Group #44399239 — subgroups

Subgroups — #44399239

Order 29056.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#28409428	C₂	1	1	yes	maximal, characteristic
2	#28409328	C₂	1	1	yes	maximal, characteristic
3	#28409239	C₂	1	1	yes	maximal, characteristic
4	#28409159	C₂	1	1	yes	maximal, characteristic
5	#28408784	C₂	1	1	yes	maximal, characteristic
6	#28408780	C₂	1	1	yes	maximal, characteristic
7	#332×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#16813874	—	2	1	no	—
9	#20300288	C₂²	1	1	yes	characteristic
10	C₄×Dic₄₅₄	C₂²	1	1	yes	characteristic
11	#20300222	—	2	1	no	—
12	#20300219	—	2	1	no	—
13	#20300219	—	2	1	no	—
14	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₂₂₇⋊C₈	—	2	1	no	—
16	#20300213	—	2	1	no	—
17	#20300212	—	2	1	no	—
18	#20300198	—	2	1	no	—
19	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₄₅₄	—	2	1	no	—
24	C₂×Dic₄₅₄	D₄	1	1	yes	characteristic
25	#16813874	—	4	2	no	—
26	C₂×C₂₂₇⋊C₈	—	2	1	no	—
27	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₂₂₇	—	2	1	no	—
30	C₄⋊Dic₂₂₇	—	2	1	no	—
31	C₄⋊Dic₂₂₇	D₄	1	1	yes	characteristic
32	Dic₂₂₇⋊C₄	—	2	1	no	—
33	C₄×Dic₂₂₇	—	2	1	no	—
34	C₂×Q₈×C₂₂₇	—	2	1	no	—
35	C₂×C₈×C₂₂₇	—	2	1	no	—
36	C₂×C₈×C₂₂₇	—	2	1	no	—
37	C₄⋊C₄×C₂₂₇	—	2	1	no	—
38	C₄⋊C₄×C₂₂₇	—	2	1	no	—
39	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
40	Dic₄₅₄	—	2	1	no	—
41	Dic₄₅₄	—	2	1	no	—
42	Dic₄₅₄	—	8	1	no	—
43	C₂×Dic₂₂₇	—	2	1	no	—
44	C₂×Dic₂₂₇	—	2	1	no	—
45	C₂×Dic₂₂₇	—	4	1	no	—
46	C₂₂₇⋊C₈	—	2	1	no	—
47	C₂₂₇⋊C₈	—	4	1	no	—
48	C₂₂₇⋊C₈	Q₁₆	1	2	yes	complemented
49	Q₈×C₂₂₇	—	4	1	no	—
50	Q₈×C₂₂₇	—	4	1	no	—
51	C₂×C₄×C₂₂₇	—	4	1	no	—
52	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
55	C₈×C₂₂₇	—	4	1	no	—
56	C₈×C₂₂₇	—	4	1	no	—
57	C₂²×C₂₂₇	#121	1	1	yes	characteristic
58	C₄×C₂₂₇	—	2	1	no	—
59	C₄×C₂₂₇	—	8	1	no	—
60	C₄×C₂₂₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₂₂₇	#137	1	1	yes	characteristic
63	C₄×C₂₂₇	#137	1	1	yes	characteristic
64	Dic₂₂₇	—	4	1	no	—
65	Dic₂₂₇	—	4	1	no	—
66	Dic₂₂₇	—	8	1	no	—
67	C₂×C₂₂₇	#462	1	1	yes	characteristic
68	C₂×C₂₂₇	#468	1	1	yes	characteristic
69	C₂×C₂₂₇	#471	1	1	yes	characteristic
70	C₂₂₇	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	227	1	no	maximal, Sylow
72	#498	—	227	1	no	—
73	#477	—	227	1	no	—
74	#462	—	227	1	no	—
75	#446	—	227	1	no	—
76	#335	—	227	1	no	—
77	#333	—	227	1	no	—
78	#332	D₂₂₇	1	1	yes	characteristic
79	C₂×Q₁₆	—	454	1	no	—
80	C₄⋊Q₈	—	227	1	no	—
81	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
82	C₄×Q₈	—	227	1	no	—
83	#107	—	454	1	no	—
84	#106	—	454	1	no	—
85	#106	—	454	1	no	—
86	C₄⋊C₈	—	454	1	no	—
87	C₄⋊C₈	—	227	1	no	—
88	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
90	#103	—	454	1	no	—
91	#103	—	454	1	no	—
92	#97	—	454	1	no	—
93	C₄×C₈	—	227	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	454	1	no	—
96	C₂×Q₈	—	227	1	no	—
97	Q₁₆	—	908	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	454	1	no	—
101	C₂×C₈	—	227	1	no	—
102	C₂×C₈	—	227	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	454	1	no	—
106	C₄⋊C₄	—	454	1	no	—
107	C₄⋊C₄	—	454	1	no	—
108	C₄⋊C₄	—	227	1	no	—
109	C₄²	—	454	1	no	—
110	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	1816	1	no	—
114	Q₈	—	454	1	no	—
115	Q₈	—	454	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	908	1	no	—
118	C₂×C₄	—	454	1	no	—
119	C₂×C₄	—	454	1	no	—
120	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
121	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
122	C₂×C₄	#16813884	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	908	1	no	—
126	C₈	—	454	1	no	—
127	C₈	—	227	2	no	—
128	C₂²	#20300261	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1816	1	no	—
132	C₄	—	908	1	no	—
133	C₄	—	908	1	no	—
134	C₄	Q₁₆×D₂₂₇	1	1	yes	characteristic
135	C₄	#20300312	1	1	yes	characteristic
136	C₄	#20300317	1	1	yes	characteristic
137	C₄	#20300317	1	1	yes	characteristic
138	C₂	#28409241	1	1	yes	characteristic
139	C₂	#28409277	1	1	yes	characteristic
140	C₂	#28409298	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409428

