Group #30637549 — subgroups

Subgroups — #30637549

Order 16256.

160 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 87 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	Q₈×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	#22549627	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
4	#22549597	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
7	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
8	C₄×C₁₂₇⋊C₈	—	2	2	no	—
9	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
13	#107×C₁₂₇	C₄	1	1	yes	characteristic
14	#103×C₁₂₇	C₄	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	1	no	—
18	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
19	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
20	Q₁₆×C₁₂₇	C₈	1	4	yes	complemented
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₂₇	—	2	2	no	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₁₂₇⋊C₈	—	2	2	no	—
28	C₁₂₇⋊C₈	—	4	1	no	—
29	C₁₂₇⋊C₈	—	4	2	no	—
30	Q₈×C₁₂₇	—	2	2	no	—
31	Q₈×C₁₂₇	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄×C₁₂₇	—	2	2	no	—
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
37	C₈×C₁₂₇	—	2	1	no	—
38	C₈×C₁₂₇	C₂×C₈	1	2	yes	—
39	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	4	no	—
41	C₄×C₁₂₇	—	2	1	no	—
42	C₄×C₁₂₇	—	4	2	no	—
43	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₂₇	#131	1	1	yes	characteristic
45	C₄×C₁₂₇	#137	1	2	yes	—
46	C₂×C₁₂₇	C₈×D₄	1	1	yes	characteristic
47	C₂×C₁₂₇	#441	1	1	yes	characteristic
48	C₂×C₁₂₇	#444	1	1	yes	characteristic
49	#1579	—	127	1	no	maximal, Sylow
50	C₁₂₇	#1579	1	1	yes	characteristic, complemented, Sylow
51	C₈×Q₈	—	127	2	no	—
52	C₄×Q₁₆	D₁₂₇	1	1	yes	characteristic
53	#334	—	127	1	no	—
54	Q₈⋊C₈	—	127	2	no	—
55	#322	—	127	1	no	—
56	C₂×Q₁₆	Dic₁₂₇	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
58	#107	Dic₁₂₇	1	1	yes	characteristic
59	C₄⋊C₈	—	127	2	no	—
60	C₄⋊C₈	—	254	2	no	—
61	#103	Dic₁₂₇	1	2	yes	—
62	C₄×C₈	—	127	1	no	—
63	C₄×C₈	—	127	1	no	—
64	C₄×C₈	—	254	2	no	—
65	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
67	Q₁₆	C₁₂₇⋊C₈	1	4	yes	complemented
68	C₂×C₈	—	127	2	no	—
69	C₂×C₈	—	254	2	no	—
70	C₂×C₈	—	254	1	no	—
71	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
82	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
83	C₂×C₄	#14523247	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	508	1	no	—
86	C₈	—	508	2	no	—
87	C₈	—	254	2	no	—
88	C₈	C₂×C₁₂₇⋊C₈	1	2	yes	—
89	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	2	1	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
94	C₄	#16861216	1	1	yes	characteristic
95	C₄	#16861241	1	2	yes	—
96	C₂	D₄×C₁₂₇⋊C₈	1	1	yes	characteristic
97	C₂	#22549983	1	1	yes	characteristic
98	C₂	#22549998	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal

#22549627

C₂

yes

maximal, characteristic

Q₈⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

#22549597

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄×C₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

C₄

yes

characteristic

C₄×Q₈×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

C₄

yes

characteristic

#103×C₁₂₇

C₄

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

Q₁₆×C₁₂₇

C₈

yes

complemented

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×C₈

yes

—

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₈

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

C₂×C₁₂₇

C₈×D₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1579

yes

characteristic, complemented, Sylow

C₈×Q₈

—

127

—

C₄×Q₁₆

D₁₂₇

yes

characteristic

#334

—

127

—

Q₈⋊C₈

—

127

—

#322

—

127

—

C₂×Q₁₆

Dic₁₂₇

yes

characteristic

C₄×Q₈

C₂×D₁₂₇

yes

—

#107

Dic₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

#103

Dic₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

C₂×Dic₁₂₇

yes

—

Q₁₆

C₁₂₇⋊C₈

yes

complemented

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

C₂×C₁₂₇⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

D₄×D₁₂₇

yes

characteristic

C₂×C₄

#14523247

yes

characteristic

C₈

—

C₈

—

508

—

C₈

—

508

—

C₈

—

254

—

C₈

C₂×C₁₂₇⋊C₈

yes

—

C₂²

D₄×Dic₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₁₂₇⋊C₈

yes

C₄

yes

C₄

yes

—

C₂

D₄×C₁₂₇⋊C₈

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₈
Frattini	C₄²	C₂²×D₁₂₇
Fitting	C₄×Q₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₄×D₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	C₄×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#1579
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₈

Frattini

C₄²

C₂²×D₁₂₇

Fitting

C₄×Q₁₆×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₄×D₄

cosocle

C₄²×C₁₂₇

C₂³

hypercenter

C₄×Q₁₆

D₁₂₇

hypocenter

C₁₂₇

#1579

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
20. Q₁₆×C₁₂₇	C₈	semidirect
50. C₁₂₇	#1579	semidirect
67. Q₁₆	C₁₂₇⋊C₈	semidirect

H ≅ G/N

type

20. Q₁₆×C₁₂₇

C₈

semidirect

50. C₁₂₇

#1579

semidirect

67. Q₁₆

C₁₂₇⋊C₈

semidirect

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

subgroup	index
Q₈×C₁₂₇⋊C₈	2
#22549627	2
Q₈⋊(C₁₂₇⋊C₈)	2
#22549597	2
C₄×Q₁₆×C₁₂₇	2
#1579	127

subgroup

index

Q₈×C₁₂₇⋊C₈

#22549627

Q₈⋊(C₁₂₇⋊C₈)

#22549597

C₄×Q₁₆×C₁₂₇

#1579

127

Subgroups — #30637549

Conjugacy classes of subgroups

Complements

Maximal subgroups