Group #47982537 — subgroups

Subgroups — #47982537

Order 35456.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#31800382	C₂	1	1	yes	maximal, characteristic
2	#31800287	C₂	1	2	yes	maximal
3	#31799755	C₂	1	1	yes	maximal, characteristic
4	#31799749	C₂	1	2	yes	maximal
5	C₈×Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₁₀₈	—	2	1	no	—
7	C₂×Dic₁₁₀₈	—	2	1	no	—
8	#22579880	C₂²	1	2	yes	—
9	#22579811	—	2	2	no	—
10	C₄⋊#14555660	C₂²	1	2	yes	—
11	#22579804	—	2	2	no	—
12	C₄×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₇₇	—	2	1	no	—
14	C₄⋊C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₇₇	—	2	1	no	—
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₄	—	2	2	no	—
18	Dic₁₁₀₈	—	4	2	no	—
19	Dic₁₁₀₈	—	4	2	no	—
20	C₂×#14555660	—	2	2	no	—
21	C₄⋊Dic₂₇₇	—	2	2	no	—
22	C₄⋊Dic₂₇₇	—	2	2	no	—
23	C₂×Q₈×C₂₇₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₇₇	—	2	1	no	—
25	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₂₇₇	—	2	1	no	—
28	C₄⋊C₄×C₂₇₇	D₄	1	1	yes	characteristic
29	C₄²×C₂₇₇	—	2	1	no	—
30	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
31	Dic₅₅₄	—	4	2	no	—
32	Dic₅₅₄	—	4	2	no	—
33	C₂×Dic₂₇₇	—	4	2	no	—
34	#14555660	—	4	2	no	—
35	Q₈×C₂₇₇	—	2	1	no	—
36	Q₈×C₂₇₇	Q₁₆	1	2	yes	complemented
37	C₂×C₄×C₂₇₇	—	2	1	no	—
38	C₂×C₄×C₂₇₇	—	2	1	no	—
39	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
42	C₈×C₂₇₇	—	2	1	no	—
43	C₈×C₂₇₇	—	4	1	no	—
44	C₈×C₂₇₇	SD₁₆	1	2	yes	—
45	C₂²×C₂₇₇	#121	1	1	yes	characteristic
46	C₄×C₂₇₇	—	2	2	no	—
47	C₄×C₂₇₇	—	2	1	no	—
48	C₄×C₂₇₇	—	4	1	no	—
49	C₄×C₂₇₇	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₂₇₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₂₇₇	#135	1	1	yes	characteristic
52	C₄×C₂₇₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₂₇₇	—	8	2	no	—
54	C₂×C₂₇₇	#460	1	1	yes	characteristic
55	C₂×C₂₇₇	#467	1	1	yes	characteristic
56	C₂×C₂₇₇	#473	1	1	yes	characteristic
57	C₂₇₇	#1670	1	1	yes	characteristic, complemented, Sylow
58	#1670	—	277	1	no	maximal, Sylow
59	#494	—	277	1	no	—
60	#475	—	277	2	no	—
61	C₈×Q₈	D₂₇₇	1	1	yes	characteristic
62	#335	—	277	1	no	—
63	#332	—	277	2	no	—
64	C₂×Q₁₆	—	554	1	no	—
65	C₂×Q₁₆	—	554	1	no	—
66	C₄⋊Q₈	—	277	2	no	—
67	C₄×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
68	#106	—	554	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	277	2	no	—
71	C₄⋊C₈	C₂×D₂₇₇	1	1	yes	characteristic
72	#103	—	554	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
75	C₂×Q₈	—	554	2	no	—
76	C₂×Q₈	D₁₁₀₈	1	1	yes	characteristic
77	Q₁₆	—	1108	2	no	—
78	Q₁₆	—	1108	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	554	2	no	—
81	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	554	2	no	—
85	C₄⋊C₄	—	554	2	no	—
86	C₄⋊C₄	D₁₁₀₈	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	1108	2	no	—
91	Q₈	—	1108	2	no	—
92	Q₈	Dic₁₁₀₈	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	1108	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
97	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
98	C₂×C₄	#17984563	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	1108	2	no	—
102	C₈	#17984550	1	2	yes	—
103	C₂²	#22579854	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	2216	2	no	—
108	C₄	C₂×#17984550	1	1	yes	characteristic
109	C₄	C₂×Dic₁₁₀₈	1	1	yes	characteristic
110	C₄	#22579902	1	1	yes	characteristic
111	C₄	#22579905	1	1	yes	characteristic
112	C₂	#31800202	1	1	yes	characteristic
113	C₂	#31800243	1	1	yes	characteristic
114	C₂	#31800277	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800382

