Group #44575762 — subgroups

Subgroups — C₁₂₇×#12089 #44575762

Order 32512.

226 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2177	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1594	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1333	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄×D₈×C₁₂₇	C₂²	1	2	yes	—
8	#436×C₁₂₇	—	2	2	no	—
9	#367×C₁₂₇	C₄	1	1	yes	characteristic
10	#357×C₁₂₇	C₄	1	2	yes	—
11	#346×C₁₂₇	C₂²	1	1	yes	characteristic
12	#346×C₁₂₇	C₂²	1	1	yes	characteristic
13	#334×C₁₂₇	C₂²	1	2	yes	—
14	D₄⋊C₈×C₁₂₇	—	2	2	no	—
15	#322×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×M₁₆×C₁₂₇	—	4	2	no	—
18	C₄×D₄×C₁₂₇	—	2	2	no	—
19	D₁₆×C₁₂₇	—	2	2	no	—
20	C₂×C₁₆×C₁₂₇	—	2	1	no	—
21	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	#107×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₄⋊C₈×C₁₂₇	—	2	2	no	—
25	#102×C₁₂₇	—	2	2	no	—
26	C₂²⋊C₈×C₁₂₇	—	4	2	no	—
27	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₁₂₇	—	2	2	no	—
30	C₂²×C₄×C₁₂₇	—	4	2	no	—
31	D₈×C₁₂₇	—	2	2	no	—
32	D₈×C₁₂₇	—	2	2	no	—
33	M₁₆×C₁₂₇	—	8	4	no	—
34	C₂×C₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	—	4	2	no	—
36	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₁₂₇	—	2	2	no	—
39	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
40	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₁₂₇	—	2	1	no	—
42	C₁₆×C₁₂₇	—	4	1	no	—
43	C₁₆×C₁₂₇	M₁₆	1	2	yes	complemented
44	C₂³×C₁₂₇	—	4	2	no	—
45	D₄×C₁₂₇	—	4	2	no	—
46	D₄×C₁₂₇	—	4	2	no	—
47	C₂×C₄×C₁₂₇	—	4	2	no	—
48	C₂×C₄×C₁₂₇	—	8	2	no	—
49	C₂×C₄×C₁₂₇	—	8	2	no	—
50	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₈×C₁₂₇	—	2	1	no	—
53	C₈×C₁₂₇	—	4	2	no	—
54	C₈×C₁₂₇	—	8	2	no	—
55	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₁₂₇	#131	1	1	yes	characteristic
57	C₂²×C₁₂₇	—	8	2	no	—
58	C₂²×C₁₂₇	—	8	2	no	—
59	C₂²×C₁₂₇	—	8	2	no	—
60	C₂²×C₁₂₇	#442	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	—	8	2	no	—
63	C₄×C₁₂₇	#436	1	1	yes	characteristic
64	C₄×C₁₂₇	#444	1	1	yes	characteristic
65	#12089	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₂₇	—	16	2	no	—
67	C₂×C₁₂₇	#1594	1	1	yes	characteristic
68	C₂×C₁₂₇	AGL(1,ℤ/16)	1	2	yes	—
69	#2177	C₂×C₁₂₇	1	1	yes	characteristic
70	#1594	C₂×C₁₂₇	1	2	yes	complemented
71	#1368	C₂×C₁₂₇	1	1	yes	characteristic, complemented
72	#1333	C₂×C₁₂₇	1	2	yes	complemented
73	#1313	C₂×C₁₂₇	1	1	yes	characteristic, complemented
74	C₁₂₇	#12089	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×D₁₆	C₄×C₁₂₇	1	1	yes	characteristic
76	C₄×D₈	C₂²×C₁₂₇	1	2	yes	—
77	#436	—	2	2	no	—
78	#367	C₄×C₁₂₇	1	1	yes	characteristic
79	#357	C₄×C₁₂₇	1	2	yes	—
80	#346	C₂²×C₁₂₇	1	1	yes	characteristic
81	#346	C₂²×C₁₂₇	1	1	yes	characteristic
82	#334	C₂²×C₁₂₇	1	2	yes	—
83	D₄⋊C₈	—	2	2	no	—
84	#322	C₂²×C₁₂₇	1	1	yes	characteristic
85	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
86	C₂×M₁₆	—	4	2	no	—
87	C₄×D₄	—	2	2	no	—
88	D₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
92	#107	C₂×C₄×C₁₂₇	1	2	yes	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	2	2	no	—
95	C₂²⋊C₈	—	4	2	no	—
96	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
97	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
98	C₂×D₄	—	2	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	2	no	—
102	M₁₆	—	8	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
106	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	M₁₆×C₁₂₇	1	2	yes	complemented
113	C₂³	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	8	2	no	—
124	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
125	C₈	#131×C₁₂₇	1	1	yes	characteristic
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	#442×C₁₂₇	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	8	2	no	—
132	C₄	#436×C₁₂₇	1	1	yes	characteristic
133	C₄	#444×C₁₂₇	1	1	yes	characteristic
134	C₂	—	16	2	no	—
135	C₂	C₁₂₇×#1594	1	1	yes	characteristic
136	C₂	C₁₂₇×AGL(1,ℤ/16)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2177

