Group #47984160 — subgroups

Subgroups — #47984160

Order 35456.

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

normal subgroups only

This group has 228 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	C₂×#22579872	C₂	1	1	yes	maximal, characteristic
2	#31800026	C₂	1	1	yes	maximal, characteristic
3	#31799770	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#17984547	C₂²	1	2	yes	—
6	#22579872	—	2	1	no	—
7	#22579872	—	2	1	no	—
8	#22579872	—	2	2	no	—
9	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
10	C₂×#17984542	C₄	1	4	yes	—
11	C₄⋊#14555660	—	2	2	no	—
12	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
14	#97×C₂₇₇	—	2	1	no	—
15	C₄×C₈×C₂₇₇	—	2	1	no	—
16	C₂²×Dic₂₇₇	C₂×C₄	1	4	yes	—
17	#17984547	—	2	4	no	—
18	C₂×#14555660	—	2	2	no	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₄⋊Dic₂₇₇	—	2	2	no	—
21	#17984542	—	2	4	no	—
22	#17984542	—	2	4	no	—
23	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₇₇	—	2	2	no	—
26	M₁₆×C₂₇₇	Q₈	1	4	yes	—
27	C₂×C₈×C₂₇₇	—	2	2	no	—
28	C₄²×C₂₇₇	D₄	1	1	yes	characteristic
29	C₄²×C₂₇₇	D₄	1	1	yes	characteristic
30	C₄²×C₂₇₇	D₄	1	2	yes	—
31	C₂×Dic₂₇₇	—	2	4	no	—
32	C₂×Dic₂₇₇	—	4	4	no	—
33	#14555660	—	4	2	no	—
34	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₇₇	—	4	2	no	—
36	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₇₇	#41	1	2	yes	—
42	C₈×C₂₇₇	—	2	4	no	—
43	C₂²×C₂₇₇	—	2	2	no	—
44	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₇₇	C₄×Q₈	1	2	yes	—
46	C₄×C₂₇₇	—	2	2	no	—
47	C₄×C₂₇₇	—	2	2	no	—
48	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₇₇	#124	1	1	yes	characteristic
50	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₂₇₇	—	4	4	no	—
52	C₂×C₂₇₇	—	2	2	no	—
53	C₂×C₂₇₇	#391	1	1	yes	characteristic
54	C₂×C₂₇₇	#413	1	2	yes	—
55	C₂₇₇	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	277	1	no	maximal, Sylow
57	C₂×#125	—	277	1	no	—
58	#423	—	277	1	no	—
59	C₄×M₁₆	D₂₇₇	1	1	yes	characteristic
60	#343	—	277	4	no	—
61	C₂×M₁₆	—	277	2	no	—
62	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
63	#125	—	554	1	no	—
64	#125	—	554	1	no	—
65	#125	—	554	2	no	—
66	C₂×C₄⋊C₄	—	277	4	no	—
67	C₂×C₄⋊C₄	—	277	2	no	—
68	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
69	C₄⋊C₈	—	554	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	277	4	no	—
73	C₂²×C₄	C₄×D₂₇₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	554	4	no	—
77	M₁₆	Dic₅₅₄	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	554	2	no	—
80	C₄⋊C₄	—	554	4	no	—
81	C₄⋊C₄	—	554	4	no	—
82	C₄⋊C₄	—	554	2	no	—
83	C₄⋊C₄	—	554	2	no	—
84	C₄²	C₂²⋊D₂₇₇	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₇₇	1	1	yes	characteristic
86	C₄²	D₁₁₀₈	1	2	yes	—
87	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	554	4	no	—
90	C₂×C₄	—	1108	4	no	—
91	C₂×C₄	#17984540	1	2	yes	—
92	C₂×C₄	#17984540	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
94	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₅₅₄	1	2	yes	—
96	C₂×C₄	#17984563	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	1108	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#17984540	1	1	yes	characteristic
101	C₂²	C₄×Dic₅₅₄	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	1108	4	no	—
105	C₄	#22579859	1	2	yes	—
106	C₄	#22579868	1	1	yes	characteristic
107	C₄	#22579880	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#31799908	1	1	yes	characteristic
110	C₂	#31799984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579872

C₂

yes

maximal, characteristic

#31800026

C₂

yes

maximal, characteristic

#31799770

C₂

yes

maximal

C₄×M₁₆×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

#22579872

—

#22579872

—

#22579872

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×#17984542

C₄

yes

—

C₄⋊#14555660

—

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#97×C₂₇₇

—

C₄×C₈×C₂₇₇

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

#17984547

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

#17984542

—

#17984542

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

Q₈

yes

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

D₄

yes

characteristic

C₄²×C₂₇₇

D₄

yes

characteristic

C₄²×C₂₇₇

D₄

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₄×Q₈

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

#124

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#391

yes

characteristic

C₂×C₂₇₇

#413

yes

—

C₂₇₇

#1998

yes

characteristic, complemented, Sylow

#1998

—

277

maximal, Sylow

C₂×#125

—

277

—

#423

—

277

—

C₄×M₁₆

D₂₇₇

yes

characteristic

#343

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

#125

—

554

—

#125

—

554

—

#125

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

554

—

#97

—

C₄×C₈

—

C₂²×C₄

—

277

—

C₂²×C₄

C₄×D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

554

—

M₁₆

Dic₅₅₄

yes

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

C₂²⋊D₂₇₇

yes

characteristic

C₄²

C₂²⋊D₂₇₇

yes

characteristic

C₄²

D₁₁₀₈

yes

—

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

—

1108

—

C₂×C₄

#17984540

yes

—

C₂×C₄

#17984540

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

C₂×C₄

C₂×Dic₅₅₄

yes

—

C₂×C₄

#17984563

yes

—

C₈

—

C₈

—

1108

—

C₂²

—

100

C₂²

C₂×#17984540

yes

characteristic

101

C₂²

C₄×Dic₅₅₄

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

1108

—

105

C₄

#22579859

yes

—

106

C₄

#22579868

yes

characteristic

107

C₄

#22579880

yes

characteristic

108

C₂

—

109

C₂

#31799908

yes

characteristic

110

C₂

#31799984

yes

—

	subgroup	G/H
center	C₂²	C₂×#17984540
commutator, prime radical	C₂³×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₄×M₁₆×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₄×M₁₆	D₂₇₇
hypocenter	C₂₇₇	#1998
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#17984540

commutator, prime radical

C₂³×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₄×M₁₆×C₂₇₇

C₂

socle

C₂²×C₂₇₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

hypercenter

C₄×M₁₆

D₂₇₇

hypocenter

C₂₇₇

#1998

LCS

G ▷ C₂³×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₂₇₇	#1998	semidirect

H ≅ G/N

type

55. C₂₇₇

#1998

semidirect

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

subgroup	index
C₂×#22579872	2
#31800026	2
#31799770	2
C₄×M₁₆×C₂₇₇	2
#1998	277

subgroup

index

C₄×M₁₆×C₂₇₇

277

Subgroups — #47984160

Conjugacy classes of subgroups

Complements

Maximal subgroups