Group #44399783 — subgroups

Subgroups — #44399783

Order 29056.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20300296	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300223	C₂	1	2	yes	maximal, complemented
3	#28409014	C₂	1	2	yes	maximal
4	C₂₂₇⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813884	C₂²	1	1	yes	characteristic
7	#20300296	—	2	4	no	—
8	C₂×#16813868	C₂²	1	2	yes	—
9	C₂×#16813867	C₄	1	2	yes	—
10	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₂₇	C₄	1	2	yes	—
12	#20300223	—	2	4	no	—
13	#20300206	—	2	2	no	—
14	#20300204	—	2	2	no	—
15	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
17	#16813884	—	2	2	no	—
18	#16813884	—	2	2	no	—
19	C₂×Dic₄₅₄	D₄	1	1	yes	characteristic
20	C₂×D₉₀₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₂₇	Q₈	1	2	yes	—
22	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
23	#16813868	—	2	2	no	—
24	#16813868	—	2	2	no	—
25	#16813867	—	2	4	no	—
26	C₂×C₂₂₇⋊C₈	—	2	2	no	—
27	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	—
30	C₈×D₂₂₇	—	2	4	no	—
31	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₂₇	—	2	2	no	—
33	M₁₆×C₂₂₇	—	2	2	no	—
34	M₁₆×C₂₂₇	—	2	2	no	—
35	C₂×C₈×C₂₂₇	—	2	2	no	—
36	C₂×C₈×C₂₂₇	C₂×C₄	1	2	yes	—
37	C₂²×D₂₂₇	C₄⋊C₄	1	2	yes	—
38	Dic₄₅₄	—	2	2	no	—
39	D₉₀₈	—	2	2	no	—
40	C₂²⋊D₂₂₇	—	2	4	no	—
41	C₂×Dic₂₂₇	C₄⋊C₄	1	2	yes	—
42	C₄×D₂₂₇	—	2	2	no	—
43	C₄×D₂₂₇	—	2	2	no	—
44	C₂₂₇⋊C₈	—	4	2	no	—
45	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₂₇	#41	1	2	yes	—
52	C₈×C₂₂₇	—	2	2	no	—
53	C₈×C₂₂₇	—	4	2	no	—
54	C₂²×C₂₂₇	—	4	1	no	—
55	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₂₂₇	—	2	2	no	—
58	C₂×D₂₂₇	—	4	2	no	—
59	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₂₇	#117	1	1	yes	characteristic
61	C₄×C₂₂₇	#121	1	2	yes	—
62	Dic₂₂₇	—	2	2	no	—
63	C₂×C₂₂₇	—	2	1	no	—
64	C₂×C₂₂₇	—	2	2	no	—
65	C₂×C₂₂₇	#380	1	1	yes	characteristic
66	D₂₂₇	—	4	2	no	—
67	C₂₂₇	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	227	1	no	maximal, Sylow
69	C₂×#131	—	227	1	no	—
70	C₂×#108	—	227	2	no	—
71	#413	—	227	2	no	—
72	#338	—	227	1	no	—
73	#338	D₂₂₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	227	1	no	—
75	#131	—	454	4	no	—
76	C₂×M₁₆	—	227	2	no	—
77	C₂×M₁₆	—	227	2	no	—
78	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
80	C₂²×C₈	—	227	2	no	—
81	C₂²×C₈	—	227	1	no	—
82	#108	—	454	4	no	—
83	#101	—	454	2	no	—
84	#100	—	454	2	no	—
85	#41	—	454	2	no	—
86	#41	—	454	2	no	—
87	C₂×Q₈	—	227	1	no	—
88	C₂×D₄	—	227	2	no	—
89	C₂×D₄	—	227	1	no	—
90	C₂²×C₄	—	227	2	no	—
91	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	454	4	no	—
96	M₁₆	—	454	2	no	—
97	M₁₆	—	454	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	454	4	no	—
100	C₂×C₈	—	454	2	no	—
101	C₂×C₈	—	227	1	no	—
102	C₂×C₈	—	227	1	no	—
103	C₂×C₈	—	227	4	no	—
104	C₂×C₈	C₄×D₂₂₇	1	2	yes	—
105	C₂³	—	227	2	no	—
106	C₂³	Q₈×D₂₂₇	1	1	yes	characteristic
107	Q₈	—	454	2	no	—
108	D₄	—	454	4	no	—
109	D₄	—	454	2	no	—
110	C₂×C₄	—	454	2	no	—
111	C₂×C₄	—	454	2	no	—
112	C₂×C₄	—	227	2	no	—
113	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
115	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
116	C₂×C₄	#16813884	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	908	2	no	—
120	C₈	—	227	2	no	—
121	C₈	—	227	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	908	2	no	—
124	C₂²	—	454	2	no	—
125	C₂²	C₄⋊C₄×D₂₂₇	1	1	yes	characteristic
126	C₂²	#20300266	1	2	yes	—
127	C₄	—	454	2	no	—
128	C₄	C₂²⋊C₄×D₂₂₇	1	1	yes	characteristic
129	C₄	#20300245	1	1	yes	characteristic
130	C₄	#20300261	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	908	2	no	—
134	C₂	#28408891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300296

