Group #16840326 — subgroups

Subgroups — #16840326

Order 3904.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519773	C₂	1	2	yes	maximal, complemented
3	#14519766	C₂	1	2	yes	maximal, complemented
4	#14519751	C₂	1	4	yes	maximal, complemented
5	#14519745	C₂	1	4	yes	maximal, complemented
6	#14519736	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₆₁	C₂²	1	2	yes	—
9	#41×C₆₁	—	2	4	no	—
10	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄×C₆₁	C₂²	1	2	yes	—
12	C₂×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂²	1	2	yes	complemented
15	C₂×C₂²⋊D₆₁	C₂²	1	8	yes	complemented
16	C₂×D₂₄₄	—	2	4	no	—
17	C₂×D₂₄₄	C₂²	1	2	yes	—
18	C₂×D₂₄₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₆₁	C₂²	1	2	yes	—
20	#11758198	C₂²	1	8	yes	complemented
21	C₄⋊Dic₆₁	C₂²	1	2	yes	complemented
22	C₄×Dic₆₁	C₂²	1	2	yes	complemented
23	C₂³×C₆₁	C₂³	1	2	yes	—
24	C₂³×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×D₆₁	—	2	4	no	—
26	C₂²×D₆₁	—	4	4	no	—
27	C₂²×D₆₁	C₂³	1	4	yes	—
28	Q₈×C₆₁	—	2	2	no	—
29	D₄×C₆₁	—	2	4	no	—
30	D₄×C₆₁	—	2	2	no	—
31	C₂×C₄×C₆₁	—	2	4	no	—
32	C₂×C₄×C₆₁	D₄	1	4	yes	complemented
33	C₂×C₄×C₆₁	C₂³	1	2	yes	—
34	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²⋊D₆₁	—	4	8	no	—
37	C₂×Dic₆₁	C₂³	1	4	yes	—
38	D₂₄₄	—	2	4	no	—
39	D₂₄₄	—	2	4	no	—
40	C₂²×C₆₁	—	2	2	no	—
41	C₂²×C₆₁	—	2	2	no	—
42	C₂²×C₆₁	—	2	4	no	—
43	C₂²×C₆₁	C₂×D₄	1	2	yes	—
44	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₆₁	—	2	4	no	—
46	C₂×D₆₁	—	2	8	no	—
47	C₂×D₆₁	—	4	8	no	—
48	C₄×C₆₁	—	2	2	no	—
49	C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₄×C₆₁	C₂×D₄	1	2	yes	—
51	Dic₆₁	—	2	4	no	—
52	C₂×C₆₁	—	2	2	no	—
53	C₂×C₆₁	—	4	2	no	—
54	C₂×C₆₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₆₁	H(5,𝔽₂)	1	2	yes	—
56	D₆₁	—	4	4	no	—
57	#535	—	61	1	no	maximal, Sylow
58	C₆₁	#535	1	1	yes	characteristic, complemented, Sylow
59	C₂×#41	D₆₁	1	1	yes	characteristic, complemented
60	C₂²×D₄	—	61	1	no	—
61	#127	—	61	2	no	—
62	#124	—	61	2	no	—
63	#121	—	61	4	no	—
64	#120	—	61	4	no	—
65	#117	—	61	1	no	—
66	C₂⁴	—	61	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic, complemented
69	C₂×D₄	—	61	8	no	—
70	C₂×D₄	—	61	2	no	—
71	C₂×D₄	—	61	2	no	—
72	C₂×D₄	—	122	4	no	—
73	C₂×D₄	C₂×D₆₁	1	2	yes	—
74	C₂×D₄	C₂×D₆₁	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₆₁	1	2	yes	complemented
77	C₄⋊C₄	—	61	2	no	—
78	C₂²⋊C₄	—	61	2	no	—
79	C₂²⋊C₄	—	61	8	no	—
80	C₄²	—	61	2	no	—
81	C₂³	—	61	4	no	—
82	C₂³	—	122	4	no	—
83	C₂³	—	244	4	no	—
84	C₂³	C₂²×D₆₁	1	2	yes	—
85	C₂³	C₂²×D₆₁	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	122	4	no	—
90	D₄	—	122	4	no	—
91	D₄	—	244	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	61	4	no	—
94	C₂×C₄	C₂²⋊D₆₁	1	4	yes	complemented
95	C₂×C₄	C₂²×D₆₁	1	2	yes	—
96	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	122	4	no	—
102	C₂²	—	122	8	no	—
103	C₂²	—	244	8	no	—
104	C₂²	C₂×C₂²⋊D₆₁	1	2	yes	—
105	C₂²	C₂³×D₆₁	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	122	4	no	—
108	C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
109	C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	244	4	no	—
113	C₂	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂	#14519823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₂₄₄

