Group #30640901 — subgroups

Subgroups — #30640901

Order 16256.

126 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22550274	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549700	C₂	1	2	yes	maximal, complemented
3	#22549697	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549686	C₂	1	2	yes	maximal
5	#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861209	—	2	1	no	—
7	#16861149	C₂²	1	2	yes	—
8	#16861141	—	2	1	no	—
9	#16861141	C₂²	1	2	yes	—
10	#135×C₁₂₇	C₂²	1	1	yes	characteristic
11	#131×C₁₂₇	—	2	1	no	—
12	#108×C₁₂₇	C₂²	1	1	yes	characteristic
13	#104×C₁₂₇	—	2	1	no	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	4	1	no	—
16	C₄⋊Dic₁₂₇	—	2	2	no	—
17	C₄⋊Dic₁₂₇	—	4	1	no	—
18	C₁₂₇⋊C₁₆	—	2	2	no	—
19	#41×C₁₂₇	—	2	1	no	—
20	Q₁₆×C₁₂₇	Q₈	1	1	yes	characteristic, complemented
21	SD₁₆×C₁₂₇	—	2	1	no	—
22	D₈×C₁₂₇	Q₈	1	1	yes	characteristic, complemented
23	M₁₆×C₁₂₇	—	2	1	no	—
24	M₁₆×C₁₂₇	—	4	1	no	—
25	C₂×C₈×C₁₂₇	—	4	1	no	—
26	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
29	Dic₂₅₄	—	8	2	no	—
30	C₂×Dic₁₂₇	—	4	2	no	—
31	Q₈×C₁₂₇	—	2	1	no	—
32	D₄×C₁₂₇	—	2	1	no	—
33	D₄×C₁₂₇	—	4	1	no	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	—	4	1	no	—
36	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₂₇	—	4	1	no	—
38	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
39	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
40	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₁₂₇	#41	1	1	yes	characteristic
42	C₂²×C₁₂₇	—	4	1	no	—
43	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
44	C₄×C₁₂₇	—	2	2	no	—
45	C₄×C₁₂₇	—	4	1	no	—
46	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₁₂₇	—	8	2	no	—
49	C₂×C₁₂₇	—	2	1	no	—
50	C₂×C₁₂₇	—	8	1	no	—
51	C₂×C₁₂₇	#476	1	1	yes	characteristic
52	#2230	—	127	1	no	maximal, Sylow
53	C₁₂₇	#2230	1	1	yes	characteristic, complemented, Sylow
54	#498	—	127	1	no	—
55	#443	D₁₂₇	1	1	yes	characteristic
56	#365	—	127	2	no	—
57	#364	—	127	1	no	—
58	#360	—	127	2	no	—
59	#135	C₂×D₁₂₇	1	1	yes	characteristic
60	#131	—	2	1	no	—
61	C₄⋊Q₈	—	254	1	no	—
62	M₃₂	—	127	2	no	—
63	#108	C₂×D₁₂₇	1	1	yes	characteristic
64	#106	—	127	2	no	—
65	#106	—	254	1	no	—
66	#104	—	2	1	no	—
67	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
68	#41	—	2	1	no	—
69	C₂×Q₈	—	508	1	no	—
70	Q₁₆	Dic₂₅₄	1	1	yes	characteristic, complemented
71	SD₁₆	—	2	1	no	—
72	D₈	Dic₂₅₄	1	1	yes	characteristic, complemented
73	M₁₆	—	2	1	no	—
74	M₁₆	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
78	C₄⋊C₄	—	508	1	no	—
79	C₄⋊C₄	—	254	2	no	—
80	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
81	C₁₆	—	254	2	no	—
82	Q₈	—	2	1	no	—
83	Q₈	—	1016	2	no	—
84	D₄	—	2	1	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	508	2	no	—
89	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	#14523236	1	1	yes	characteristic
92	C₈	#14523236	1	1	yes	characteristic
93	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
94	C₈	#14523248	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#16861234	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	4	1	no	—
99	C₄	—	1016	2	no	—
100	C₄	#16861188	1	1	yes	characteristic
101	C₄	C₂×#14523236	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	8	1	no	—
104	C₂	#22550166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550274

C₂

yes

maximal, characteristic, complemented

#22549700

C₂

yes

maximal, complemented

#22549697

C₂

yes

maximal, characteristic, complemented

#22549686

C₂

yes

maximal

#443×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

—

#16861149

C₂²

yes

—

#16861141

—

#16861141

C₂²

yes

—

#135×C₁₂₇

C₂²

yes

characteristic

#131×C₁₂₇

—

#108×C₁₂₇

C₂²

yes

characteristic

#104×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₁₂₇⋊C₁₆

—

#41×C₁₂₇

—

Q₁₆×C₁₂₇

Q₈

yes

characteristic, complemented

SD₁₆×C₁₂₇

—

D₈×C₁₂₇

Q₈

yes

characteristic, complemented

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

SD₁₆

yes

characteristic

C₈×C₁₂₇

SD₁₆

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#476

yes

characteristic

#2230

—

127

maximal, Sylow

C₁₂₇

#2230

yes

characteristic, complemented, Sylow

#498

—

127

—

#443

D₁₂₇

yes

characteristic

#365

—

127

—

#364

—

127

—

#360

—

127

—

#135

C₂×D₁₂₇

yes

characteristic

#131

—

C₄⋊Q₈

—

254

—

M₃₂

—

127

—

#108

C₂×D₁₂₇

yes

characteristic

#106

—

127

—

#106

—

254

—

#104

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

#41

—

C₂×Q₈

—

508

—

Q₁₆

Dic₂₅₄

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₂₅₄

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄²

C₂²⋊D₁₂₇

yes

characteristic

C₁₆

—

254

—

Q₈

—

Q₈

—

1016

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₂₅₄

yes

characteristic

C₈

#14523248

yes

characteristic

C₂²

—

C₂²

#16861234

yes

characteristic

C₄

—

C₄

—

C₄

—

1016

—

100

C₄

#16861188

yes

characteristic

101

C₄

C₂×#14523236

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#22550166

yes

characteristic

	subgroup	G/H
center	C₂	#22550166
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	#476
prime radical	#135×C₁₂₇	C₂²
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	#2230
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#22550166

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#443×C₁₂₇

C₂

socle

C₂×C₁₂₇

#476

prime radical

#135×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #22550274	C₂	semidirect
2. #22549700	C₂	semidirect
3. #22549697	C₂	semidirect
20. Q₁₆×C₁₂₇	Q₈	semidirect
22. D₈×C₁₂₇	Q₈	semidirect
53. C₁₂₇	#2230	semidirect
70. Q₁₆	Dic₂₅₄	semidirect
72. D₈	Dic₂₅₄	semidirect

H ≅ G/N

type

1. #22550274

C₂

semidirect

2. #22549700

C₂

semidirect

3. #22549697

C₂

semidirect

20. Q₁₆×C₁₂₇

Q₈

semidirect

22. D₈×C₁₂₇

Q₈

semidirect

53. C₁₂₇

#2230

semidirect

70. Q₁₆

Dic₂₅₄

semidirect

72. D₈

Dic₂₅₄

semidirect

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

subgroup	index
#22550274	2
#22549700	2
#22549697	2
#22549686	2
#443×C₁₂₇	2
#2230	127

subgroup

index

#443×C₁₂₇

127

Subgroups — #30640901

Conjugacy classes of subgroups

Complements

Maximal subgroups