Group #44400657 — subgroups

Subgroups — #44400657

Order 29056.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#20300288	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#20300211	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₅₄	C₂²	1	2	yes	—
6	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
7	#20300288	—	2	2	no	—
8	#20300288	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₂₇	—	2	1	no	—
10	C₂×C₄⋊Dic₂₂₇	—	2	1	no	—
11	C₄⋊(C₂₂₇⋊C₈)	—	2	2	no	—
12	#20300211	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₂₇	—	2	1	no	—
15	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
16	#95×C₂₂₇	—	2	1	no	—
17	C₂×Dic₄₅₄	—	2	2	no	—
18	C₂×Dic₄₅₄	—	2	4	no	—
19	C₂×Dic₄₅₄	—	2	4	no	—
20	C₂×Dic₄₅₄	C₂×C₄	1	4	yes	—
21	C₂²×Dic₂₂₇	—	2	2	no	—
22	C₂×C₂₂₇⋊C₈	—	2	4	no	—
23	C₂×C₂₂₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₂₇	—	2	4	no	—
25	C₄⋊Dic₂₂₇	—	2	4	no	—
26	C₂²×C₄×C₂₂₇	—	2	2	no	—
27	C₂²×C₄×C₂₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₂₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₂₇	—	2	2	no	—
31	C₄⋊C₄×C₂₂₇	D₄	1	4	yes	—
32	C₄²×C₂₂₇	—	4	2	no	—
33	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
34	C₄²×C₂₂₇	D₄	1	2	yes	—
35	Dic₄₅₄	—	2	8	no	—
36	Dic₄₅₄	—	4	2	no	—
37	Dic₄₅₄	—	4	2	no	—
38	C₂×Dic₂₂₇	—	2	4	no	—
39	C₂×Dic₂₂₇	—	4	4	no	—
40	C₂₂₇⋊C₈	—	4	2	no	—
41	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₇	—	2	2	no	—
43	C₂×C₄×C₂₂₇	—	2	4	no	—
44	C₂×C₄×C₂₂₇	—	4	2	no	—
45	C₂×C₄×C₂₂₇	—	4	2	no	—
46	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₂₇	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₂₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₂₇	#41	1	2	yes	—
54	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₂₇	#121	1	2	yes	—
56	C₂²×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₂₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₂₂₇	#137	1	2	yes	—
59	C₄×C₂₂₇	—	2	2	no	—
60	C₄×C₂₂₇	—	4	4	no	—
61	C₄×C₂₂₇	#103	1	4	yes	complemented
62	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
63	C₄×C₂₂₇	#124	1	1	yes	characteristic
64	C₄×C₂₂₇	#127	1	1	yes	characteristic
65	Dic₂₂₇	—	4	4	no	—
66	C₂×C₂₂₇	#390	1	1	yes	characteristic
67	C₂×C₂₂₇	C₂×#103	1	1	yes	characteristic
68	C₂×C₂₂₇	#419	1	1	yes	characteristic
69	C₂×C₂₂₇	#461	1	2	yes	—
70	C₂×C₂₂₇	#462	1	2	yes	—
71	C₂₂₇	#1981	1	1	yes	characteristic, complemented, Sylow
72	#1981	—	227	1	no	maximal, Sylow
73	C₂×C₄⋊Q₈	—	227	1	no	—
74	C₂×C₄⋊C₈	—	227	1	no	—
75	C₂×#103	—	227	4	no	—
76	C₄×C₄⋊C₄	D₂₂₇	1	1	yes	characteristic
77	C₂²×Q₈	—	227	2	no	—
78	C₂²×C₈	—	227	2	no	—
79	C₄⋊Q₈	—	454	2	no	—
80	C₄⋊Q₈	—	227	4	no	—
81	C₂×C₄⋊C₄	—	454	1	no	—
82	C₂×C₄⋊C₄	—	454	1	no	—
83	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
86	C₄⋊C₈	—	454	2	no	—
87	#103	—	454	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	454	4	no	—
90	C₂×Q₈	—	454	4	no	—
91	C₂×Q₈	—	454	2	no	—
92	C₂×Q₈	—	227	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	454	2	no	—
95	C₂²×C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
98	C₂×C₈	—	454	4	no	—
99	C₂×C₈	—	454	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	454	4	no	—
102	C₄⋊C₄	—	454	4	no	—
103	C₄⋊C₄	D₉₀₈	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₂₂₇	1	2	yes	—
106	C₄²	D₉₀₈	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
108	Q₈	—	908	2	no	—
109	Q₈	—	908	2	no	—
110	Q₈	—	454	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	908	4	no	—
116	C₂×C₄	—	454	4	no	—
117	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
119	C₂×C₄	#16813872	1	2	yes	—
120	C₂×C₄	#16813874	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₉₀₈	1	2	yes	—
123	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
124	C₂×C₄	#16813884	1	2	yes	—
125	C₈	—	908	2	no	—
126	C₂²	C₂×(C₂×D₂₂₇)⋊C₄	1	1	yes	characteristic
127	C₂²	#20300262	1	2	yes	—
128	C₂²	C₂×#16813872	1	1	yes	characteristic
129	C₂²	C₂×#16813874	1	1	yes	characteristic
130	C₂²	#20300318	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	908	4	no	—
134	C₄	#20300211	1	4	yes	complemented
135	C₄	C₄×D₉₀₈	1	2	yes	—
136	C₄	#20300276	1	1	yes	characteristic
137	C₄	#20300285	1	1	yes	characteristic
138	C₂	#28408934	1	1	yes	characteristic
139	C₂	C₂×#20300211	1	1	yes	characteristic
140	C₂	#28409040	1	1	yes	characteristic
141	C₂	#28409237	1	2	yes	—
142	C₂	#28409242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300288

