Group #47982793 — subgroups

Subgroups — #47982793

Order 35456.

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

normal subgroups only

This group has 220 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	#31800400	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800374	C₂	1	2	yes	maximal, complemented
3	#31800348	C₂	1	2	yes	maximal, complemented
4	#31799723	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984551	—	2	4	no	—
7	#22579880	C₂²	1	2	yes	—
8	#22579812	—	2	2	no	—
9	#22579801	—	2	4	no	—
10	C₄×#14555660	C₂²	1	2	yes	—
11	C₂×D₈×C₂₇₇	—	2	2	no	—
12	#127×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×Dic₅₅₄	—	2	2	no	—
16	#17984551	—	4	8	no	—
17	C₂×#14555660	D₄	1	4	yes	—
18	C₄⋊Dic₂₇₇	—	2	2	no	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₂×D₄×C₂₇₇	—	2	1	no	—
21	C₂×D₄×C₂₇₇	—	2	1	no	—
22	C₂×D₄×C₂₇₇	—	2	2	no	—
23	C₂×D₄×C₂₇₇	—	2	2	no	—
24	D₈×C₂₇₇	—	4	4	no	—
25	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
26	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
27	Dic₅₅₄	—	4	4	no	—
28	C₂×Dic₂₇₇	—	4	2	no	—
29	#14555660	—	2	4	no	—
30	C₂³×C₂₇₇	—	4	1	no	—
31	C₂³×C₂₇₇	—	4	1	no	—
32	D₄×C₂₇₇	—	4	2	no	—
33	D₄×C₂₇₇	—	4	2	no	—
34	D₄×C₂₇₇	—	4	4	no	—
35	D₄×C₂₇₇	—	4	4	no	—
36	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₇₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₂₇₇	—	8	1	no	—
40	C₂²×C₂₇₇	—	8	1	no	—
41	C₂²×C₂₇₇	—	8	2	no	—
42	C₂²×C₂₇₇	—	8	2	no	—
43	C₂²×C₂₇₇	#127	1	1	yes	characteristic
44	C₄×C₂₇₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₂₇₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₂₇₇	—	8	2	no	—
47	C₂×C₂₇₇	—	16	1	no	—
48	C₂×C₂₇₇	—	16	1	no	—
49	C₂×C₂₇₇	#492	1	1	yes	characteristic
50	C₂×C₂₇₇	#496	1	2	yes	—
51	C₂₇₇	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	277	1	no	maximal, Sylow
53	#498	—	277	1	no	—
54	#493	D₂₇₇	1	1	yes	characteristic
55	#492	—	277	2	no	—
56	#486	—	277	2	no	—
57	#322	—	277	1	no	—
58	C₂×SD₁₆	—	554	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	277	2	no	—
61	#127	C₂×D₂₇₇	1	1	yes	characteristic
62	#127	C₂×D₂₇₇	1	1	yes	characteristic
63	#106	—	554	2	no	—
64	#102	—	554	4	no	—
65	C₄×C₈	—	277	2	no	—
66	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
67	C₂×Q₈	—	554	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	1108	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	277	4	no	—
75	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
76	C₄⋊C₄	—	554	2	no	—
77	C₄⋊C₄	—	554	2	no	—
78	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	1108	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	1108	2	no	—
87	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
89	C₈	—	554	4	no	—
90	C₈	#17984551	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#22579876	1	1	yes	characteristic
96	C₄	—	2216	2	no	—
97	C₄	C₂×#17984551	1	2	yes	—
98	C₄	#22579904	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#31800374	1	1	yes	characteristic
102	C₂	#31800390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800400

C₂

yes

maximal, characteristic, complemented

#31800374

C₂

yes

maximal, complemented

#31800348

C₂

yes

maximal, complemented

#31799723

C₂

yes

maximal, characteristic, complemented

#493×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984551

—

#22579880

C₂²

yes

—

#22579812

—

#22579801

—

C₄×#14555660

C₂²

yes

—

C₂×D₈×C₂₇₇

—

#127×C₂₇₇

C₂²

yes

characteristic

#127×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

#17984551

—

C₂×#14555660

D₄

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

D₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

SD₁₆

yes

complemented

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#127

yes

characteristic

C₄×C₂₇₇

C₂×SD₁₆

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#492

yes

characteristic

C₂×C₂₇₇

#496

yes

—

C₂₇₇

#1715

yes

characteristic, complemented, Sylow

#1715

—

277

maximal, Sylow

#498

—

277

—

#493

D₂₇₇

yes

characteristic

#492

—

277

—

#486

—

277

—

#322

—

277

—

C₂×SD₁₆

—

554

—

C₂×D₈

—

C₄⋊Q₈

—

277

—

#127

C₂×D₂₇₇

yes

characteristic

#127

C₂×D₂₇₇

yes

characteristic

#106

—

554

—

#102

—

554

—

C₄×C₈

—

277

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

554

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

1108

—

D₈

—

C₂×C₈

—

277

—

C₂×C₈

C₂²⋊D₂₇₇

yes

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

1108

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

1108

—

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₈

—

554

—

C₈

#17984551

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#22579876

yes

characteristic

C₄

—

2216

—

C₄

C₂×#17984551

yes

—

C₄

#22579904

yes

—

C₂

—

100

C₂

—

101

C₂

#31800374

yes

characteristic

102

C₂

#31800390

yes

—

	subgroup	G/H
center	C₂²	#22579876
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting, prime radical	#493×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#127
hypercenter	#493	D₂₇₇
hypocenter	C₂₇₇	#1715
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22579876

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting, prime radical

#493×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #31800400	C₂	semidirect
2. #31800374	C₂	semidirect
3. #31800348	C₂	semidirect
4. #31799723	C₂	semidirect
38. C₈×C₂₇₇	SD₁₆	semidirect
51. C₂₇₇	#1715	semidirect
90. C₈	#17984551	semidirect

H ≅ G/N

type

1. #31800400

C₂

semidirect

2. #31800374

C₂

semidirect

3. #31800348

C₂

semidirect

4. #31799723

C₂

semidirect

38. C₈×C₂₇₇

SD₁₆

semidirect

51. C₂₇₇

#1715

semidirect

90. C₈

#17984551

semidirect

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

subgroup	index
#31800400	2
#31800374	2
#31800348	2
#31799723	2
#493×C₂₇₇	2
#1715	277

subgroup

index

#493×C₂₇₇

277

Subgroups — #47982793

Conjugacy classes of subgroups

Complements

Maximal subgroups