Group #44577193 — subgroups

Subgroups — C₁₂₇×#13520 #44577193

Order 32512.

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

normal subgroups only

This group has 238 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×#2151	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1953	C₂	1	2	yes	maximal
4	C₁₂₇×#1385	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1381	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	#430×C₁₂₇	—	2	2	no	—
9	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
10	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
11	#405×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	—	2	2	no	—
13	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
14	#344×C₁₂₇	—	2	2	no	—
15	C₂×M₁₆×C₁₂₇	—	2	2	no	—
16	C₂×M₁₆×C₁₂₇	—	2	2	no	—
17	C₂×M₁₆×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
19	#117×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	2	2	no	—
21	C₂×C₁₆×C₁₂₇	—	2	1	no	—
22	C₂×C₁₆×C₁₂₇	—	2	2	no	—
23	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
24	#108×C₁₂₇	—	2	2	no	—
25	#108×C₁₂₇	—	2	2	no	—
26	#108×C₁₂₇	—	2	2	no	—
27	#108×C₁₂₇	—	4	4	no	—
28	#108×C₁₂₇	Q₈	1	4	yes	—
29	#97×C₁₂₇	—	2	1	no	—
30	C₄×C₈×C₁₂₇	—	2	1	no	—
31	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₁₂₇	—	2	4	no	—
33	M₁₆×C₁₂₇	—	2	2	no	—
34	M₁₆×C₁₂₇	—	4	2	no	—
35	M₁₆×C₁₂₇	—	4	2	no	—
36	M₁₆×C₁₂₇	—	4	2	no	—
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	—	4	2	no	—
40	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₁₂₇	#41	1	2	yes	—
43	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₁₂₇	—	2	1	no	—
45	C₂²⋊C₄×C₁₂₇	Q₁₆	1	2	yes	—
46	C₄²×C₁₂₇	—	2	1	no	—
47	C₁₆×C₁₂₇	—	2	2	no	—
48	C₁₆×C₁₂₇	—	4	1	no	—
49	C₂³×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₈×C₁₂₇	—	4	1	no	—
56	C₈×C₁₂₇	—	4	4	no	—
57	C₈×C₁₂₇	—	8	2	no	—
58	C₈×C₁₂₇	#123	1	1	yes	characteristic
59	C₈×C₁₂₇	#125	1	2	yes	—
60	C₈×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₂₇	—	4	1	no	—
62	C₂²×C₁₂₇	#467	1	1	yes	characteristic
63	C₂²×C₁₂₇	#477	1	2	yes	—
64	C₄×C₁₂₇	—	4	2	no	—
65	C₄×C₁₂₇	#398	1	1	yes	characteristic
66	C₄×C₁₂₇	#469	1	1	yes	characteristic
67	C₄×C₁₂₇	#478	1	2	yes	—
68	#13520	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₁₂₇	—	2	1	no	—
70	C₂×C₁₂₇	—	2	2	no	—
71	C₂×C₁₂₇	#2087	1	1	yes	characteristic
72	#2161	C₂×C₁₂₇	1	1	yes	characteristic
73	#2151	C₂×C₁₂₇	1	1	yes	characteristic
74	#1953	C₂×C₁₂₇	1	2	yes	—
75	#1385	C₂×C₁₂₇	1	1	yes	characteristic
76	#1381	C₂×C₁₂₇	1	2	yes	—
77	C₁₂₇	#13520	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
79	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
80	#430	—	2	2	no	—
81	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
82	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
83	#405	C₂²×C₁₂₇	1	1	yes	characteristic
84	#368	—	2	2	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	#344	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	D₄×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
91	#117	D₄×C₁₂₇	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	2	2	no	—
99	#108	—	4	4	no	—
100	#108	Q₈×C₁₂₇	1	4	yes	—
101	#97	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
114	C₂×C₈	#41×C₁₂₇	1	2	yes	—
115	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Q₁₆×C₁₂₇	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
125	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	4	4	no	—
129	C₈	—	8	2	no	—
130	C₈	#123×C₁₂₇	1	1	yes	characteristic
131	C₈	#125×C₁₂₇	1	2	yes	—
132	C₈	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	#467×C₁₂₇	1	1	yes	characteristic
135	C₂²	#477×C₁₂₇	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	#398×C₁₂₇	1	1	yes	characteristic
138	C₄	#469×C₁₂₇	1	1	yes	characteristic
139	C₄	#478×C₁₂₇	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	C₁₂₇×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2161

