Group #21453527 — subgroups

Subgroups — #21453527

Order 7808.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16839679	C₂	1	3	yes	maximal
2	#16839673	C₂	1	2	yes	maximal, complemented
3	#16839654	C₂	1	6	yes	maximal, complemented
4	#16839627	C₂	1	3	yes	maximal, complemented
5	C₆₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
8	C₂×#11758198	C₂²	1	6	yes	—
9	C₂×#11758198	C₂²	1	6	yes	—
10	C₂×C₄×Dic₆₁	C₂²	1	3	yes	—
11	#14519684	C₂²	1	3	yes	—
12	#14519684	C₂²	1	6	yes	—
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
15	#95×C₆₁	C₂²	1	3	yes	—
16	C₂³×D₆₁	C₂³	1	2	yes	—
17	C₂×Q₈×C₆₁	—	4	3	no	—
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
21	C₂²×Dic₆₁	C₂³	1	6	yes	—
22	C₂×D₂₄₄	—	4	3	no	—
23	C₄⋊C₄×C₆₁	—	4	3	no	—
24	#11758198	—	2	12	no	—
25	#11758198	—	4	6	no	—
26	C₄⋊Dic₆₁	—	4	3	no	—
27	C₄×Dic₆₁	—	2	6	no	—
28	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
29	C₂²×D₆₁	—	2	6	no	—
30	C₂²×D₆₁	—	2	6	no	—
31	C₂²×D₆₁	—	2	2	no	—
32	Q₈×C₆₁	—	4	4	no	—
33	C₂×C₄×C₆₁	—	2	3	no	—
34	C₂×C₄×C₆₁	—	2	3	no	—
35	C₂×C₄×C₆₁	—	2	6	no	—
36	C₂×C₄×C₆₁	—	2	3	no	—
37	C₂×C₄×C₆₁	—	2	3	no	—
38	C₂×C₄×C₆₁	#41	1	6	yes	—
39	C₂×Dic₆₁	—	2	6	no	—
40	C₂×Dic₆₁	—	2	6	no	—
41	C₂×Dic₆₁	—	2	6	no	—
42	D₂₄₄	—	4	4	no	—
43	C₂²×C₆₁	C₂×#41	1	3	yes	—
44	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₆₁	—	4	6	no	—
47	C₂×D₆₁	—	4	6	no	—
48	C₂×D₆₁	—	4	2	no	—
49	C₄×C₆₁	—	2	6	no	—
50	C₄×C₆₁	—	4	1	no	—
51	C₄×C₆₁	—	4	3	no	—
52	Dic₆₁	—	4	6	no	—
53	#2785	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	#539	1	3	yes	—
55	C₂×C₆₁	#555	1	3	yes	—
56	C₂×C₆₁	H(3,𝔽₄)	1	1	yes	characteristic
57	D₆₁	—	8	2	no	—
58	#395	D₆₁	1	1	yes	characteristic, complemented
59	#394	—	61	3	no	—
60	#393	—	61	2	no	—
61	#388	—	61	6	no	—
62	#382	—	61	3	no	—
63	C₆₁	#2785	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
65	C₂²×D₄	—	61	1	no	—
66	C₂×C₄⋊C₄	—	61	3	no	—
67	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
68	C₂×C₂²⋊C₄	—	61	6	no	—
69	C₂×C₂²⋊C₄	—	61	6	no	—
70	C₂×C₄²	—	61	3	no	—
71	#95	—	61	3	no	—
72	#95	—	61	6	no	—
73	#95	C₂×D₆₁	1	3	yes	—
74	C₂⁴	—	61	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	244	3	no	—
77	C₂²×C₄	—	61	6	no	—
78	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
80	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	244	3	no	—
83	C₂²⋊C₄	—	122	12	no	—
84	C₂²⋊C₄	—	244	6	no	—
85	C₄²	—	122	6	no	—
86	C₂³	—	122	6	no	—
87	C₂³	—	122	6	no	—
88	C₂³	—	122	2	no	—
89	C₂³	C₂³×D₆₁	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	244	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	122	6	no	—
98	C₂×C₄	—	122	6	no	—
99	C₂×C₄	—	122	6	no	—
100	C₂×C₄	#11758226	1	6	yes	—
101	C₂²	—	244	6	no	—
102	C₂²	—	244	2	no	—
103	C₂²	—	244	6	no	—
104	C₂²	C₂×#11758226	1	3	yes	—
105	C₂²	#14519822	1	3	yes	—
106	C₂²	#14519824	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	244	6	no	—
111	C₂	—	488	2	no	—
112	C₂	#16840361	1	3	yes	—
113	C₂	#16840478	1	3	yes	—
114	C₂	#16840505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839679

C₂

yes

maximal

#16839673

C₂

yes

maximal, complemented

#16839654

C₂

yes

maximal, complemented

#16839627

C₂

yes

maximal, complemented

C₆₁×#395

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#95×C₆₁

C₂²

yes

—

C₂³×D₆₁

C₂³

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×D₂₄₄

—

C₄⋊C₄×C₆₁

—

#11758198

—

#11758198

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₂²×C₆₁

C₂×#41

yes

—

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2785

—

maximal, Sylow

C₂×C₆₁

#539

yes

—

C₂×C₆₁

#555

yes

—

C₂×C₆₁

H(3,𝔽₄)

yes

characteristic

D₆₁

—

#395

D₆₁

yes

characteristic, complemented

—

—

—

—

C₆₁

#2785

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₆₁

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

244

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₂²⋊C₄

—

122

—

C₂²⋊C₄

—

244

—

C₄²

—

122

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

100

C₂×C₄

#11758226

yes

—

101

C₂²

—

244

—

102

C₂²

—

244

—

103

C₂²

—

244

—

104

C₂²

C₂×#11758226

yes

—

105

C₂²

#14519822

yes

—

106

C₂²

#14519824

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

244

—

111

C₂

—

488

—

112

C₂

#16840361

yes

—

113

C₂

#16840478

yes

—

114

C₂

#16840505

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#395	C₂
hypercenter	#395	D₆₁
hypocenter	C₆₁	#2785
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#395

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #16839673	C₂	semidirect
3. #16839654	C₂	semidirect
4. #16839627	C₂	semidirect
5. C₆₁×#395	C₂	semidirect
58. #395	D₆₁	semidirect
63. C₆₁	#2785	semidirect

H ≅ G/N

type

2. #16839673

C₂

semidirect

3. #16839654

C₂

semidirect

4. #16839627

C₂

semidirect

5. C₆₁×#395

C₂

semidirect

58. #395

D₆₁

semidirect

63. C₆₁

#2785

semidirect

Subgroups — #21453527

Conjugacy classes of subgroups

Complements