Group #44577373 — subgroups

Subgroups — C₁₂₇×#13700 #44577373

Order 32512.

188 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₂×D₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#2258	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#2221	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1425	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1419	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₁₂₇	C₂²	1	2	yes	—
7	#504×C₁₂₇	C₂²	1	1	yes	characteristic
8	#472×C₁₂₇	—	2	2	no	—
9	D₃₂×C₁₂₇	—	2	1	no	—
10	D₃₂×C₁₂₇	—	2	1	no	—
11	M₆₄×C₁₂₇	—	2	1	no	—
12	M₆₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₁₂₇	—	2	1	no	—
14	C₂×C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
15	#368×C₁₂₇	C₂²	1	2	yes	—
16	#361×C₁₂₇	—	2	2	no	—
17	AGL(1,ℤ/8)×C₁₂₇	—	4	2	no	—
18	C₂×D₈×C₁₂₇	—	2	2	no	—
19	#131×C₁₂₇	D₄	1	1	yes	characteristic
20	D₁₆×C₁₂₇	—	2	2	no	—
21	D₁₆×C₁₂₇	—	2	2	no	—
22	M₃₂×C₁₂₇	—	2	1	no	—
23	M₃₂×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₂₇	—	2	1	no	—
25	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
26	#108×C₁₂₇	—	2	2	no	—
27	#100×C₁₂₇	—	4	2	no	—
28	C₃₂×C₁₂₇	—	2	1	no	—
29	C₃₂×C₁₂₇	D₄	1	1	yes	characteristic
30	C₃₂×C₁₂₇	D₄	1	1	yes	characteristic
31	#41×C₁₂₇	D₈	1	1	yes	characteristic
32	C₂×D₄×C₁₂₇	—	4	2	no	—
33	SD₁₆×C₁₂₇	—	8	2	no	—
34	D₈×C₁₂₇	—	4	2	no	—
35	D₈×C₁₂₇	—	4	2	no	—
36	D₈×C₁₂₇	—	8	2	no	—
37	M₁₆×C₁₂₇	—	2	1	no	—
38	M₁₆×C₁₂₇	—	4	2	no	—
39	M₁₆×C₁₂₇	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₁₂₇	—	2	1	no	—
43	C₁₆×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₁₂₇	#41	1	1	yes	characteristic
45	C₂³×C₁₂₇	—	8	2	no	—
46	Q₈×C₁₂₇	D₁₆	1	1	yes	characteristic
47	D₄×C₁₂₇	—	2	1	no	—
48	D₄×C₁₂₇	—	8	2	no	—
49	D₄×C₁₂₇	—	8	2	no	—
50	D₄×C₁₂₇	D₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	1	no	—
54	C₈×C₁₂₇	—	8	2	no	—
55	C₈×C₁₂₇	#121	1	1	yes	characteristic
56	C₈×C₁₂₇	#135	1	1	yes	characteristic
57	C₂²×C₁₂₇	—	2	1	no	—
58	C₂²×C₁₂₇	—	16	2	no	—
59	C₂²×C₁₂₇	—	32	2	no	—
60	C₂²×C₁₂₇	#508	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	#466	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
64	#13700	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₂₇	—	2	1	no	—