C₂

yes

maximal, characteristic

#28409328

C₂

yes

maximal, characteristic

#28409239

C₂

yes

maximal, characteristic

#28409159

C₂

yes

maximal, characteristic

#28408784

C₂

yes

maximal, characteristic

#28408780

C₂

yes

maximal, characteristic

#332×C₂₂₇

C₂

yes

maximal, characteristic

C₂×#16813874

—

#20300288

C₂²

yes

characteristic

C₄×Dic₄₅₄

C₂²

yes

—

—

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

characteristic

Dic₂₂₇⋊C₈

—

#20300213

—

#20300212

—

#20300198

—

C₄×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

C₂²

yes

characteristic

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

D₄

yes

characteristic

#16813874

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

D₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

D₄

yes

characteristic

Dic₂₂₇⋊C₄

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

Q₁₆

yes

complemented

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

characteristic

C₈×C₂₂₇

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

#121

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×Q₁₆

yes

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

C₂₂₇

yes

characteristic, complemented, Sylow

#1692

—

227

maximal, Sylow

#498

—

227

—

#477

—

227

—

#462

—

227

—

#446

—

227

—

#335

—

227

—

#333

—

227

—

#332

D₂₂₇

yes

characteristic

C₂×Q₁₆

—

454

—

C₄⋊Q₈

—

227

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

C₄×Q₈

—

227

—

#107

—

454

—

#106

—

454

—

#106

—

454

—

C₄⋊C₈

—

454

—

C₄⋊C₈

—

227

—

C₄⋊C₈

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

C₂×D₂₂₇

yes

characteristic

#103

—

454

—

#103

—

454

—

#97

—

454

—

C₄×C₈

—

227

—

C₂×Q₈

—

C₂×Q₈

—

454

—

C₂×Q₈

—

227

—

Q₁₆

—

908

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

454

—

101

C₂×C₈

—

227

—

102

C₂×C₈

—

227

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

454

—

106

C₄⋊C₄

—

454

—

107

C₄⋊C₄

—

454

—

108

C₄⋊C₄

—

227

—

109

C₄²

—

454

—

110

C₄²

C₂²×D₂₂₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

1816

—

114

Q₈

—

454

—

115

Q₈

—

454

—

116

C₂×C₄

—

117

C₂×C₄

—

908

—

118

C₂×C₄

—

454

—

119

C₂×C₄

—

454

—

120

C₂×C₄

D₄×D₂₂₇

yes

characteristic

121

C₂×C₄

D₄×D₂₂₇

yes

characteristic

122

C₂×C₄

#16813884

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

908

—

126

C₈

—

454

—

127

C₈

—

227

—

128

C₂²

#20300261

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1816

—

132

C₄

—

908

—

133

C₄

—

908

—

134

C₄

Q₁₆×D₂₂₇

yes

characteristic

135

C₄

#20300312

yes

characteristic

136

C₄

#20300317

yes

characteristic

137

C₄

#20300317

yes

characteristic

138

C₂

#28409241

yes

characteristic

139

C₂

#28409277

yes

characteristic

140

C₂

#28409298

yes

characteristic

	subgroup	G/H
center	C₂²	#20300261
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#332×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	#121
hypercenter	#332	D₂₂₇
hypocenter	C₂₂₇	#1692
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#20300261

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#332×C₂₂₇

C₂

C₂²×C₂₂₇

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
48. C₂₂₇⋊C₈	Q₁₆	semidirect
70. C₂₂₇	#1692	semidirect

H ≅ G/N

type

48. C₂₂₇⋊C₈

Q₁₆

semidirect

70. C₂₂₇

#1692

semidirect

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

subgroup	index
#28409428	2
#28409328	2
#28409239	2
#28409159	2
#28408784	2
#28408780	2
#332×C₂₂₇	2
#1692	227

subgroup

index

#332×C₂₂₇

227

Subgroups — #44399239

Conjugacy classes of subgroups

Complements

Maximal subgroups