Group #44577192 — subgroups

Subgroups — C₁₂₇×#13519 #44577192

Order 32512.

252 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₁₂₇×#2161	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1945	C₂	1	2	yes	maximal
4	C₁₂₇×#1389	C₂	1	2	yes	maximal
5	C₁₂₇×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#427×C₁₂₇	C₂²	1	2	yes	—
10	#424×C₁₂₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	—	2	2	no	—
13	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
14	#339×C₁₂₇	—	2	2	no	—
15	C₂×M₁₆×C₁₂₇	—	2	2	no	—
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
18	#117×C₁₂₇	—	2	2	no	—
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	2	2	no	—
21	M₃₂×C₁₂₇	—	2	2	no	—
22	C₂×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
24	#108×C₁₂₇	—	2	2	no	—
25	#108×C₁₂₇	—	2	2	no	—
26	#107×C₁₂₇	—	2	2	no	—
27	#106×C₁₂₇	Q₈	1	4	yes	—
28	C₄⋊C₈×C₁₂₇	—	4	4	no	—
29	C₄×C₈×C₁₂₇	—	2	1	no	—
30	C₄×C₈×C₁₂₇	—	2	1	no	—
31	C₂²×C₄×C₁₂₇	—	2	1	no	—
32	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	4	2	no	—
34	M₁₆×C₁₂₇	—	4	2	no	—
35	C₂×C₈×C₁₂₇	—	2	1	no	—
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₂×C₈×C₁₂₇	—	4	2	no	—
38	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₁₂₇	#41	1	2	yes	—
41	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₁₂₇	—	2	4	no	—
43	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
44	C₄²×C₁₂₇	—	2	1	no	—
45	C₄²×C₁₂₇	—	4	2	no	—
46	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
47	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
48	C₁₆×C₁₂₇	—	2	2	no	—
49	C₁₆×C₁₂₇	—	4	1	no	—
50	C₂³×C₁₂₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	4	2	no	—
54	C₂×C₄×C₁₂₇	—	4	4	no	—
55	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₈×C₁₂₇	—	4	1	no	—
60	C₈×C₁₂₇	—	8	2	no	—
61	C₈×C₁₂₇	#123	1	1	yes	characteristic
62	C₈×C₁₂₇	#125	1	2	yes	—
63	C₈×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₂₇	—	4	1	no	—
65	C₂²×C₁₂₇	#467	1	1	yes	characteristic
66	C₂²×C₁₂₇	#478	1	2	yes	—
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	4	1	no	—
69	C₄×C₁₂₇	—	8	4	no	—
70	C₄×C₁₂₇	#398	1	1	yes	characteristic
71	C₄×C₁₂₇	#469	1	1	yes	characteristic
72	C₄×C₁₂₇	#477	1	2	yes	—
73	#13519	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₁₂₇	—	2	1	no	—
75	C₂×C₁₂₇	—	2	2	no	—
76	C₂×C₁₂₇	#2087	1	1	yes	characteristic
77	#2161	C₂×C₁₂₇	1	1	yes	characteristic
78	#2150	C₂×C₁₂₇	1	1	yes	characteristic
79	#1945	C₂×C₁₂₇	1	2	yes	—
80	#1389	C₂×C₁₂₇	1	2	yes	—
81	#1384	C₂×C₁₂₇	1	1	yes	characteristic
82	C₁₂₇	#13519	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
84	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
85	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
86	#427	C₂²×C₁₂₇	1	2	yes	—
87	#424	—	2	2	no	—
88	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
89	#368	—	2	2	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	#339	—	2	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
95	#117	—	2	2	no	—
96	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
97	M₃₂	—	2	2	no	—
98	M₃₂	—	2	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	2	2	no	—
103	#107	—	2	2	no	—
104	#106	Q₈×C₁₂₇	1	4	yes	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
117	C₂×C₈	#41×C₁₂₇	1	2	yes	—
118	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	4	2	no	—
123	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
124	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	—	4	1	no	—
127	C₂³	#135×C₁₂₇	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
134	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
136	C₈	—	4	1	no	—
137	C₈	—	8	2	no	—
138	C₈	#123×C₁₂₇	1	1	yes	characteristic
139	C₈	#125×C₁₂₇	1	2	yes	—
140	C₈	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂²	—	4	1	no	—
142	C₂²	#467×C₁₂₇	1	1	yes	characteristic
143	C₂²	#478×C₁₂₇	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	8	4	no	—
147	C₄	#398×C₁₂₇	1	1	yes	characteristic
148	C₄	#469×C₁₂₇	1	1	yes	characteristic
149	C₄	#477×C₁₂₇	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	C₁₂₇×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2161

