Group #21461532 — subgroups

Subgroups — #21461532

Order 7808.

296 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 126 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×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×#6923	C₂	1	1	yes	maximal, characteristic
3	(C₂×#6923)⋊C₈	C₂	1	4	yes	maximal
4	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#6923	C₄	1	2	yes	—
7	C₂×C₄×#6923	C₄	1	2	yes	—
8	C₂×C₄×#6923	C₂²	1	2	yes	—
9	C₄×#92420	C₄	1	2	yes	complemented
10	C₄×#92420	C₄	1	2	yes	complemented
11	C₂×C₈×D₆₁	C₂²	1	2	yes	—
12	C₄²×D₆₁	C₂²	1	1	yes	characteristic
13	C₄×#92418	C₄	1	1	yes	characteristic, complemented
14	C₈×Dic₆₁	C₄	1	2	yes	complemented
15	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6923	C₂×C₄	1	4	yes	—
17	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
20	C₂×#92420	C₂×C₄	1	4	yes	—
21	C₄×#6923	—	2	4	no	—
22	C₄×#6923	—	2	4	no	—
23	C₄×#6923	C₈	1	8	yes	complemented
24	D₆₁⋊C₈	—	2	4	no	—
25	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×#92418	C₂×C₄	1	2	yes	—
31	C₈×D₆₁	—	2	4	no	—
32	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×#6923	—	2	4	no	—
34	C₂×#6923	—	2	4	no	—
35	C₂×#6923	C₂×C₈	1	8	yes	—
36	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₆₁	C₄²	1	1	yes	characteristic
40	C₂×Dic₆₁	C₄²	1	1	yes	characteristic
41	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄×D₆₁	C₂×C₈	1	4	yes	—
43	C₄×D₆₁	M₁₆	1	4	yes	—
44	C₄×D₆₁	#41	1	4	yes	—
45	#92420	—	2	4	no	—
46	C₈×C₆₁	—	2	2	no	—
47	#92418	—	2	2	no	—
48	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
49	C₂×D₆₁	#117	1	2	yes	—
50	C₂×D₆₁	C₂²×C₈	1	2	yes	—
51	C₂×D₆₁	C₂×M₁₆	1	2	yes	—
52	#6923	—	2	8	no	—
53	C₄×C₆₁	C₄×C₈	1	2	yes	—
54	C₄×C₆₁	#97	1	2	yes	—
55	C₄×C₆₁	#117	1	2	yes	—
56	Dic₆₁	C₄×C₈	1	2	yes	—
57	Dic₆₁	#97	1	2	yes	—
58	Dic₆₁	#117	1	2	yes	—
59	#1744	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	#376	1	1	yes	characteristic
61	C₂×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂×C₆₁	C₂×#97	1	1	yes	characteristic
63	D₆₁	#431	1	4	yes	—
64	C₂×C₄×C₈	—	61	1	no	—
65	C₂×C₄×C₈	—	61	1	no	—
66	C₄³	—	61	1	no	—
67	#336	—	61	4	no	—
68	C₆₁	#1744	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	61	2	no	—
70	C₂²×C₈	—	61	2	no	—
71	C₂×C₄²	—	61	2	no	—
72	C₂×C₄²	—	61	1	no	—
73	C₂×C₄²	—	61	2	no	—
74	C₂×C₄²	—	61	2	no	—
75	C₄×C₈	—	61	2	no	—
76	C₄×C₈	—	61	2	no	—
77	C₄×C₈	—	61	2	no	—
78	C₄×C₈	—	61	1	no	—
79	C₄×C₈	#6923	1	1	yes	characteristic, complemented
80	C₂²×C₄	—	61	1	no	—
81	C₂²×C₄	—	61	1	no	—
82	C₂²×C₄	—	61	1	no	—
83	C₂²×C₄	—	61	4	no	—
84	C₂×C₈	—	61	4	no	—
85	C₂×C₈	—	61	2	no	—
86	C₂×C₈	—	122	4	no	—
87	C₂×C₈	—	122	4	no	—
88	C₂×C₈	C₂×#6923	1	2	yes	—
89	C₄²	—	61	8	no	—
90	C₄²	—	61	1	no	—
91	C₄²	—	61	1	no	—
92	C₄²	—	61	1	no	—
93	C₄²	—	122	4	no	—
94	C₄²	—	122	4	no	—
95	C₄²	C₂×#6923	1	1	yes	characteristic
96	C₂³	—	61	1	no	—
97	C₂×C₄	—	61	1	no	—
98	C₂×C₄	—	61	1	no	—
99	C₂×C₄	—	61	1	no	—
100	C₂×C₄	—	61	4	no	—
101	C₂×C₄	—	61	4	no	—
102	C₂×C₄	—	61	4	no	—
103	C₂×C₄	—	61	8	no	—
104	C₂×C₄	—	122	4	no	—
105	C₂×C₄	—	122	4	no	—
106	C₂×C₄	C₄×#6923	1	1	yes	characteristic
107	C₂×C₄	C₄×#6923	1	1	yes	characteristic
108	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	122	4	no	—
111	C₈	—	122	2	no	—
112	C₂²	—	61	2	no	—
113	C₂²	—	61	2	no	—
114	C₂²	—	61	2	no	—
115	C₂²	C₂×C₄×#6923	1	1	yes	characteristic
116	C₄	—	61	2	no	—
117	C₄	—	61	2	no	—
118	C₄	—	61	2	no	—
119	C₄	—	122	8	no	—
120	C₄	C₈×#6923	1	2	yes	—
121	C₄	#14519831	1	2	yes	—
122	C₄	#14519856	1	2	yes	—
123	C₂	—	61	4	no	—
124	C₂	#16840698	1	1	yes	characteristic
125	C₂	C₂×C₈×#6923	1	1	yes	characteristic
126	C₂	C₂×#14519831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₆₁⋊C₈

