Group #21461752 — subgroups

Subgroups — #21461752

Order 7808.

300 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₂×#14519844	C₂	1	4	yes	maximal
2	C₂×C₄⋊#92420	C₂	1	2	yes	maximal
3	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92420	C₂²	1	4	yes	—
5	#14519844	—	2	8	no	—
6	C₄⋊#92420	—	2	4	no	—
7	C₂×C₄×Dic₆₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
16	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
17	C₂×#92420	—	2	4	no	—
18	C₂×#92420	—	2	8	no	—
19	C₄²×C₆₁	—	2	2	no	—
20	C₄×Dic₆₁	—	2	4	no	—
21	C₄×Dic₆₁	C₈	1	4	yes	complemented
22	C₄×Dic₆₁	D₄	1	2	yes	—
23	C₄×Dic₆₁	D₄	1	4	yes	—
24	C₄×Dic₆₁	Q₈	1	2	yes	—
25	C₄×Dic₆₁	Q₈	1	4	yes	—
26	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₆₁	—	2	2	no	—
28	C₂×C₄×C₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
33	C₂×Dic₆₁	—	2	2	no	—
34	C₂×Dic₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₆₁	C₄⋊C₄	1	4	yes	—
36	C₂×Dic₆₁	C₂×C₈	1	4	yes	—
37	C₂×Dic₆₁	M₁₆	1	4	yes	—
38	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
40	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
41	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
42	#92420	—	4	4	no	—
43	C₂²×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
44	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
47	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
48	C₄×C₆₁	—	2	4	no	—
49	C₄×C₆₁	C₄⋊C₈	1	4	yes	complemented
50	Dic₆₁	—	2	2	no	—
51	Dic₆₁	C₄⋊C₈	1	8	yes	complemented
52	Dic₆₁	#127	1	1	yes	characteristic
53	Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₆₁	C₄⋊Q₈	1	2	yes	—
55	#1842	—	61	1	no	maximal, Sylow
56	C₂×C₆₁	#384	1	1	yes	characteristic
57	C₂×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
58	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₆₁	#423	1	2	yes	—
60	C₂×C₆₁	#423	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	61	2	no	—
62	C₂×C₄⋊C₈	—	61	4	no	—
63	C₄³	—	61	1	no	—
64	C₆₁	#1842	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	61	4	no	—
66	C₂×C₄²	—	61	2	no	—
67	C₂×C₄²	—	61	1	no	—
68	C₂×C₄²	—	61	1	no	—
69	C₂×C₄²	—	61	2	no	—
70	C₂×C₄²	#6923	1	1	yes	characteristic
71	C₄⋊C₈	—	122	4	no	—
72	C₄⋊C₈	—	122	8	no	—
73	C₂²×C₄	—	61	1	no	—
74	C₂²×C₄	—	61	2	no	—
75	C₂²×C₄	—	61	1	no	—
76	C₂²×C₄	C₂×#6923	1	2	yes	—
77	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
78	C₂×C₈	—	122	4	no	—
79	C₂×C₈	—	122	8	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	61	4	no	—
82	C₄²	—	61	4	no	—
83	C₄²	—	61	2	no	—
84	C₄²	—	61	4	no	—
85	C₄²	—	61	2	no	—
86	C₄²	—	122	4	no	—
87	C₂³	C₂²×#6923	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	61	4	no	—
91	C₂×C₄	—	61	2	no	—
92	C₂×C₄	—	61	1	no	—
93	C₂×C₄	—	61	2	no	—
94	C₂×C₄	—	61	1	no	—
95	C₂×C₄	—	61	4	no	—
96	C₂×C₄	—	61	4	no	—
97	C₂×C₄	—	61	2	no	—
98	C₂×C₄	—	122	2	no	—
99	C₂×C₄	D₆₁⋊C₈	1	2	yes	—
100	C₂×C₄	C₄⋊#6923	1	4	yes	—
101	C₂×C₄	C₄⋊#6923	1	2	yes	—
102	C₂×C₄	#11758217	1	2	yes	—
103	C₈	—	244	4	no	—
104	C₂²	C₂×C₄⋊#6923	1	2	yes	—
105	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
106	C₂²	C₂×D₆₁⋊C₈	1	1	yes	characteristic
107	C₂²	C₂×#11758217	1	1	yes	characteristic
108	C₂²	D₆₁⋊M₁₆	1	2	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	61	1	no	—
111	C₄	—	61	2	no	—
112	C₄	—	61	1	no	—
113	C₄	—	61	8	no	—
114	C₄	—	122	2	no	—
115	C₄	#14519844	1	4	yes	complemented
116	C₂	#16840708	1	1	yes	characteristic
117	C₂	D₆₁⋊(C₄⋊C₈)	1	2	yes	—
118	C₂	C₂×#14519844	1	1	yes	characteristic
119	C₂	#16840765	1	2	yes	—
120	C₂	#16840767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519844

C₂

yes

maximal

C₂×C₄⋊#92420

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

—

#14519844

—

C₄⋊#92420

—

C₂×C₄×Dic₆₁

C₄

yes

—

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×#92420

—

C₂×#92420

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₈

yes

complemented

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂×C₈

yes

—

C₂×Dic₆₁

M₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

characteristic

#92420

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

C₂²×C₈

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

complemented

Dic₆₁

—

Dic₆₁

C₄⋊C₈

yes

complemented

Dic₆₁

#127

yes

characteristic

Dic₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

C₄⋊Q₈

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊C₈

yes

—

C₂×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₆₁

#423

yes

—

C₂×C₆₁

#423

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₄³

—

C₆₁

#1842

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#6923

yes

characteristic

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

—

C₂²×C₄

C₂×#6923

yes

characteristic

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₂³

C₂²×#6923

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

D₆₁⋊C₈

yes

—

100

C₂×C₄

C₄⋊#6923

yes

—

101

C₂×C₄

C₄⋊#6923

yes

—

102

C₂×C₄

#11758217

yes

—

103

C₈

—

244

—

104

C₂²

C₂×C₄⋊#6923

yes

—

105

C₂²

C₂×C₄⋊#6923

yes

characteristic

106

C₂²

C₂×D₆₁⋊C₈

yes

characteristic

107

C₂²

C₂×#11758217

yes

characteristic

108

C₂²

D₆₁⋊M₁₆

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

122

—

115

C₄

#14519844

yes

complemented

116

C₂

#16840708

yes

characteristic

117

C₂

D₆₁⋊(C₄⋊C₈)

yes

—

118

C₂

C₂×#14519844

yes

characteristic

119

C₂

#16840765

yes

—

120

C₂

#16840767

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6923
commutator	C₂²×C₆₁	C₂²×C₈
Fitting	C₂×C₄²×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂×C₄²	#6923
hypocenter	C₆₁	#1842
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#6923

commutator

C₂²×C₆₁

C₂²×C₈

Fitting

C₂×C₄²×C₆₁

C₄

socle

C₂³×C₆₁

C₂²×C₄

C₂×C₄²

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
21. C₄×Dic₆₁	C₈	semidirect
49. C₄×C₆₁	C₄⋊C₈	semidirect
51. Dic₆₁	C₄⋊C₈	semidirect
64. C₆₁	#1842	semidirect
115. C₄	#14519844	semidirect

H ≅ G/N

type

21. C₄×Dic₆₁

C₈

semidirect

49. C₄×C₆₁

C₄⋊C₈

semidirect

51. Dic₆₁

C₄⋊C₈

semidirect

64. C₆₁

#1842

semidirect

115. C₄

#14519844

semidirect

Subgroups — #21461752

Conjugacy classes of subgroups

Complements