Group #30636565 — subgroups

Subgroups — #30636565

Order 16256.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₂₇	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	4	no	—
10	C₂×C₁₂₇⋊C₈	—	4	2	no	—
11	C₂×C₁₂₇⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	—	2	2	no	—
16	C₂×C₈×C₁₂₇	—	2	2	no	—
17	C₂×C₈×C₁₂₇	C₈	1	4	yes	complemented
18	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₂₇	D₄	1	2	yes	—
20	C₁₂₇⋊C₈	—	4	4	no	—
21	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
29	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
31	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	4	no	—
33	C₂²×C₁₂₇	#95	1	1	yes	characteristic
34	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₂₇	#106	1	1	yes	characteristic
36	C₂²×C₁₂₇	#107	1	1	yes	characteristic
37	C₂²×C₁₂₇	#108	1	2	yes	—
38	C₄×C₁₂₇	—	2	2	no	—
39	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
40	C₄×C₁₂₇	#97	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
42	C₄×C₁₂₇	#102	1	2	yes	—
43	C₄×C₁₂₇	#103	1	2	yes	—
44	C₂×C₁₂₇	#334	1	2	yes	—
45	C₂×C₁₂₇	#335	1	2	yes	—
46	C₂×C₁₂₇	#336	1	1	yes	characteristic
47	C₂×C₁₂₇	#340	1	1	yes	characteristic
48	C₂×C₁₂₇	#341	1	1	yes	characteristic
49	#1277	—	127	1	no	maximal, Sylow
50	C₁₂₇	#1277	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄⋊C₈	—	127	2	no	—
52	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
53	C₂²×C₈	—	254	2	no	—
54	C₂²×C₈	Dic₁₂₇	1	2	yes	—
55	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
56	C₄⋊C₈	—	127	4	no	—
57	C₄⋊C₈	—	254	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
61	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	508	2	no	—
65	C₂×C₈	—	508	2	no	—
66	C₂×C₈	—	254	4	no	—
67	C₂×C₈	C₁₂₇⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₁₂₇	1	2	yes	—
69	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
70	C₂³	C₄⋊Dic₁₂₇	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
75	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
76	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
77	C₂×C₄	#14523232	1	2	yes	—
78	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
79	C₂×C₄	#14523237	1	2	yes	—
80	C₂×C₄	Dic₅₀₈	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	508	4	no	—
83	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
84	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
85	C₂²	#16861141	1	1	yes	characteristic
86	C₂²	#16861143	1	1	yes	characteristic
87	C₂²	#16861145	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
90	C₄	#16861120	1	1	yes	characteristic
91	C₄	C₂²⋊(C₁₂₇⋊C₈)	1	2	yes	—
92	C₄	#16861131	1	2	yes	—
93	C₄	#16861134	1	2	yes	—
94	C₂	#22549627	1	2	yes	—
95	C₂	#22549629	1	2	yes	—
96	C₂	#22549631	1	1	yes	characteristic
97	C₂	#22549639	1	1	yes	characteristic
98	C₂	#22549641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

—

C₂²×C₈×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₈

yes

complemented

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

D₈

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#95

yes

characteristic

C₂²×C₁₂₇

C₄⋊C₈

yes

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×C₈

yes

characteristic

C₄×C₁₂₇

#97

yes

characteristic

C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₄×C₁₂₇

#102

yes

—

C₄×C₁₂₇

#103

yes

—

C₂×C₁₂₇

#334

yes

—

C₂×C₁₂₇

#335

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1277

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₂²×C₈

—

254

—

C₂²×C₈

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

508

—

C₂×C₈

—

508

—

C₂×C₈

—

254

—

C₂×C₈

C₁₂₇⋊C₈

yes

complemented

C₄²

C₄×D₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₁₂₇

yes

characteristic

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₂×C₁₂₇⋊C₈

yes

—

C₂×C₄

#14523232

yes

—

C₂×C₄

D₁₀₁₆

yes

characteristic

C₂×C₄

#14523237

yes

—

C₂×C₄

Dic₅₀₈

yes

characteristic

C₈

—

C₈

—

508

—

C₂²

C₁₂₇⋊#95

yes

characteristic

C₂²

C₄⋊(C₁₂₇⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₄×C₁₂₇⋊C₈

yes

characteristic

C₄

#16861120

yes

characteristic

C₄

C₂²⋊(C₁₂₇⋊C₈)

yes

—

C₄

#16861131

yes

—

C₄

#16861134

yes

—

C₂

#22549627

yes

—

C₂

#22549629

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₅₀₈
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1277
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₅₀₈

commutator

C₄×C₁₂₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₂₇

C₂²

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

#1277

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₂×C₈×C₁₂₇	C₈	semidirect
50. C₁₂₇	#1277	semidirect
67. C₂×C₈	C₁₂₇⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₈×C₁₂₇

C₈

semidirect

50. C₁₂₇

#1277

semidirect

67. C₂×C₈

C₁₂₇⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×C₄×C₈×C₁₂₇	2
#1277	127

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×C₄×C₈×C₁₂₇

#1277

127

Subgroups — #30636565

Conjugacy classes of subgroups

Complements

Maximal subgroups