₇

C₂²

yes

characteristic

C₂×D₉₀₈

—

C₂×D₉₀₈

D₄

yes

—

C₂×C₄×D₂₂₇

—

#16813869

—

C₂×C₂₂₇⋊C₈

—

C₄⋊Dic₂₂₇

D₄

yes

—

(C₂×D₂₂₇)⋊C₄

—

(C₂×D₂₂₇)⋊C₄

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

D₄

yes

—

C₂×D₄×C₂₂₇

—

C₂×D₄×C₂₂₇

—

SD₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

C₂²×D₂₂₇

—

D₉₀₈

—

D₉₀₈

—

D₉₀₈

—

C₂×Dic₂₂₇

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

—

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

#120

yes

characteristic

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#135

yes

—

C₄×C₂₂₇

AGL(1,ℤ/8)

yes

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#449

yes

—

C₂×C₂₂₇

#453

yes

characteristic

D₂₂₇

—

C₂₂₇

#1624

yes

characteristic, complemented, Sylow

#1624

—

227

maximal, Sylow

#555

—

227

—

#492

D₂₂₇

yes

characteristic, complemented

#459

—

227

—

#333

—

227

—

D₄⋊C₈

—

227

—

C₂×SD₁₆

—

C₂×D₈

—

454

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

#127

C₂×D₂₂₇

yes

characteristic

#124

—

454

—

#121

—

454

—

C₄×D₄

—

227

—

#117

—

454

—

C₄⋊C₈

—

227

—

#102

—

454

—

C₄×C₈

C₂×D₂₂₇

yes

characteristic

C₂×Q₈

C₂²⋊D₂₂₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

454

—

C₂×D₄

—

227

—

C₂²×C₄

—

454

—

SD₁₆

—

D₈

—

908

—

C₂×C₈

—

C₂×C₈

—

454

—

C₄⋊C₄

—

C₄⋊C₄

—

227

—

C₂²⋊C₄

—

454

—

C₂²⋊C₄

—

454

—

C₄²

—

454

—

C₄²

C₂²×D₂₂₇

yes

characteristic

C₂³

—

100

C₂³

—

454

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

1816

—

106

D₄

—

454

—

107

D₄

—

454

—

108

C₂×C₄

—

109

C₂×C₄

—

908

—

110

C₂×C₄

—

908

—

111

C₂×C₄

—

454

—

112

C₂×C₄

D₄×D₂₂₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

114

C₈

—

115

C₈

—

908

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

908

—

119

C₂²

—

908

—

120

C₂²

—

908

—

121

C₂²

#20300256

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

908

—

125

C₄

#20300309

yes

—

126

C₄

#20300315

yes

—

127

C₂

—

128

C₂

—

1816

—

129

C₂

#28409176

yes

—

130

C₂

#28409199

yes

characteristic

	subgroup	G/H
center	C₂²	#20300256
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#492×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#120
hypercenter	#492	D₂₂₇
hypocenter	C₂₂₇	#1624
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20300256

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. #28409748	C₂	semidirect
2. #28409224	C₂	semidirect
3. #28408781	C₂	semidirect
4. #28408761	C₂	semidirect
5. #492×C₂₂₇	C₂	semidirect
65. C₂₂₇	#1624	semidirect
68. #492	D₂₂₇	semidirect

H ≅ G/N

type

1. #28409748

C₂

semidirect

2. #28409224

C₂

semidirect

3. #28408781

C₂

semidirect

4. #28408761

C₂

semidirect

5. #492×C₂₂₇

C₂

semidirect

65. C₂₂₇

#1624

semidirect

68. #492

D₂₂₇

semidirect

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

subgroup	index
#28409748	2
#28409224	2
#28408781	2
#28408761	2
#492×C₂₂₇	2
#1624	227

subgroup

index

#492×C₂₂₇

227

Subgroups — #44398815

Conjugacy classes of subgroups

Complements

Maximal subgroups