Group #44399782 — subgroups

Subgroups — #44399782

Order 29056.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20300295	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300223	C₂	1	2	yes	maximal, complemented
3	#28409016	C₂	1	1	yes	maximal, characteristic
4	C₂₂₇⋊#338	C₂	1	2	yes	maximal, complemented
5	#413×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	#20300295	—	2	4	no	—
7	C₂×#16813868	C₄	1	2	yes	—
8	C₂×#16813868	C₂²	1	2	yes	—
9	C₂×#16813868	C₂²	1	1	yes	characteristic
10	C₂²×C₂₂₇⋊C₈	C₄	1	2	yes	—
11	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	#20300223	—	2	4	no	—
13	#20300207	—	2	1	no	—
14	#20300205	—	2	1	no	—
15	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
17	#101×C₂₂₇	—	2	1	no	—
18	#100×C₂₂₇	—	2	1	no	—
19	#16813868	—	2	4	no	—
20	#16813868	—	2	2	no	—
21	#16813868	—	2	2	no	—
22	#16813868	—	2	2	no	—
23	C₂×C₂₂₇⋊C₈	—	2	4	no	—
24	C₂×C₂₂₇⋊C₈	—	2	2	no	—
25	C₂×C₂₂₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	—
29	#41×C₂₂₇	—	2	2	no	—
30	#41×C₂₂₇	—	2	2	no	—
31	C₂×Q₈×C₂₂₇	D₄	1	1	yes	characteristic
32	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
33	C₂×D₄×C₂₂₇	Q₈	1	2	yes	—
34	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₂₇	—	2	2	no	—
37	M₁₆×C₂₂₇	—	2	2	no	—
38	C₂×C₈×C₂₂₇	—	2	2	no	—
39	C₂₂₇⋊C₈	—	2	2	no	—
40	C₂₂₇⋊C₈	—	4	2	no	—
41	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
42	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₂₂₇	C₄⋊C₄	1	2	yes	—
44	C₂³×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
45	Q₈×C₂₂₇	—	2	2	no	—
46	D₄×C₂₂₇	—	2	4	no	—
47	D₄×C₂₂₇	—	2	2	no	—
48	C₂×C₄×C₂₂₇	—	2	2	no	—
49	C₂×C₄×C₂₂₇	—	2	2	no	—
50	C₂×C₄×C₂₂₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₂₇	#41	1	2	yes	—
55	C₈×C₂₂₇	—	4	2	no	—
56	C₂²×C₂₂₇	—	2	2	no	—
57	C₂²×C₂₂₇	—	4	1	no	—
58	C₂²×C₂₂₇	—	4	2	no	—
59	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₂₇	—	2	2	no	—
62	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₂₂₇	#117	1	1	yes	characteristic
64	C₄×C₂₂₇	#121	1	2	yes	—
65	C₂×C₂₂₇	—	2	1	no	—
66	C₂×C₂₂₇	—	2	2	no	—
67	C₂×C₂₂₇	—	4	2	no	—
68	C₂×C₂₂₇	#380	1	1	yes	characteristic
69	C₂₂₇	#1813	1	1	yes	characteristic, complemented, Sylow
70	#1813	—	227	1	no	maximal, Sylow
71	C₂×#131	—	227	1	no	—
72	C₂×#108	—	227	2	no	—
73	#413	—	227	1	no	—
74	#413	D₂₂₇	1	1	yes	characteristic
75	#338	—	227	2	no	—
76	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
77	#131	—	454	4	no	—
78	C₂×M₁₆	—	227	2	no	—
79	C₂×M₁₆	—	227	2	no	—
80	C₂×M₁₆	—	227	1	no	—
81	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
82	C₂²×C₈	—	227	2	no	—
83	C₂²×C₈	—	227	1	no	—
84	#108	—	454	4	no	—
85	#101	—	2	1	no	—
86	#101	—	454	1	no	—
87	#100	—	2	1	no	—
88	#100	—	454	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₂₂₇	1	1	yes	characteristic
92	C₂×D₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
93	C₂×D₄	Dic₄₅₄	1	2	yes	—
94	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	454	2	no	—
99	M₁₆	—	454	2	no	—
100	M₁₆	—	454	2	no	—
101	M₁₆	—	454	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	454	2	no	—
104	C₂×C₈	—	454	4	no	—
105	C₂×C₈	—	227	2	no	—
106	C₂×C₈	—	227	1	no	—
107	C₂×C₈	—	227	1	no	—
108	C₂×C₈	—	227	4	no	—
109	C₂³	Dic₂₂₇⋊C₄	1	2	yes	—
110	C₂³	C₂×Dic₄₅₄	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	Dic₂₂₇⋊C₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
120	C₂×C₄	#16813883	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	908	2	no	—
123	C₈	—	454	2	no	—
124	C₈	—	227	2	no	—
125	C₈	—	227	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×Dic₂₂₇⋊C₄	1	1	yes	characteristic
130	C₂²	#20300265	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₂₂₇	1	1	yes	characteristic
133	C₄	#20300244	1	1	yes	characteristic
134	C₄	#20300260	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#28408890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20300295

