Group #47982831 — subgroups

Subgroups — #47982831

Order 35456.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31800379	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800378	C₂	1	2	yes	maximal, complemented
3	#31800350	C₂	1	2	yes	maximal, complemented
4	#31799723	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₂₁₆	—	2	2	no	—
7	C₂×#17984548	—	2	4	no	—
8	#22579877	C₂²	1	2	yes	—
9	#22579800	—	2	4	no	—
10	C₄×#14555660	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₇₇	—	2	2	no	—
12	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
13	#127×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₁₀₈	—	2	2	no	—
16	C₂×D₁₁₀₈	—	2	2	no	—
17	C₂×D₁₁₀₈	—	2	2	no	—
18	D₂₂₁₆	—	4	4	no	—
19	#17984548	—	4	8	no	—
20	C₂×#14555660	D₄	1	4	yes	—
21	C₂×Q₈×C₂₇₇	—	2	1	no	—
22	C₂×D₄×C₂₇₇	—	2	1	no	—
23	C₂×D₄×C₂₇₇	—	2	2	no	—
24	SD₁₆×C₂₇₇	—	4	4	no	—
25	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₇₇	—	2	2	no	—
27	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
28	C₂²×D₂₇₇	—	4	2	no	—
29	D₁₁₀₈	—	4	4	no	—
30	D₁₁₀₈	—	4	4	no	—
31	D₁₁₀₈	—	4	4	no	—
32	#14555660	—	2	4	no	—
33	C₂³×C₂₇₇	—	4	1	no	—
34	Q₈×C₂₇₇	—	4	2	no	—
35	D₄×C₂₇₇	—	4	2	no	—
36	D₄×C₂₇₇	—	4	4	no	—
37	C₂×C₄×C₂₇₇	—	4	1	no	—
38	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₇₇	D₈	1	4	yes	complemented
41	C₂²×C₂₇₇	—	8	1	no	—
42	C₂²×C₂₇₇	—	8	2	no	—
43	C₂²×C₂₇₇	#127	1	1	yes	characteristic
44	C₂×D₂₇₇	—	8	2	no	—
45	C₂×D₂₇₇	—	8	2	no	—
46	C₂×D₂₇₇	—	8	2	no	—
47	C₄×C₂₇₇	—	8	1	no	—
48	C₄×C₂₇₇	C₂×D₈	1	2	yes	—
49	C₄×C₂₇₇	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₂₇₇	—	16	1	no	—
51	C₂×C₂₇₇	#493	1	1	yes	characteristic
52	C₂×C₂₇₇	#496	1	2	yes	—
53	D₂₇₇	—	16	2	no	—
54	C₂₇₇	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	277	1	no	maximal, Sylow
56	#493	—	277	2	no	—
57	#493	—	277	1	no	—
58	#492	D₂₇₇	1	1	yes	characteristic, complemented
59	#486	—	277	2	no	—
60	#322	—	277	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	554	4	no	—
63	C₂×D₈	—	554	2	no	—
64	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
65	#127	—	277	2	no	—
66	#127	C₂×D₂₇₇	1	1	yes	characteristic
67	#102	—	554	4	no	—
68	C₄×C₈	—	277	2	no	—
69	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	554	2	no	—
74	C₂×D₄	—	554	2	no	—
75	C₂×D₄	—	554	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	1108	8	no	—
78	D₈	—	1108	4	no	—
79	C₂×C₈	—	277	4	no	—
80	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	1108	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	1108	4	no	—
89	D₄	—	1108	4	no	—
90	D₄	—	1108	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
94	C₈	—	554	4	no	—
95	C₈	#17984548	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	2216	2	no	—
99	C₂²	—	2216	2	no	—
100	C₂²	—	2216	2	no	—
101	C₂²	#22579876	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#17984548	1	2	yes	—
104	C₄	#22579907	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	4432	2	no	—
107	C₂	#31800378	1	1	yes	characteristic
108	C₂	#31800388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800379

C₂

yes

maximal, characteristic, complemented

#31800378

C₂

yes

maximal, complemented

#31800350

C₂

yes

maximal, complemented

#31799723

C₂

yes

maximal, characteristic, complemented

#492×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×D₂₂₁₆

—

C₂×#17984548

—

#22579877

C₂²

yes

—

#22579800

—

C₄×#14555660

C₂²

yes

—

C₂×SD₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

#127×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

D₂₂₁₆

—

#17984548

—

C₂×#14555660

D₄

yes

—

C₂×Q₈×C₂₇₇

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

SD₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂²×D₂₇₇

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

#14555660

—

C₂³×C₂₇₇

—

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

D₈

yes

complemented

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#127

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×D₈

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#493

yes

characteristic

C₂×C₂₇₇

#496

yes

—

D₂₇₇

—

C₂₇₇

#1721

yes

characteristic, complemented, Sylow

#1721

—

277

maximal, Sylow

#493

—

277

—

#493

—

277

—

#492

D₂₇₇

yes

characteristic, complemented

#486

—

277

—

#322

—

277

—

C₂×SD₁₆

—

C₂×D₈

—

554

—

C₂×D₈

—

554

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#127

—

277

—

#127

C₂×D₂₇₇

yes

characteristic

#102

—

554

—

C₄×C₈

—

277

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

SD₁₆

—

D₈

—

1108

—

D₈

—

1108

—

C₂×C₈

—

277

—

C₂×C₈

C₂²⋊D₂₇₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

C₂³

—

1108

—

Q₈

—

D₄

—

D₄

—

D₄

—

1108

—

D₄

—

1108

—

D₄

—

1108

—

C₂×C₄

—

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₈

—

554

—

C₈

#17984548

yes

complemented

C₂²

—

C₂²

—

C₂²

—

2216

—

C₂²

—

2216

—

100

C₂²

—

2216

—

101

C₂²

#22579876

yes

characteristic

102

C₄

—

103

C₄

C₂×#17984548

yes

—

104

C₄

#22579907

yes

—

105

C₂

—

106

C₂

—

4432

—

107

C₂

#31800378

yes

characteristic

108

C₂

#31800388

yes

—

	subgroup	G/H
center	C₂²	#22579876
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#492×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#127
hypercenter	#492	D₂₇₇
hypocenter	C₂₇₇	#1721
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22579876

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#492×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31800379	C₂	semidirect
2. #31800378	C₂	semidirect
3. #31800350	C₂	semidirect
4. #31799723	C₂	semidirect
5. #492×C₂₇₇	C₂	semidirect
40. C₈×C₂₇₇	D₈	semidirect
54. C₂₇₇	#1721	semidirect
58. #492	D₂₇₇	semidirect
95. C₈	#17984548	semidirect

H ≅ G/N

type

1. #31800379

C₂

semidirect

2. #31800378

C₂

semidirect

3. #31800350

C₂

semidirect

4. #31799723

C₂

semidirect

5. #492×C₂₇₇

C₂

semidirect

40. C₈×C₂₇₇

D₈

semidirect

54. C₂₇₇

#1721

semidirect

58. #492

D₂₇₇

semidirect

95. C₈

#17984548

semidirect

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

subgroup	index
#31800379	2
#31800378	2
#31800350	2
#31799723	2
#492×C₂₇₇	2
#1721	277

subgroup

index

#492×C₂₇₇

277

Subgroups — #47982831

Conjugacy classes of subgroups

Complements

Maximal subgroups