Group #30639696 — subgroups

Subgroups — #30639696

Order 16256.

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

normal subgroups only

This group has 252 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	1	yes	maximal, characteristic
2	#22549742	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal
4	C₂×#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
7	C₄×C₁₂₇⋊C₈	—	2	2	no	—
8	C₁₂₇⋊#95	—	2	2	no	—
9	#125×C₁₂₇	—	2	2	no	—
10	#125×C₁₂₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₁₂₇	—	2	4	no	—
19	C₂²×C₄×C₁₂₇	—	2	2	no	—
20	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₂₇	—	2	8	no	—
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₂₇	D₄	1	2	yes	—
27	C₂×Dic₁₂₇	—	4	2	no	—
28	C₂×Dic₁₂₇	—	4	4	no	—
29	C₁₂₇⋊C₈	—	2	4	no	—
30	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	—	2	4	no	—
32	C₂×C₄×C₁₂₇	—	2	4	no	—
33	C₂×C₄×C₁₂₇	—	4	4	no	—
34	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₂₇	#41	1	2	yes	—
39	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₂₇	#124	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₂₇	#135	1	4	yes	—
43	C₄×C₁₂₇	—	2	4	no	—
44	C₄×C₁₂₇	—	4	4	no	—
45	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₂₇	—	8	2	no	—
48	C₂×C₁₂₇	#391	1	1	yes	characteristic
49	C₂×C₁₂₇	#416	1	2	yes	—
50	C₂×C₁₂₇	#488	1	2	yes	—
51	C₂×C₁₂₇	#499	1	2	yes	—
52	#1991	—	127	1	no	maximal, Sylow
53	C₁₂₇	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	D₁₂₇	1	1	yes	characteristic
55	C₂×C₄×C₈	—	127	1	no	—
56	#382	—	127	1	no	—
57	#340	—	127	4	no	—
58	C₂²×C₈	—	127	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₁₂₇	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	127	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
64	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
65	C₄×C₈	—	254	2	no	—
66	#95	—	254	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	254	2	no	—
69	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
71	C₂×C₈	—	127	4	no	—
72	C₂×C₈	—	254	2	no	—
73	C₂×C₈	—	254	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	254	4	no	—
77	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
78	C₄²	C₂×Dic₁₂₇	1	2	yes	—
79	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	508	2	no	—
85	C₂×C₄	—	508	4	no	—
86	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
90	C₂×C₄	#14523249	1	2	yes	—
91	C₈	—	254	4	no	—
92	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
93	C₂²	#16861198	1	1	yes	characteristic
94	C₂²	#16861207	1	1	yes	characteristic
95	C₂²	#16861229	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	1016	2	no	—
99	C₄	Q₈×Dic₁₂₇	1	2	yes	—
100	C₄	#16861185	1	2	yes	—
101	C₂	#22549783	1	1	yes	characteristic
102	C₂	#22549872	1	2	yes	—
103	C₂	#22550233	1	2	yes	—
104	C₂	#22550275	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549742

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

C₂×#125×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₄×C₁₂₇⋊C₈

—

C₁₂₇⋊#95

—

#125×C₁₂₇

—

#125×C₁₂₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#124

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#135

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#391

yes

characteristic

C₂×C₁₂₇

#416

yes

—

C₂×C₁₂₇

#488

yes

—

C₂×C₁₂₇

#499

yes

—

#1991

—

127

maximal, Sylow

C₁₂₇

#1991

yes

characteristic, complemented, Sylow

C₂×#125

D₁₂₇

yes

characteristic

C₂×C₄×C₈

—

127

—

#382

—

127

—

#340

—

127

—

C₂²×C₈

—

127

—

#125

—

#125

Dic₁₂₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄×C₈

—

254

—

#95

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

Q₈×D₁₂₇

yes

—

C₂×C₄

#14523249

yes

—

C₈

—

254

—

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

1016

—

C₄

Q₈×Dic₁₂₇

yes

—

100

C₄

#16861185

yes

—

101

C₂

#22549783

yes

characteristic

102

C₂

#22549872

yes

—

103

C₂

#22550233

yes

—

104

C₂

#22550275

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#125×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#125	D₁₂₇
hypocenter	C₁₂₇	#1991
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#125×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₂₇	C₄	semidirect
53. C₁₂₇	#1991	semidirect
60. #125	Dic₁₂₇	semidirect

H ≅ G/N

type

10. #125×C₁₂₇

C₄

semidirect

53. C₁₂₇

#1991

semidirect

60. #125

Dic₁₂₇

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
#22549742	2
#22549638	2
C₂×#125×C₁₂₇	2
#1991	127

subgroup

index

C₂×C₄×C₁₂₇⋊C₈

#22549742

#22549638

C₂×#125×C₁₂₇

#1991

127

Subgroups — #30639696

Conjugacy classes of subgroups

Complements

Maximal subgroups