Group #47982833 — subgroups

Subgroups — #47982833

Order 35456.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	2	yes	maximal, complemented
2	#31800379	C₂	1	1	yes	maximal, characteristic, complemented
3	#31800375	C₂	1	1	yes	maximal, characteristic, complemented
4	#31800349	C₂	1	2	yes	maximal, complemented
5	#322×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17984552	—	2	2	no	—
7	C₂×D₂₂₁₆	—	2	4	no	—
8	C₂×D₂₂₁₆	—	2	2	no	—
9	#22579880	C₂²	1	1	yes	characteristic
10	#22579877	C₂²	1	2	yes	—
11	#22579877	C₂²	1	1	yes	characteristic
12	#22579802	—	2	4	no	—
13	C₄×C₈×C₂₇₇	C₂²	1	2	yes	—
14	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₅₄	—	2	1	no	—
16	C₂×D₁₁₀₈	—	2	2	no	—
17	C₂×D₁₁₀₈	—	2	2	no	—
18	C₂×D₁₁₀₈	—	2	2	no	—
19	C₂×D₁₁₀₈	—	2	2	no	—
20	C₂×D₁₁₀₈	—	2	1	no	—
21	#17984552	—	4	4	no	—
22	D₂₂₁₆	—	4	8	no	—
23	D₂₂₁₆	—	4	4	no	—
24	C₄⋊Dic₂₇₇	—	2	2	no	—
25	C₂×C₈×C₂₇₇	D₄	1	4	yes	—
26	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
27	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
28	C₂²×D₂₇₇	—	4	2	no	—
29	C₂²×D₂₇₇	—	4	1	no	—
30	Dic₅₅₄	—	4	2	no	—
31	D₁₁₀₈	—	4	4	no	—
32	D₁₁₀₈	—	4	2	no	—
33	D₁₁₀₈	—	4	4	no	—
34	D₁₁₀₈	—	4	4	no	—
35	D₁₁₀₈	—	4	4	no	—
36	C₂×Dic₂₇₇	—	4	1	no	—
37	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
39	C₈×C₂₇₇	—	2	4	no	—
40	C₈×C₂₇₇	D₈	1	4	yes	complemented
41	C₂²×C₂₇₇	#127	1	1	yes	characteristic
42	C₂×D₂₇₇	—	8	2	no	—
43	C₂×D₂₇₇	—	8	1	no	—
44	C₂×D₂₇₇	—	8	2	no	—
45	C₂×D₂₇₇	—	8	2	no	—
46	C₂×D₂₇₇	—	8	2	no	—
47	C₄×C₂₇₇	C₂×D₈	1	2	yes	—
48	C₄×C₂₇₇	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₇₇	—	8	1	no	—
50	C₂×C₂₇₇	#493	1	1	yes	characteristic
51	C₂×C₂₇₇	#496	1	2	yes	—
52	D₂₇₇	—	16	2	no	—
53	D₂₇₇	—	16	1	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	—	277	1	no	—
59	#486	—	277	2	no	—
60	#322	D₂₇₇	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	554	2	no	—
62	C₂×D₈	—	554	4	no	—
63	C₂×D₈	—	554	2	no	—
64	C₄⋊Q₈	—	277	1	no	—
65	#127	—	277	2	no	—
66	#127	—	277	1	no	—
67	#102	—	554	4	no	—
68	C₄×C₈	C₂×D₂₇₇	1	2	yes	—
69	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
70	C₂×Q₈	—	554	1	no	—
71	C₂×D₄	—	554	2	no	—
72	C₂×D₄	—	554	2	no	—
73	C₂×D₄	—	554	2	no	—
74	C₂×D₄	—	554	2	no	—
75	C₂×D₄	—	554	1	no	—
76	SD₁₆	—	1108	4	no	—
77	D₈	—	1108	8	no	—
78	D₈	—	1108	4	no	—
79	C₂×C₈	D₁₁₀₈	1	4	yes	—
80	C₂×C₈	D₁₁₀₈	1	2	yes	—
81	C₄⋊C₄	—	554	2	no	—
82	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
83	C₂³	—	1108	2	no	—
84	C₂³	—	1108	1	no	—
85	Q₈	—	1108	2	no	—
86	D₄	—	1108	4	no	—
87	D₄	—	1108	2	no	—
88	D₄	—	1108	4	no	—
89	D₄	—	1108	4	no	—
90	D₄	—	1108	4	no	—
91	C₂×C₄	—	1108	1	no	—
92	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₁₀₈	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₂₂₁₆	1	4	yes	complemented
96	C₂²	—	2216	2	no	—
97	C₂²	—	2216	1	no	—
98	C₂²	—	2216	2	no	—
99	C₂²	—	2216	2	no	—
100	C₂²	—	2216	2	no	—
101	C₂²	#22579877	1	1	yes	characteristic
102	C₄	—	2216	1	no	—
103	C₄	C₂×D₂₂₁₆	1	2	yes	—
104	C₄	#22579908	1	4	yes	—
105	C₂	—	4432	2	no	—
106	C₂	—	4432	1	no	—
107	C₂	#31800379	1	1	yes	characteristic
108	C₂	#31800391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800379

