Group #30638356 — subgroups

Subgroups — #30638356

Order 16256.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22550248	C₂	1	2	yes	maximal, complemented
3	#22550222	C₂	1	2	yes	maximal, complemented
4	#22549597	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523236	—	2	4	no	—
7	#16861209	C₂²	1	2	yes	—
8	#16861141	—	2	2	no	—
9	#16861130	—	2	4	no	—
10	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×D₈×C₁₂₇	—	2	2	no	—
12	#127×C₁₂₇	C₂²	1	1	yes	characteristic
13	#127×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	2	no	—
16	#14523236	—	4	8	no	—
17	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₁₂₇	—	2	2	no	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	C₂×D₄×C₁₂₇	—	2	2	no	—
21	C₂×D₄×C₁₂₇	—	2	1	no	—
22	C₂×D₄×C₁₂₇	—	2	1	no	—
23	C₂×D₄×C₁₂₇	—	2	2	no	—
24	D₈×C₁₂₇	—	4	4	no	—
25	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	Dic₂₅₄	—	4	4	no	—
28	C₂×Dic₁₂₇	—	4	2	no	—
29	C₁₂₇⋊C₈	—	2	4	no	—
30	C₂³×C₁₂₇	—	4	1	no	—
31	C₂³×C₁₂₇	—	4	1	no	—
32	D₄×C₁₂₇	—	4	2	no	—
33	D₄×C₁₂₇	—	4	2	no	—
34	D₄×C₁₂₇	—	4	4	no	—
35	D₄×C₁₂₇	—	4	4	no	—
36	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₂₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₂₇	—	8	1	no	—
40	C₂²×C₁₂₇	—	8	1	no	—
41	C₂²×C₁₂₇	—	8	2	no	—
42	C₂²×C₁₂₇	—	8	2	no	—
43	C₂²×C₁₂₇	#127	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₂₇	—	8	2	no	—
47	C₂×C₁₂₇	—	16	1	no	—
48	C₂×C₁₂₇	—	16	1	no	—
49	C₂×C₁₂₇	#492	1	1	yes	characteristic
50	C₂×C₁₂₇	#496	1	2	yes	—
51	#1715	—	127	1	no	maximal, Sylow
52	C₁₂₇	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	—	127	1	no	—
54	#493	D₁₂₇	1	1	yes	characteristic
55	#492	—	127	2	no	—
56	#486	—	127	2	no	—
57	#322	—	127	1	no	—
58	C₂×SD₁₆	—	254	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	127	2	no	—
61	#127	C₂×D₁₂₇	1	1	yes	characteristic
62	#127	C₂×D₁₂₇	1	1	yes	characteristic
63	#106	—	254	2	no	—
64	#102	—	254	4	no	—
65	C₄×C₈	—	127	2	no	—
66	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂×Q₈	—	254	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	508	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	127	4	no	—
75	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
76	C₄⋊C₄	—	254	2	no	—
77	C₄⋊C₄	—	254	2	no	—
78	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	508	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	508	2	no	—
87	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₈	—	254	4	no	—
90	C₈	#14523236	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#16861205	1	1	yes	characteristic
96	C₄	—	1016	2	no	—
97	C₄	C₂×#14523236	1	2	yes	—
98	C₄	#16861233	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#22550248	1	1	yes	characteristic
102	C₂	#22550264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550274

C₂

yes

maximal, characteristic, complemented

#22550248

C₂

yes

maximal, complemented

#22550222

C₂

yes

maximal, complemented

#22549597

C₂

yes

maximal, characteristic, complemented

#493×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523236

—

#16861209

C₂²

yes

—

#16861141

—

#16861130

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂×D₈×C₁₂₇

—

#127×C₁₂₇

C₂²

yes

characteristic

#127×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

#14523236

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#127

yes

characteristic

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#492

yes

characteristic

C₂×C₁₂₇

#496

yes

—

#1715

—

127

maximal, Sylow

C₁₂₇

#1715

yes

characteristic, complemented, Sylow

#498

—

127

—

#493

D₁₂₇

yes

characteristic

#492

—

127

—

#486

—

127

—

#322

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

C₄⋊Q₈

—

127

—

#127

C₂×D₁₂₇

yes

characteristic

#127

C₂×D₁₂₇

yes

characteristic

#106

—

254

—

#102

—

254

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

508

—

D₈

—

C₂×C₈

—

127

—

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

508

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

508

—

C₂×C₄

D₄×D₁₂₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

#14523236

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#16861205

yes

characteristic

C₄

—

1016

—

C₄

C₂×#14523236

yes

—

C₄

#16861233

yes

—

C₂

—

100

C₂

—

101

C₂

#22550248

yes

characteristic

102

C₂

#22550264

yes

—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting, prime radical	#493×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#1715
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16861205

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting, prime radical

#493×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22550274	C₂	semidirect
2. #22550248	C₂	semidirect
3. #22550222	C₂	semidirect
4. #22549597	C₂	semidirect
38. C₈×C₁₂₇	SD₁₆	semidirect
52. C₁₂₇	#1715	semidirect
90. C₈	#14523236	semidirect

H ≅ G/N

type

1. #22550274

C₂

semidirect

2. #22550248

C₂

semidirect

3. #22550222

C₂

semidirect

4. #22549597

C₂

semidirect

38. C₈×C₁₂₇

SD₁₆

semidirect

52. C₁₂₇

#1715

semidirect

90. C₈

#14523236

semidirect

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

subgroup	index
#22550274	2
#22550248	2
#22550222	2
#22549597	2
#493×C₁₂₇	2
#1715	127

subgroup

index

#493×C₁₂₇

127

Subgroups — #30638356

Conjugacy classes of subgroups

Complements

Maximal subgroups