Group #30640483 — subgroups

Subgroups — C₂×C₄⋊(C₁₂₇⋊C₁₆) #30640483

Order 16256.

184 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×C₁₂₇⋊C₁₆	C₂	1	2	yes	maximal
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	—	2	4	no	—
5	C₂×C₁₂₇⋊C₁₆	C₂²	1	4	yes	—
6	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₁₂₇⋊C₁₆	—	2	4	no	—
12	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	—	2	2	no	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
19	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
20	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₂₇	C₈	1	2	yes	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	2	no	—
33	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
37	C₂²×C₁₂₇	C₂×C₁₆	1	2	yes	—
38	C₂²×C₁₂₇	M₃₂	1	2	yes	—
39	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
42	C₄×C₁₂₇	C₂×C₁₆	1	4	yes	complemented
43	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₁₂₇	C₄⋊C₁₆	1	4	yes	complemented
46	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
48	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	—	127	1	no	maximal, Sylow
50	C₁₂₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	—	127	2	no	—
52	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
53	C₄⋊C₁₆	—	127	4	no	—
54	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
57	C₂×C₁₆	—	127	4	no	—
58	C₂×C₁₆	—	254	4	no	—
59	C₄×C₈	Dic₁₂₇	1	2	yes	—
60	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
61	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
62	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
66	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
67	C₂×C₈	D₅₀₈	1	2	yes	—
68	C₂×C₈	Dic₂₅₄	1	2	yes	—
69	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₄²	C₁₂₇⋊C₈	1	2	yes	—
71	C₄²	C₂×Dic₁₂₇	1	2	yes	—
72	C₁₆	—	254	4	no	—
73	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	C₁₂₇⋊C₁₆	1	4	yes	complemented
76	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
77	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
78	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
79	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
80	C₂×C₄	#14523232	1	2	yes	—
81	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
84	C₈	C₂×D₅₀₈	1	1	yes	characteristic
85	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
86	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
87	C₂²	C₂×C₁₂₇⋊C₁₆	1	2	yes	—
88	C₂²	#16861149	1	2	yes	—
89	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
92	C₄	C₂×C₁₂₇⋊C₁₆	1	4	yes	complemented
93	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
94	C₄	C₂×#14523232	1	1	yes	characteristic
95	C₂	C₄⋊(C₁₂₇⋊C₁₆)	1	4	yes	direct product factor
96	C₂	C₂×C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
97	C₂	C₂²×C₁₂₇⋊C₁₆	1	1	yes	characteristic
98	C₂	C₂×#16861149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₂₇⋊C₁₆

C₂

yes

maximal

C₄⋊(C₁₂₇⋊C₁₆)

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₄

yes

characteristic

C₄×C₈×C₁₂₇

C₄

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₈

yes

characteristic

C₂²×C₄×C₁₂₇

C₈

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₈

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₁₆

yes

complemented

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

C₄⋊C₈

yes

—

C₂²×C₁₂₇

C₂×C₁₆

yes

—

C₂²×C₁₂₇

M₃₂

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

C₂×C₁₆

yes

complemented

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂×C₁₂₇

C₄⋊C₁₆

yes

complemented

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₂₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

C₂×C₄⋊C₁₆

—

127

maximal, Sylow

C₁₂₇

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₄⋊C₁₆

—

127

—

C₂²×C₈

Dic₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

Dic₁₂₇

yes

characteristic

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

C₄×C₈

Dic₁₂₇

yes

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₂²×C₄

C₁₂₇⋊C₈

yes

characteristic

C₂²×C₄

C₁₂₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

Dic₂₅₄

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

C₁₂₇⋊C₈

yes

—

C₄²

C₂×Dic₁₂₇

yes

—

C₁₆

—

254

—

C₂³

C₂×C₁₂₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₁₂₇⋊C₁₆

yes

complemented

C₂×C₄

C₄⋊Dic₁₂₇

yes

—

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

C₂×C₄

#14523232

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₈

—

C₈

C₄⋊Dic₁₂₇

yes

—

C₈

C₂×D₅₀₈

yes

characteristic

C₈

C₂×Dic₂₅₄

yes

characteristic

C₂²

C₄⋊(C₁₂₇⋊C₈)

yes

—

C₂²

C₂×C₁₂₇⋊C₁₆

yes

—

C₂²

#16861149

yes

—

C₂²

C₂²×C₁₂₇⋊C₈

yes

characteristic

C₄

—

C₄

C₄⋊(C₁₂₇⋊C₈)

yes

—

C₄

C₂×C₁₂₇⋊C₁₆

yes

complemented

C₄

C₂×C₄⋊Dic₁₂₇

yes

characteristic

C₄

C₂×#14523232

yes

characteristic

C₂

C₄⋊(C₁₂₇⋊C₁₆)

yes

direct product factor

C₂

C₂×C₄⋊(C₁₂₇⋊C₈)

yes

characteristic

C₂

C₂²×C₁₂₇⋊C₁₆

yes

characteristic

C₂

C₂×#16861149

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. C₄⋊(C₁₂₇⋊C₁₆)	C₂	direct (N × H)
25. C₂×C₄×C₁₂₇	C₁₆	semidirect
42. C₄×C₁₂₇	C₂×C₁₆	semidirect
45. C₂×C₁₂₇	C₄⋊C₁₆	semidirect
50. C₁₂₇	C₂×C₄⋊C₁₆	semidirect
75. C₂×C₄	C₁₂₇⋊C₁₆	semidirect
92. C₄	C₂×C₁₂₇⋊C₁₆	semidirect
95. C₂	C₄⋊(C₁₂₇⋊C₁₆)	direct (N × H)

H ≅ G/N

type

2. C₄⋊(C₁₂₇⋊C₁₆)

C₂

direct (N × H)

25. C₂×C₄×C₁₂₇

C₁₆

semidirect

42. C₄×C₁₂₇

C₂×C₁₆

semidirect

45. C₂×C₁₂₇

C₄⋊C₁₆

semidirect

50. C₁₂₇

C₂×C₄⋊C₁₆

semidirect

75. C₂×C₄

C₁₂₇⋊C₁₆

semidirect

92. C₄

C₂×C₁₂₇⋊C₁₆

semidirect

95. C₂

C₄⋊(C₁₂₇⋊C₁₆)

direct (N × H)

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

subgroup	index
C₂²×C₁₂₇⋊C₁₆	2
C₄⋊(C₁₂₇⋊C₁₆)	2
C₂×C₄×C₈×C₁₂₇	2
C₂×C₄⋊C₁₆	127

subgroup

index

C₂²×C₁₂₇⋊C₁₆

C₄⋊(C₁₂₇⋊C₁₆)

C₂×C₄×C₈×C₁₂₇

C₂×C₄⋊C₁₆

127

Subgroups — C₂×C₄⋊(C₁₂₇⋊C₁₆) #30640483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C4⋊(C127⋊C16) #30640483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₁₂₇⋊C₁₆) #30640483