Group #47983955 — subgroups

Subgroups — #47983955

Order 35456.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#31800050	C₂	1	2	yes	maximal
2	C₂×#22579816	C₂	1	1	yes	maximal, characteristic
3	#31799957	C₂	1	1	yes	maximal, characteristic
4	#31799773	C₂	1	2	yes	maximal
5	C₂×#108×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984547	C₂²	1	2	yes	—
7	C₂×#17984547	C₂²	1	1	yes	characteristic
8	#22579839	C₂²	1	1	yes	characteristic
9	#22579816	C₄	1	4	yes	complemented
10	#22579815	—	2	4	no	—
11	#22579790	—	2	1	no	—
12	C₈×Dic₂₇₇	—	2	1	no	—
13	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	#108×C₂₇₇	—	2	1	no	—
16	#108×C₂₇₇	—	2	1	no	—
17	#17984547	—	2	1	no	—
18	#17984547	—	2	2	no	—
19	#17984547	C₂×C₄	1	4	yes	—
20	#17984547	D₄	1	2	yes	—
21	C₂×#14555660	—	2	1	no	—
22	C₂×#14555660	—	2	2	no	—
23	C₄⋊Dic₂₇₇	—	2	1	no	—
24	C₂²⋊Dic₂₇₇	Q₈	1	2	yes	—
25	C₄×Dic₂₇₇	—	2	1	no	—
26	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₇₇	—	2	2	no	—
28	M₁₆×C₂₇₇	—	2	2	no	—
29	C₂×C₈×C₂₇₇	—	2	2	no	—
30	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₇₇	Q₈	1	2	yes	—
34	C₂×Dic₂₇₇	—	2	2	no	—
35	#14555660	—	2	4	no	—
36	#14555660	—	2	2	no	—
37	C₂³×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
44	C₈×C₂₇₇	—	4	2	no	—
45	C₈×C₂₇₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₇₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₇₇	—	4	1	no	—
48	C₂²×C₂₇₇	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₂₇₇	C₄×D₄	1	1	yes	characteristic
53	C₄×C₂₇₇	#121	1	1	yes	characteristic
54	C₄×C₂₇₇	#125	1	1	yes	characteristic
55	Dic₂₇₇	—	4	2	no	—
56	C₂×C₂₇₇	—	2	1	no	—
57	C₂×C₂₇₇	—	2	1	no	—
58	C₂×C₂₇₇	—	2	1	no	—
59	C₂×C₂₇₇	#389	1	1	yes	characteristic
60	C₂₇₇	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	277	1	no	maximal, Sylow
62	#430	—	277	2	no	—
63	C₂×#108	—	277	1	no	—
64	C₂×#108	D₂₇₇	1	1	yes	characteristic
65	#405	—	277	1	no	—
66	#344	—	277	2	no	—
67	C₂×M₁₆	—	277	2	no	—
68	C₂×M₁₆	—	277	1	no	—
69	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
70	C₂²×C₈	C₂×D₂₇₇	1	1	yes	characteristic
71	#117	—	277	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	277	4	no	—
75	#108	—	554	4	no	—
76	#97	—	554	1	no	—
77	C₄×C₈	—	554	1	no	—
78	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	277	4	no	—
82	M₁₆	—	277	2	no	—
83	M₁₆	—	554	1	no	—
84	M₁₆	—	554	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	554	2	no	—
87	C₂×C₈	—	554	1	no	—
88	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
91	C₂×C₈	Dic₅₅₄	1	2	yes	—
92	C₄⋊C₄	—	554	1	no	—
93	C₂²⋊C₄	—	277	2	no	—
94	C₄²	—	554	1	no	—
95	C₂³	Q₈×D₂₇₇	1	1	yes	characteristic
96	C₂×C₄	—	554	2	no	—
97	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
98	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
101	C₂×C₄	#17984562	1	1	yes	characteristic
102	C₂×C₄	#17984564	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	554	4	no	—
105	C₈	—	554	2	no	—
106	C₈	#17984542	1	2	yes	—
107	C₈	#17984542	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#22579845	1	1	yes	characteristic
110	C₂²	#22579856	1	1	yes	characteristic
111	C₂²	#22579879	1	1	yes	characteristic
112	C₄	—	1108	2	no	—
113	C₄	C₂×#17984542	1	1	yes	characteristic
114	C₄	#22579842	1	1	yes	characteristic
115	C₄	#22579851	1	1	yes	characteristic
116	C₄	#22579871	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#31799898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800050

