Group #30634340 — subgroups

Subgroups — C₁₂₇×#1388 #30634340

Order 16256.

124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₁₂₇	—	2	1	no	—
5	C₂×M₁₆×C₁₂₇	—	2	1	no	—
6	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₂₇	—	2	1	no	—
8	M₃₂×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₂₇	—	2	1	no	—
10	#108×C₁₂₇	—	2	1	no	—
11	#108×C₁₂₇	—	2	1	no	—
12	#108×C₁₂₇	C₄	1	2	yes	—
13	#108×C₁₂₇	C₄	1	2	yes	—
14	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	M₁₆×C₁₂₇	—	2	2	no	—
16	M₁₆×C₁₂₇	—	2	2	no	—
17	M₁₆×C₁₂₇	—	4	1	no	—
18	M₁₆×C₁₂₇	—	4	1	no	—
19	C₂×C₈×C₁₂₇	—	4	1	no	—
20	C₂×C₈×C₁₂₇	—	4	1	no	—
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
27	C₁₆×C₁₂₇	—	2	2	no	—
28	C₂³×C₁₂₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
35	C₈×C₁₂₇	—	4	2	no	—
36	C₈×C₁₂₇	—	4	2	no	—
37	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	#103	1	1	yes	characteristic
43	C₂²×C₁₂₇	#103	1	1	yes	characteristic
44	C₂²×C₁₂₇	#107	1	1	yes	characteristic
45	C₄×C₁₂₇	#95	1	1	yes	characteristic
46	C₄×C₁₂₇	#102	1	1	yes	characteristic
47	C₄×C₁₂₇	#102	1	1	yes	characteristic
48	C₄×C₁₂₇	#106	1	1	yes	characteristic
49	C₂×C₁₂₇	—	2	1	no	—
50	C₂×C₁₂₇	—	2	1	no	—
51	C₂×C₁₂₇	—	2	1	no	—
52	C₂×C₁₂₇	#340	1	1	yes	characteristic
53	#1388	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₁₂₇	#1388	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
56	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic
57	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	2	1	no	—
60	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
61	M₃₂	—	2	1	no	—
62	M₃₂	C₄×C₁₂₇	1	2	yes	—
63	C₂×C₁₆	—	2	1	no	—
64	#108	—	2	1	no	—
65	#108	—	2	1	no	—
66	#108	C₄×C₁₂₇	1	2	yes	—
67	#108	C₄×C₁₂₇	1	2	yes	—
68	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	4	1	no	—
72	M₁₆	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
76	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
77	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
78	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
79	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
80	C₂×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
81	C₁₆	—	2	2	no	—
82	C₂³	Q₁₆×C₁₂₇	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
85	C₂×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
86	C₂×C₄	D₈×C₁₂₇	1	1	yes	characteristic
87	C₂×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
88	C₂×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	4	2	no	—
91	C₈	C₄²×C₁₂₇	1	1	yes	characteristic
92	C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
93	C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
94	C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#103×C₁₂₇	1	1	yes	characteristic
97	C₂²	#103×C₁₂₇	1	1	yes	characteristic
98	C₂²	#107×C₁₂₇	1	1	yes	characteristic
99	C₄	#95×C₁₂₇	1	1	yes	characteristic
100	C₄	#102×C₁₂₇	1	1	yes	characteristic
101	C₄	#102×C₁₂₇	1	1	yes	characteristic
102	C₄	#106×C₁₂₇	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	#340×C₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₃₂×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

—

M₃₂×C₁₂₇

C₄

yes

—

C₂×C₁₆×C₁₂₇

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#108×C₁₂₇

C₄

yes

—

#108×C₁₂₇

C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

characteristic

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

Q₁₆

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

D₈

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄²

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#340

yes

characteristic

#1388

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1388

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂×C₁₂₇

yes

characteristic

C₂×#108

C₂×C₁₂₇

yes

characteristic

C₂×#108

C₂×C₁₂₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×C₁₂₇

yes

characteristic

M₃₂

—

M₃₂

C₄×C₁₂₇

yes

—

C₂×C₁₆

—

#108

—

#108

—

#108

C₄×C₁₂₇

yes

—

#108

C₄×C₁₂₇

yes

—

C₂²×C₄

D₄×C₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₈

D₄×C₁₂₇

yes

characteristic

C₂×C₈

D₄×C₁₂₇

yes

characteristic

C₂×C₈

Q₈×C₁₂₇

yes

characteristic

C₁₆

—

C₂³

Q₁₆×C₁₂₇

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₂×C₄

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂×C₄

D₈×C₁₂₇

yes

characteristic

C₂×C₄

SD₁₆×C₁₂₇

yes

characteristic

C₂×C₄

SD₁₆×C₁₂₇

yes

characteristic

C₈

—

C₈

—

C₈

C₄²×C₁₂₇

yes

characteristic

C₈

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₈

C₄⋊C₄×C₁₂₇

yes

characteristic

C₈

C₄⋊C₄×C₁₂₇

yes

characteristic

C₂²

—

C₂²

#103×C₁₂₇

yes

characteristic

C₂²

#103×C₁₂₇

yes

characteristic

C₂²

#107×C₁₂₇

yes

characteristic

C₄

#95×C₁₂₇

yes

characteristic

100

C₄

#102×C₁₂₇

yes

characteristic

101

C₄

#102×C₁₂₇

yes

characteristic

102

C₄

#106×C₁₂₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#340×C₁₂₇

yes

characteristic

	subgroup	G/H
center	C₄×C₁₂₇	#102
commutator	C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₄×C₁₂₇

#102

commutator

C₈

C₄²×C₁₂₇

Frattini, cosocle

C₂²×C₈

C₂²×C₁₂₇

socle

C₂×C₁₂₇

#340

prime radical

C₂³×C₁₂₇

Q₁₆

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
53. #1388	C₁₂₇	direct (N × H)
54. C₁₂₇	#1388	direct (N × H)

H ≅ G/N

type

53. #1388

C₁₂₇

direct (N × H)

54. C₁₂₇

#1388

direct (N × H)

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

subgroup	index
C₂×M₃₂×C₁₂₇	2
C₂×#108×C₁₂₇	2
#1388	127

subgroup

index

C₂×M₃₂×C₁₂₇

C₂×#108×C₁₂₇

#1388

127

Subgroups — C₁₂₇×#1388 #30634340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1388 #30634340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1388 #30634340