Group #21453859 — subgroups

Subgroups — #21453859

Order 7808.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14519774	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839679	C₂	1	6	yes	maximal, complemented
3	#16839671	C₂	1	6	yes	maximal, complemented
4	#16839630	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#126×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₄₄	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₆₁	C₂²	1	4	yes	—
8	#14519774	—	2	4	no	—
9	C₂×#11758198	C₂²	1	12	yes	—
10	#14519684	C₂²	1	6	yes	—
11	#126×C₆₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
13	C₂×C₂²⋊C₄×C₆₁	C₂²	1	3	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
16	C₂³×D₆₁	C₂³	1	4	yes	—
17	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
18	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
19	C₂×C₂²⋊D₆₁	—	4	12	no	—
20	C₂²×Dic₆₁	C₂³	1	4	yes	—
21	C₂×D₂₄₄	—	2	12	no	—
22	C₂×D₂₄₄	—	4	12	no	—
23	C₄⋊C₄×C₆₁	—	2	6	no	—
24	C₂²⋊C₄×C₆₁	—	2	6	no	—
25	C₄²×C₆₁	D₄	1	4	yes	complemented
26	#11758198	—	4	12	no	—
27	C₂³×C₆₁	—	2	1	no	—
28	C₂³×C₆₁	—	2	6	no	—
29	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
30	C₂²×D₆₁	—	2	4	no	—
31	C₂²×D₆₁	—	2	12	no	—
32	C₂²×D₆₁	—	2	12	no	—
33	C₂×C₄×C₆₁	—	2	6	no	—
34	C₂×C₄×C₆₁	—	2	3	no	—
35	C₂×C₄×C₆₁	—	2	6	no	—
36	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
37	C₂²⋊D₆₁	—	4	16	no	—
38	C₂×Dic₆₁	—	2	12	no	—
39	D₂₄₄	—	4	24	no	—
40	C₂²×C₆₁	—	4	3	no	—
41	C₂²×C₆₁	—	4	4	no	—
42	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
43	C₂²×C₆₁	H(5,𝔽₂)	1	6	yes	—
44	C₂×D₆₁	—	4	12	no	—
45	C₂×D₆₁	—	4	4	no	—
46	C₂×D₆₁	—	4	12	no	—
47	C₄×C₆₁	—	2	6	no	—
48	C₄×C₆₁	—	4	3	no	—
49	Dic₆₁	—	4	4	no	—
50	#2818	—	61	1	no	maximal, Sylow
51	C₂×C₆₁	—	8	1	no	—
52	C₂×C₆₁	#535	1	3	yes	—
53	C₂×C₆₁	#560	1	4	yes	—
54	D₆₁	—	8	4	no	—
55	C₂×#127	—	61	1	no	—
56	C₂×#126	D₆₁	1	1	yes	characteristic, complemented
57	#394	—	61	6	no	—
58	#392	—	61	6	no	—
59	#383	—	61	1	no	—
60	C₆₁	#2818	1	1	yes	characteristic, complemented, Sylow
61	C₂²×D₄	—	61	4	no	—
62	C₂²×D₄	—	61	6	no	—
63	#127	—	122	4	no	—
64	#126	—	2	4	no	—
65	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
66	C₂×C₂²⋊C₄	—	61	12	no	—
67	C₂×C₂²⋊C₄	C₂×D₆₁	1	3	yes	—
68	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
69	#95	—	61	6	no	—
70	C₂⁴	—	61	4	no	—
71	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
72	C₂×D₄	—	122	12	no	—
73	C₂×D₄	—	244	12	no	—
74	C₂×D₄	—	244	12	no	—
75	C₂²×C₄	—	61	4	no	—
76	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
77	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
78	C₄⋊C₄	—	2	6	no	—
79	C₂²⋊C₄	—	2	6	no	—
80	C₂²⋊C₄	—	244	12	no	—
81	C₄²	C₂²⋊D₆₁	1	4	yes	complemented
82	C₂³	—	2	1	no	—
83	C₂³	—	2	6	no	—
84	C₂³	—	122	4	no	—
85	C₂³	—	122	12	no	—
86	C₂³	—	122	12	no	—
87	C₂³	C₂³×D₆₁	1	1	yes	characteristic
88	D₄	—	244	24	no	—
89	D₄	—	244	16	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	122	12	no	—
94	C₂×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
95	C₂²	—	4	4	no	—
96	C₂²	—	4	3	no	—
97	C₂²	—	244	4	no	—
98	C₂²	—	244	12	no	—
99	C₂²	—	244	12	no	—
100	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₂²	#14519823	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	4	3	no	—
104	C₄	—	244	4	no	—
105	C₂	—	8	1	no	—
106	C₂	—	488	4	no	—
107	C₂	#16840326	1	3	yes	—
108	C₂	#16840503	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519774

C₂

yes

maximal, characteristic, complemented

#16839679

C₂

yes

maximal, complemented

#16839671

C₂

yes

maximal, complemented

#16839630

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₂²

yes

—

C₂²×C₂²⋊D₆₁

C₂²

yes

—

#14519774

—

C₂×#11758198

C₂²

yes

—

#14519684

C₂²

yes

—

#126×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂³×D₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

complemented

#11758198

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2818

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#535

yes

—

C₂×C₆₁

#560

yes

—

D₆₁

—

C₂×#127

—

C₂×#126

D₆₁

yes

characteristic, complemented

#394

—

#392

—

#383

—

C₆₁

#2818

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

#127

—

122

—

#126

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#95

—

C₂⁴

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

244

—

C₄²

C₂²⋊D₆₁

yes

complemented

C₂³

—

C₂³

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

C₂³×D₆₁

yes

characteristic

D₄

—

244

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂²

—

C₂²

—

C₂²

—

244

—

C₂²

—

244

—

C₂²

—

244

—

100

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

101

C₂²

#14519823

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

244

—

105

C₂

—

106

C₂

—

488

—

107

C₂

#16840326

yes

—

108

C₂

#16840503

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂×#126×C₆₁	C₂
hypercenter	C₂×#126	D₆₁
hypocenter	C₆₁	#2818
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂×#126×C₆₁

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#14519774	C₂	semidirect
2. #16839679	C₂	semidirect
3. #16839671	C₂	semidirect
4. #16839630	C₂	semidirect
5. C₂×#126×C₆₁	C₂	semidirect
25. C₄²×C₆₁	D₄	semidirect
56. C₂×#126	D₆₁	semidirect
60. C₆₁	#2818	semidirect
81. C₄²	C₂²⋊D₆₁	semidirect

H ≅ G/N

type

1. C₂×#14519774

C₂

semidirect

2. #16839679

C₂

semidirect

3. #16839671

C₂

semidirect

4. #16839630

C₂

semidirect

5. C₂×#126×C₆₁

C₂

semidirect

25. C₄²×C₆₁

D₄

semidirect

56. C₂×#126

D₆₁

semidirect

60. C₆₁

#2818

semidirect

81. C₄²

C₂²⋊D₆₁

semidirect

Subgroups — #21453859

Conjugacy classes of subgroups

Complements