C₂

yes

maximal

C₂×#22579816

C₂

yes

maximal, characteristic

#31799957

C₂

yes

maximal, characteristic

#31799773

C₂

yes

maximal

C₂×#108×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

C₂×#17984547

C₂²

yes

characteristic

#22579839

C₂²

yes

characteristic

#22579816

C₄

yes

complemented

#22579815

—

#22579790

—

C₈×Dic₂₇₇

—

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

characteristic

#108×C₂₇₇

—

#108×C₂₇₇

—

#17984547

—

#17984547

—

#17984547

C₂×C₄

yes

—

#17984547

D₄

yes

—

C₂×#14555660

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

Q₈

yes

—

C₄×Dic₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

Q₈

yes

—

C₂×Dic₂₇₇

—

#14555660

—

#14555660

—

C₂³×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

C₂×C₄×C₂₇₇

yes

C₂×C₄×C₂₇₇

yes

C₈×C₂₇₇

—

C₈×C₂₇₇

C₄⋊C₄

yes

—

C₈×C₂₇₇

C₄⋊C₄

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₄×Q₈

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₇₇

C₄⋊Q₈

yes

characteristic

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₇

C₄×D₄

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#389

yes

characteristic

C₂₇₇

#1953

yes

characteristic, complemented, Sylow

#1953

—

277

maximal, Sylow

#430

—

277

—

C₂×#108

—

277

—

C₂×#108

D₂₇₇

yes

characteristic

#405

—

277

—

#344

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₂²×C₈

C₂×D₂₇₇

yes

characteristic

#117

—

277

—

#108

—

#108

—

#108

—

277

—

#108

—

554

—

#97

—

554

—

C₄×C₈

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

277

—

M₁₆

—

277

—

M₁₆

—

554

—

M₁₆

—

554

—

C₂×C₈

—

C₂×C₈

—

554

—

C₂×C₈

—

554

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₂×C₈

Dic₅₅₄

yes

—

C₄⋊C₄

—

554

—

C₂²⋊C₄

—

277

—

C₄²

—

554

—

C₂³

Q₈×D₂₇₇

yes

characteristic

C₂×C₄

—

554

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

D₄×D₂₇₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

100

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

101

C₂×C₄

#17984562

yes

characteristic

102

C₂×C₄

#17984564

yes

characteristic

103

C₈

—

104

C₈

—

554

—

105

C₈

—

554

—

106

C₈

#17984542

yes

—

107

C₈

#17984542

yes

—

108

C₂²

—

109

C₂²

#22579845

yes

characteristic

110

C₂²

#22579856

yes

characteristic

111

C₂²

#22579879

yes

characteristic

112

C₄

—

1108

—

113

C₄

C₂×#17984542

yes

characteristic

114

C₄

#22579842

yes

characteristic

115

C₄

#22579851

yes

characteristic

116

C₄

#22579871

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#31799898

yes

characteristic

	subgroup	G/H
center	C₄	#22579851
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₂×#108×C₂₇₇	C₂
socle	C₂×C₂₇₇	#389
prime radical	C₂³×C₂₇₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×#108	D₂₇₇
hypocenter	C₂₇₇	#1953
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#22579851

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₂×#108×C₂₇₇

C₂

socle

C₂×C₂₇₇

#389

prime radical

C₂³×C₂₇₇

C₂×Q₈

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22579816	C₄	semidirect
60. C₂₇₇	#1953	semidirect

H ≅ G/N

type

9. #22579816

C₄

semidirect

60. C₂₇₇

#1953

semidirect

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

subgroup	index
#31800050	2
C₂×#22579816	2
#31799957	2
#31799773	2
C₂×#108×C₂₇₇	2
#1953	277

subgroup

index

C₂×#108×C₂₇₇

277

Subgroups — #47983955

Conjugacy classes of subgroups

Complements

Maximal subgroups