Group #30638484 — subgroups

Subgroups — #30638484

Order 16256.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22549834	C₂	1	1	yes	maximal, characteristic
2	#22549832	C₂	1	2	yes	maximal
3	C₁₂₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#22549632	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16861168	C₂²	1	1	yes	characteristic
9	#16861120	C₄	1	4	yes	complemented
10	#16861119	C₄	1	2	yes	complemented
11	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₁₂₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	#117×C₁₂₇	C₂²	1	2	yes	—
15	#14523232	—	2	2	no	—
16	#14523232	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₂₇	—	2	1	no	—
23	C₂²⋊Dic₁₂₇	—	2	1	no	—
24	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₂₇	—	2	2	no	—
29	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₂₇	—	2	2	no	—
32	C₁₂₇⋊C₈	—	2	4	no	—
33	C₁₂₇⋊C₈	C₄²	1	1	yes	characteristic
34	C₁₂₇⋊C₈	C₄²	1	1	yes	characteristic
35	C₁₂₇⋊C₈	C₄²	1	2	yes	—
36	C₂³×C₁₂₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	#41	1	2	yes	—
41	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
42	C₈×C₁₂₇	—	2	2	no	—
43	C₂²×C₁₂₇	—	4	1	no	—
44	C₂²×C₁₂₇	#117	1	2	yes	—
45	C₂²×C₁₂₇	#117	1	1	yes	characteristic
46	C₄×C₁₂₇	—	4	4	no	—
47	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₂₇	#117	1	1	yes	characteristic
49	C₄×C₁₂₇	#117	1	2	yes	—
50	Dic₁₂₇	—	4	2	no	—
51	C₂×C₁₂₇	—	2	1	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	#376	1	1	yes	characteristic
54	#1747	—	127	1	no	maximal, Sylow
55	C₁₂₇	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	127	1	no	—
57	#405	—	127	2	no	—
58	#338	—	127	1	no	—
59	H(3,ℤ/4)	—	127	2	no	—
60	H(3,ℤ/4)	D₁₂₇	1	1	yes	characteristic
61	C₂×M₁₆	—	127	2	no	—
62	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂²×C₈	—	127	1	no	—
64	#117	—	127	1	no	—
65	#117	C₂×D₁₂₇	1	2	yes	—
66	#97	—	127	2	no	—
67	#97	—	127	4	no	—
68	C₄×C₈	—	127	2	no	—
69	C₄×C₈	—	127	4	no	—
70	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	254	2	no	—
73	M₁₆	—	254	2	no	—
74	C₂×C₈	—	127	1	no	—
75	C₂×C₈	—	127	2	no	—
76	C₂×C₈	—	127	1	no	—
77	C₂×C₈	—	127	2	no	—
78	C₂×C₈	—	127	4	no	—
79	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	254	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	254	1	no	—
84	C₄²	—	127	2	no	—
85	C₄²	C₄×D₁₂₇	1	4	yes	—
86	C₂³	#14523247	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	254	2	no	—
89	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
91	C₂×C₄	#14523247	1	2	yes	—
92	C₂×C₄	#14523249	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	127	1	no	—
95	C₈	—	127	1	no	—
96	C₈	—	127	2	no	—
97	C₈	—	254	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#16861165	1	2	yes	—
100	C₂²	#16861167	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	508	2	no	—
103	C₄	C₄²×D₁₂₇	1	1	yes	characteristic
104	C₄	#16861165	1	1	yes	characteristic
105	C₄	#16861167	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#22549721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549834

C₂

yes

maximal, characteristic

#22549832

C₂

yes

maximal

C₁₂₇⋊#338

C₂

yes

maximal, characteristic

#22549632

C₂

yes

maximal

H(3,ℤ/4)×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

#16861120

C₄

yes

complemented

#16861119

C₄

yes

complemented

C₄×C₁₂₇⋊C₈

C₄

yes

complemented

C₈×Dic₁₂₇

C₄

yes

complemented

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄²

yes

characteristic

C₁₂₇⋊C₈

C₄²

yes

characteristic

C₁₂₇⋊C₈

C₄²

yes

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#117

yes

—

C₂²×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄²

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#376

yes

characteristic

#1747

—

127

maximal, Sylow

C₁₂₇

#1747

yes

characteristic, complemented, Sylow

#405

—

127

—

#405

—

127

—

#338

—

127

—

H(3,ℤ/4)

—

127

—

H(3,ℤ/4)

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

#97

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

—

C₂³

#14523247

yes

characteristic

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

C₂×C₄

#14523247

yes

—

C₂×C₄

#14523249

yes

characteristic

C₈

—

C₈

—

127

—

C₈

—

127

—

C₈

—

127

—

C₈

—

254

—

C₂²

—

C₂²

#16861165

yes

—

100

C₂²

#16861167

yes

characteristic

101

C₄

—

102

C₄

—

508

—

103

C₄

C₄²×D₁₂₇

yes

characteristic

104

C₄

#16861165

yes

characteristic

105

C₄

#16861167

yes

—

106

C₂

—

107

C₂

—

108

C₂

#22549721

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	H(3,ℤ/4)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#376
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₂₇
hypocenter	C₁₂₇	#1747
LCS	G ▷ C₄×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₂₇

commutator

C₄×C₁₂₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

H(3,ℤ/4)×C₁₂₇

C₂

C₂×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16861120	C₄	semidirect
10. #16861119	C₄	semidirect
11. C₄×C₁₂₇⋊C₈	C₄	semidirect
12. C₈×Dic₁₂₇	C₄	semidirect
55. C₁₂₇	#1747	semidirect

H ≅ G/N

type

9. #16861120

C₄

semidirect

10. #16861119

C₄

semidirect

11. C₄×C₁₂₇⋊C₈

C₄

semidirect

12. C₈×Dic₁₂₇

C₄

semidirect

55. C₁₂₇

#1747

semidirect

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

subgroup	index
#22549834	2
#22549832	2
C₁₂₇⋊#338	2
#22549632	2
H(3,ℤ/4)×C₁₂₇	2
#1747	127

subgroup

index

C₁₂₇⋊#338

H(3,ℤ/4)×C₁₂₇

127

Subgroups — #30638484

Conjugacy classes of subgroups

Complements

Maximal subgroups