66	C₂×C₁₂₇	—	4	1	no	—
67	C₂×C₁₂₇	—	32	2	no	—
68	C₂×C₁₂₇	#2215	1	1	yes	characteristic
69	C₂×D₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
70	#2258	C₂×C₁₂₇	1	1	yes	characteristic, complemented
71	#2221	C₂×C₁₂₇	1	2	yes	complemented
72	#1425	C₂×C₁₂₇	1	1	yes	characteristic, complemented
73	#1419	C₂×C₁₂₇	1	2	yes	complemented
74	C₁₂₇	#13700	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×D₁₆	C₂²×C₁₂₇	1	2	yes	—
76	#504	C₂²×C₁₂₇	1	1	yes	characteristic
77	#472	—	2	2	no	—
78	D₃₂	—	2	1	no	—
79	D₃₂	—	2	1	no	—
80	M₆₄	—	2	1	no	—
81	M₆₄	C₂²×C₁₂₇	1	1	yes	characteristic
82	C₂×C₃₂	—	2	1	no	—
83	C₂×C₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
84	#368	C₂²×C₁₂₇	1	2	yes	—
85	#361	—	2	2	no	—
86	AGL(1,ℤ/8)	—	4	2	no	—
87	C₂×D₈	—	2	2	no	—
88	#131	D₄×C₁₂₇	1	1	yes	characteristic
89	D₁₆	—	2	2	no	—
90	D₁₆	—	2	2	no	—
91	M₃₂	—	2	1	no	—
92	M₃₂	D₄×C₁₂₇	1	1	yes	characteristic
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
95	#108	—	2	2	no	—
96	#100	—	4	2	no	—
97	C₃₂	—	2	1	no	—
98	C₃₂	D₄×C₁₂₇	1	1	yes	characteristic
99	C₃₂	D₄×C₁₂₇	1	1	yes	characteristic
100	#41	D₈×C₁₂₇	1	1	yes	characteristic
101	C₂×D₄	—	4	2	no	—
102	SD₁₆	—	8	2	no	—
103	D₈	—	4	2	no	—
104	D₈	—	4	2	no	—
105	D₈	—	8	2	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	D₈×C₁₂₇	1	1	yes	characteristic
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	C₂×D₄×C₁₂₇	1	1	yes	characteristic
113	C₁₆	#41×C₁₂₇	1	1	yes	characteristic
114	C₂³	—	8	2	no	—
115	Q₈	D₁₆×C₁₂₇	1	1	yes	characteristic
116	D₄	—	2	1	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	2	no	—
119	D₄	D₁₆×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	8	2	no	—
124	C₈	#121×C₁₂₇	1	1	yes	characteristic
125	C₈	#135×C₁₂₇	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	16	2	no	—
128	C₂²	—	32	2	no	—
129	C₂²	#508×C₁₂₇	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	#466×C₁₂₇	1	1	yes	characteristic
132	C₄	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	4	1	no	—
135	C₂	—	32	2	no	—
136	C₂	C₁₂₇×#2215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₂×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₁₂₇×#2258

