Group #44400455 — subgroups

Subgroups — #44400455

Order 29056.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₂₂₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×#20300199	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
4	#28408786	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₂₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
10	Dic₂₂₇⋊C₈	—	2	4	no	—
11	#20300199	—	2	2	no	—
12	C₂₂₇⋊#95	C₄	1	1	yes	characteristic
13	C₂₂₇⋊#95	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₂₇	—	2	2	no	—
17	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₂₂₇⋊C₈	—	2	1	no	—
20	C₂×C₂₂₇⋊C₈	—	2	1	no	—
21	C₂×C₂₂₇⋊C₈	—	2	1	no	—
22	C₂×C₂₂₇⋊C₈	—	2	1	no	—
23	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
24	C₂×C₂₂₇⋊C₈	Q₈	1	2	yes	—
25	C₄⋊Dic₂₂₇	—	2	4	no	—
26	C₄×Dic₂₂₇	—	2	2	no	—
27	C₄×Dic₂₂₇	—	2	2	no	—
28	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₂₇	—	2	2	no	—
32	C₂×C₈×C₂₂₇	—	2	2	no	—
33	C₂×C₈×C₂₂₇	—	2	2	no	—
34	C₄²×C₂₂₇	D₄	1	2	yes	—
35	C₄²×C₂₂₇	Q₈	1	2	yes	—
36	C₂×Dic₂₂₇	—	2	2	no	—
37	C₂×Dic₂₂₇	—	2	2	no	—
38	C₂×Dic₂₂₇	—	2	2	no	—
39	C₂×Dic₂₂₇	—	2	2	no	—
40	C₂×Dic₂₂₇	—	2	2	no	—
41	C₂×Dic₂₂₇	M₁₆	1	4	yes	—
42	C₂₂₇⋊C₈	—	2	2	no	—
43	C₂₂₇⋊C₈	—	2	2	no	—
44	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₂₇	—	2	2	no	—
46	C₂×C₄×C₂₂₇	—	2	2	no	—
47	C₂×C₄×C₂₂₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₂₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
55	C₈×C₂₂₇	—	4	2	no	—
56	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₂₂₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₂₂₇	C₂×M₁₆	1	2	yes	—
60	C₂²×C₂₂₇	#131	1	2	yes	—
61	C₄×C₂₂₇	—	2	2	no	—
62	C₄×C₂₂₇	—	2	2	no	—
63	C₄×C₂₂₇	#121	1	1	yes	characteristic
64	C₄×C₂₂₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₂₂₇	#125	1	1	yes	characteristic
66	C₄×C₂₂₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₂₂₇	—	2	4	no	—
68	Dic₂₂₇	—	4	2	no	—
69	C₂×C₂₂₇	#389	1	1	yes	characteristic
70	C₂×C₂₂₇	#423	1	1	yes	characteristic
71	C₂×C₂₂₇	#424	1	1	yes	characteristic
72	C₂×C₂₂₇	#435	1	2	yes	—
73	C₂×C₂₂₇	#446	1	2	yes	—
74	C₂₂₇	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	227	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	227	2	no	—
77	C₂×C₄⋊C₈	D₂₂₇	1	1	yes	characteristic
78	C₂×#97	—	227	1	no	—
79	C₄×C₄⋊C₄	—	227	1	no	—
80	#336	—	227	2	no	—
81	C₂²×C₈	—	227	1	no	—
82	C₂²×C₈	—	227	1	no	—
83	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
84	C₂×C₄⋊C₄	—	227	2	no	—
85	C₂×C₄²	—	227	2	no	—
86	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	454	4	no	—
89	#97	—	454	2	no	—
90	#95	—	227	1	no	—
91	#95	—	227	1	no	—
92	C₂²×C₄	—	227	2	no	—
93	C₂²×C₄	—	227	2	no	—
94	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	454	1	no	—
101	C₂×C₈	—	454	1	no	—
102	C₂×C₈	—	454	1	no	—
103	C₂×C₈	—	454	1	no	—
104	C₂×C₈	—	227	2	no	—
105	C₂×C₈	—	227	2	no	—
106	C₄⋊C₄	—	454	4	no	—
107	C₄²	—	454	2	no	—
108	C₄²	—	454	2	no	—
109	C₄²	C₂²⋊D₂₂₇	1	2	yes	—
110	C₄²	Dic₄₅₄	1	2	yes	—
111	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	454	2	no	—
115	C₂×C₄	—	454	2	no	—
116	C₂×C₄	—	454	2	no	—
117	C₂×C₄	—	454	2	no	—
118	C₂×C₄	—	454	2	no	—
119	C₂×C₄	—	227	4	no	—
120	C₂×C₄	Dic₂₂₇⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₂₂₇⋊C₄	1	2	yes	—
122	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₂₂₇	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₄₅₄	1	1	yes	characteristic
126	C₂×C₄	#16813883	1	1	yes	characteristic
127	C₂×C₄	#16813885	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	454	2	no	—
130	C₈	—	454	2	no	—
131	C₂²	C₂×Dic₂₂₇⋊C₄	1	1	yes	characteristic
132	C₂²	#20300250	1	1	yes	characteristic
133	C₂²	#20300253	1	1	yes	characteristic
134	C₂²	M₁₆×D₂₂₇	1	2	yes	—
135	C₂²	#20300294	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	908	2	no	—
139	C₄	—	454	4	no	—
140	C₄	#20300259	1	1	yes	characteristic
141	C₄	#20300264	1	1	yes	characteristic
142	C₄	#20300279	1	1	yes	characteristic
143	C₄	#20300287	1	1	yes	characteristic
144	C₂	#28408928	1	1	yes	characteristic
145	C₂	#28409055	1	1	yes	characteristic
146	C₂	#28409058	1	1	yes	characteristic
147	C₂	#28409102	1	2	yes	—
148	C₂	#28409161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₂₇⋊C₈

