Group #47983320 — subgroups

Subgroups — #47983320

Order 35456.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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⋊#14555660	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579809	C₂	1	4	yes	maximal
3	#31799848	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14555660	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
7	C₄⋊#14555660	—	2	2	no	—
8	#22579809	—	2	8	no	—
9	#22579787	C₄	1	2	yes	—
10	#22579787	C₄	1	2	yes	—
11	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₇₇	—	2	2	no	—
14	C₂²×Dic₂₇₇	C₂×C₄	1	4	yes	—
15	C₂×#14555660	—	2	4	no	—
16	C₂×#14555660	—	2	2	no	—
17	C₄×Dic₂₇₇	D₄	1	4	yes	—
18	C₄×Dic₂₇₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₇₇	—	2	2	no	—
23	C₂×C₈×C₂₇₇	—	2	4	no	—
24	C₄²×C₂₇₇	D₄	1	2	yes	—
25	C₄²×C₂₇₇	Q₈	1	2	yes	—
26	C₂×Dic₂₇₇	—	2	4	no	—
27	C₂×Dic₂₇₇	—	2	4	no	—
28	C₂×Dic₂₇₇	C₄⋊C₄	1	8	yes	—
29	#14555660	—	4	2	no	—
30	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₇₇	—	2	2	no	—
32	C₂×C₄×C₂₇₇	—	2	2	no	—
33	C₂×C₄×C₂₇₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₇₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₂₇₇	—	4	2	no	—
40	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₂₇₇	#131	1	2	yes	—
43	C₂²×C₂₇₇	#131	1	2	yes	—
44	C₄×C₂₇₇	—	2	2	no	—
45	C₄×C₂₇₇	—	2	2	no	—
46	C₄×C₂₇₇	#127	1	1	yes	characteristic
47	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₂₇₇	C₄⋊Q₈	1	2	yes	—
49	Dic₂₇₇	—	2	8	no	—
50	C₂×C₂₇₇	#384	1	1	yes	characteristic
51	C₂×C₂₇₇	#424	1	4	yes	—
52	C₂×C₂₇₇	#424	1	1	yes	characteristic
53	C₂×C₂₇₇	#424	1	1	yes	characteristic
54	C₂₇₇	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	277	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	277	4	no	—
57	C₂×C₄⋊C₈	—	277	1	no	—
58	C₂×C₄⋊C₈	D₂₇₇	1	1	yes	characteristic
59	#376	—	277	1	no	—
60	C₂²×C₈	—	277	2	no	—
61	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
62	C₂×C₄²	—	277	2	no	—
63	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	554	8	no	—
66	C₄⋊C₈	—	554	2	no	—
67	#95	—	277	2	no	—
68	#95	—	277	2	no	—
69	C₂²×C₄	—	277	4	no	—
70	C₂²×C₄	C₄×D₂₇₇	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₂₇₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	554	4	no	—
76	C₂×C₈	—	554	2	no	—
77	C₄²	—	277	4	no	—
78	C₄²	—	277	4	no	—
79	C₄²	C₂²⋊D₂₇₇	1	2	yes	—
80	C₄²	Dic₅₅₄	1	2	yes	—
81	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	277	8	no	—
85	C₂×C₄	—	554	4	no	—
86	C₂×C₄	—	554	4	no	—
87	C₂×C₄	#17984542	1	2	yes	—
88	C₂×C₄	#17984542	1	2	yes	—
89	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	1108	2	no	—
95	C₂²	C₄⋊C₄×D₂₇₇	1	2	yes	—
96	C₂²	C₂×#17984542	1	1	yes	characteristic
97	C₂²	#22579886	1	2	yes	—
98	C₂²	#22579888	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	554	8	no	—
102	C₄	#22579876	1	1	yes	characteristic
103	C₄	#22579878	1	1	yes	characteristic
104	C₄	#22579879	1	2	yes	—
105	C₂	#31799871	1	1	yes	characteristic
106	C₂	#31800027	1	4	yes	—
107	C₂	#31800028	1	1	yes	characteristic
108	C₂	#31800030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14555660

C₂

yes

maximal, characteristic

C₂×#22579809

C₂

yes

maximal

#31799848

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

C₄⋊#14555660

—

#22579809

—

#22579787

C₄

yes

—

#22579787

C₄

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₇₇

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂×#14555660

—

C₂×#14555660

—

C₄×Dic₂₇₇

D₄

yes

—

C₄×Dic₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

Q₈

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

#14555660

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₇

#131

yes

—

C₂²×C₂₇₇

#131

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#127

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#384

yes

characteristic

C₂×C₂₇₇

#424

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1843

—

277

maximal, Sylow

C₂×C₄⋊C₈

—

277

—

C₂×C₄⋊C₈

—

277

—

C₂×C₄⋊C₈

D₂₇₇

yes

characteristic

#376

—

277

—

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

554

—

C₄⋊C₈

—

554

—

#95

—

277

—

#95

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₄×D₂₇₇

yes

characteristic

C₂²×C₄

C₄×D₂₇₇

yes

characteristic

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₄²

—

277

—

C₄²

—

277

—

C₄²

C₂²⋊D₂₇₇

yes

—

C₄²

Dic₅₅₄

yes

—

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

277

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

C₂×C₄

#17984542

yes

—

C₂×C₄

#17984542

yes

—

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

Q₈×D₂₇₇

yes

—

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

C₈

—

C₈

—

1108

—

C₂²

C₄⋊C₄×D₂₇₇

yes

—

C₂²

C₂×#17984542

yes

characteristic

C₂²

#22579886

yes

—

C₂²

#22579888

yes

—

C₄

—

100

C₄

—

101

C₄

—

554

—

102

C₄

#22579876

yes

characteristic

103

C₄

#22579878

yes

characteristic

104

C₄

#22579879

yes

—

105

C₂

#31799871

yes

characteristic

106

C₂

#31800027

yes

—

107

C₂

#31800028

yes

characteristic

108

C₂

#31800030

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₇
commutator, socle, prime radical	C₂³×C₂₇₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₇₇	C₂
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₇₇
hypocenter	C₂₇₇	#1843
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₇₇

commutator, socle, prime radical

C₂³×C₂₇₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₇₇

C₂

cosocle

C₂²×C₄×C₂₇₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₂₇₇	#1843	semidirect

H ≅ G/N

type

54. C₂₇₇

#1843

semidirect

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

subgroup	index
C₂×C₄⋊#14555660	2
C₂×#22579809	2
#31799848	2
C₂×C₄⋊C₈×C₂₇₇	2
#1843	277

subgroup

index

C₂×C₄⋊C₈×C₂₇₇

277

Subgroups — #47983320

Conjugacy classes of subgroups

Complements

Maximal subgroups