Group #21461733 — subgroups

Subgroups — #21461733

Order 7808.

236 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 62 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×#14519838	C₂	1	2	yes	maximal
2	#16840643	C₂	1	4	yes	maximal
3	#16840239	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758217	C₂²	1	4	yes	—
5	#14519838	—	2	2	no	—
6	#14519838	—	2	2	no	—
7	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	C₄×Dic₁₂₂	—	2	2	no	—
9	#14519742	—	2	2	no	—
10	#14519736	C₄	1	1	yes	characteristic
11	#14519733	C₄	1	2	yes	—
12	#14519733	C₂²	1	2	yes	—
13	#117×C₆₁	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₂₂	—	2	4	no	—
18	C₂×Dic₁₂₂	D₄	1	2	yes	—
19	C₂×Dic₁₂₂	D₄	1	2	yes	—
20	#11758217	—	2	4	no	—
21	#11758217	—	2	4	no	—
22	C₂×#92420	—	2	4	no	—
23	C₄⋊C₄×C₆₁	—	2	1	no	—
24	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₄²×C₆₁	—	2	1	no	—
26	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₆₁	—	2	1	no	—
28	#11758196	—	2	2	no	—
29	#11758196	—	2	2	no	—
30	C₄×Dic₆₁	—	2	1	no	—
31	C₄×Dic₆₁	—	2	2	no	—
32	C₄×Dic₆₁	—	2	2	no	—
33	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	2	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₆₁	—	2	2	no	—
39	C₂×Dic₆₁	—	2	2	no	—
40	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
42	C₂×Dic₆₁	#41	1	4	yes	—
43	Dic₁₂₂	—	2	4	no	—
44	Dic₁₂₂	—	2	4	no	—
45	#92420	—	4	4	no	—
46	C₂²×C₆₁	—	4	1	no	—
47	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₆₁	#117	1	2	yes	—
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	—	4	2	no	—
52	Dic₆₁	—	2	2	no	—
53	Dic₆₁	—	4	2	no	—
54	Dic₆₁	#123	1	4	yes	—
55	#1830	—	61	1	no	maximal, Sylow
56	C₂×C₆₁	—	2	1	no	—
57	C₂×C₆₁	—	2	2	no	—
58	C₂×C₆₁	#381	1	1	yes	characteristic
59	#519	—	61	1	no	—
60	C₂×#101	—	61	2	no	—
61	#344	—	61	4	no	—
62	C₆₁	#1830	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	61	1	no	—
64	C₂×M₁₆	—	61	4	no	—
65	C₄×Q₈	—	122	2	no	—
66	C₄×Q₈	—	122	2	no	—
67	#117	—	61	1	no	—
68	#117	—	61	2	no	—
69	#117	—	61	2	no	—
70	#117	#6923	1	1	yes	characteristic
71	#101	—	122	2	no	—
72	#101	—	122	2	no	—
73	C₂×Q₈	—	61	2	no	—
74	C₂×Q₈	—	61	2	no	—
75	C₂×Q₈	—	122	4	no	—
76	C₂²×C₄	—	61	1	no	—
77	C₂²×C₄	—	61	1	no	—
78	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
79	M₁₆	—	122	4	no	—
80	M₁₆	—	122	4	no	—
81	C₂×C₈	—	122	4	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	122	2	no	—
84	C₄⋊C₄	—	122	2	no	—
85	C₄⋊C₄	—	122	1	no	—
86	C₂²⋊C₄	—	61	2	no	—
87	C₂²⋊C₄	C₂×#6923	1	2	yes	—
88	C₄²	—	2	1	no	—
89	C₄²	—	122	2	no	—
90	C₄²	—	122	2	no	—
91	C₄²	—	122	1	no	—
92	C₂³	C₂²×#6923	1	1	yes	characteristic
93	Q₈	—	122	4	no	—
94	Q₈	—	122	4	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	61	2	no	—
98	C₂×C₄	—	61	4	no	—
99	C₂×C₄	—	61	2	no	—
100	C₂×C₄	—	122	2	no	—
101	C₂×C₄	—	122	2	no	—
102	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
104	C₈	—	244	4	no	—
105	C₂²	—	4	1	no	—
106	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
107	C₂²	#14519856	1	2	yes	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	2	no	—
111	C₄	—	61	4	no	—
112	C₄	—	122	2	no	—
113	C₄	—	244	2	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	#16840705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519838

C₂

yes

maximal

#16840643

C₂

yes

maximal

#16840239

C₂

yes

maximal, characteristic

C₂×#11758217

C₂²

yes

—

#14519838

—

#14519838

—

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₄×Dic₁₂₂

—

#14519742

—

#14519736

C₄

yes

characteristic

#14519733

C₄

yes

—

#14519733

C₂²

yes

—

#117×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

—

C₂×Dic₁₂₂

D₄

yes

—

#11758217

—

#11758217

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

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

—

C₂×Dic₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#123

yes

—

#1830

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

—

C₆₁

#1830

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

—

—

—

yes

—

122

—

#101

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

M₁₆

—

122

—

M₁₆

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×#6923

yes

—

C₄²

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

—

122

—

C₂³

C₂²×#6923

yes

characteristic

Q₈

—

122

—

Q₈

—

122

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

122

—

101

C₂×C₄

—

122

—

102

C₂×C₄

C₂²⋊#6923

yes

characteristic

103

C₂×C₄

C₂²⋊#6923

yes

characteristic

104

C₈

—

244

—

105

C₂²

—

106

C₂²

C₂×C₂²⋊#6923

yes

characteristic

107

C₂²

#14519856

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

122

—

113

C₄

—

244

—

114

C₂

—

115

C₂

—

116

C₂

#16840705

yes

characteristic

	subgroup	G/H
center	C₂	#16840705
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂³	C₂²×#6923
Fitting	#117×C₆₁	C₄
socle	C₂×C₆₁	#381
hypercenter	#117	#6923
hypocenter	C₆₁	#1830
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117

subgroup

G/H

center

C₂

#16840705

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂³

C₂²×#6923

Fitting

#117×C₆₁

C₄

C₂×C₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ #117

N	H ≅ G/N	type
62. C₆₁	#1830	semidirect

H ≅ G/N

type

62. C₆₁

#1830

semidirect

Subgroups — #21461733

Conjugacy classes of subgroups

Complements