Group #49093186 — subgroups

Subgroups — #49093186

Order 35968.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×C₄⋊#14555886	C₂	1	2	yes	maximal
2	#384×C₂₈₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14555886	—	2	2	no	—
4	C₄⋊#14555886	—	2	2	no	—
5	C₄⋊#14555886	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₂₈₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₈₁	C₄	1	2	yes	—
8	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
9	C₂×#14555886	—	2	4	no	—
10	C₂×#14555886	—	4	2	no	—
11	C₂×#14555886	—	4	2	no	—
12	C₂²×C₄×C₂₈₁	—	2	2	no	—
13	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
15	C₄⋊C₄×C₂₈₁	—	4	2	no	—
16	C₄⋊C₄×C₂₈₁	—	4	2	no	—
17	C₄⋊C₄×C₂₈₁	—	4	2	no	—
18	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
19	C₄²×C₂₈₁	D₄	1	1	yes	characteristic
20	C₄²×C₂₈₁	Q₈	1	1	yes	characteristic
21	#14555886	—	4	4	no	—
22	C₂³×C₂₈₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₈₁	—	2	1	no	—
24	C₂×C₄×C₂₈₁	—	2	1	no	—
25	C₂×C₄×C₂₈₁	—	4	2	no	—
26	C₂×C₄×C₂₈₁	—	4	4	no	—
27	C₂×C₄×C₂₈₁	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₈₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₈₁	D₈	1	2	yes	—
32	C₂×C₄×C₂₈₁	SD₁₆	1	2	yes	—
33	C₂×C₄×C₂₈₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₂₈₁	Q₁₆	1	2	yes	—
35	C₂²×C₂₈₁	#95	1	1	yes	characteristic
36	C₂²×C₂₈₁	#100	1	1	yes	characteristic
37	C₂²×C₂₈₁	#101	1	1	yes	characteristic
38	C₂²×C₂₈₁	#102	1	2	yes	—
39	C₂²×C₂₈₁	#103	1	2	yes	—
40	C₄×C₂₈₁	—	2	1	no	—
41	C₄×C₂₈₁	—	2	1	no	—
42	C₄×C₂₈₁	—	8	2	no	—
43	C₄×C₂₈₁	#102	1	2	yes	—
44	C₄×C₂₈₁	#103	1	2	yes	—
45	C₄×C₂₈₁	#106	1	2	yes	—
46	C₄×C₂₈₁	#107	1	2	yes	—
47	C₂×C₂₈₁	#331	1	1	yes	characteristic
48	C₂×C₂₈₁	#332	1	2	yes	—
49	C₂×C₂₈₁	#333	1	1	yes	characteristic
50	C₂×C₂₈₁	#340	1	2	yes	—
51	C₂×C₂₈₁	#343	1	1	yes	characteristic
52	C₂₈₁	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	281	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	281	2	no	—
55	#384	D₂₈₁	1	1	yes	characteristic
56	C₂²×C₈	—	562	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₂₈₁	1	2	yes	—
59	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
60	C₄⋊C₈	—	281	4	no	—
61	C₄⋊C₈	—	562	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
65	C₂×C₈	—	562	4	no	—
66	C₂×C₈	—	1124	2	no	—
67	C₂×C₈	—	1124	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₂₈₁	1	2	yes	—
72	C₄²	D₁₁₂₄	1	1	yes	characteristic
73	C₄²	Dic₅₆₂	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₂₈₁	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₂₈₁	1	1	yes	characteristic
80	C₂×C₄	#17985115	1	2	yes	—
81	C₂×C₄	#17985117	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₂₈₁	1	1	yes	characteristic
83	C₂×C₄	#17985123	1	2	yes	—
84	C₂×C₄	#17985125	1	2	yes	—
85	C₂×C₄	#17985126	1	2	yes	—
86	C₂×C₄	#17985128	1	2	yes	—
87	C₈	—	1124	4	no	—
88	C₂²	#22582786	1	1	yes	characteristic
89	C₂²	#22582796	1	1	yes	characteristic
90	C₂²	#22582798	1	1	yes	characteristic
91	C₂²	#22582800	1	2	yes	—
92	C₂²	#22582803	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#22582799	1	2	yes	—
97	C₄	#22582802	1	2	yes	—
98	C₄	#22582810	1	2	yes	—
99	C₄	#22582812	1	2	yes	—
100	C₂	#31823207	1	1	yes	characteristic
101	C₂	#31823209	1	2	yes	—
102	C₂	#31823212	1	1	yes	characteristic
103	C₂	#31823225	1	2	yes	—
104	C₂	#31823232	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14555886