C₂

yes

maximal, characteristic

C₁₂₇×#1594

C₂

yes

maximal, complemented

C₁₂₇×#1368

C₂

yes

maximal, characteristic, complemented

C₁₂₇×#1333

C₂

yes

maximal, complemented

C₁₂₇×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₁₂₇

C₄

yes

characteristic

C₄×D₈×C₁₂₇

C₂²

yes

—

#436×C₁₂₇

—

#367×C₁₂₇

C₄

yes

characteristic

#357×C₁₂₇

C₄

yes

—

#346×C₁₂₇

C₂²

yes

characteristic

#346×C₁₂₇

C₂²

yes

characteristic

#334×C₁₂₇

C₂²

yes

—

D₄⋊C₈×C₁₂₇

—

#322×C₁₂₇

C₂²

yes

characteristic

C₂×D₈×C₁₂₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₂₇

—

C₄×D₄×C₁₂₇

—

D₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

characteristic

#107×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₈×C₁₂₇

—

#102×C₁₂₇

—

C₂²⋊C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₄×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₂₇

#41

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×D₄

yes

characteristic

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

M₁₆

yes

complemented

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×M₁₆

yes

characteristic

C₈×C₁₂₇

#131

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#442

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#1594

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/16)

yes

—

#2177

C₂×C₁₂₇

yes

characteristic

#1594

C₂×C₁₂₇

yes

complemented

#1368

C₂×C₁₂₇

yes

characteristic, complemented

#1333

C₂×C₁₂₇

yes

complemented

#1313

C₂×C₁₂₇

yes

characteristic, complemented

C₁₂₇

#12089

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₄×C₁₂₇

yes

characteristic

C₄×D₈

C₂²×C₁₂₇

yes

—

#436

—

#367

C₄×C₁₂₇

yes

characteristic

#357

C₄×C₁₂₇

yes

—

#346

C₂²×C₁₂₇

yes

characteristic

#346

C₂²×C₁₂₇

yes

characteristic

#334

C₂²×C₁₂₇

yes

—

D₄⋊C₈

—

#322

C₂²×C₁₂₇

yes

characteristic

C₂×D₈

C₂×C₄×C₁₂₇

yes

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

characteristic

#107

C₂×C₄×C₁₂₇

yes

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

D₄×C₁₂₇

yes

—

C₄×C₈

C₂³×C₁₂₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×C₄×C₁₂₇

yes

characteristic

106

C₂×C₈

#41×C₁₂₇

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂×D₄×C₁₂₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

M₁₆×C₁₂₇

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄×D₄×C₁₂₇

yes

characteristic

120

C₂×C₄

AGL(1,ℤ/8)×C₁₂₇

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×M₁₆×C₁₂₇

yes

characteristic

125

C₈

#131×C₁₂₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#442×C₁₂₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#436×C₁₂₇

yes

characteristic

133

C₄

#444×C₁₂₇

yes

characteristic

134

C₂

—

135

C₂

C₁₂₇×#1594

yes

characteristic

136

C₂

C₁₂₇×AGL(1,ℤ/16)

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#442
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂×D₁₆×C₁₂₇	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ #322×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₂₇

#442

commutator

C₂×C₈

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₄×C₈

C₂³×C₁₂₇

prime radical

C₂×D₁₆×C₁₂₇

C₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ #322×C₁₂₇ ◁ G

N	H ≅ G/N	type
2. C₁₂₇×#1594	C₂	semidirect
3. C₁₂₇×#1368	C₂	semidirect
4. C₁₂₇×#1333	C₂	semidirect
5. C₁₂₇×#1313	C₂	semidirect
43. C₁₆×C₁₂₇	M₁₆	semidirect
65. #12089	C₁₂₇	direct (N × H)
70. #1594	C₂×C₁₂₇	semidirect
71. #1368	C₂×C₁₂₇	semidirect
72. #1333	C₂×C₁₂₇	semidirect
73. #1313	C₂×C₁₂₇	semidirect
74. C₁₂₇	#12089	direct (N × H)
112. C₁₆	M₁₆×C₁₂₇	semidirect

H ≅ G/N

type

2. C₁₂₇×#1594

C₂

semidirect

3. C₁₂₇×#1368

C₂

semidirect

4. C₁₂₇×#1333

C₂

semidirect

5. C₁₂₇×#1313

C₂

semidirect

43. C₁₆×C₁₂₇

M₁₆

semidirect

65. #12089

C₁₂₇

direct (N × H)

70. #1594

C₂×C₁₂₇

semidirect

71. #1368

C₂×C₁₂₇

semidirect

72. #1333

C₂×C₁₂₇

semidirect

73. #1313

C₂×C₁₂₇

semidirect

74. C₁₂₇

#12089

direct (N × H)

112. C₁₆

M₁₆×C₁₂₇

semidirect

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

subgroup	index
C₁₂₇×#2177	2
C₁₂₇×#1594	2
C₁₂₇×#1368	2
C₁₂₇×#1333	2
C₁₂₇×#1313	2
#12089	127

subgroup

index

C₁₂₇×#2177

C₁₂₇×#1594

C₁₂₇×#1368

C₁₂₇×#1333

C₁₂₇×#1313

#12089

127

Subgroups — C₁₂₇×#12089 #44575762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#12089 #44575762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12089 #44575762