C₂

yes

maximal, characteristic

#31800287

C₂

yes

maximal

#31799755

C₂

yes

maximal, characteristic

#31799749

C₂

yes

maximal

C₈×Q₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂×Dic₁₁₀₈

—

C₂×Dic₁₁₀₈

—

#22579880

C₂²

yes

—

#22579811

—

C₄⋊#14555660

C₂²

yes

—

#22579804

—

C₄×Q₈×C₂₇₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₇₇

—

C₄⋊C₈×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

—

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

Dic₁₁₀₈

—

Dic₁₁₀₈

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

D₄

yes

characteristic

C₄²×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

#14555660

—

Q₈×C₂₇₇

—

Q₈×C₂₇₇

Q₁₆

yes

complemented

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

characteristic

C₈×C₂₇₇

—

C₈×C₂₇₇

—

C₈×C₂₇₇

SD₁₆

yes

—

C₂²×C₂₇₇

#121

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×SD₁₆

yes

characteristic

C₄×C₂₇₇

C₂×Q₁₆

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

Dic₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#1670

—

277

maximal, Sylow

#494

—

277

—

#475

—

277

—

C₈×Q₈

D₂₇₇

yes

characteristic

#335

—

277

—

#332

—

277

—

C₂×Q₁₆

—

554

—

C₂×Q₁₆

—

554

—

C₄⋊Q₈

—

277

—

C₄×Q₈

C₂×D₂₇₇

yes

characteristic

#106

—

554

—

C₄⋊C₈

—

C₄⋊C₈

—

277

—

C₄⋊C₈

C₂×D₂₇₇

yes

characteristic

#103

—

554

—

C₄×C₈

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

554

—

C₂×Q₈

D₁₁₀₈

yes

characteristic

Q₁₆

—

1108

—

Q₁₆

—

1108

—

C₂×C₈

—

C₂×C₈

—

554

—

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

D₁₁₀₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₂₇₇

yes

characteristic

Q₈

—

Q₈

—

1108

—

Q₈

—

1108

—

Q₈

Dic₁₁₀₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1108

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

C₂×C₄

#17984563

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

1108

—

102

C₈

#17984550

yes

—

103

C₂²

#22579854

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

2216

—

108

C₄

C₂×#17984550

yes

characteristic

109

C₄

C₂×Dic₁₁₀₈

yes

characteristic

110

C₄

#22579902

yes

characteristic

111

C₄

#22579905

yes

characteristic

112

C₂

#31800202

yes

characteristic

113

C₂

#31800243

yes

characteristic

114

C₂

#31800277

yes

characteristic

	subgroup	G/H
center	C₂²	#22579854
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	C₈×Q₈×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	#121
hypercenter	C₈×Q₈	D₂₇₇
hypocenter	C₂₇₇	#1670
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#22579854

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

C₈×Q₈×C₂₇₇

C₂

socle, prime radical

C₂²×C₂₇₇

#121

hypercenter

C₈×Q₈

D₂₇₇

hypocenter

C₂₇₇

#1670

LCS

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
36. Q₈×C₂₇₇	Q₁₆	semidirect
57. C₂₇₇	#1670	semidirect
92. Q₈	Dic₁₁₀₈	semidirect

H ≅ G/N

type

36. Q₈×C₂₇₇

Q₁₆

semidirect

57. C₂₇₇

#1670

semidirect

92. Q₈

Dic₁₁₀₈

semidirect

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

subgroup	index
#31800382	2
#31800287	2
#31799755	2
#31799749	2
C₈×Q₈×C₂₇₇	2
#1670	277

subgroup

index

C₈×Q₈×C₂₇₇

277

Subgroups — #47982537

Conjugacy classes of subgroups

Complements

Maximal subgroups