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₂₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#20300211

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₅₄

C₂²

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

#20300288

—

#20300288

C₄

yes

complemented

C₂×C₄⋊Dic₂₂₇

—

C₂×C₄⋊Dic₂₂₇

—

C₄⋊(C₂₂₇⋊C₈)

—

#20300211

—

C₂×C₄⋊C₄×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

#95×C₂₂₇

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

C₂×C₄

yes

—

C₂²×Dic₂₂₇

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

D₄

yes

characteristic

C₂²×C₄×C₂₂₇

D₄

yes

characteristic

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

D₄

yes

—

C₄²×C₂₂₇

—

C₄²×C₂₂₇

C₂×C₄

yes

—

C₄²×C₂₂₇

D₄

yes

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

SD₁₆

yes

—

C₂×C₄×C₂₂₇

Q₁₆

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

characteristic

C₂²×C₂₂₇

#121

yes

—

C₂²×C₂₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₂₂₇

C₂×Q₁₆

yes

characteristic

C₂²×C₂₂₇

#137

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#103

yes

complemented

C₄×C₂₂₇

C₄×D₄

yes

—

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

—

C₂×C₂₂₇

#462

yes

—

C₂₂₇

#1981

yes

characteristic, complemented, Sylow

#1981

—

227

maximal, Sylow

C₂×C₄⋊Q₈

—

227

—

C₂×C₄⋊C₈

—

227

—

C₂×#103

—

227

—

C₄×C₄⋊C₄

D₂₂₇

yes

characteristic

C₂²×Q₈

—

227

—

C₂²×C₈

—

227

—

C₄⋊Q₈

—

454

—

C₄⋊Q₈

—

227

—

C₂×C₄⋊C₄

—

454

—

C₂×C₄⋊C₄

—

454

—

C₂×C₄⋊C₄

C₂×D₂₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

454

—

#103

—

454

—

#95

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

C₂×Q₈

—

227

—

C₂²×C₄

—

C₂²×C₄

—

454

—

C₂²×C₄

C₂²⋊D₂₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₂₂₇

yes

characteristic

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

C₂×C₈

—

454

—

C₂×C₈

—

454

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

454

—

102

C₄⋊C₄

—

454

—

103

C₄⋊C₄

D₉₀₈

yes

—

104

C₄²

—

105

C₄²

C₄×D₂₂₇

yes

—

106

C₄²

D₉₀₈

yes

—

107

C₂³

C₂×C₂²⋊D₂₂₇

yes

characteristic

108

Q₈

—

908

—

109

Q₈

—

908

—

110

Q₈

—

454

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

908

—

116

C₂×C₄

—

454

—

117

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

119

C₂×C₄

#16813872

yes

—

120

C₂×C₄

#16813874

yes

—

121

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

122

C₂×C₄

C₂×D₉₀₈

yes

—

123

C₂×C₄

C₂×D₉₀₈

yes

characteristic

124

C₂×C₄

#16813884

yes

—

125

C₈

—

908

—

126

C₂²

C₂×(C₂×D₂₂₇)⋊C₄

yes

characteristic

127

C₂²

#20300262

yes

—

128

C₂²

C₂×#16813872

yes

characteristic

129

C₂²

C₂×#16813874

yes

characteristic

130

C₂²

#20300318

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

908

—

134

C₄

#20300211

yes

complemented

135

C₄

C₄×D₉₀₈

yes

—

136

C₄

#20300276

yes

characteristic

137

C₄

#20300285

yes

characteristic

138

C₂

#28408934

yes

characteristic

139

C₂

C₂×#20300211

yes

characteristic

140

C₂

#28409040

yes

characteristic

141

C₂

#28409237

yes

—

142

C₂

#28409242

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₂₇
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	C₄×C₄⋊C₄×C₂₂₇	C₂
socle, prime radical	C₂³×C₂₂₇	C₂×D₄
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₂₇
hypocenter	C₂₂₇	#1981
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₂₇

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

C₄×C₄⋊C₄×C₂₂₇

C₂

socle, prime radical

C₂³×C₂₂₇

C₂×D₄

cosocle

C₂²×C₄×C₂₂₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₂₇

hypocenter

C₂₂₇

#1981

LCS

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #20300288	C₄	semidirect
61. C₄×C₂₂₇	#103	semidirect
71. C₂₂₇	#1981	semidirect
134. C₄	#20300211	semidirect

H ≅ G/N

type

8. #20300288

C₄

semidirect

61. C₄×C₂₂₇

#103

semidirect

71. C₂₂₇

#1981

semidirect

134. C₄

#20300211

semidirect

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

subgroup	index
C₂×#20300288	2
C₂×C₄⋊(C₂₂₇⋊C₈)	2
C₂×#20300211	2
C₄×C₄⋊C₄×C₂₂₇	2
#1981	227

subgroup

index

C₂×#20300288

C₂×C₄⋊(C₂₂₇⋊C₈)

C₂×#20300211

C₄×C₄⋊C₄×C₂₂₇

#1981

227

Subgroups — #44400657

Conjugacy classes of subgroups

Complements

Maximal subgroups