Group #21461310 — subgroups

Subgroups — #21461310

Order 7808.

194 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 58 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	#16840783	C₂	1	2	yes	maximal
2	#16840619	C₂	1	4	yes	maximal, complemented
3	#16840315	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#92420	—	2	2	no	—
5	C₂²⋊#92420	—	2	2	no	—
6	#14519832	C₂²	1	4	yes	—
7	#14519776	C₄	1	2	yes	—
8	#14519768	C₂²	1	2	yes	—
9	#14519754	—	2	2	no	—
10	C₄×Dic₁₂₂	—	2	2	no	—
11	#14519736	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	#117×C₆₁	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₆₁	D₄	1	2	yes	—
16	C₂×Dic₁₂₂	—	2	2	no	—
17	C₂×#92420	—	2	4	no	—
18	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
19	C₂²⋊C₄×C₆₁	—	2	1	no	—
20	C₄²×C₆₁	—	2	1	no	—
21	C₂²⋊Dic₆₁	—	2	1	no	—
22	C₄⋊Dic₆₁	—	2	2	no	—
23	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
24	#11758196	—	2	2	no	—
25	#11758196	—	2	2	no	—
26	C₄×Dic₆₁	—	2	1	no	—
27	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₆₁	D₄	1	2	yes	—
29	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
30	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₆₁	—	2	2	no	—
36	C₂×Dic₆₁	—	4	2	no	—
37	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
39	Dic₁₂₂	—	4	4	no	—
40	#92420	—	4	4	no	—
41	C₂²×C₆₁	—	2	1	no	—
42	C₂²×C₆₁	—	2	2	no	—
43	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₄×C₆₁	—	2	1	no	—
45	C₄×C₆₁	—	4	2	no	—
46	C₄×C₆₁	#101	1	2	yes	—
47	Dic₆₁	—	2	2	no	—
48	Dic₆₁	—	2	2	no	—
49	Dic₆₁	—	4	2	no	—
50	#1537	—	61	1	no	maximal, Sylow
51	C₂×C₆₁	—	4	1	no	—
52	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
53	C₂×C₆₁	#421	1	2	yes	—
54	#533	—	61	1	no	—
55	#432	—	61	2	no	—
56	#330	—	61	4	no	—
57	C₆₁	#1537	1	1	yes	characteristic, complemented, Sylow
58	C₄⋊Q₈	—	61	2	no	—
59	#125	—	61	2	no	—
60	C₂²⋊Q₈	—	122	2	no	—
61	C₄×Q₈	—	122	2	no	—
62	#117	—	61	1	no	—
63	#117	#6923	1	1	yes	characteristic
64	C₂×C₄²	—	61	1	no	—
65	C₄⋊C₈	—	122	2	no	—
66	C₂²⋊C₈	—	122	2	no	—
67	#97	—	61	4	no	—
68	C₂×Q₈	—	122	2	no	—
69	C₂²×C₄	—	61	2	no	—
70	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
71	C₂×C₈	—	122	4	no	—
72	C₄⋊C₄	—	61	2	no	—
73	C₄⋊C₄	—	122	2	no	—
74	C₄⋊C₄	—	122	2	no	—
75	C₄⋊C₄	—	122	2	no	—
76	C₄⋊C₄	C₂×#6923	1	2	yes	—
77	C₂²⋊C₄	—	2	1	no	—
78	C₂²⋊C₄	—	122	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	—	61	2	no	—
81	C₄²	—	61	1	no	—
82	C₄²	—	61	1	no	—
83	C₄²	—	122	1	no	—
84	C₂³	C₂²⋊#6923	1	1	yes	characteristic
85	Q₈	—	244	4	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	61	2	no	—
89	C₂×C₄	—	61	2	no	—
90	C₂×C₄	—	122	2	no	—
91	C₂×C₄	—	244	2	no	—
92	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
93	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
94	C₈	—	244	4	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	4	2	no	—
100	C₄	—	122	2	no	—
101	C₄	—	122	2	no	—
102	C₄	—	244	2	no	—
103	C₄	#14519838	1	2	yes	—
104	C₂	—	4	1	no	—
105	C₂	C₂×#14519838	1	1	yes	characteristic
106	C₂	#16840759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840783

C₂

yes

maximal

#16840619

C₂

yes

maximal, complemented

#16840315

C₂

yes

maximal, characteristic

C₄⋊#92420

—

C₂²⋊#92420

—

#14519832

C₂²

yes

—

#14519776

C₄

yes

—

#14519768

C₂²

yes

—

#14519754

—

C₄×Dic₁₂₂

—

#14519736

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#117×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×#92420

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#101

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#1537

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#101

yes

characteristic

C₂×C₆₁

#421

yes

—

#533

—

#432

—

#330

—

C₆₁

#1537

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

—

yes

C₂×C₄²

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂×#6923

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₂³

C₂²⋊#6923

yes

characteristic

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂²⋊#6923

yes

characteristic

C₂×C₄

C₂²×#6923

yes

characteristic

C₈

—

244

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#6923

yes

characteristic

C₄

—

C₄

—

100

C₄

—

122

—

101

C₄

—

122

—

102

C₄

—

244

—

103

C₄

#14519838

yes

—

104

C₂

—

105

C₂

C₂×#14519838

yes

characteristic

106

C₂

#16840759

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	#117×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#117	#6923
hypocenter	C₆₁	#1537
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

#117×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
2. #16840619	C₂	semidirect
57. C₆₁	#1537	semidirect

H ≅ G/N

type

2. #16840619

C₂

semidirect

57. C₆₁

#1537

semidirect

Subgroups — #21461310

Conjugacy classes of subgroups

Complements