Group #44575694 — subgroups

Subgroups — C₁₂₇×#12021 #44575694

Order 32512.

216 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1568	C₂	1	2	yes	maximal
3	C₁₂₇×#1368	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₂₇	C₄	1	2	yes	—
5	#432×C₁₂₇	—	2	2	no	—
6	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	#346×C₁₂₇	C₂²	1	2	yes	—
8	C₄×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
10	#335×C₁₂₇	—	2	2	no	—
11	#334×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
15	M₃₂×C₁₂₇	—	2	4	no	—
16	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₄⋊C₈×C₁₂₇	—	2	4	no	—
19	C₂²⋊C₈×C₁₂₇	—	4	2	no	—
20	#97×C₁₂₇	—	4	2	no	—
21	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	4	4	no	—
30	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
36	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₁₂₇	—	2	2	no	—
38	C₁₆×C₁₂₇	M₁₆	1	4	yes	—
39	C₂³×C₁₂₇	#106	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	—	2	1	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	#106	1	2	yes	—
44	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
47	C₈×C₁₂₇	—	2	1	no	—
48	C₈×C₁₂₇	—	2	1	no	—
49	C₈×C₁₂₇	—	8	4	no	—
50	C₈×C₁₂₇	#108	1	2	yes	—
51	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	#423	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
60	#12021	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₁₂₇	—	4	1	no	—
62	C₂×C₁₂₇	#1566	1	1	yes	characteristic
63	C₂×C₁₂₇	#2155	1	2	yes	—
64	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
65	#1568	C₂×C₁₂₇	1	2	yes	—
66	#1368	C₂×C₁₂₇	1	4	yes	complemented
67	C₁₂₇	#12021	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×M₃₂	C₄×C₁₂₇	1	2	yes	—
69	#432	—	2	2	no	—
70	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
71	#346	C₂²×C₁₂₇	1	2	yes	—
72	C₄×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
73	C₄×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
74	#335	—	2	2	no	—
75	#334	C₂²×C₁₂₇	1	4	yes	—
76	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
77	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
78	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
79	M₃₂	—	2	4	no	—
80	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
82	C₄⋊C₈	—	2	4	no	—
83	C₂²⋊C₈	—	4	2	no	—
84	#97	—	4	2	no	—
85	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
86	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
87	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
88	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
95	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
98	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
99	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
100	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	M₁₆×C₁₂₇	1	4	yes	—
103	C₂³	#106×C₁₂₇	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	#106×C₁₂₇	1	2	yes	—
108	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	8	4	no	—
114	C₈	#108×C₁₂₇	1	2	yes	—
115	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	2	no	—
121	C₄	—	2	2	no	—
122	C₄	#423×C₁₂₇	1	1	yes	characteristic
123	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	C₁₂₇×#1566	1	1	yes	characteristic
126	C₂	C₁₂₇×#2155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₁₂₇

C₂

yes

maximal, characteristic

C₁₂₇×#1568

C₂

yes

maximal

C₁₂₇×#1368

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₂₇

C₄

yes

—

#432×C₁₂₇

—

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

#346×C₁₂₇

C₂²

yes

—

C₄×C₁₆×C₁₂₇

C₄

yes

characteristic

C₄×C₁₆×C₁₂₇

C₄

yes

characteristic

#335×C₁₂₇

—

#334×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₂₇

Q₈

yes

characteristic

C₂×C₄²×C₁₂₇

D₄

yes

characteristic

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₈×C₁₂₇

—

C₂²⋊C₈×C₁₂₇

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₂₇

D₄

yes

characteristic

C₄×C₈×C₁₂₇

Q₈

yes

characteristic

C₄×C₈×C₁₂₇

C₂³

yes

characteristic

C₂²×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₄²×C₁₂₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₂₇

SD₁₆

yes

—

C₄²×C₁₂₇

C₂×D₄

yes

characteristic

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

M₁₆

yes

—

C₂³×C₁₂₇

#106

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#106

yes

—

C₂×C₄×C₁₂₇

#106

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#108

yes

—

C₈×C₁₂₇

C₂×M₁₆

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×#106

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#1566

yes

characteristic

C₂×C₁₂₇

#2155

yes

—

C₄×M₃₂

C₂×C₁₂₇

yes

characteristic

#1568

C₂×C₁₂₇

yes

—

#1368

C₂×C₁₂₇

yes

complemented

C₁₂₇

#12021

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₁₂₇

yes

—

#432

—

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

#346

C₂²×C₁₂₇

yes

—

C₄×C₁₆

C₄×C₁₂₇

yes

characteristic

C₄×C₁₆

C₄×C₁₂₇

yes

characteristic

#335

—

#334

C₂²×C₁₂₇

yes

—

C₂²×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₈

Q₈×C₁₂₇

yes

characteristic

C₂×C₄²

D₄×C₁₂₇

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

—

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₄×C₈

D₄×C₁₂₇

yes

characteristic

C₄×C₈

Q₈×C₁₂₇

yes

characteristic

C₄×C₈

C₂³×C₁₂₇

yes

characteristic

C₂²×C₄

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂²×C₄

SD₁₆×C₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₂²×C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₂₇

yes

characteristic

C₄²

C₄⋊C₄×C₁₂₇

yes

characteristic

C₄²

SD₁₆×C₁₂₇

yes

—

100

C₄²

C₂×D₄×C₁₂₇

yes

characteristic

101

C₁₆

—

102

C₁₆

M₁₆×C₁₂₇

yes

—

103

C₂³

#106×C₁₂₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#106×C₁₂₇

yes

—

108

C₂×C₄

#106×C₁₂₇

yes

characteristic

109

C₂×C₄

C₂×C₄⋊C₄×C₁₂₇

yes

characteristic

110

C₂×C₄

C₂×SD₁₆×C₁₂₇

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#108×C₁₂₇

yes

—

115

C₈

C₂×M₁₆×C₁₂₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×#106×C₁₂₇

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#423×C₁₂₇

yes

characteristic

123

C₄

C₂×#108×C₁₂₇

yes

characteristic

124

C₂

—

125

C₂

C₁₂₇×#1566

yes

characteristic

126

C₂

C₁₂₇×#2155

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₂₇

C₂×#106

commutator

C₂×C₈

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₄×C₈

C₂³×C₁₂₇

prime radical

C₂³×C₁₂₇

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
3. C₁₂₇×#1368	C₂	semidirect
60. #12021	C₁₂₇	direct (N × H)
66. #1368	C₂×C₁₂₇	semidirect
67. C₁₂₇	#12021	direct (N × H)

H ≅ G/N

type

3. C₁₂₇×#1368

C₂

semidirect

60. #12021

C₁₂₇

direct (N × H)

66. #1368

C₂×C₁₂₇

semidirect

67. C₁₂₇

#12021

direct (N × H)

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

subgroup	index
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1568	2
C₁₂₇×#1368	2
#12021	127

subgroup

index

C₄×M₃₂×C₁₂₇

C₁₂₇×#1568

C₁₂₇×#1368

#12021

127

Subgroups — C₁₂₇×#12021 #44575694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#12021 #44575694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12021 #44575694