Group #47982921 — subgroups

Subgroups — #47982921

Order 35456.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#31799960	C₂	1	1	yes	maximal, characteristic
2	#31799958	C₂	1	2	yes	maximal
3	#31799760	C₂	1	1	yes	maximal, characteristic
4	#31799758	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984547	C₂²	1	2	yes	—
7	C₂²×#14555660	C₂²	1	1	yes	characteristic
8	#22579839	C₂²	1	1	yes	characteristic
9	#22579791	C₄	1	4	yes	complemented
10	#22579790	C₄	1	2	yes	complemented
11	C₄×#14555660	C₄	1	4	yes	complemented
12	C₈×Dic₂₇₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₂₇₇	C₂²	1	1	yes	characteristic
14	#117×C₂₇₇	C₂²	1	2	yes	—
15	#17984547	—	2	2	no	—
16	#17984547	—	2	2	no	—
17	C₂×#14555660	C₂×C₄	1	4	yes	—
18	C₂×#14555660	C₂×C₄	1	1	yes	characteristic
19	C₂×#14555660	C₂×C₄	1	2	yes	—
20	C₂×#14555660	C₂×C₄	1	1	yes	characteristic
21	C₂×#14555660	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₂₇₇	—	2	1	no	—
23	C₂²⋊Dic₂₇₇	—	2	1	no	—
24	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₇₇	—	2	2	no	—
27	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₂₇₇	—	2	2	no	—
29	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
30	C₄²×C₂₇₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₂₇₇	—	2	2	no	—
32	#14555660	—	2	4	no	—
33	#14555660	C₄²	1	1	yes	characteristic
34	#14555660	C₄²	1	1	yes	characteristic
35	#14555660	C₄²	1	2	yes	—
36	C₂³×C₂₇₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₂₇₇	—	2	4	no	—
38	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₇	#41	1	2	yes	—
41	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
42	C₈×C₂₇₇	—	2	2	no	—
43	C₂²×C₂₇₇	—	4	1	no	—
44	C₂²×C₂₇₇	#117	1	2	yes	—
45	C₂²×C₂₇₇	#117	1	1	yes	characteristic
46	C₄×C₂₇₇	—	4	4	no	—
47	C₄×C₂₇₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₂₇₇	#117	1	1	yes	characteristic
49	C₄×C₂₇₇	#117	1	2	yes	—
50	Dic₂₇₇	—	4	2	no	—
51	C₂×C₂₇₇	—	2	1	no	—
52	C₂×C₂₇₇	—	2	2	no	—
53	C₂×C₂₇₇	#376	1	1	yes	characteristic
54	C₂₇₇	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	277	1	no	maximal, Sylow
56	#405	—	277	1	no	—
57	#405	—	277	2	no	—
58	#338	—	277	1	no	—
59	H(3,ℤ/4)	—	277	2	no	—
60	H(3,ℤ/4)	D₂₇₇	1	1	yes	characteristic
61	C₂×M₁₆	—	277	2	no	—
62	C₂×M₁₆	C₂×D₂₇₇	1	1	yes	characteristic
63	C₂²×C₈	—	277	1	no	—
64	#117	—	277	1	no	—
65	#117	C₂×D₂₇₇	1	2	yes	—
66	#97	—	277	2	no	—
67	#97	—	277	4	no	—
68	C₄×C₈	—	277	2	no	—
69	C₄×C₈	—	277	4	no	—
70	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	554	2	no	—
73	M₁₆	—	554	2	no	—
74	C₂×C₈	—	277	1	no	—
75	C₂×C₈	—	277	2	no	—
76	C₂×C₈	—	277	1	no	—
77	C₂×C₈	—	277	2	no	—
78	C₂×C₈	—	277	4	no	—
79	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	554	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	554	1	no	—
84	C₄²	—	277	2	no	—
85	C₄²	C₄×D₂₇₇	1	4	yes	—
86	C₂³	#17984562	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	554	2	no	—
89	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
91	C₂×C₄	#17984562	1	2	yes	—
92	C₂×C₄	#17984564	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	277	1	no	—
95	C₈	—	277	1	no	—
96	C₈	—	277	2	no	—
97	C₈	—	554	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#22579836	1	2	yes	—
100	C₂²	#22579838	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	1108	2	no	—
103	C₄	C₄²×D₂₇₇	1	1	yes	characteristic
104	C₄	#22579836	1	1	yes	characteristic
105	C₄	#22579838	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#31799847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31799960