C₂

yes

maximal

#384×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

—

C₄⋊#14555886

—

C₄⋊#14555886

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₈₁

—

C₂×C₄⋊C₄×C₂₈₁

C₄

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

—

—

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

D₄

yes

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

characteristic

C₄²×C₂₈₁

Q₈

yes

characteristic

#14555886

—

C₂³×C₂₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₄²

yes

characteristic

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

D₈

yes

—

C₂×C₄×C₂₈₁

SD₁₆

yes

—

C₂×C₄×C₂₈₁

SD₁₆

yes

—

C₂×C₄×C₂₈₁

Q₁₆

yes

—

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

—

C₂²×C₂₈₁

#103

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#102

yes

—

C₄×C₂₈₁

#103

yes

—

C₄×C₂₈₁

#106

yes

—

C₄×C₂₈₁

#107

yes

—

C₂×C₂₈₁

#331

yes

characteristic

C₂×C₂₈₁

#332

yes

—

C₂×C₂₈₁

#333

yes

characteristic

C₂×C₂₈₁

#340

yes

—

C₂×C₂₈₁

#343

yes

characteristic

C₂₈₁

#1296

yes

characteristic, complemented, Sylow

#1296

—

281

maximal, Sylow

C₂×C₄⋊C₈

—

281

—

#384

D₂₈₁

yes

characteristic

C₂²×C₈

—

562

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₂₈₁

yes

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

C₄⋊C₈

—

281

—

C₄⋊C₈

—

562

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₈₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₈₁

yes

—

C₂×C₈

—

562

—

C₂×C₈

—

1124

—

C₂×C₈

—

1124

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₂₈₁

yes

—

C₄²

D₁₁₂₄

yes

characteristic

C₄²

Dic₅₆₂

yes

characteristic

C₂³

C₂²⋊Dic₂₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₈₁

yes

characteristic

C₂×C₄

#17985115

yes

—

C₂×C₄

#17985117

yes

—

C₂×C₄

C₄⋊Dic₂₈₁

yes

characteristic

C₂×C₄

#17985123

yes

—

C₂×C₄

#17985125

yes

—

C₂×C₄

#17985126

yes

—

C₂×C₄

#17985128

yes

—

C₈

—

1124

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#22582803

yes

—

C₄

—

C₄

—

C₄

—

C₄

#22582799

yes

—

C₄

#22582802

yes

—

C₄

#22582810

yes

—

C₄

#22582812

yes

—

100

C₂

#31823207

yes

characteristic

101

C₂

#31823209

yes

—

102

C₂

#31823212

yes

characteristic

103

C₂

#31823225

yes

—

104

C₂

#31823232

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₂₈₁
commutator	C₂×C₄×C₂₈₁	C₄²
Frattini	C₂×C₄²	C₂×D₂₈₁
Fitting	#384×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₈₁	C₂²
hypercenter	#384	D₂₈₁
hypocenter	C₂₈₁	#1296
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₂₈₁

commutator

C₂×C₄×C₂₈₁

C₄²

Frattini

C₂×C₄²

C₂×D₂₈₁

Fitting

#384×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂²⋊C₄

cosocle

C₂×C₄²×C₂₈₁

C₂²

D₂₈₁

C₂₈₁

G ▷ C₂×C₄×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#14555886	C₄	semidirect
52. C₂₈₁	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#14555886

C₄

semidirect

52. C₂₈₁

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#14555886	2
#384×C₂₈₁	2
#1296	281

subgroup

index

C₂×C₄⋊#14555886

#384×C₂₈₁

#1296

281

Subgroups — #49093186

Conjugacy classes of subgroups

Complements

Maximal subgroups