Group #47983302 — subgroups

Subgroups — #47983302

Order 35456.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22579791	C₂	1	1	yes	maximal, characteristic
2	#31799868	C₂	1	1	yes	maximal, characteristic
3	#31799764	C₂	1	4	yes	maximal
4	#384×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14555660	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
7	#22579791	C₄	1	4	yes	complemented
8	#22579787	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₇₇	—	2	2	no	—
13	C₂×#14555660	—	2	2	no	—
14	C₂×#14555660	—	2	2	no	—
15	C₂×#14555660	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₂₇₇	—	2	4	no	—
17	C₂²×C₄×C₂₇₇	—	2	2	no	—
18	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
19	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₇₇	—	2	4	no	—
21	C₄⋊C₄×C₂₇₇	—	4	4	no	—
22	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
23	C₄²×C₂₇₇	Q₈	1	2	yes	—
24	C₂×Dic₂₇₇	—	4	2	no	—
25	C₂×Dic₂₇₇	—	4	2	no	—
26	C₂×Dic₂₇₇	—	4	2	no	—
27	#14555660	—	2	4	no	—
28	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₇₇	—	2	2	no	—
30	C₂×C₄×C₂₇₇	—	2	2	no	—
31	C₂×C₄×C₂₇₇	—	4	2	no	—
32	C₂×C₄×C₂₇₇	—	4	4	no	—
33	C₂×C₄×C₂₇₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₇₇	#41	1	2	yes	—
37	C₂×C₄×C₂₇₇	#41	1	2	yes	—
38	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₇₇	#124	1	1	yes	characteristic
40	C₂²×C₂₇₇	#124	1	1	yes	characteristic
41	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₂₇₇	#137	1	2	yes	—
43	C₄×C₂₇₇	—	2	4	no	—
44	C₄×C₂₇₇	—	8	2	no	—
45	C₄×C₂₇₇	#117	1	2	yes	—
46	C₄×C₂₇₇	#125	1	2	yes	—
47	Dic₂₇₇	—	8	2	no	—
48	C₂×C₂₇₇	#382	1	1	yes	characteristic
49	C₂×C₂₇₇	#428	1	2	yes	—
50	C₂×C₂₇₇	#489	1	2	yes	—
51	C₂×C₂₇₇	#490	1	1	yes	characteristic
52	C₂×C₂₇₇	#491	1	1	yes	characteristic
53	C₂₇₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	277	1	no	maximal, Sylow
55	C₂×#97	—	277	1	no	—
56	#384	D₂₇₇	1	1	yes	characteristic
57	#382	—	277	1	no	—
58	#340	—	277	4	no	—
59	C₂²×C₈	—	277	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	277	2	no	—
62	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
63	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
64	#97	—	277	4	no	—
65	#95	—	554	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	554	2	no	—
68	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
70	C₂×C₈	—	277	4	no	—
71	C₂×C₈	—	554	2	no	—
72	C₂×C₈	—	554	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	554	4	no	—
76	C₄²	C₄×D₂₇₇	1	2	yes	—
77	C₄²	Dic₅₅₄	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	1108	2	no	—
84	C₂×C₄	—	1108	2	no	—
85	C₂×C₄	—	1108	2	no	—
86	C₂×C₄	#17984542	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
89	C₂×C₄	#17984562	1	2	yes	—
90	C₂×C₄	#17984564	1	2	yes	—
91	C₈	—	554	4	no	—
92	C₂²	C₂×#17984542	1	1	yes	characteristic
93	C₂²	#22579867	1	1	yes	characteristic
94	C₂²	#22579869	1	1	yes	characteristic
95	C₂²	#22579906	1	2	yes	—
96	C₂²	#22579911	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	2216	2	no	—
100	C₄	#22579838	1	2	yes	—
101	C₄	#22579871	1	2	yes	—
102	C₂	#31799867	1	1	yes	characteristic
103	C₂	#31800042	1	2	yes	—
104	C₂	#31800364	1	2	yes	—
105	C₂	#31800366	1	1	yes	characteristic
106	C₂	#31800370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579791

C₂

yes

maximal, characteristic

#31799868

C₂

yes

maximal, characteristic

#31799764

C₂

yes

maximal

#384×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

#22579791

C₄

yes

complemented

#22579787

—

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂²×Dic₂₇₇

—

C₂×#14555660

—

C₂×#14555660

—

C₂×#14555660

C₂×C₄

yes

—

C₄⋊Dic₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

Q₈

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₂²×C₂₇₇

#137

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#117

yes

—

C₄×C₂₇₇

#125

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#382

yes

characteristic

C₂×C₂₇₇

#428

yes

—

C₂×C₂₇₇

#489

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1836

—

277

maximal, Sylow

C₂×#97

—

277

—

#384

D₂₇₇

yes

characteristic

#382

—

277

—

#340

—

277

—

C₂²×C₈

—

277

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#97

—

277

—

#95

—

554

—

C₂²×C₄

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

277

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄²

C₄×D₂₇₇

yes

—

C₄²

Dic₅₅₄

yes

—

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1108

—

C₂×C₄

—

1108

—

C₂×C₄

—

1108

—

C₂×C₄

#17984542

yes

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

C₂×C₄

#17984562

yes

—

C₂×C₄

#17984564

yes

—

C₈

—

554

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#22579911

yes

—

C₄

—

C₄

—

C₄

—

2216

—

100

C₄

#22579838

yes

—

101

C₄

#22579871

yes

—

102

C₂

#31799867

yes

characteristic

103

C₂

#31800042

yes

—

104

C₂

#31800364

yes

—

105

C₂

#31800366

yes

characteristic

106

C₂

#31800370

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#384×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#384	D₂₇₇
hypocenter	C₂₇₇	#1836
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#384×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #22579791	C₄	semidirect
53. C₂₇₇	#1836	semidirect

H ≅ G/N

type

7. #22579791

C₄

semidirect

53. C₂₇₇

#1836

semidirect

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

subgroup	index
C₂×#22579791	2
#31799868	2
#31799764	2
#384×C₂₇₇	2
#1836	277

subgroup

index

#384×C₂₇₇

277

Subgroups — #47983302

Conjugacy classes of subgroups

Complements

Maximal subgroups