C₂

yes

maximal, characteristic

C₂×#20300223

C₂

yes

maximal, complemented

#28409016

C₂

yes

maximal, characteristic

C₂₂₇⋊#338

C₂

yes

maximal, complemented

#413×C₂₂₇

C₂

yes

maximal, characteristic

#20300295

—

C₂×#16813868

C₄

yes

—

C₂×#16813868

C₂²

yes

—

C₂×#16813868

C₂²

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₄

yes

—

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

—

—

C₂×#41×C₂₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₂₇

C₂²

yes

—

#101×C₂₂₇

—

#100×C₂₂₇

—

—

—

—

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

—

#41×C₂₂₇

—

#41×C₂₂₇

—

C₂×Q₈×C₂₂₇

D₄

yes

characteristic

C₂×D₄×C₂₂₇

D₄

yes

characteristic

C₂×D₄×C₂₂₇

Q₈

yes

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₂₇

C₄⋊C₄

yes

—

C₂³×C₂₂₇

C₂×Q₈

yes

characteristic

Q₈×C₂₂₇

—

D₄×C₂₂₇

—

D₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₄⋊C₄

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

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₂²⋊Q₈

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#380

yes

characteristic

C₂₂₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

227

maximal, Sylow

C₂×#131

—

227

—

C₂×#108

—

227

—

#413

—

227

—

#413

D₂₂₇

yes

characteristic

#338

—

227

—

C₂×#41

C₂×D₂₂₇

yes

characteristic

#131

—

454

—

C₂×M₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂×M₁₆

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₂²×C₈

—

227

—

C₂²×C₈

—

227

—

#108

—

454

—

#101

—

#101

—

454

—

#100

—

#100

—

454

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₂₂₇

yes

characteristic

C₂×D₄

C₂²⋊D₂₂₇

yes

characteristic

C₂×D₄

Dic₄₅₄

yes

—

C₂²×C₄

C₄×D₂₂₇

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

454

—

M₁₆

—

454

—

100

M₁₆

—

454

—

101

M₁₆

—

454

—

102

C₂×C₈

—

103

C₂×C₈

—

454

—

104

C₂×C₈

—

454

—

105

C₂×C₈

—

227

—

106

C₂×C₈

—

227

—

107

C₂×C₈

—

227

—

108

C₂×C₈

—

227

—

109

C₂³

Dic₂₂₇⋊C₄

yes

—

110

C₂³

C₂×Dic₄₅₄

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₂₂₇⋊C₄

yes

—

117

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

118

C₂×C₄

D₄×D₂₂₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

120

C₂×C₄

#16813883

yes

—

121

C₈

—

122

C₈

—

908

—

123

C₈

—

454

—

124

C₈

—

227

—

125

C₈

—

227

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₂₂₇⋊C₄

yes

characteristic

130

C₂²

#20300265

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₂₂₇

yes

characteristic

133

C₄

#20300244

yes

characteristic

134

C₄

#20300260

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#28408890

yes

characteristic

	subgroup	G/H
center	C₄	#20300244
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#413×C₂₂₇	C₂
socle	C₂×C₂₂₇	#380
prime radical	C₂×#41×C₂₂₇	C₂²
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#413	D₂₂₇
hypocenter	C₂₂₇	#1813
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#20300244

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#413×C₂₂₇

C₂

socle

C₂×C₂₂₇

#380

prime radical

C₂×#41×C₂₂₇

C₂²

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#20300223	C₂	semidirect
4. C₂₂₇⋊#338	C₂	semidirect
69. C₂₂₇	#1813	semidirect

H ≅ G/N

type

2. C₂×#20300223

C₂

semidirect

4. C₂₂₇⋊#338

C₂

semidirect

69. C₂₂₇

#1813

semidirect

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

subgroup	index
C₂×#20300295	2
C₂×#20300223	2
#28409016	2
C₂₂₇⋊#338	2
#413×C₂₂₇	2
#1813	227

subgroup

index

C₂×#20300295

C₂×#20300223

#28409016

C₂₂₇⋊#338

#413×C₂₂₇

#1813

227

Subgroups — #44399782

Conjugacy classes of subgroups

Complements

Maximal subgroups