C₂

yes

maximal, characteristic

C₁₂₇×#2151

C₂

yes

maximal, characteristic

C₁₂₇×#1953

C₂

yes

maximal

C₁₂₇×#1385

C₂

yes

maximal, characteristic

C₁₂₇×#1381

C₂

yes

maximal

C₂×M₃₂×C₁₂₇

C₂²

yes

characteristic

C₂²×C₁₆×C₁₂₇

C₂²

yes

characteristic

#430×C₁₂₇

—

C₂×#108×C₁₂₇

C₂²

yes

—

C₂×#108×C₁₂₇

C₂²

yes

—

#405×C₁₂₇

C₂²

yes

characteristic

#368×C₁₂₇

—

C₄⋊C₁₆×C₁₂₇

—

#344×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

D₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

#117×C₁₂₇

D₄

yes

characteristic

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

D₄

yes

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#108×C₁₂₇

Q₈

yes

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

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

Q₁₆

yes

—

C₄²×C₁₂₇

—

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#121

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₂₇

#135

yes

characteristic

C₂×C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₂₇

—

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

#477

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#13520

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#2087

yes

characteristic

#2161

C₂×C₁₂₇

yes

characteristic

#2151

C₂×C₁₂₇

yes

characteristic

#1953

C₂×C₁₂₇

yes

—

#1385

C₂×C₁₂₇

yes

characteristic

#1381

C₂×C₁₂₇

yes

—

C₁₂₇

#13520

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₁₂₇

yes

characteristic

C₂²×C₁₆

C₂²×C₁₂₇

yes

characteristic

#430

—

C₂×#108

C₂²×C₁₂₇

yes

—

C₂×#108

C₂²×C₁₂₇

yes

—

#405

C₂²×C₁₂₇

yes

characteristic

#368

—

C₄⋊C₁₆

—

#344

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₁₂₇

yes

characteristic

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

#117

D₄×C₁₂₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₂₇

yes

—

—

—

—

—

100

#108

Q₈×C₁₂₇

yes

—

101

#97

—

102

C₄×C₈

—

103

C₂²×C₄

C₂×D₄×C₁₂₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₁₂₇

yes

—

114

C₂×C₈

#41×C₁₂₇

yes

—

115

C₂×C₈

#41×C₁₂₇

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Q₁₆×C₁₂₇

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Q₁₆×C₁₂₇

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

#121×C₁₂₇

yes

characteristic

124

C₂×C₄

C₂²⋊Q₈×C₁₂₇

yes

—

125

C₂×C₄

#135×C₁₂₇

yes

characteristic

126

C₂×C₄

AGL(1,ℤ/8)×C₁₂₇

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#123×C₁₂₇

yes

characteristic

131

C₈

#125×C₁₂₇

yes

—

132

C₈

C₄⋊Q₈×C₁₂₇

yes

characteristic

133

C₂²

—

134

C₂²

#467×C₁₂₇

yes

characteristic

135

C₂²

#477×C₁₂₇

yes

—

136

C₄

—

137

C₄

#398×C₁₂₇

yes

characteristic

138

C₄

#469×C₁₂₇

yes

characteristic

139

C₄

#478×C₁₂₇

yes

—

140

C₂

—

141

C₂

—

142

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	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

subgroup

G/H

center

C₄×C₁₂₇

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

socle

C₂×C₁₂₇

#2087

prime radical

C₂³×C₁₂₇

C₂×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

N	H ≅ G/N	type
68. #13520	C₁₂₇	direct (N × H)
77. C₁₂₇	#13520	direct (N × H)

H ≅ G/N

type

68. #13520

C₁₂₇

direct (N × H)

77. C₁₂₇

#13520

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×#2151	2
C₁₂₇×#1953	2
C₁₂₇×#1385	2
C₁₂₇×#1381	2
#13520	127

subgroup

index

C₁₂₇×#2161

C₁₂₇×#2151

C₁₂₇×#1953

C₁₂₇×#1385

C₁₂₇×#1381

#13520

127

Subgroups — C₁₂₇×#13520 #44577193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#13520 #44577193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13520 #44577193