Group #21444100 — subgroups

Subgroups — #21444100

Order 7808.

224 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₄×#11758194	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
3	#16839484	C₂	1	2	yes	maximal, complemented
4	C₈×#92418	C₂	1	2	yes	maximal, complemented
5	C₆₁×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₄	1	2	yes	—
7	C₂²×#92418	C₄	1	2	yes	—
8	#14519688	C₄	1	2	yes	—
9	C₄×#92418	C₄	1	2	yes	—
10	C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄×#92418	C₂²	1	1	yes	characteristic
12	C₄×#92418	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	C₄	1	2	yes	—
15	C₂²⋊C₈×C₆₁	C₄	1	2	yes	—
16	C₄×C₈×C₆₁	C₂²	1	2	yes	—
17	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	—
21	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	#11758194	C₈	1	8	yes	complemented
26	C₂×#92418	—	2	4	no	—
27	C₂×#92418	C₂×C₄	1	2	yes	—
28	C₂×#92418	C₂×C₄	1	4	yes	—
29	C₂×#92418	C₂×C₄	1	2	yes	—
30	C₂³×C₆₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₁	—	2	4	no	—
41	#92418	—	2	2	no	—
42	#92418	C₂×C₈	1	8	yes	complemented
43	#92418	M₁₆	1	4	yes	complemented
44	C₂²×C₆₁	—	2	1	no	—
45	C₂²×C₆₁	—	2	1	no	—
46	C₂²×C₆₁	C₄×C₈	1	2	yes	—
47	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₆₁	—	2	1	no	—
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	C₄×C₈	1	2	yes	—
51	C₄×C₆₁	C₂²×C₈	1	2	yes	—
52	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
53	C₄×C₆₁	#131	1	2	yes	—
54	C₈×M₁₆	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	—	2	2	no	—
56	C₂×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂×C₆₁	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₆₁	#405	1	1	yes	characteristic
59	#431	D₆₁	1	1	yes	characteristic
60	C₄×M₁₆	—	61	1	no	—
61	C₂×C₄×C₈	—	61	1	no	—
62	#322	—	61	2	no	—
63	C₈²	—	61	2	no	—
64	C₆₁	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	61	2	no	—
66	C₂²×C₈	—	61	2	no	—
67	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
68	C₄⋊C₈	Dic₆₁	1	2	yes	—
69	C₂²⋊C₈	Dic₆₁	1	2	yes	—
70	#97	—	61	2	no	—
71	C₄×C₈	—	61	2	no	—
72	C₄×C₈	—	61	1	no	—
73	C₄×C₈	—	61	2	no	—
74	C₄×C₈	—	61	1	no	—
75	C₄×C₈	C₂×D₆₁	1	2	yes	—
76	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
79	M₁₆	—	61	8	no	—
80	C₂×C₈	—	61	2	no	—
81	C₂×C₈	—	61	4	no	—
82	C₂×C₈	—	61	2	no	—
83	C₂×C₈	—	122	4	no	—
84	C₂×C₈	C₂×Dic₆₁	1	4	yes	—
85	C₄²	C₄×D₆₁	1	1	yes	characteristic
86	C₄²	C₄×D₆₁	1	1	yes	characteristic
87	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
88	C₄²	C₂²×D₆₁	1	1	yes	characteristic
89	C₂³	C₄×Dic₆₁	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₈×D₆₁	1	4	yes	—
93	C₂×C₄	C₄×Dic₆₁	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₆₁	1	1	yes	characteristic
95	C₂×C₄	C₄×Dic₆₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	61	8	no	—
101	C₈	—	61	4	no	—
102	C₈	—	122	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	C₈×Dic₆₁	1	2	yes	—
106	C₂²	C₂×C₄×Dic₆₁	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	C₈×Dic₆₁	1	2	yes	—
110	C₄	C₂×C₈×D₆₁	1	2	yes	—
111	C₄	M₁₆×D₆₁	1	2	yes	—
112	C₄	#14519784	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×C₈×Dic₆₁	1	1	yes	characteristic
115	C₂	M₁₆×Dic₆₁	1	1	yes	characteristic
116	C₂	#16839718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11758194

C₂

yes

maximal, characteristic

C₂×C₄×#92418

C₂

yes

maximal, characteristic

#16839484

C₂

yes

maximal, complemented

C₈×#92418

C₂

yes

maximal, complemented

C₆₁×#431

C₂

yes

maximal, characteristic

C₂×#11758194

C₄

yes

—

C₂²×#92418

C₄

yes

—

#14519688

C₄

yes

—

C₄×#92418

C₄

yes

—

C₄×#92418

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₄

yes

—

C₂²⋊C₈×C₆₁

C₄

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758194

C₈

yes

complemented

C₂×#92418

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

—

#92418

—

#92418

C₂×C₈

yes

complemented

#92418

M₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₈

yes

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×C₈

yes

—

C₄×C₆₁

C₂²×C₈

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

#131

yes

—

C₈×M₁₆

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×C₆₁

C₄×M₁₆

yes

C₂×C₆₁

yes

D₆₁

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₆₁

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

Dic₆₁

yes

—

C₂²⋊C₈

Dic₆₁

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₄²

C₄×D₆₁

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

C₄×Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₆₁

yes

—

C₂×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

122

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₈×Dic₆₁

yes

—

106

C₂²

C₂×C₄×Dic₆₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₈×Dic₆₁

yes

—

110

C₄

C₂×C₈×D₆₁

yes

—

111

C₄

M₁₆×D₆₁

yes

—

112

C₄

#14519784

yes

—

113

C₂

—

114

C₂

C₂×C₈×Dic₆₁

yes

characteristic

115

C₂

M₁₆×Dic₆₁

yes

characteristic

116

C₂

#16839718

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#431	C₂
socle	C₂²×C₆₁	C₂×C₄²
hypercenter	#431	D₆₁
hypocenter	C₆₁	C₈×M₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₆₁

commutator

C₂×C₆₁

C₂×C₄×C₈

Fitting

C₆₁×#431

C₂

socle

C₂²×C₆₁

C₂×C₄²

hypercenter

#431

D₆₁

hypocenter

C₆₁

C₈×M₁₆

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #16839484	C₂	semidirect
4. C₈×#92418	C₂	semidirect
25. #11758194	C₈	semidirect
42. #92418	C₂×C₈	semidirect
43. #92418	M₁₆	semidirect
64. C₆₁	C₈×M₁₆	semidirect

H ≅ G/N

type

3. #16839484

C₂

semidirect

4. C₈×#92418

C₂

semidirect

25. #11758194

C₈

semidirect

42. #92418

C₂×C₈

semidirect

43. #92418

M₁₆

semidirect

64. C₆₁

C₈×M₁₆

semidirect

Subgroups — #21444100

Conjugacy classes of subgroups

Complements