Group #44570822 — subgroups

Subgroups — C₁₂₇×#7149 #44570822

Order 32512.

266 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2163	C₂	1	2	yes	maximal
2	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₁₂₇	—	2	2	no	—
4	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₄³×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₁₂₇	C₄	1	4	yes	complemented
8	C₂²⋊C₁₆×C₁₂₇	—	2	2	no	—
9	#346×C₁₂₇	—	2	2	no	—
10	C₂²×C₈×C₁₂₇	—	2	1	no	—
11	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
12	C₂×C₄²×C₁₂₇	—	2	1	no	—
13	C₂×C₄²×C₁₂₇	—	2	1	no	—
14	C₂×C₄²×C₁₂₇	C₈	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	—	2	4	no	—
17	#97×C₁₂₇	—	2	8	no	—
18	C₄×C₈×C₁₂₇	—	2	1	no	—
19	C₄×C₈×C₁₂₇	C₈	1	2	yes	—
20	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	—	2	4	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	4	no	—
30	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₂₇	—	2	4	no	—
33	C₄²×C₁₂₇	—	4	2	no	—
34	C₄²×C₁₂₇	—	4	2	no	—
35	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
36	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
39	C₁₆×C₁₂₇	—	4	4	no	—
40	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	—	4	2	no	—
46	C₂×C₄×C₁₂₇	—	4	2	no	—
47	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#97	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	4	no	—
54	C₈×C₁₂₇	—	2	2	no	—
55	C₈×C₁₂₇	#104	1	4	yes	—
56	C₂²×C₁₂₇	—	2	1	no	—
57	C₂²×C₁₂₇	—	2	1	no	—
58	C₂²×C₁₂₇	#336	1	1	yes	characteristic
59	C₂²×C₁₂₇	#349	1	2	yes	—
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	—	2	1	no	—
63	C₄×C₁₂₇	—	2	1	no	—
64	C₄×C₁₂₇	—	4	4	no	—
65	C₄×C₁₂₇	#339	1	2	yes	—
66	C₄×C₁₂₇	#347	1	2	yes	—
67	#7149	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₁₂₇	—	2	2	no	—
69	C₂×C₁₂₇	#1276	1	1	yes	characteristic
70	C₂×C₁₂₇	#1375	1	1	yes	characteristic
71	C₂×C₁₂₇	#1377	1	1	yes	characteristic
72	#2163	C₂×C₁₂₇	1	2	yes	—
73	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic
74	C₁₂₇	#7149	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×#97	—	2	2	no	—
76	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
78	C₄³	C₄×C₁₂₇	1	1	yes	characteristic
79	C₄⋊C₁₆	C₄×C₁₂₇	1	4	yes	complemented
80	C₂²⋊C₁₆	—	2	2	no	—
81	#346	—	2	2	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₈×C₁₂₇	1	2	yes	—
87	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
88	C₂×C₁₆	—	2	4	no	—
89	#97	—	2	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₈×C₁₂₇	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
93	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
94	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
98	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
103	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
104	C₄²	—	2	4	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
108	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
109	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
110	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
111	C₁₆	—	4	4	no	—
112	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄	#97×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	2	2	no	—
127	C₈	#104×C₁₂₇	1	4	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#336×C₁₂₇	1	1	yes	characteristic
131	C₂²	#349×C₁₂₇	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	4	no	—
137	C₄	#339×C₁₂₇	1	2	yes	—
138	C₄	#347×C₁₂₇	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	C₁₂₇×#1276	1	1	yes	characteristic
141	C₂	C₁₂₇×#1375	1	1	yes	characteristic
142	C₂	C₁₂₇×#1377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2163