C₂

yes

maximal, characteristic

C₄²×#6923

C₂

yes

maximal, characteristic

(C₂×#6923)⋊C₈

C₂

yes

maximal

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

C₂²

yes

—

C₂×C₄×#6923

C₄

yes

—

C₂×C₄×#6923

C₄

yes

—

C₂×C₄×#6923

C₂²

yes

—

C₄×#92420

C₄

yes

complemented

C₄×#92420

C₄

yes

complemented

C₂×C₈×D₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄×#92418

C₄

yes

characteristic, complemented

C₈×Dic₆₁

C₄

yes

complemented

C₄×C₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×#92420

C₂×C₄

yes

—

C₄×#6923

—

C₄×#6923

—

C₄×#6923

C₈

yes

complemented

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

—

C₈×D₆₁

—

C₂²×D₆₁

C₂²×C₄

yes

—

—

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₄²

yes

characteristic

C₂×Dic₆₁

C₄²

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₄×D₆₁

C₂×C₈

yes

—

C₄×D₆₁

M₁₆

yes

—

C₄×D₆₁

#41

yes

—

#92420

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂×D₆₁

#117

yes

—

C₂×D₆₁

C₂²×C₈

yes

—

C₂×D₆₁

C₂×M₁₆

yes

—

#6923

—

C₄×C₆₁

C₄×C₈

yes

—

C₄×C₆₁

#97

yes

—

C₄×C₆₁

#117

yes

—

Dic₆₁

C₄×C₈

yes

—

Dic₆₁

#97

yes

—

Dic₆₁

#117

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×C₆₁

C₂×#97

yes

characteristic

D₆₁

#431

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

#336

—

C₆₁

#1744

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6923

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

C₂×#6923

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

122

—

106

C₂×C₄

C₄×#6923

yes

characteristic

107

C₂×C₄

C₄×#6923

yes

characteristic

108

C₂×C₄

C₂²×#6923

yes

characteristic

109

C₈

—

110

C₈

—

122

—

111

C₈

—

122

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄×#6923

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

122

—

120

C₄

C₈×#6923

yes

—

121

C₄

#14519831

yes

—

122

C₄

#14519856

yes

—

123

C₂

—

124

C₂

#16840698

yes

characteristic

125

C₂

C₂×C₈×#6923

yes

characteristic

126

C₂

C₂×#14519831

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#6923
commutator	C₂×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄×C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄²
hypercenter	C₄×C₈	#6923
hypocenter	C₆₁	#1744
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#6923

commutator

C₂×C₆₁

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄×C₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₄²

C₄×C₈

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
9. C₄×#92420	C₄	semidirect
10. C₄×#92420	C₄	semidirect
13. C₄×#92418	C₄	semidirect
14. C₈×Dic₆₁	C₄	semidirect
15. C₄×C₈×C₆₁	C₄	semidirect
23. C₄×#6923	C₈	semidirect
68. C₆₁	#1744	semidirect
79. C₄×C₈	#6923	semidirect

H ≅ G/N

type

9. C₄×#92420

C₄

semidirect

10. C₄×#92420

C₄

semidirect

13. C₄×#92418

C₄

semidirect

14. C₈×Dic₆₁

C₄

semidirect

15. C₄×C₈×C₆₁

C₄

semidirect

23. C₄×#6923

C₈

semidirect

68. C₆₁

#1744

semidirect

79. C₄×C₈

#6923

semidirect

Subgroups — #21461532

Conjugacy classes of subgroups

Complements