Group #44570828 — subgroups

Subgroups — C₁₂₇×#7155 #44570828

Order 32512.

222 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 134 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×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1564	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
4	#431×C₁₂₇	—	2	2	no	—
5	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₁₂₇	—	2	2	no	—
7	#335×C₁₂₇	C₄	1	4	yes	—
8	#334×C₁₂₇	—	2	2	no	—
9	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	—	2	2	no	—
13	C₂×C₁₆×C₁₂₇	—	2	2	no	—
14	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	complemented
15	C₄⋊C₈×C₁₂₇	—	2	4	no	—
16	C₂²⋊C₈×C₁₂₇	—	4	2	no	—
17	C₄×C₈×C₁₂₇	—	4	2	no	—
18	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
20	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
22	C₂×C₈×C₁₂₇	—	2	2	no	—
23	C₂×C₈×C₁₂₇	—	4	4	no	—
24	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
29	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
31	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
32	C₁₆×C₁₂₇	—	2	4	no	—
33	C₂³×C₁₂₇	#106	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	4	no	—
35	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	#102	1	2	yes	—
37	C₂×C₄×C₁₂₇	#103	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
39	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	#108	1	2	yes	—
41	C₈×C₁₂₇	—	2	2	no	—
42	C₈×C₁₂₇	—	8	4	no	—
43	C₈×C₁₂₇	C₄×C₈	1	1	yes	characteristic
44	C₈×C₁₂₇	#97	1	1	yes	characteristic
45	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
46	C₂²×C₁₂₇	—	2	2	no	—
47	C₂²×C₁₂₇	#334	1	2	yes	—
48	C₂²×C₁₂₇	#340	1	1	yes	characteristic
49	C₄×C₁₂₇	—	2	2	no	—
50	C₄×C₁₂₇	#335	1	2	yes	—
51	C₄×C₁₂₇	#336	1	1	yes	characteristic
52	C₄×C₁₂₇	#341	1	1	yes	characteristic
53	C₄×C₁₂₇	#359	1	4	yes	—
54	#7155	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	#1277	1	1	yes	characteristic
57	C₂×C₁₂₇	#1382	1	2	yes	—
58	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
59	#1564	C₂×C₁₂₇	1	2	yes	—
60	C₁₂₇	#7155	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
62	#431	—	2	2	no	—
63	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
64	C₄×C₁₆	—	2	2	no	—
65	#335	C₄×C₁₂₇	1	4	yes	—
66	#334	—	2	2	no	—
67	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
68	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
69	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	complemented
73	C₄⋊C₈	—	2	4	no	—
74	C₂²⋊C₈	—	4	2	no	—
75	C₄×C₈	—	4	2	no	—
76	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
77	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
78	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
79	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
84	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
85	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
86	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
87	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
88	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
89	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
90	C₁₆	—	2	4	no	—
91	C₂³	#106×C₁₂₇	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
94	C₂×C₄	#102×C₁₂₇	1	2	yes	—
95	C₂×C₄	#103×C₁₂₇	1	2	yes	—
96	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
97	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
98	C₂×C₄	#108×C₁₂₇	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	8	4	no	—
101	C₈	C₄×C₈×C₁₂₇	1	1	yes	characteristic
102	C₈	#97×C₁₂₇	1	1	yes	characteristic
103	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	#334×C₁₂₇	1	2	yes	—
106	C₂²	#340×C₁₂₇	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	#335×C₁₂₇	1	2	yes	—
109	C₄	#336×C₁₂₇	1	1	yes	characteristic
110	C₄	#341×C₁₂₇	1	1	yes	characteristic
111	C₄	#359×C₁₂₇	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	C₁₂₇×#1277	1	1	yes	characteristic
114	C₂	C₁₂₇×#1382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₁₂₇×#1564