C₂

yes

maximal

C₂×#20300199

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₂₇

C₂

yes

maximal, characteristic

#28408786

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₂₇

C₄

yes

—

C₂×C₄×Dic₂₂₇

C₂²

yes

—

Dic₂₂₇⋊C₈

—

#20300199

—

C₂₂₇⋊#95

C₄

yes

characteristic

C₂₂₇⋊#95

C₄

yes

characteristic

C₂²×C₈×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

—

C₂²×Dic₂₂₇

C₂×C₄

yes

—

C₂²×Dic₂₂₇

C₂×C₄

yes

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×C₂₂₇⋊C₈

Q₈

yes

—

C₄⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄²×C₂₂₇

D₄

yes

—

C₄²×C₂₂₇

Q₈

yes

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

M₁₆

yes

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂³×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₂₇

C₂×Q₈

yes

C₂×C₄×C₂₂₇

yes

C₂×C₄×C₂₂₇

yes

C₈×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₄×D₄

yes

characteristic

C₂²×C₂₂₇

C₄×Q₈

yes

characteristic

C₂²×C₂₂₇

C₂×M₁₆

yes

—

C₂²×C₂₂₇

#131

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#121

yes

characteristic

C₄×C₂₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₂₂₇

#125

yes

characteristic

C₄×C₂₂₇

C₄⋊Q₈

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

—

C₂×C₂₂₇

#446

yes

—

C₂₂₇

#1940

yes

characteristic, complemented, Sylow

#1940

—

227

maximal, Sylow

C₂×C₄⋊C₈

—

227

—

C₂×C₄⋊C₈

D₂₂₇

yes

characteristic

C₂×#97

—

227

—

C₄×C₄⋊C₄

—

227

—

#336

—

227

—

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

C₂²×C₈

C₂×D₂₂₇

yes

—

C₂×C₄⋊C₄

—

227

—

C₂×C₄²

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

454

—

#97

—

454

—

#95

—

227

—

#95

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

—

227

—

C₂²×C₄

C₄×D₂₂₇

yes

characteristic

C₂²×C₄

C₄×D₂₂₇

yes

characteristic

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

454

—

101

C₂×C₈

—

454

—

102

C₂×C₈

—

454

—

103

C₂×C₈

—

454

—

104

C₂×C₈

—

227

—

105

C₂×C₈

—

227

—

106

C₄⋊C₄

—

454

—

107

C₄²

—

454

—

108

C₄²

—

454

—

109

C₄²

C₂²⋊D₂₂₇

yes

—

110

C₄²

Dic₄₅₄

yes

—

111

C₂³

C₂×C₄×D₂₂₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

454

—

115

C₂×C₄

—

454

—

116

C₂×C₄

—

454

—

117

C₂×C₄

—

454

—

118

C₂×C₄

—

454

—

119

C₂×C₄

—

227

—

120

C₂×C₄

Dic₂₂₇⋊C₄

yes

—

121

C₂×C₄

Dic₂₂₇⋊C₄

yes

—

122

C₂×C₄

D₄×D₂₂₇

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

124

C₂×C₄

Q₈×D₂₂₇

yes

characteristic

125

C₂×C₄

C₂×Dic₄₅₄

yes

characteristic

126

C₂×C₄

#16813883

yes

characteristic

127

C₂×C₄

#16813885

yes

characteristic

128

C₈

—

129

C₈

—

454

—

130

C₈

—

454

—

131

C₂²

C₂×Dic₂₂₇⋊C₄

yes

characteristic

132

C₂²

#20300250

yes

characteristic

133

C₂²

#20300253

yes

characteristic

134

C₂²

M₁₆×D₂₂₇

yes

—

135

C₂²

#20300294

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

908

—

139

C₄

—

454

—

140

C₄

#20300259

yes

characteristic

141

C₄

#20300264

yes

characteristic

142

C₄

#20300279

yes

characteristic

143

C₄

#20300287

yes

characteristic

144

C₂

#28408928

yes

characteristic

145

C₂

#28409055

yes

characteristic

146

C₂

#28409058

yes

characteristic

147

C₂

#28409102

yes

—

148

C₂

#28409161

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₂₇
commutator, socle, prime radical	C₂³×C₂₂₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₂₇	C₂
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₂₇
hypocenter	C₂₂₇	#1940
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₂₇

commutator, socle, prime radical

C₂³×C₂₂₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₂₇

C₂

cosocle

C₂²×C₄×C₂₂₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₂₇

C₂₂₇

G ▷ C₂³×C₂₂₇ ▷ C₂₂₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
74. C₂₂₇	#1940	semidirect

H ≅ G/N

type

74. C₂₂₇

#1940

semidirect

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

subgroup	index
C₂×Dic₂₂₇⋊C₈	2
C₂×#20300199	2
C₄×C₄⋊Dic₂₂₇	2
#28408786	2
C₂×C₄⋊C₈×C₂₂₇	2
#1940	227

subgroup

index

C₂×Dic₂₂₇⋊C₈

C₂×#20300199

C₄×C₄⋊Dic₂₂₇

#28408786

C₂×C₄⋊C₈×C₂₂₇

#1940

227

Subgroups — #44400455

Conjugacy classes of subgroups

Complements

Maximal subgroups