Group #49094441 — subgroups

Subgroups — #49094441

Order 35968.

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	#31824193	C₂	1	1	yes	maximal, characteristic
2	#31823755	C₂	1	2	yes	maximal
3	#31823212	C₂	1	1	yes	maximal, characteristic
4	#31823195	C₂	1	2	yes	maximal
5	#494×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	#22582878	—	2	2	no	—
7	#22582877	—	2	2	no	—
8	C₄×Dic₅₆₂	C₂²	1	2	yes	—
9	#22582844	—	2	2	no	—
10	#22582812	—	2	2	no	—
11	C₄⋊#14555886	C₂²	1	2	yes	—
12	#22582802	—	2	2	no	—
13	C₂×Q₁₆×C₂₈₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₆₂	—	2	2	no	—
18	C₂×Dic₅₆₂	D₄	1	2	yes	—
19	C₂×#14555886	—	2	2	no	—
20	C₄⋊Dic₂₈₁	D₄	1	2	yes	—
21	#17985117	—	2	4	no	—
22	#17985117	—	2	2	no	—
23	C₄×Dic₂₈₁	—	2	2	no	—
24	C₄×Dic₂₈₁	—	2	2	no	—
25	C₂×Q₈×C₂₈₁	—	2	1	no	—
26	C₂×Q₈×C₂₈₁	D₄	1	2	yes	—
27	Q₁₆×C₂₈₁	—	4	4	no	—
28	C₂×C₈×C₂₈₁	—	2	1	no	—
29	C₄⋊C₄×C₂₈₁	—	2	2	no	—
30	C₄⋊C₄×C₂₈₁	—	2	2	no	—
31	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
32	Dic₅₆₂	—	2	2	no	—
33	Dic₅₆₂	—	8	2	no	—
34	Dic₅₆₂	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₈₁	—	2	2	no	—
36	C₂×Dic₂₈₁	—	2	2	no	—
37	#14555886	—	4	2	no	—
38	Q₈×C₂₈₁	—	4	2	no	—
39	Q₈×C₂₈₁	—	4	2	no	—
40	C₂×C₄×C₂₈₁	—	2	2	no	—
41	C₂×C₄×C₂₈₁	—	4	1	no	—
42	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₈₁	—	2	2	no	—
45	C₂²×C₂₈₁	#120	1	1	yes	characteristic
46	C₄×C₂₈₁	—	2	1	no	—
47	C₄×C₂₈₁	—	4	2	no	—
48	C₄×C₂₈₁	—	8	1	no	—
49	C₄×C₂₈₁	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₈₁	#137	1	2	yes	—
51	Dic₂₈₁	—	2	4	no	—
52	Dic₂₈₁	—	4	2	no	—
53	C₂×C₂₈₁	#451	1	2	yes	—
54	C₂×C₂₈₁	#453	1	1	yes	characteristic
55	C₂₈₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	281	1	no	maximal, Sylow
57	Q₈²	—	281	1	no	—
58	#494	D₂₈₁	1	1	yes	characteristic
59	#477	—	281	2	no	—
60	#333	—	281	1	no	—
61	Q₈⋊C₈	—	281	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	562	2	no	—
64	C₄⋊Q₈	—	562	2	no	—
65	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
67	C₄×Q₈	—	281	2	no	—
68	C₄×Q₈	—	562	2	no	—
69	#107	—	562	2	no	—
70	C₄⋊C₈	—	281	2	no	—
71	#103	—	562	2	no	—
72	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	281	2	no	—
75	C₂×Q₈	—	562	2	no	—
76	C₂×Q₈	C₂²⋊D₂₈₁	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	562	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	281	2	no	—
83	C₄⋊C₄	—	562	4	no	—
84	C₄⋊C₄	—	562	2	no	—
85	C₄²	—	562	2	no	—
86	C₄²	—	562	2	no	—
87	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	281	4	no	—
91	Q₈	—	562	2	no	—
92	Q₈	—	2248	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	562	2	no	—
96	C₂×C₄	—	562	2	no	—
97	C₂×C₄	D₄×D₂₈₁	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	1124	2	no	—
101	C₂²	#22582847	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	562	4	no	—
106	C₄	—	1124	2	no	—
107	C₄	Q₁₆×D₂₈₁	1	2	yes	—
108	C₄	#22582911	1	2	yes	—
109	C₂	#31823617	1	2	yes	—
110	C₂	#31823630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31824193

C₂

yes

maximal, characteristic

#31823755

C₂

yes

maximal

#31823212

C₂

yes

maximal, characteristic

#31823195

C₂

yes

maximal

#494×C₂₈₁

C₂

yes

maximal, characteristic

#22582878

—

#22582877

—

C₄×Dic₅₆₂

C₂²

yes

—

#22582844

—

#22582812

—

C₄⋊#14555886

C₂²

yes

—

#22582802

—

C₂×Q₁₆×C₂₈₁

—

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

D₄

yes

—

C₂×#14555886

—

C₄⋊Dic₂₈₁

D₄

yes

—

#17985117

—

#17985117

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

D₄

yes

—

Q₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

Dic₅₆₂

—

Dic₅₆₂

Q₁₆

yes

complemented

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

—

C₂²×C₂₈₁

#120

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×Q₁₆

yes

—

C₄×C₂₈₁

#137

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

#451

yes

—

C₂×C₂₈₁

#453

yes

characteristic

C₂₈₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

281

maximal, Sylow

Q₈²

—

281

—

#494

D₂₈₁

yes

characteristic

#477

—

281

—

#333

—

281

—

Q₈⋊C₈

—

281

—

C₂×Q₁₆

—

C₄⋊Q₈

—

562

—

C₄⋊Q₈

—

562

—

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

C₄×Q₈

—

281

—

C₄×Q₈

—

562

—

#107

—

562

—

C₄⋊C₈

—

281

—

#103

—

562

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

281

—

C₂×Q₈

—

562

—

C₂×Q₈

C₂²⋊D₂₈₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

562

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄²

—

562

—

C₄²

—

562

—

C₄²

C₂²×D₂₈₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

281

—

Q₈

—

562

—

Q₈

—

2248

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

—

562

—

C₂×C₄

D₄×D₂₈₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₈

—

100

C₈

—

1124

—

101

C₂²

#22582847

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

562

—

106

C₄

—

1124

—

107

C₄

Q₁₆×D₂₈₁

yes

—

108

C₄

#22582911

yes

—

109

C₂

#31823617

yes

—

110

C₂

#31823630

yes

characteristic

	subgroup	G/H
center	C₂²	#22582847
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting	#494×C₂₈₁	C₂
socle, prime radical	C₂²×C₂₈₁	#120
hypercenter	#494	D₂₈₁
hypocenter	C₂₈₁	#1633
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22582847

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting

#494×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₅₆₂	Q₁₆	semidirect
55. C₂₈₁	#1633	semidirect

H ≅ G/N

type

34. Dic₅₆₂

Q₁₆

semidirect

55. C₂₈₁

#1633

semidirect

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

subgroup	index
#31824193	2
#31823755	2
#31823212	2
#31823195	2
#494×C₂₈₁	2
#1633	281

subgroup

index

#494×C₂₈₁

281

Subgroups — #49094441

Conjugacy classes of subgroups

Complements

Maximal subgroups