Group #21462006 — subgroups

Subgroups — #21462006

Order 7808.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#16840766	C₂	1	1	yes	maximal, characteristic
2	#16840698	C₂	1	1	yes	maximal, characteristic
3	#16840632	C₂	1	4	yes	maximal
4	#16840318	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758213	C₂²	1	2	yes	—
6	C₂×C₄×#6923	C₂²	1	2	yes	—
7	C₄⋊#92420	—	2	2	no	—
8	(C₂×#6923)⋊C₄	—	2	2	no	—
9	#14519776	C₄	1	2	yes	complemented
10	#14519769	C₄	1	1	yes	characteristic, complemented
11	#14519755	—	2	2	no	—
12	#14519742	—	2	2	no	—
13	#14519735	C₂²	1	2	yes	—
14	C₄²×D₆₁	C₂²	1	1	yes	characteristic
15	#125×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6923	—	2	2	no	—
17	C₂×C₄×D₆₁	D₄	1	2	yes	—
18	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×#92420	—	2	2	no	—
21	C₄×#6923	—	2	2	no	—
22	C₄×#6923	Q₈	1	4	yes	complemented
23	#11758213	—	2	4	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
26	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	#11758198	—	2	2	no	—
28	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	#11758196	—	2	2	no	—
30	#11758196	—	2	2	no	—
31	C₄×Dic₆₁	—	2	2	no	—
32	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₆₁	D₄	1	2	yes	—
34	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×#6923	—	2	4	no	—
36	C₂×#6923	—	4	4	no	—
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₆₁	—	2	2	no	—
41	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₄×D₆₁	—	4	2	no	—
44	C₄×D₆₁	C₂×Q₈	1	2	yes	—
45	C₄×D₆₁	#41	1	2	yes	—
46	Dic₁₂₂	—	4	4	no	—
47	#92420	—	4	2	no	—
48	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₆₁	—	2	2	no	—
50	C₂×D₆₁	C₂²⋊Q₈	1	2	yes	—
51	#6923	—	4	4	no	—
52	C₄×C₆₁	—	2	2	no	—
53	C₄×C₆₁	—	4	2	no	—
54	C₄×C₆₁	C₄×Q₈	1	2	yes	—
55	Dic₆₁	—	2	2	no	—
56	Dic₆₁	—	4	2	no	—
57	Dic₆₁	#124	1	1	yes	characteristic
58	Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
59	#1995	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	#391	1	1	yes	characteristic
61	C₂×C₆₁	#421	1	2	yes	—
62	D₆₁	—	2	2	no	—
63	#533	—	61	1	no	—
64	#423	—	61	1	no	—
65	#376	—	61	1	no	—
66	#339	—	61	4	no	—
67	C₆₁	#1995	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	61	2	no	—
69	C₄⋊Q₈	—	61	2	no	—
70	#125	—	61	1	no	—
71	#125	#6923	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	122	2	no	—
73	C₄×Q₈	—	122	2	no	—
74	#117	—	61	2	no	—
75	C₂×C₄²	—	61	1	no	—
76	C₂×C₄²	—	61	2	no	—
77	C₄⋊C₈	—	122	2	no	—
78	#95	—	122	2	no	—
79	C₂×Q₈	—	122	2	no	—
80	C₂²×C₄	—	61	1	no	—
81	C₂²×C₄	—	61	2	no	—
82	C₂²×C₄	—	122	2	no	—
83	M₁₆	—	122	4	no	—
84	C₂×C₈	—	122	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	61	2	no	—
87	C₄⋊C₄	—	122	2	no	—
88	C₄⋊C₄	—	122	2	no	—
89	C₄⋊C₄	C₂×#6923	1	2	yes	—
90	C₂²⋊C₄	—	122	2	no	—
91	C₄²	—	61	2	no	—
92	C₄²	—	61	1	no	—
93	C₄²	—	61	4	no	—
94	C₄²	—	122	2	no	—
95	C₄²	—	122	2	no	—
96	C₄²	C₂×#6923	1	1	yes	characteristic
97	C₂³	—	61	1	no	—
98	Q₈	—	244	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	61	1	no	—
101	C₂×C₄	—	61	2	no	—
102	C₂×C₄	—	61	2	no	—
103	C₂×C₄	—	61	2	no	—
104	C₂×C₄	—	122	4	no	—
105	C₂×C₄	—	122	2	no	—
106	C₂×C₄	—	244	2	no	—
107	C₂×C₄	—	244	4	no	—
108	C₂×C₄	C₂²⋊#6923	1	2	yes	—
109	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
110	C₈	—	244	2	no	—
111	C₂²	—	61	2	no	—
112	C₂²	—	122	2	no	—
113	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	61	1	no	—
117	C₄	—	61	1	no	—
118	C₄	—	122	2	no	—
119	C₄	—	244	4	no	—
120	C₄	—	244	2	no	—
121	C₄	Q₈×#6923	1	2	yes	—
122	C₂	—	122	2	no	—
123	C₂	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
124	C₂	#16840759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840766

C₂

yes

maximal, characteristic

#16840698

C₂

yes

maximal, characteristic

#16840632

C₂

yes

maximal

#16840318

C₂

yes

maximal, characteristic

C₂×#11758213

C₂²

yes

—

C₂×C₄×#6923

C₂²

yes

—

C₄⋊#92420

—

(C₂×#6923)⋊C₄

—

#14519776

C₄

yes

complemented

#14519769

C₄

yes

characteristic, complemented

#14519755

—

#14519742

—

#14519735

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#125×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×#92420

—

C₄×#6923

—

C₄×#6923

Q₈

yes

complemented

#11758213

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

C₂×#6923

—

C₂×#6923

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×Q₈

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₂²⋊Q₈

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#124

yes

characteristic

Dic₆₁

C₄⋊Q₈

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

—

—

—

—

—

C₆₁

#1995

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#6923

yes

characteristic, complemented

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

122

—

#95

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

M₁₆

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂×#6923

yes

—

C₂²⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

Q₈

—

244

—

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

—

244

—

107

C₂×C₄

—

244

—

108

C₂×C₄

C₂²⋊#6923

yes

—

109

C₂×C₄

C₂²×#6923

yes

characteristic

110

C₈

—

244

—

111

C₂²

—

112

C₂²

—

122

—

113

C₂²

C₂×C₂²⋊#6923

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

122

—

119

C₄

—

244

—

120

C₄

—

244

—

121

C₄

Q₈×#6923

yes

—

122

C₂

—

122

—

123

C₂

(C₂×#6923)⋊Q₈

yes

characteristic

124

C₂

#16840759

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	#125×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#125	#6923
hypocenter	C₆₁	#1995
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

#125×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #14519776	C₄	semidirect
10. #14519769	C₄	semidirect
15. #125×C₆₁	C₄	semidirect
22. C₄×#6923	Q₈	semidirect
67. C₆₁	#1995	semidirect
71. #125	#6923	semidirect

H ≅ G/N

type

9. #14519776

C₄

semidirect

10. #14519769

C₄

semidirect

15. #125×C₆₁

C₄

semidirect

22. C₄×#6923

Q₈

semidirect

67. C₆₁

#1995

semidirect

71. #125

#6923

semidirect

Subgroups — #21462006

Conjugacy classes of subgroups

Complements