C₂

yes

maximal, characteristic

C₂×#20300223

C₂

yes

maximal, complemented

#28409014

C₂

yes

maximal

C₂₂₇⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₂₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#16813884

C₂²

yes

characteristic

#20300296

—

C₂×#16813868

C₂²

yes

—

C₂×#16813867

C₄

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₂₇

C₄

yes

—

#20300223

—

#20300206

—

#20300204

—

C₂×M₁₆×C₂₂₇

C₂²

yes

—

C₂×M₁₆×C₂₂₇

C₂²

yes

characteristic

#16813884

—

#16813884

—

C₂×Dic₄₅₄

D₄

yes

characteristic

C₂×D₉₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₂₂₇

Q₈

yes

—

C₂×C₄×D₂₂₇

C₂×C₄

yes

—

#16813868

—

#16813868

—

#16813867

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₈×D₂₂₇

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

C₂×C₄

yes

—

C₂²×D₂₂₇

C₄⋊C₄

yes

—

Dic₄₅₄

—

D₉₀₈

—

C₂²⋊D₂₂₇

—

C₂×Dic₂₂₇

C₄⋊C₄

yes

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₂₇

C₂×Q₈

yes

characteristic

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

—

C₈×C₂₂₇

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₂²⋊Q₈

yes

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#380

yes

characteristic

D₂₂₇

—

C₂₂₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

227

maximal, Sylow

C₂×#131

—

227

—

C₂×#108

—

227

—

#413

—

227

—

#338

—

227

—

#338

D₂₂₇

yes

characteristic, complemented

C₂×#41

—

227

—

#131

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₂×M₁₆

C₂×D₂₂₇

yes

characteristic

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

#108

—

454

—

#101

—

454

—

#100

—

454

—

#41

—

454

—

#41

—

454

—

C₂×Q₈

—

227

—

C₂×D₄

—

227

—

C₂×D₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

454

—

M₁₆

—

454

—

M₁₆

—

454

—

C₂×C₈

—

C₂×C₈

—

454

—

100

C₂×C₈

—

454

—

101

C₂×C₈

—

227

—

102

C₂×C₈

—

227

—

103

C₂×C₈

—

227

—

104

C₂×C₈

C₄×D₂₂₇

yes

—

105

C₂³

—

227

—

106

C₂³

Q₈×D₂₂₇

yes

characteristic

107

Q₈

—

454

—

108

D₄

—

454

—

109

D₄

—

454

—

110

C₂×C₄

—

454

—

111

C₂×C₄

—

454

—

112

C₂×C₄

—

227

—

113

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

114

C₂×C₄

D₄×D₂₂₇

yes

—

115

C₂×C₄

D₄×D₂₂₇

yes

characteristic

116

C₂×C₄

#16813884

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

908

—

120

C₈

—

227

—

121

C₈

—

227

—

122

C₂²

—

123

C₂²

—

908

—

124

C₂²

—

454

—

125

C₂²

C₄⋊C₄×D₂₂₇

yes

characteristic

126

C₂²

#20300266

yes

—

127

C₄

—

454

—

128

C₄

C₂²⋊C₄×D₂₂₇

yes

characteristic

129

C₄

#20300245

yes

characteristic

130

C₄

#20300261

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

908

—

134

C₂

#28408891

yes

characteristic

	subgroup	G/H
center	C₄	#20300245
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#338×C₂₂₇	C₂
socle	C₂×C₂₂₇	#380
prime radical	C₂×#16813884	C₂²
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#338	D₂₂₇
hypocenter	C₂₂₇	#1813
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#20300245

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#338×C₂₂₇

C₂

C₂×C₂₂₇

C₂²

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#20300223	C₂	semidirect
4. C₂₂₇⋊#338	C₂	semidirect
5. #338×C₂₂₇	C₂	semidirect
67. C₂₂₇	#1813	semidirect
73. #338	D₂₂₇	semidirect

H ≅ G/N

type

2. C₂×#20300223

C₂

semidirect

4. C₂₂₇⋊#338

C₂

semidirect

5. #338×C₂₂₇

C₂

semidirect

67. C₂₂₇

#1813

semidirect

73. #338

D₂₂₇

semidirect

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

subgroup	index
C₂×#20300296	2
C₂×#20300223	2
#28409014	2
C₂₂₇⋊#338	2
#338×C₂₂₇	2
#1813	227

subgroup

index

C₂×#20300296

C₂×#20300223

#28409014

C₂₂₇⋊#338

#338×C₂₂₇

#1813

227

Subgroups — #44399783

Conjugacy classes of subgroups

Complements

Maximal subgroups