Group #21461468 — subgroups

Subgroups — #21461468

Order 7808.

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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	#16840796	C₂	1	2	yes	maximal, complemented
2	#16840623	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840610	C₂	1	2	yes	maximal, complemented
4	#16840603	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839844	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758213	—	2	2	no	—
7	C₄⋊#92420	C₂²	1	2	yes	—
8	#14519834	—	2	2	no	—
9	C₄×#92420	C₂²	1	1	yes	characteristic
10	C₄×#92420	C₂²	1	1	yes	characteristic
11	C₂×D₄₈₈	C₄	1	1	yes	characteristic
12	#14519739	C₂²	1	2	yes	—
13	#14519697	C₄	1	2	yes	—
14	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	#107×C₆₁	C₄	1	1	yes	characteristic
16	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₆₁	—	2	2	no	—
18	C₂×#92420	—	2	2	no	—
19	C₂×#92420	—	2	1	no	—
20	C₂×#92420	D₄	1	2	yes	—
21	#11758213	—	4	4	no	—
22	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
24	#11758198	—	2	2	no	—
25	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×#92418	C₂×C₄	1	1	yes	characteristic
27	D₄₈₈	—	2	2	no	—
28	C₂²×D₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	—	2	2	no	—
30	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₆₁	#41	1	1	yes	characteristic
33	D₂₄₄	—	2	2	no	—
34	D₂₄₄	—	2	2	no	—
35	C₄×D₆₁	—	4	2	no	—
36	C₄×D₆₁	—	4	2	no	—
37	#92420	—	2	2	no	—
38	#92420	—	4	1	no	—
39	#92420	—	4	2	no	—
40	C₈×C₆₁	M₁₆	1	2	yes	complemented
41	#92418	—	2	1	no	—
42	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
43	C₂×D₆₁	—	4	2	no	—
44	C₂×D₆₁	—	4	2	no	—
45	C₂×D₆₁	—	4	2	no	—
46	C₄×C₆₁	—	4	2	no	—
47	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₆₁	#131	1	1	yes	characteristic
49	Dic₆₁	—	2	1	no	—
50	Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
51	#1594	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	#436	1	1	yes	characteristic
53	C₂×C₆₁	#442	1	1	yes	characteristic
54	C₂×C₆₁	#444	1	1	yes	characteristic
55	D₆₁	—	8	2	no	—
56	C₄×D₈	—	61	1	no	—
57	#436	—	61	2	no	—
58	#334	—	61	1	no	—
59	D₄⋊C₈	—	61	2	no	—
60	#322	—	61	1	no	—
61	C₆₁	#1594	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	61	1	no	—
63	C₂×M₁₆	—	122	2	no	—
64	C₄×D₄	—	61	2	no	—
65	#107	#6923	1	1	yes	characteristic
66	C₄⋊C₈	—	61	2	no	—
67	#102	—	61	2	no	—
68	C₂²⋊C₈	—	122	2	no	—
69	C₄×C₈	—	61	1	no	—
70	C₄×C₈	—	61	1	no	—
71	C₄×C₈	—	61	1	no	—
72	C₂×D₄	—	61	2	no	—
73	C₂²×C₄	—	122	2	no	—
74	D₈	—	122	2	no	—
75	M₁₆	—	244	4	no	—
76	C₂×C₈	—	61	2	no	—
77	C₂×C₈	—	61	1	no	—
78	C₂×C₈	—	122	1	no	—
79	C₂×C₈	—	122	2	no	—
80	C₂×C₈	C₂×#6923	1	1	yes	characteristic
81	C₄⋊C₄	C₂×#6923	1	2	yes	—
82	C₂²⋊C₄	—	122	2	no	—
83	C₄²	—	61	1	no	—
84	C₂³	—	122	2	no	—
85	D₄	—	122	2	no	—
86	D₄	—	122	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	61	1	no	—
89	C₂×C₄	—	61	1	no	—
90	C₂×C₄	—	244	2	no	—
91	C₂×C₄	—	244	2	no	—
92	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
93	C₈	—	122	2	no	—
94	C₈	—	122	1	no	—
95	C₈	—	244	2	no	—
96	C₈	—	244	1	no	—
97	C₈	#11758213	1	2	yes	complemented
98	C₂²	—	244	2	no	—
99	C₂²	—	244	2	no	—
100	C₂²	—	244	2	no	—
101	C₂²	D₄×#6923	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	61	2	no	—
104	C₄	—	122	1	no	—
105	C₄	C₂×#11758213	1	1	yes	characteristic
106	C₄	#14519865	1	1	yes	characteristic
107	C₂	—	488	2	no	—
108	C₂	#16840796	1	1	yes	characteristic
109	C₂	#16840803	1	1	yes	characteristic
110	C₂	#16840808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840796

C₂

yes

maximal, complemented

#16840623

C₂

yes

maximal, characteristic, complemented

#16840610

C₂

yes

maximal, complemented

#16840603

C₂

yes

maximal, characteristic, complemented

#16839844

C₂

yes

maximal, characteristic

C₂×#11758213

—

C₄⋊#92420

C₂²

yes

—

#14519834

—

C₄×#92420

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

characteristic

C₂×D₄₈₈

C₄

yes

characteristic

#14519739

C₂²

yes

—

#14519697

C₄

yes

—

C₈×Dic₆₁

C₂²

yes

characteristic

#107×C₆₁

C₄

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

D₄

yes

—

#11758213

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

#11758198

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

D₄₈₈

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

#41

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

#92420

—

#92420

—

C₈×C₆₁

M₁₆

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

Dic₆₁

—

Dic₆₁

AGL(1,ℤ/8)

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₄×D₈

—

#436

—

#334

—

D₄⋊C₈

—

#322

—

C₆₁

#1594

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

122

—

C₄×D₄

—

#107

#6923

yes

characteristic

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

122

—

D₈

—

122

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

C₂×#6923

yes

characteristic

C₄⋊C₄

C₂×#6923

yes

—

C₂²⋊C₄

—

122

—

C₄²

—

C₂³

—

122

—

D₄

—

122

—

D₄

—

122

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂²×#6923

yes

characteristic

C₈

—

122

—

C₈

—

122

—

C₈

—

244

—

C₈

—

244

—

C₈

#11758213

yes

complemented

C₂²

—

244

—

C₂²

—

244

—

100

C₂²

—

244

—

101

C₂²

D₄×#6923

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

122

—

105

C₄

C₂×#11758213

yes

characteristic

106

C₄

#14519865

yes

characteristic

107

C₂

—

488

—

108

C₂

#16840796

yes

characteristic

109

C₂

#16840803

yes

characteristic

110

C₂

#16840808

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	#107×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#107	#6923
hypocenter	C₆₁	#1594
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

#107×C₆₁

C₄

socle

C₂²×C₆₁

C₄×D₄

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. #16840796	C₂	semidirect
2. #16840623	C₂	semidirect
3. #16840610	C₂	semidirect
4. #16840603	C₂	semidirect
40. C₈×C₆₁	M₁₆	semidirect
61. C₆₁	#1594	semidirect
97. C₈	#11758213	semidirect

H ≅ G/N

type

1. #16840796

C₂

semidirect

2. #16840623

C₂

semidirect

3. #16840610

C₂

semidirect

4. #16840603

C₂

semidirect

40. C₈×C₆₁

M₁₆

semidirect

61. C₆₁

#1594

semidirect

97. C₈

#11758213

semidirect

Subgroups — #21461468

Conjugacy classes of subgroups

Complements