C₂

yes

maximal, characteristic

#31799958

C₂

yes

maximal

#31799760

C₂

yes

maximal, characteristic

#31799758

C₂

yes

maximal

H(3,ℤ/4)×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

C₂²×#14555660

C₂²

yes

characteristic

#22579839

C₂²

yes

characteristic

#22579791

C₄

yes

complemented

#22579790

C₄

yes

complemented

C₄×#14555660

C₄

yes

complemented

C₈×Dic₂₇₇

C₄

yes

complemented

C₂×M₁₆×C₂₇₇

C₂²

yes

characteristic

#117×C₂₇₇

C₂²

yes

—

#17984547

—

#17984547

—

C₂×#14555660

C₂×C₄

yes

—

C₂×#14555660

C₂×C₄

yes

characteristic

C₂×#14555660

C₂×C₄

yes

—

C₂×#14555660

C₂×C₄

yes

characteristic

C₂×#14555660

C₂×C₄

yes

—

C₄⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₂×Dic₂₇₇

—

#14555660

—

#14555660

C₄²

yes

characteristic

#14555660

C₄²

yes

characteristic

#14555660

C₄²

yes

—

C₂³×C₂₇₇

#41

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂×C₄×C₂₇₇

#41

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#117

yes

—

C₂²×C₂₇₇

#117

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×C₄²

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#376

yes

characteristic

C₂₇₇

#1747

yes

characteristic, complemented, Sylow

#1747

—

277

maximal, Sylow

#405

—

277

—

#405

—

277

—

#338

—

277

—

H(3,ℤ/4)

—

277

—

H(3,ℤ/4)

D₂₇₇

yes

characteristic

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

#117

—

277

—

#117

C₂×D₂₇₇

yes

—

#97

—

277

—

#97

—

277

—

C₄×C₈

—

277

—

C₄×C₈

—

277

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

554

—

M₁₆

—

554

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₂²⋊C₄

—

C₂²⋊C₄

—

554

—

C₄²

—

277

—

C₄²

C₄×D₂₇₇

yes

—

C₂³

#17984562

yes

characteristic

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

C₂×C₄×D₂₇₇

yes

—

C₂×C₄

#17984562

yes

—

C₂×C₄

#17984564

yes

characteristic

C₈

—

C₈

—

277

—

C₈

—

277

—

C₈

—

277

—

C₈

—

554

—

C₂²

—

C₂²

#22579836

yes

—

100

C₂²

#22579838

yes

characteristic

101

C₄

—

102

C₄

—

1108

—

103

C₄

C₄²×D₂₇₇

yes

characteristic

104

C₄

#22579836

yes

characteristic

105

C₄

#22579838

yes

—

106

C₂

—

107

C₂

—

108

C₂

#31799847

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₂₇₇
commutator	C₄×C₂₇₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	H(3,ℤ/4)×C₂₇₇	C₂
socle	C₂×C₂₇₇	#376
prime radical	C₂³×C₂₇₇	#41
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	H(3,ℤ/4)	D₂₇₇
hypocenter	C₂₇₇	#1747
LCS	G ▷ C₄×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₂₇₇

commutator

C₄×C₂₇₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

H(3,ℤ/4)×C₂₇₇

C₂

C₂×C₂₇₇

C₂³×C₂₇₇

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₄×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #22579791	C₄	semidirect
10. #22579790	C₄	semidirect
11. C₄×#14555660	C₄	semidirect
12. C₈×Dic₂₇₇	C₄	semidirect
54. C₂₇₇	#1747	semidirect

H ≅ G/N

type

9. #22579791

C₄

semidirect

10. #22579790

C₄

semidirect

11. C₄×#14555660

C₄

semidirect

12. C₈×Dic₂₇₇

C₄

semidirect

54. C₂₇₇

#1747

semidirect

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

subgroup	index
#31799960	2
#31799958	2
#31799760	2
#31799758	2
H(3,ℤ/4)×C₂₇₇	2
#1747	277

subgroup

index

H(3,ℤ/4)×C₂₇₇

277

Subgroups — #47982921

Conjugacy classes of subgroups

Complements

Maximal subgroups