C₂

yes

maximal, characteristic

C₁₂₇×#2150

C₂

yes

maximal, characteristic

C₁₂₇×#1945

C₂

yes

maximal

C₁₂₇×#1389

C₂

yes

maximal

C₁₂₇×#1384

C₂

yes

maximal, characteristic

C₂×M₃₂×C₁₂₇

C₂²

yes

characteristic

C₂×M₃₂×C₁₂₇

C₂²

yes

characteristic

C₂×#108×C₁₂₇

C₂²

yes

—

#427×C₁₂₇

C₂²

yes

—

#424×C₁₂₇

—

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

#368×C₁₂₇

—

C₄⋊C₁₆×C₁₂₇

—

#339×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

D₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

#117×C₁₂₇

—

C₂×C₄²×C₁₂₇

D₄

yes

characteristic

M₃₂×C₁₂₇

—

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

D₄

yes

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#107×C₁₂₇

—

#106×C₁₂₇

Q₈

yes

—

C₄⋊C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×Q₈

yes

—

C₂×C₈×C₁₂₇

#41

yes

—

C₂×C₈×C₁₂₇

#41

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

Q₁₆

yes

characteristic

C₄²×C₁₂₇

Q₁₆

yes

characteristic

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

#135

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#121

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#123

yes

characteristic

C₈×C₁₂₇

#125

yes

—

C₈×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#467

yes

characteristic

C₂²×C₁₂₇

#478

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#13519

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#2087

yes

characteristic

#2161

C₂×C₁₂₇

yes

characteristic

#2150

C₂×C₁₂₇

yes

characteristic

#1945

C₂×C₁₂₇

yes

—

#1389

C₂×C₁₂₇

yes

—

#1384

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#13519

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₁₂₇

yes

characteristic

C₂×M₃₂

C₂²×C₁₂₇

yes

characteristic

C₂×#108

C₂²×C₁₂₇

yes

—

#427

C₂²×C₁₂₇

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

#368

—

C₄⋊C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₂₇

yes

characteristic

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

#117

—

C₂×C₄²

D₄×C₁₂₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

D₄×C₁₂₇

yes

—

101

#108

—

102

#108

—

103

#107

—

104

#106

Q₈×C₁₂₇

yes

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×D₄×C₁₂₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

116

C₂×C₈

C₂×Q₈×C₁₂₇

yes

—

117

C₂×C₈

#41×C₁₂₇

yes

—

118

C₂×C₈

#41×C₁₂₇

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

Q₁₆×C₁₂₇

yes

characteristic

124

C₄²

Q₁₆×C₁₂₇

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₂³

#135×C₁₂₇

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#121×C₁₂₇

yes

characteristic

133

C₂×C₄

C₂²⋊Q₈×C₁₂₇

yes

—

134

C₂×C₄

C₂×Q₁₆×C₁₂₇

yes

characteristic

135

C₂×C₄

AGL(1,ℤ/8)×C₁₂₇

yes

—

136

C₈

—

137

C₈

—

138

C₈

#123×C₁₂₇

yes

characteristic

139

C₈

#125×C₁₂₇

yes

—

140

C₈

C₄⋊Q₈×C₁₂₇

yes

characteristic

141

C₂²

—

142

C₂²

#467×C₁₂₇

yes

characteristic

143

C₂²

#478×C₁₂₇

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#398×C₁₂₇

yes

characteristic

148

C₄

#469×C₁₂₇

yes

characteristic

149

C₄

#477×C₁₂₇

yes

—

150

C₂

—

151

C₂

—

152

C₂

C₁₂₇×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₁₂₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#2087
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₄×C₁₂₇

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

C₂×C₁₂₇

C₂³×C₁₂₇

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
73. #13519	C₁₂₇	direct (N × H)
82. C₁₂₇	#13519	direct (N × H)

H ≅ G/N

type

73. #13519

C₁₂₇

direct (N × H)

82. C₁₂₇

#13519

direct (N × H)

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

subgroup	index
C₁₂₇×#2161	2
C₁₂₇×#2150	2
C₁₂₇×#1945	2
C₁₂₇×#1389	2
C₁₂₇×#1384	2
#13519	127

subgroup

index

C₁₂₇×#2161

C₁₂₇×#2150

C₁₂₇×#1945

C₁₂₇×#1389

C₁₂₇×#1384

#13519

127

Subgroups — C₁₂₇×#13519 #44577192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#13519 #44577192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13519 #44577192