Group #30640482 — subgroups

Subgroups — C₂×Dic₁₂₇⋊C₁₆ #30640482

Order 16256.

184 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 117 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	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	Dic₁₂₇⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₂₇	C₄	1	2	yes	—
10	C₈×Dic₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	—	2	2	no	—
13	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₂₇	C₈	1	2	yes	—
19	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
20	C₁₂₇⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
25	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₂₇	—	2	2	no	—
27	C₂×Dic₁₂₇	—	2	2	no	—
28	C₂×Dic₁₂₇	C₁₆	1	4	yes	complemented
29	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
31	C₁₂₇⋊C₈	—	2	2	no	—
32	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
36	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
38	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₂₇	C₂×C₁₆	1	2	yes	—
42	C₂²×C₁₂₇	M₃₂	1	2	yes	—
43	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
47	Dic₁₂₇	—	2	2	no	—
48	Dic₁₂₇	C₂×C₁₆	1	4	yes	complemented
49	C₂×C₁₂₇	C₄⋊C₁₆	1	4	yes	complemented
50	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
53	C₂×C₄⋊C₁₆	—	127	1	no	maximal, Sylow
54	C₁₂₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	127	1	no	—
56	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
57	C₂×C₄×C₈	—	127	1	no	—
58	C₄⋊C₁₆	—	127	4	no	—
59	C₂²×C₈	—	127	1	no	—
60	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×C₄²	—	127	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	127	2	no	—
64	C₂×C₁₆	—	254	2	no	—
65	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
66	C₄×C₈	—	127	2	no	—
67	C₄×C₈	—	127	2	no	—
68	C₂²×C₄	—	127	1	no	—
69	C₂²×C₄	—	127	1	no	—
70	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
71	C₂×C₈	—	127	2	no	—
72	C₂×C₈	—	254	2	no	—
73	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
74	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
75	C₂×C₈	Dic₂₅₄	1	2	yes	—
76	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
77	C₄²	—	127	2	no	—
78	C₄²	—	127	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	254	2	no	—
81	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
82	C₂×C₄	—	127	2	no	—
83	C₂×C₄	—	127	4	no	—
84	C₂×C₄	—	127	2	no	—
85	C₂×C₄	—	254	2	no	—
86	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
87	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	#14523231	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
90	C₈	—	254	2	no	—
91	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
92	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
93	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
94	C₂²	Dic₁₂₇⋊C₈	1	2	yes	—
95	C₂²	C₁₆×D₁₂₇	1	2	yes	—
96	C₂²	#16861148	1	2	yes	—
97	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
98	C₄	—	127	4	no	—
99	C₄	—	254	2	no	—
100	C₄	Dic₁₂₇⋊C₈	1	2	yes	—
101	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
102	C₄	C₂×#14523231	1	1	yes	characteristic
103	C₂	Dic₁₂₇⋊C₁₆	1	4	yes	direct product factor
104	C₂	C₂×Dic₁₂₇⋊C₈	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₁₂₇	1	1	yes	characteristic
106	C₂	C₂×#16861148	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₂₇

C₄

yes

characteristic

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₂₇

C₄

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₈

yes

characteristic

C₂²×Dic₁₂₇

C₈

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₈

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₁₆

yes

complemented

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

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⋊C₈

yes

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂×C₁₆

yes

complemented

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

—

C₂²×C₁₆

D₁₂₇

yes

characteristic

C₂×C₄×C₈

—

127

—

C₄⋊C₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

—

127

—

C₂×C₁₆

—

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₂×C₈

Dic₂₅₄

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₁₆

—

C₁₆

—

254

—

C₂³

C₈×D₁₂₇

yes

characteristic

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₈×D₁₂₇

yes

characteristic

C₂×C₄

#14523231

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

Dic₁₂₇⋊C₄

yes

—

C₈

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₈

C₂×Dic₂₅₄

yes

characteristic

C₂²

Dic₁₂₇⋊C₈

yes

—

C₂²

C₁₆×D₁₂₇

yes

—

C₂²

#16861148

yes

—

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

C₄

—

127

—

C₄

—

254

—

100

C₄

Dic₁₂₇⋊C₈

yes

—

101

C₄

C₂×Dic₁₂₇⋊C₄

yes

characteristic

102

C₄

C₂×#14523231

yes

characteristic

103

C₂

Dic₁₂₇⋊C₁₆

yes

direct product factor

104

C₂

C₂×Dic₁₂₇⋊C₈

yes

characteristic

105

C₂

C₂×C₁₆×D₁₂₇

yes

characteristic

106

C₂

C₂×#16861148

yes

characteristic

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

subgroup

G/H

center

C₂²×C₈

C₂×D₁₂₇

commutator

C₂×C₁₂₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

C₂²×C₁₆×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂²×C₁₆

D₁₂₇

hypocenter

C₁₂₇

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. Dic₁₂₇⋊C₁₆	C₂	direct (N × H)
28. C₂×Dic₁₂₇	C₁₆	semidirect
48. Dic₁₂₇	C₂×C₁₆	semidirect
49. C₂×C₁₂₇	C₄⋊C₁₆	semidirect
54. C₁₂₇	C₂×C₄⋊C₁₆	semidirect
103. C₂	Dic₁₂₇⋊C₁₆	direct (N × H)

H ≅ G/N

type

3. Dic₁₂₇⋊C₁₆

C₂

direct (N × H)

28. C₂×Dic₁₂₇

C₁₆

semidirect

48. Dic₁₂₇

C₂×C₁₆

semidirect

49. C₂×C₁₂₇

C₄⋊C₁₆

semidirect

54. C₁₂₇

C₂×C₄⋊C₁₆

semidirect

103. C₂

Dic₁₂₇⋊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
C₂×C₈×Dic₁₂₇	2
Dic₁₂₇⋊C₁₆	2
C₂²×C₁₆×C₁₂₇	2
C₂×C₄⋊C₁₆	127

subgroup

index

C₂²×C₁₂₇⋊C₁₆

C₂×C₈×Dic₁₂₇

Dic₁₂₇⋊C₁₆

C₂²×C₁₆×C₁₂₇

C₂×C₄⋊C₁₆

127

Subgroups — C₂×Dic₁₂₇⋊C₁₆ #30640482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Dic127⋊C16 #30640482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₁₂₇⋊C₁₆ #30640482