C₂

yes

maximal

C₄×#97×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#97×C₁₂₇

—

C₂×C₄×C₈×C₁₂₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄³×C₁₂₇

C₄

yes

characteristic

C₄⋊C₁₆×C₁₂₇

C₄

yes

complemented

C₂²⋊C₁₆×C₁₂₇

—

#346×C₁₂₇

—

C₂²×C₈×C₁₂₇

—

C₂²×C₈×C₁₂₇

D₄

yes

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₈

yes

—

C₂×C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

C₈

yes

—

C₄×C₈×C₁₂₇

C₂×C₄

yes

—

C₄×C₈×C₁₂₇

D₄

yes

characteristic

C₄×C₈×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₂₇

M₁₆

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₄²

yes

characteristic

C₄²×C₁₂₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₈

yes

characteristic

C₄²×C₁₂₇

M₁₆

yes

characteristic

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#95

yes

characteristic

C₂×C₄×C₁₂₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₂₇

#97

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#104

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#336

yes

characteristic

C₂²×C₁₂₇

#349

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#339

yes

—

C₄×C₁₂₇

#347

yes

—

#7149

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₄×#97

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#7149

yes

characteristic, direct product factor, Sylow

C₂×#97

—

C₂×C₄×C₈

C₄×C₁₂₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄³

C₄×C₁₂₇

yes

characteristic

C₄⋊C₁₆

C₄×C₁₂₇

yes

complemented

C₂²⋊C₁₆

—

#346

—

C₂²×C₈

—

C₂²×C₈

D₄×C₁₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₈×C₁₂₇

yes

—

C₂×C₄²

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

C₈×C₁₂₇

yes

—

C₄×C₈

C₂×C₄×C₁₂₇

yes

—

C₄×C₈

D₄×C₁₂₇

yes

characteristic

C₄×C₈

Q₈×C₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₂₇

yes

characteristic

C₂²×C₄

M₁₆×C₁₂₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊C₄×C₁₂₇

yes

—

103

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₄²×C₁₂₇

yes

characteristic

108

C₄²

C₄⋊C₄×C₁₂₇

yes

characteristic

109

C₄²

C₂×C₈×C₁₂₇

yes

characteristic

110

C₄²

M₁₆×C₁₂₇

yes

characteristic

111

C₁₆

—

112

C₂³

C₂²⋊C₈×C₁₂₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#95×C₁₂₇

yes

characteristic

120

C₂×C₄

C₄×C₈×C₁₂₇

yes

characteristic

121

C₂×C₄

#97×C₁₂₇

yes

characteristic

122

C₂×C₄

C₂²⋊C₈×C₁₂₇

yes

characteristic

123

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

characteristic

124

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

#104×C₁₂₇

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

#336×C₁₂₇

yes

characteristic

131

C₂²

#349×C₁₂₇

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#339×C₁₂₇

yes

—

138

C₄

#347×C₁₂₇

yes

—

139

C₂

—

140

C₂

C₁₂₇×#1276

yes

characteristic

141

C₂

C₁₂₇×#1375

yes

characteristic

142

C₂

C₁₂₇×#1377

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#95
commutator	C₂×C₄	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#336
prime radical	C₂³×C₁₂₇	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₂₇

#95

commutator

C₂×C₄

C₄×C₈×C₁₂₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₂₇

socle

C₂²×C₁₂₇

#336

prime radical

C₂³×C₁₂₇

C₂²⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
7. C₄⋊C₁₆×C₁₂₇	C₄	semidirect
67. #7149	C₁₂₇	direct (N × H)
74. C₁₂₇	#7149	direct (N × H)
79. C₄⋊C₁₆	C₄×C₁₂₇	semidirect

H ≅ G/N

type

7. C₄⋊C₁₆×C₁₂₇

C₄

semidirect

67. #7149

C₁₂₇

direct (N × H)

74. C₁₂₇

#7149

direct (N × H)

79. C₄⋊C₁₆

C₄×C₁₂₇

semidirect

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

subgroup	index
C₁₂₇×#2163	2
C₄×#97×C₁₂₇	2
#7149	127

subgroup

index

C₁₂₇×#2163

C₄×#97×C₁₂₇

#7149

127

Subgroups — C₁₂₇×#7149 #44570822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#7149 #44570822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7149 #44570822