C₂

yes

maximal, characteristic, complemented

C₁₂₇×#2221

C₂

yes

maximal, complemented

C₁₂₇×#1425

C₂

yes

maximal, characteristic, complemented

C₁₂₇×#1419

C₂

yes

maximal, complemented

C₂×D₁₆×C₁₂₇

C₂²

yes

—

#504×C₁₂₇

C₂²

yes

characteristic

#472×C₁₂₇

—

D₃₂×C₁₂₇

—

D₃₂×C₁₂₇

—

M₆₄×C₁₂₇

—

M₆₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₃₂×C₁₂₇

—

C₂×C₃₂×C₁₂₇

C₂²

yes

characteristic

#368×C₁₂₇

C₂²

yes

—

#361×C₁₂₇

—

AGL(1,ℤ/8)×C₁₂₇

—

C₂×D₈×C₁₂₇

—

#131×C₁₂₇

D₄

yes

characteristic

D₁₆×C₁₂₇

—

D₁₆×C₁₂₇

—

M₃₂×C₁₂₇

—

M₃₂×C₁₂₇

D₄

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂³

yes

characteristic

#108×C₁₂₇

—

#100×C₁₂₇

—

C₃₂×C₁₂₇

—

C₃₂×C₁₂₇

D₄

yes

characteristic

C₃₂×C₁₂₇

D₄

yes

characteristic

#41×C₁₂₇

D₈

yes

characteristic

C₂×D₄×C₁₂₇

—

SD₁₆×C₁₂₇

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

D₈

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

C₂×D₄

yes

characteristic

C₁₆×C₁₂₇

#41

yes

characteristic

C₂³×C₁₂₇

—

Q₈×C₁₂₇

D₁₆

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

D₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

yes

C₈×C₁₂₇

yes

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#508

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#466

yes

characteristic

C₄×C₁₂₇

C₂×D₁₆

yes

characteristic

#13700

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#2215

yes

characteristic

C₂×D₃₂

C₂×C₁₂₇

yes

characteristic, complemented

#2258

C₂×C₁₂₇

yes

characteristic, complemented

#2221

C₂×C₁₂₇

yes

complemented

#1425

C₂×C₁₂₇

yes

characteristic, complemented

#1419

C₂×C₁₂₇

yes

complemented

C₁₂₇

#13700

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₂²×C₁₂₇

yes

—

#504

C₂²×C₁₂₇

yes

characteristic

#472

—

D₃₂

—

D₃₂

—

M₆₄

—

M₆₄

C₂²×C₁₂₇

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₂₇

yes

characteristic

#368

C₂²×C₁₂₇

yes

—

#361

—

AGL(1,ℤ/8)

—

C₂×D₈

—

#131

D₄×C₁₂₇

yes

characteristic

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

D₄×C₁₂₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂³×C₁₂₇

yes

characteristic

#108

—

#100

—

C₃₂

—

C₃₂

D₄×C₁₂₇

yes

characteristic

C₃₂

D₄×C₁₂₇

yes

characteristic

100

#41

D₈×C₁₂₇

yes

characteristic

101

C₂×D₄

—

102

SD₁₆

—

103

D₈

—

104

D₈

—

105

D₈

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

D₈×C₁₂₇

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

111

C₁₆

—

112

C₁₆

C₂×D₄×C₁₂₇

yes

characteristic

113

C₁₆

#41×C₁₂₇

yes

characteristic

114

C₂³

—

115

Q₈

D₁₆×C₁₂₇

yes

characteristic

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

D₁₆×C₁₂₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×D₈×C₁₂₇

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#121×C₁₂₇

yes

characteristic

125

C₈

#135×C₁₂₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#508×C₁₂₇

yes

characteristic

130

C₄

—

131

C₄

#466×C₁₂₇

yes

characteristic

132

C₄

C₂×D₁₆×C₁₂₇

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₁₂₇×#2215

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #131×C₁₂₇ ◁ #504×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂×C₁₂₇

#2215

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₂₇

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #131×C₁₂₇ ◁ #504×C₁₂₇ ◁ G

N	H ≅ G/N	type
1. C₂×D₃₂×C₁₂₇	C₂	semidirect
2. C₁₂₇×#2258	C₂	semidirect
3. C₁₂₇×#2221	C₂	semidirect
4. C₁₂₇×#1425	C₂	semidirect
5. C₁₂₇×#1419	C₂	semidirect
64. #13700	C₁₂₇	direct (N × H)
69. C₂×D₃₂	C₂×C₁₂₇	semidirect
70. #2258	C₂×C₁₂₇	semidirect
71. #2221	C₂×C₁₂₇	semidirect
72. #1425	C₂×C₁₂₇	semidirect
73. #1419	C₂×C₁₂₇	semidirect
74. C₁₂₇	#13700	direct (N × H)

H ≅ G/N

type

1. C₂×D₃₂×C₁₂₇

C₂

semidirect

2. C₁₂₇×#2258

C₂

semidirect

3. C₁₂₇×#2221

C₂

semidirect

4. C₁₂₇×#1425

C₂

semidirect

5. C₁₂₇×#1419

C₂

semidirect

64. #13700

C₁₂₇

direct (N × H)

69. C₂×D₃₂

C₂×C₁₂₇

semidirect

70. #2258

C₂×C₁₂₇

semidirect

71. #2221

C₂×C₁₂₇

semidirect

72. #1425

C₂×C₁₂₇

semidirect

73. #1419

C₂×C₁₂₇

semidirect

74. C₁₂₇

#13700

direct (N × H)

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

subgroup	index
C₂×D₃₂×C₁₂₇	2
C₁₂₇×#2258	2
C₁₂₇×#2221	2
C₁₂₇×#1425	2
C₁₂₇×#1419	2
#13700	127

subgroup

index

C₂×D₃₂×C₁₂₇

C₁₂₇×#2258

C₁₂₇×#2221

C₁₂₇×#1425

C₁₂₇×#1419

#13700

127

Subgroups — C₁₂₇×#13700 #44577373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#13700 #44577373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13700 #44577373