Group #21447607 — subgroups

Subgroups — #21447607

Order 7808.

156 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16839800	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519711	C₂	1	1	yes	maximal, characteristic
3	#16839590	C₂	1	2	yes	maximal, complemented
4	#16839588	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519736	—	2	1	no	—
7	C₂×C₄⋊Dic₆₁	—	2	1	no	—
8	#14519711	—	2	1	no	—
9	#14519711	C₂²	1	2	yes	—
10	#14519711	C₂²	1	1	yes	characteristic
11	#14519711	C₂²	1	1	yes	characteristic
12	#14519709	—	2	1	no	—
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	M₃₂×C₆₁	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×Dic₆₁	—	4	1	no	—
18	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	Q₈	1	2	yes	—
21	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²⋊Dic₆₁	—	4	1	no	—
23	C₄⋊Dic₆₁	—	2	1	no	—
24	C₄⋊Dic₆₁	—	2	1	no	—
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	4	1	no	—
27	C₄×Dic₆₁	—	4	1	no	—
28	C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
29	C₂³×C₆₁	D₈	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₆₁	—	4	1	no	—
35	C₂×Dic₆₁	—	4	1	no	—
36	C₂×Dic₆₁	—	4	2	no	—
37	C₂×Dic₆₁	—	8	2	no	—
38	C₈×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₆₁	—	2	1	no	—
42	C₂²×C₆₁	—	2	1	no	—
43	C₂²×C₆₁	#107	1	2	yes	—
44	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
45	C₄×C₆₁	#107	1	2	yes	—
46	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₆₁	—	8	1	no	—
49	Dic₆₁	—	8	2	no	—
50	Dic₆₁	—	8	1	no	—
51	#2159	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	—	2	2	no	—
53	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
54	C₂×C₆₁	#509	1	1	yes	characteristic
55	C₂×C₆₁	#510	1	1	yes	characteristic
56	C₂×M₃₂	D₆₁	1	1	yes	characteristic
57	#427	—	61	1	no	—
58	C₂×#107	—	61	1	no	—
59	#367	—	61	2	no	—
60	#366	—	61	2	no	—
61	C₆₁	#2159	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
63	#117	—	122	1	no	—
64	C₂×C₄⋊C₄	—	122	1	no	—
65	M₃₂	Dic₆₁	1	4	yes	complemented
66	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
67	#107	—	61	1	no	—
68	#107	—	61	2	no	—
69	#107	—	61	1	no	—
70	#107	—	122	1	no	—
71	#106	—	122	1	no	—
72	C₂²×C₄	—	244	1	no	—
73	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
74	C₂×C₈	Dic₁₂₂	1	2	yes	—
75	C₂×C₈	D₂₄₄	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
77	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
78	C₄⋊C₄	—	122	1	no	—
79	C₄⋊C₄	—	122	1	no	—
80	C₄⋊C₄	—	122	2	no	—
81	C₄⋊C₄	—	244	1	no	—
82	C₂²⋊C₄	—	244	1	no	—
83	C₄²	—	244	1	no	—
84	C₁₆	C₂×Dic₆₁	1	4	yes	complemented
85	C₂³	D₄₈₈	1	1	yes	characteristic
86	C₂×C₄	—	488	2	no	—
87	C₂×C₄	—	244	1	no	—
88	C₂×C₄	—	244	1	no	—
89	C₂×C₄	—	244	2	no	—
90	C₂×C₄	D₄₈₈	1	1	yes	characteristic
91	C₂×C₄	Dic₂₄₄	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
93	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
94	C₈	C₄⋊Dic₆₁	1	2	yes	—
95	C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
96	C₈	C₂²×Dic₆₁	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#14519711	1	2	yes	—
100	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
101	C₄	—	488	2	no	—
102	C₄	—	488	1	no	—
103	C₄	—	488	1	no	—
104	C₄	#14519711	1	2	yes	—
105	C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
106	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#14519711	1	1	yes	characteristic
109	C₂	#16840200	1	1	yes	characteristic
110	C₂	#16840205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839800

C₂

yes

maximal, characteristic

C₂×#14519711

C₂

yes

maximal, characteristic

#16839590

C₂

yes

maximal, complemented

#16839588

C₂

yes

maximal, complemented

C₂×M₃₂×C₆₁

C₂

yes

maximal, characteristic

#14519736

—

C₂×C₄⋊Dic₆₁

—

#14519711

—

#14519711

C₂²

yes

—

#14519711

C₂²

yes

characteristic

#14519711

C₂²

yes

characteristic

#14519709

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

M₃₂×C₆₁

C₄

yes

complemented

C₂×C₁₆×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₁₆×C₆₁

C₂×C₄

yes

complemented

C₂³×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

C₄⋊C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#107

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

#107

yes

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2159

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×M₃₂

D₆₁

yes

—

—

—

—

C₆₁

#2159

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

122

—

C₂×C₄⋊C₄

—

122

—

M₃₂

Dic₆₁

yes

complemented

C₂×C₁₆

C₂×D₆₁

yes

—

—

—

—

—

122

—

#106

—

122

—

C₂²×C₄

—

244

—

C₂²×C₄

D₂₄₄

yes

characteristic

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₂²⋊C₄

—

244

—

C₄²

—

244

—

C₁₆

C₂×Dic₆₁

yes

complemented

C₂³

D₄₈₈

yes

characteristic

C₂×C₄

—

488

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

D₄₈₈

yes

characteristic

C₂×C₄

Dic₂₄₄

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₈

C₄⋊Dic₆₁

yes

—

C₈

C₂×Dic₁₂₂

yes

characteristic

C₈

C₂²×Dic₆₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

#14519711

yes

—

100

C₂²

C₂×D₄₈₈

yes

characteristic

101

C₄

—

488

—

102

C₄

—

488

—

103

C₄

—

488

—

104

C₄

#14519711

yes

—

105

C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

106

C₄

C₂×Dic₂₄₄

yes

characteristic

107

C₂

—

108

C₂

C₂×#14519711

yes

characteristic

109

C₂

#16840200

yes

characteristic

110

C₂

#16840205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂×M₃₂	D₆₁
hypocenter	C₆₁	#2159
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₄₈₈

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×M₃₂×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₂×M₃₂

D₆₁

hypocenter

C₆₁

#2159

LCS

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16839590	C₂	semidirect
4. #16839588	C₂	semidirect
14. M₃₂×C₆₁	C₄	semidirect
28. C₁₆×C₆₁	C₂×C₄	semidirect
61. C₆₁	#2159	semidirect
65. M₃₂	Dic₆₁	semidirect
84. C₁₆	C₂×Dic₆₁	semidirect

H ≅ G/N

type

3. #16839590

C₂

semidirect

4. #16839588

C₂

semidirect

14. M₃₂×C₆₁

C₄

semidirect

28. C₁₆×C₆₁

C₂×C₄

semidirect

61. C₆₁

#2159

semidirect

65. M₃₂

Dic₆₁

semidirect

84. C₁₆

C₂×Dic₆₁

semidirect

Subgroups — #21447607

Conjugacy classes of subgroups

Complements