C₂

yes

maximal, complemented

#31800379

C₂

yes

maximal, characteristic, complemented

#31800375

C₂

yes

maximal, characteristic, complemented

#31800349

C₂

yes

maximal, complemented

#322×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984552

—

C₂×D₂₂₁₆

—

C₂×D₂₂₁₆

—

#22579880

C₂²

yes

characteristic

#22579877

C₂²

yes

—

#22579877

C₂²

yes

characteristic

#22579802

—

C₄×C₈×C₂₇₇

C₂²

yes

—

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

#17984552

—

D₂₂₁₆

—

D₂₂₁₆

—

C₄⋊Dic₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

Dic₅₅₄

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₈×C₂₇₇

—

C₈×C₂₇₇

D₈

yes

complemented

C₂²×C₂₇₇

#127

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

C₂×D₈

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#493

yes

characteristic

C₂×C₂₇₇

#496

yes

—

D₂₇₇

—

D₂₇₇

—

C₂₇₇

#1721

yes

characteristic, complemented, Sylow

#1721

—

277

maximal, Sylow

#493

—

277

—

#493

—

277

—

#492

—

277

—

#486

—

277

—

#322

D₂₇₇

yes

characteristic, complemented

C₂×SD₁₆

—

554

—

C₂×D₈

—

554

—

C₂×D₈

—

554

—

C₄⋊Q₈

—

277

—

#127

—

277

—

#127

—

277

—

#102

—

554

—

C₄×C₈

C₂×D₂₇₇

yes

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

SD₁₆

—

1108

—

D₈

—

1108

—

D₈

—

1108

—

C₂×C₈

D₁₁₀₈

yes

—

C₂×C₈

D₁₁₀₈

yes

—

C₄⋊C₄

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

1108

—

C₂³

—

1108

—

Q₈

—

1108

—

D₄

—

1108

—

D₄

—

1108

—

D₄

—

1108

—

D₄

—

1108

—

D₄

—

1108

—

C₂×C₄

—

1108

—

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

C₂×C₄

C₂×D₁₁₀₈

yes

—

C₈

—

C₈

D₂₂₁₆

yes

complemented

C₂²

—

2216

—

C₂²

—

2216

—

C₂²

—

2216

—

C₂²

—

2216

—

100

C₂²

—

2216

—

101

C₂²

#22579877

yes

characteristic

102

C₄

—

2216

—

103

C₄

C₂×D₂₂₁₆

yes

—

104

C₄

#22579908

yes

—

105

C₂

—

4432

—

106

C₂

—

4432

—

107

C₂

#31800379

yes

characteristic

108

C₂

#31800391

yes

—

	subgroup	G/H
center	C₂²	#22579877
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#322×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#127
hypercenter	#322	D₂₇₇
hypocenter	C₂₇₇	#1721
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22579877

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#322×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #31800379	C₂	semidirect
2. #31800379	C₂	semidirect
3. #31800375	C₂	semidirect
4. #31800349	C₂	semidirect
5. #322×C₂₇₇	C₂	semidirect
40. C₈×C₂₇₇	D₈	semidirect
54. C₂₇₇	#1721	semidirect
60. #322	D₂₇₇	semidirect
95. C₈	D₂₂₁₆	semidirect

H ≅ G/N

type

1. #31800379

C₂

semidirect

2. #31800379

C₂

semidirect

3. #31800375

C₂

semidirect

4. #31800349

C₂

semidirect

5. #322×C₂₇₇

C₂

semidirect

40. C₈×C₂₇₇

D₈

semidirect

54. C₂₇₇

#1721

semidirect

60. #322

D₂₇₇

semidirect

95. C₈

D₂₂₁₆

semidirect

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

subgroup	index
#31800379	2
#31800375	2
#31800349	2
#322×C₂₇₇	2
#1721	277

subgroup

index

#322×C₂₇₇

277

Subgroups — #47982833

Conjugacy classes of subgroups

Complements

Maximal subgroups