Group #47982305 — subgroups

Subgroups — #47982305

Order 35456.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31800732	C₂	1	1	yes	maximal, characteristic
2	#31800294	C₂	1	2	yes	maximal
3	#31799751	C₂	1	1	yes	maximal, characteristic
4	#31799734	C₂	1	2	yes	maximal
5	#494×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	#22579879	—	2	2	no	—
7	#22579878	—	2	2	no	—
8	C₄×Dic₅₅₄	C₂²	1	2	yes	—
9	#22579845	—	2	2	no	—
10	#22579813	—	2	2	no	—
11	C₄⋊#14555660	C₂²	1	2	yes	—
12	#22579803	—	2	2	no	—
13	C₂×Q₁₆×C₂₇₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₄	—	2	2	no	—
18	C₂×Dic₅₅₄	D₄	1	2	yes	—
19	C₂×#14555660	—	2	2	no	—
20	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
21	#17984542	—	2	4	no	—
22	#17984542	—	2	2	no	—
23	C₄×Dic₂₇₇	—	2	2	no	—
24	C₄×Dic₂₇₇	—	2	2	no	—
25	C₂×Q₈×C₂₇₇	—	2	1	no	—
26	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
27	Q₁₆×C₂₇₇	—	4	4	no	—
28	C₂×C₈×C₂₇₇	—	2	1	no	—
29	C₄⋊C₄×C₂₇₇	—	2	2	no	—
30	C₄⋊C₄×C₂₇₇	—	2	2	no	—
31	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
32	Dic₅₅₄	—	2	2	no	—
33	Dic₅₅₄	—	8	2	no	—
34	Dic₅₅₄	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₇₇	—	2	2	no	—
36	C₂×Dic₂₇₇	—	2	2	no	—
37	#14555660	—	4	2	no	—
38	Q₈×C₂₇₇	—	4	2	no	—
39	Q₈×C₂₇₇	—	4	2	no	—
40	C₂×C₄×C₂₇₇	—	2	2	no	—
41	C₂×C₄×C₂₇₇	—	4	1	no	—
42	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₇₇	—	2	2	no	—
45	C₂²×C₂₇₇	#120	1	1	yes	characteristic
46	C₄×C₂₇₇	—	2	1	no	—
47	C₄×C₂₇₇	—	4	2	no	—
48	C₄×C₂₇₇	—	8	1	no	—
49	C₄×C₂₇₇	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₇₇	#137	1	2	yes	—
51	Dic₂₇₇	—	2	4	no	—
52	Dic₂₇₇	—	4	2	no	—
53	C₂×C₂₇₇	#451	1	2	yes	—
54	C₂×C₂₇₇	#453	1	1	yes	characteristic
55	C₂₇₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	277	1	no	maximal, Sylow
57	Q₈²	—	277	1	no	—
58	#494	D₂₇₇	1	1	yes	characteristic
59	#477	—	277	2	no	—
60	#333	—	277	1	no	—
61	Q₈⋊C₈	—	277	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	554	2	no	—
64	C₄⋊Q₈	—	554	2	no	—
65	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
67	C₄×Q₈	—	277	2	no	—
68	C₄×Q₈	—	554	2	no	—
69	#107	—	554	2	no	—
70	C₄⋊C₈	—	277	2	no	—
71	#103	—	554	2	no	—
72	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	277	2	no	—
75	C₂×Q₈	—	554	2	no	—
76	C₂×Q₈	C₂²⋊D₂₇₇	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	554	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	277	2	no	—
83	C₄⋊C₄	—	554	4	no	—
84	C₄⋊C₄	—	554	2	no	—
85	C₄²	—	554	2	no	—
86	C₄²	—	554	2	no	—
87	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	277	4	no	—
91	Q₈	—	554	2	no	—
92	Q₈	—	2216	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	554	2	no	—
96	C₂×C₄	—	554	2	no	—
97	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	1108	2	no	—
101	C₂²	#22579848	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	554	4	no	—
106	C₄	—	1108	2	no	—
107	C₄	Q₁₆×D₂₇₇	1	2	yes	—
108	C₄	#22579912	1	2	yes	—
109	C₂	#31800156	1	2	yes	—
110	C₂	#31800169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800732

C₂

yes

maximal, characteristic

#31800294

C₂

yes

maximal

#31799751

C₂

yes

maximal, characteristic

#31799734

C₂

yes

maximal

#494×C₂₇₇

C₂

yes

maximal, characteristic

#22579879

—

#22579878

—

C₄×Dic₅₅₄

C₂²

yes

—

#22579845

—

#22579813

—

C₄⋊#14555660

C₂²

yes

—

#22579803

—

C₂×Q₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

D₄

yes

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

D₄

yes

—

#17984542

—

#17984542

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

—

Q₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

Dic₅₅₄

—

Dic₅₅₄

Q₁₆

yes

complemented

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×Q₁₆

yes

—

C₄×C₂₇₇

#137

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#451

yes

—

C₂×C₂₇₇

#453

yes

characteristic

C₂₇₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

277

maximal, Sylow

Q₈²

—

277

—

#494

D₂₇₇

yes

characteristic

#477

—

277

—

#333

—

277

—

Q₈⋊C₈

—

277

—

C₂×Q₁₆

—

C₄⋊Q₈

—

554

—

C₄⋊Q₈

—

554

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

C₄×Q₈

—

277

—

C₄×Q₈

—

554

—

#107

—

554

—

C₄⋊C₈

—

277

—

#103

—

554

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

C₂²⋊D₂₇₇

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

—

554

—

C₄²

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

277

—

Q₈

—

554

—

Q₈

—

2216

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

554

—

C₂×C₄

—

554

—

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₈

—

100

C₈

—

1108

—

101

C₂²

#22579848

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

554

—

106

C₄

—

1108

—

107

C₄

Q₁₆×D₂₇₇

yes

—

108

C₄

#22579912

yes

—

109

C₂

#31800156

yes

—

110

C₂

#31800169

yes

characteristic

	subgroup	G/H
center	C₂²	#22579848
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#494×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	#120
hypercenter	#494	D₂₇₇
hypocenter	C₂₇₇	#1633
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22579848

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#494×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₅₅₄	Q₁₆	semidirect
55. C₂₇₇	#1633	semidirect

H ≅ G/N

type

34. Dic₅₅₄

Q₁₆

semidirect

55. C₂₇₇

#1633

semidirect

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

subgroup	index
#31800732	2
#31800294	2
#31799751	2
#31799734	2
#494×C₂₇₇	2
#1633	277

subgroup

index

#494×C₂₇₇

277

Subgroups — #47982305

Conjugacy classes of subgroups

Complements

Maximal subgroups