C₂

yes

maximal

C₂²×C₁₆×C₁₂₇

C₄

yes

—

#431×C₁₂₇

—

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₂₇

—

#335×C₁₂₇

C₄

yes

—

#334×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₂₇

Q₈

yes

characteristic

C₂×C₄²×C₁₂₇

D₄

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₈

yes

complemented

C₄⋊C₈×C₁₂₇

—

C₂²⋊C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂×C₄

yes

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₄²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₈

yes

—

C₂×C₈×C₁₂₇

M₁₆

yes

—

C₄²×C₁₂₇

C₂²⋊C₄

yes

—

C₄²×C₁₂₇

D₈

yes

characteristic

C₄²×C₁₂₇

Q₁₆

yes

characteristic

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

#106

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#95

yes

characteristic

C₂×C₄×C₁₂₇

#102

yes

—

C₂×C₄×C₁₂₇

#103

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₂₇

#107

yes

characteristic

C₂×C₄×C₁₂₇

#108

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄×C₈

yes

characteristic

C₈×C₁₂₇

#97

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₈

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#334

yes

—

C₂²×C₁₂₇

#340

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#335

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#7155

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#1277

yes

characteristic

C₂×C₁₂₇

#1382

yes

—

C₂×C₄×C₁₆

C₂×C₁₂₇

yes

characteristic

#1564

C₂×C₁₂₇

yes

—

C₁₂₇

#7155

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₄×C₁₂₇

yes

—

#431

—

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄×C₁₆

—

#335

C₄×C₁₂₇

yes

—

#334

—

C₂²×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₈

Q₈×C₁₂₇

yes

characteristic

C₂×C₄²

D₄×C₁₂₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₁₂₇

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₁₂₇

yes

—

C₄×C₈

D₄×C₁₂₇

yes

—

C₂²×C₄

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂²×C₄

SD₁₆×C₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

—

C₂×C₈

C₂×C₈×C₁₂₇

yes

—

C₂×C₈

M₁₆×C₁₂₇

yes

—

C₄²

C₂²⋊C₄×C₁₂₇

yes

—

C₄²

D₈×C₁₂₇

yes

characteristic

C₄²

Q₁₆×C₁₂₇

yes

characteristic

C₁₆

—

C₂³

#106×C₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₁₂₇

yes

characteristic

C₂×C₄

#102×C₁₂₇

yes

—

C₂×C₄

#103×C₁₂₇

yes

—

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

—

C₂×C₄

#107×C₁₂₇

yes

characteristic

C₂×C₄

#108×C₁₂₇

yes

—

C₈

—

100

C₈

—

101

C₈

C₄×C₈×C₁₂₇

yes

characteristic

102

C₈

#97×C₁₂₇

yes

characteristic

103

C₈

C₂²⋊C₈×C₁₂₇

yes

—

104

C₂²

—

105

C₂²

#334×C₁₂₇

yes

—

106

C₂²

#340×C₁₂₇

yes

characteristic

107

C₄

—

108

C₄

#335×C₁₂₇

yes

—

109

C₄

#336×C₁₂₇

yes

characteristic

110

C₄

#341×C₁₂₇

yes

characteristic

111

C₄

#359×C₁₂₇

yes

—

112

C₂

—

113

C₂

C₁₂₇×#1277

yes

characteristic

114

C₂

C₁₂₇×#1382

yes

—

	subgroup	G/H
center	C₄²×C₁₂₇	D₈
commutator	C₈	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₄²×C₁₂₇

D₈

commutator

C₈

C₄×C₈×C₁₂₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₂₇

C₂²×C₁₂₇

C₂³×C₁₂₇

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
14. C₂×C₁₆×C₁₂₇	C₈	semidirect
54. #7155	C₁₂₇	direct (N × H)
60. C₁₂₇	#7155	direct (N × H)
72. C₂×C₁₆	C₈×C₁₂₇	semidirect

H ≅ G/N

type

14. C₂×C₁₆×C₁₂₇

C₈

semidirect

54. #7155

C₁₂₇

direct (N × H)

60. C₁₂₇

#7155

direct (N × H)

72. C₂×C₁₆

C₈×C₁₂₇

semidirect

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

subgroup	index
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1564	2
#7155	127

subgroup

index

C₂×C₄×C₁₆×C₁₂₇

C₁₂₇×#1564

#7155

127

Subgroups — C₁₂₇×#7155 #44570828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#7155 #44570828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7155 #44570828