Group #47982496 — subgroups

Subgroups — #47982496

Order 35456.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂₇₇⋊#564	C₂	1	1	yes	maximal, characteristic
2	#31800411	C₂	1	2	yes	maximal
3	#31800372	C₂	1	1	yes	maximal, characteristic
4	#31799743	C₂	1	2	yes	maximal
5	#330×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	#22579880	C₂²	1	1	yes	characteristic
7	#22579872	C₂²	1	2	yes	—
8	#22579871	—	2	2	no	—
9	#22579870	—	2	2	no	—
10	#22579870	—	2	2	no	—
11	#22579814	—	2	2	no	—
12	#22579812	—	2	2	no	—
13	#22579803	—	2	2	no	—
14	#22579791	C₂²	1	1	yes	characteristic
15	#125×C₂₇₇	C₂²	1	1	yes	characteristic
16	#97×C₂₇₇	C₂²	1	2	yes	—
17	C₂×Dic₅₅₄	—	2	1	no	—
18	C₂×#14555660	—	2	1	no	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₄⋊Dic₂₇₇	—	2	2	no	—
21	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
22	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
23	#17984542	—	2	2	no	—
24	#17984542	—	2	2	no	—
25	#17984542	—	2	2	no	—
26	#17984542	—	2	2	no	—
27	C₄×Dic₂₇₇	—	2	2	no	—
28	C₂×C₈×C₂₇₇	—	2	2	no	—
29	C₄⋊C₄×C₂₇₇	—	2	2	no	—
30	C₄⋊C₄×C₂₇₇	D₄	1	2	yes	—
31	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
32	Dic₅₅₄	—	4	2	no	—
33	C₂×Dic₂₇₇	—	2	2	no	—
34	C₂×Dic₂₇₇	—	2	2	no	—
35	C₂×Dic₂₇₇	—	4	1	no	—
36	#14555660	—	4	1	no	—
37	C₂×C₄×C₂₇₇	—	2	2	no	—
38	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₇₇	—	2	4	no	—
41	C₂²×C₂₇₇	#120	1	1	yes	characteristic
42	C₄×C₂₇₇	—	2	1	no	—
43	C₄×C₂₇₇	—	2	2	no	—
44	C₄×C₂₇₇	—	4	2	no	—
45	Dic₂₇₇	—	4	2	no	—
46	Dic₂₇₇	—	4	2	no	—
47	Dic₂₇₇	—	8	1	no	—
48	C₂×C₂₇₇	#454	1	1	yes	characteristic
49	C₂×C₂₇₇	#456	1	2	yes	—
50	C₂₇₇	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	277	1	no	maximal, Sylow
52	#564	—	277	1	no	—
53	#501	—	277	2	no	—
54	#491	—	277	1	no	—
55	#330	—	277	2	no	—
56	#330	D₂₇₇	1	1	yes	characteristic
57	C₄⋊Q₈	—	277	1	no	—
58	#125	—	277	2	no	—
59	#125	—	554	2	no	—
60	#125	—	554	2	no	—
61	#125	—	554	2	no	—
62	#125	C₂×D₂₇₇	1	1	yes	characteristic
63	#107	—	554	2	no	—
64	#106	—	554	2	no	—
65	#103	—	554	2	no	—
66	#97	—	277	1	no	—
67	#97	C₂×D₂₇₇	1	2	yes	—
68	C₂×Q₈	—	554	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	554	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	277	2	no	—
73	C₄⋊C₄	—	277	2	no	—
74	C₄⋊C₄	—	554	2	no	—
75	C₄⋊C₄	—	554	2	no	—
76	C₄⋊C₄	—	554	2	no	—
77	C₄⋊C₄	—	554	2	no	—
78	C₄⋊C₄	—	554	2	no	—
79	C₄⋊C₄	—	554	2	no	—
80	C₄⋊C₄	D₁₁₀₈	1	2	yes	—
81	C₄²	—	554	2	no	—
82	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
83	Q₈	—	1108	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	554	2	no	—
86	C₂×C₄	—	554	2	no	—
87	C₂×C₄	—	1108	1	no	—
88	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
89	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	1108	1	no	—
92	C₂²	#22579850	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	1108	2	no	—
97	C₄	—	1108	2	no	—
98	C₄	—	2216	1	no	—
99	C₂	#31800177	1	1	yes	characteristic
100	C₂	#31800186	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₇₇⋊#564

C₂

yes

maximal, characteristic

#31800411

C₂

yes

maximal

#31800372

C₂

yes

maximal, characteristic

#31799743

C₂

yes

maximal

#330×C₂₇₇

C₂

yes

maximal, characteristic

#22579880

C₂²

yes

characteristic

#22579872

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₂₇₇

C₂²

yes

characteristic

#97×C₂₇₇

C₂²

yes

—

C₂×Dic₅₅₄

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

D₄

yes

—

C₄⋊Dic₂₇₇

D₄

yes

—

—

—

—

—

C₄×Dic₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#454

yes

characteristic

C₂×C₂₇₇

#456

yes

—

C₂₇₇

#1663

yes

characteristic, complemented, Sylow

#1663

—

277

maximal, Sylow

#564

—

277

—

#501

—

277

—

#491

—

277

—

#330

—

277

—

#330

D₂₇₇

yes

characteristic

C₄⋊Q₈

—

277

—

#125

—

277

—

#125

—

554

—

#125

—

554

—

#125

—

554

—

#125

C₂×D₂₇₇

yes

characteristic

#107

—

554

—

#106

—

554

—

#103

—

554

—

#97

—

277

—

#97

C₂×D₂₇₇

yes

—

C₂×Q₈

—

554

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

D₁₁₀₈

yes

—

C₄²

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

Q₈

—

1108

—

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

C₂×C₄

—

1108

—

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

C₈

—

C₈

—

1108

—

C₂²

#22579850

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

1108

—

C₄

—

1108

—

C₄

—

2216

—

C₂

#31800177

yes

characteristic

100

C₂

#31800186

yes

—

	subgroup	G/H
center	C₂²	#22579850
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#330×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	#120
hypercenter	#330	D₂₇₇
hypocenter	C₂₇₇	#1663
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#22579850

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#330×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
50. C₂₇₇	#1663	semidirect

H ≅ G/N

type

50. C₂₇₇

#1663

semidirect

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

subgroup	index
C₂₇₇⋊#564	2
#31800411	2
#31800372	2
#31799743	2
#330×C₂₇₇	2
#1663	277

subgroup

index

C₂₇₇⋊#564

#330×C₂₇₇

277

Subgroups — #47982496

Conjugacy classes of subgroups

Complements

Maximal subgroups