C₂

yes

maximal, characteristic, complemented

#14519773

C₂

yes

maximal, complemented

#14519766

C₂

yes

maximal, complemented

#14519751

C₂

yes

maximal, complemented

#14519745

C₂

yes

maximal, complemented

#14519736

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₆₁

C₂²

yes

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₆₁

C₂²

yes

—

C₂×D₄×C₆₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂²

yes

complemented

C₂×C₂²⋊D₆₁

C₂²

yes

complemented

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂²

yes

—

C₂×D₂₄₄

C₂²

yes

—

C₂²⋊Dic₆₁

C₂²

yes

—

#11758198

C₂²

yes

complemented

C₄⋊Dic₆₁

C₂²

yes

complemented

C₄×Dic₆₁

C₂²

yes

complemented

C₂³×C₆₁

C₂³

yes

—

C₂³×C₆₁

C₂³

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂³

yes

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₄

yes

complemented

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

C₂³

yes

—

D₂₄₄

—

D₂₄₄

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×D₄

yes

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₄

yes

—

C₄×C₆₁

C₂×D₄

yes

—

Dic₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂²×D₄

yes

characteristic

C₂×C₆₁

H(5,𝔽₂)

yes

—

D₆₁

—

#535

—

maximal, Sylow

C₆₁

#535

yes

characteristic, complemented, Sylow

C₂×#41

D₆₁

yes

characteristic, complemented

C₂²×D₄

—

—

—

—

—

—

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₆₁

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

122

—

C₂×D₄

C₂×D₆₁

yes

—

C₂×D₄

C₂×D₆₁

yes

characteristic, complemented

C₂²×C₄

C₂×D₆₁

yes

characteristic

C₂²×C₄

C₂×D₆₁

yes

complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

122

—

C₂³

—

244

—

C₂³

C₂²×D₆₁

yes

—

C₂³

C₂²×D₆₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

122

—

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₆₁

yes

complemented

C₂×C₄

C₂²×D₆₁

yes

—

C₂×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₄

C₂²×D₆₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

122

—

102

C₂²

—

122

—

103

C₂²

—

244

—

104

C₂²

C₂×C₂²⋊D₆₁

yes

—

105

C₂²

C₂³×D₆₁

yes

characteristic

106

C₄

—

107

C₄

—

122

—

108

C₄

C₂×C₂²⋊D₆₁

yes

—

109

C₄

C₂×C₂²⋊D₆₁

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

244

—

113

C₂

C₂²×C₂²⋊D₆₁

yes

characteristic

114

C₂

#14519823

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₂×#41×C₆₁	C₂
hypercenter	C₂×#41	D₆₁
hypocenter	C₆₁	#535
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

C₂×#41×C₆₁

C₂

hypercenter

C₂×#41

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×D₂₄₄	C₂	semidirect
2. #14519773	C₂	semidirect
3. #14519766	C₂	semidirect
4. #14519751	C₂	semidirect
5. #14519745	C₂	semidirect
6. #14519736	C₂	semidirect
7. C₂×#41×C₆₁	C₂	semidirect
10. C₂×Q₈×C₆₁	C₂²	semidirect
12. C₂×D₄×C₆₁	C₂²	semidirect
14. C₂²×C₄×C₆₁	C₂²	semidirect
15. C₂×C₂²⋊D₆₁	C₂²	semidirect
20. #11758198	C₂²	semidirect
21. C₄⋊Dic₆₁	C₂²	semidirect
22. C₄×Dic₆₁	C₂²	semidirect
32. C₂×C₄×C₆₁	D₄	semidirect
58. C₆₁	#535	semidirect
59. C₂×#41	D₆₁	semidirect
68. C₂×Q₈	C₂×D₆₁	semidirect
74. C₂×D₄	C₂×D₆₁	semidirect
76. C₂²×C₄	C₂×D₆₁	semidirect
94. C₂×C₄	C₂²⋊D₆₁	semidirect

H ≅ G/N

type

1. C₂²×D₂₄₄

C₂

semidirect

2. #14519773

C₂

semidirect

3. #14519766

C₂

semidirect

4. #14519751

C₂

semidirect

5. #14519745

C₂

semidirect

6. #14519736

C₂

semidirect

7. C₂×#41×C₆₁

C₂

semidirect

10. C₂×Q₈×C₆₁

C₂²

semidirect

12. C₂×D₄×C₆₁

C₂²

semidirect

14. C₂²×C₄×C₆₁

C₂²

semidirect

15. C₂×C₂²⋊D₆₁

C₂²

semidirect

20. #11758198

C₂²

semidirect

21. C₄⋊Dic₆₁

C₂²

semidirect

22. C₄×Dic₆₁

C₂²

semidirect

32. C₂×C₄×C₆₁

D₄

semidirect

58. C₆₁

#535

semidirect

59. C₂×#41

D₆₁

semidirect

68. C₂×Q₈

C₂×D₆₁

semidirect

74. C₂×D₄

C₂×D₆₁

semidirect

76. C₂²×C₄

C₂×D₆₁

semidirect

94. C₂×C₄

C₂²⋊D₆₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₂₄₄	2
#14519773	2
#14519766	2
#14519751	2
#14519745	2
#14519736	2
C₂×#41×C₆₁	2
#535	61

subgroup

index

C₂²×D₂₄₄

C₂×#41×C₆₁

Subgroups — #16840326

Conjugacy classes of subgroups

Complements

Maximal subgroups