Group #30640538 — subgroups

Subgroups — #30640538

Order 16256.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22549830	C₂	1	1	yes	maximal, characteristic
2	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₁₂₇)⋊C₁₆	C₂	1	2	yes	maximal
4	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523231	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	#16861166	C₄	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
10	#16861120	C₄	1	1	yes	characteristic
11	#16861119	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
16	#14523231	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₈×D₁₂₇	—	2	2	no	—
20	Dic₁₂₇⋊C₄	C₈	1	2	yes	—
21	(C₂×D₁₂₇)⋊C₄	C₈	1	2	yes	—
22	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₁₂₇⋊C₁₆	—	2	2	no	—
24	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₁₂₇	—	2	2	no	—
28	C₂²×D₁₂₇	C₂×C₈	1	1	yes	characteristic
29	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₄×D₁₂₇	—	2	1	no	—
32	C₄×D₁₂₇	—	2	1	no	—
33	C₁₂₇⋊C₈	—	2	1	no	—
34	C₁₂₇⋊C₈	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₂₇	M₁₆	1	2	yes	—
38	C₈×C₁₂₇	#41	1	2	yes	—
39	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₁₂₇	—	2	1	no	—
41	C₂×D₁₂₇	—	2	2	no	—
42	C₄×C₁₂₇	—	2	2	no	—
43	C₄×C₁₂₇	#117	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₂₇	—	2	1	no	—
46	Dic₁₂₇	—	2	2	no	—
47	C₂×C₁₂₇	#431	1	1	yes	characteristic
48	C₂×C₁₂₇	#504	1	2	yes	—
49	D₁₂₇	—	4	1	no	—
50	#2164	—	127	1	no	maximal, Sylow
51	C₁₂₇	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	—	127	1	no	—
53	C₄⋊C₁₆	—	127	2	no	—
54	C₂²⋊C₁₆	—	127	2	no	—
55	#346	D₁₂₇	1	1	yes	characteristic, complemented
56	C₄×C₁₆	—	127	1	no	—
57	C₂×M₁₆	—	127	1	no	—
58	C₂²×C₈	—	127	1	no	—
59	#117	—	127	1	no	—
60	C₂×C₁₆	—	127	2	no	—
61	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
62	#97	—	127	1	no	—
63	#97	—	127	1	no	—
64	C₄×C₈	—	127	1	no	—
65	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂²×C₄	—	127	1	no	—
67	M₁₆	—	254	2	no	—
68	C₂×C₈	—	127	1	no	—
69	C₂×C₈	—	127	1	no	—
70	C₂×C₈	—	254	2	no	—
71	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	—	127	2	no	—
74	C₂²⋊C₄	—	127	2	no	—
75	C₄²	—	127	1	no	—
76	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	254	2	no	—
79	C₂³	—	127	1	no	—
80	C₂×C₄	—	127	2	no	—
81	C₂×C₄	—	127	1	no	—
82	C₂×C₄	—	254	1	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
86	C₈	—	254	1	no	—
87	C₈	—	254	1	no	—
88	C₈	#14523231	1	2	yes	—
89	C₈	#14523248	1	2	yes	—
90	C₂²	—	254	2	no	—
91	C₂²	—	254	1	no	—
92	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	254	2	no	—
95	C₄	—	254	1	no	—
96	C₄	#16861166	1	1	yes	characteristic
97	C₄	C₂×#14523231	1	1	yes	characteristic
98	C₂	—	508	1	no	—
99	C₂	#22549927	1	1	yes	characteristic
100	C₂	#22550294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549830

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₁₂₇)⋊C₁₆

C₂

yes

maximal

C₄×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#346×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523231

C₄

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

#16861166

C₄

yes

characteristic

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

#16861120

C₄

yes

characteristic

#16861119

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₂₇

—

Dic₁₂₇⋊C₄

C₈

yes

—

(C₂×D₁₂₇)⋊C₄

C₈

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

M₁₆

yes

—

C₈×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#431

yes

characteristic

C₂×C₁₂₇

#504

yes

—

D₁₂₇

—

#2164

—

127

maximal, Sylow

C₁₂₇

#2164

yes

characteristic, complemented, Sylow

#405

—

127

—

C₄⋊C₁₆

—

127

—

C₂²⋊C₁₆

—

127

—

#346

D₁₂₇

yes

characteristic, complemented

C₄×C₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

#117

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

#97

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₁₆

—

C₁₆

—

254

—

C₂³

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₈×D₁₂₇

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

—

254

—

C₈

#14523231

yes

—

C₈

#14523248

yes

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

yes

C₄

yes

C₂

—

508

—

C₂

#22549927

yes

characteristic

100

C₂

#22550294

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	#346×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#2164
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₈

Fitting

#346×C₁₂₇

C₂

prime radical

C₂²×D₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₁₂₇⋊C₁₆	C₂	semidirect
4. C₄×C₁₂₇⋊C₁₆	C₂	semidirect
5. #346×C₁₂₇	C₂	semidirect
51. C₁₂₇	#2164	semidirect
55. #346	D₁₂₇	semidirect

H ≅ G/N

type

2. Dic₁₂₇⋊C₁₆

C₂

semidirect

4. C₄×C₁₂₇⋊C₁₆

C₂

semidirect

5. #346×C₁₂₇

C₂

semidirect

51. C₁₂₇

#2164

semidirect

55. #346

D₁₂₇

semidirect

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

subgroup	index
#22549830	2
Dic₁₂₇⋊C₁₆	2
(C₂×D₁₂₇)⋊C₁₆	2
C₄×C₁₂₇⋊C₁₆	2
#346×C₁₂₇	2
#2164	127

subgroup

index

#22549830

Dic₁₂₇⋊C₁₆

(C₂×D₁₂₇)⋊C₁₆

C₄×C₁₂₇⋊C₁₆

#346×C₁₂₇

#2164

127

Subgroups — #30640538

Conjugacy classes of subgroups

Complements

Maximal subgroups