Group #21443716 — subgroups

Subgroups — #21443716

Order 7808.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16839815	C₂	1	1	yes	maximal, characteristic
2	#16839786	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	1	no	—
5	C₄⋊#92418	—	2	1	no	—
6	C₄⋊#92418	—	2	1	no	—
7	C₄⋊#92418	C₄	1	2	yes	—
8	C₂²⋊#92418	C₄	1	2	yes	—
9	C₄×#92418	—	2	1	no	—
10	C₂²×C₈×C₆₁	—	2	1	no	—
11	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₆₁	—	2	1	no	—
13	C₄⋊C₈×C₆₁	C₄	1	2	yes	—
14	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₆₁	—	2	2	no	—
18	C₂×C₈×C₆₁	—	4	1	no	—
19	C₂×C₈×C₆₁	—	4	1	no	—
20	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₁	D₄	1	1	yes	characteristic
23	C₄²×C₆₁	D₄	1	1	yes	characteristic
24	#11758194	—	4	1	no	—
25	#11758194	—	4	1	no	—
26	C₂×#92418	—	2	2	no	—
27	C₂×#92418	—	2	2	no	—
28	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	—	2	2	no	—
30	C₂×C₄×C₆₁	—	4	1	no	—
31	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
38	C₂×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
41	C₈×C₆₁	—	4	2	no	—
42	#92418	—	4	2	no	—
43	#92418	—	4	2	no	—
44	C₂²×C₆₁	—	2	1	no	—
45	C₂²×C₆₁	—	2	1	no	—
46	C₂²×C₆₁	#95	1	1	yes	characteristic
47	C₂²×C₆₁	#106	1	1	yes	characteristic
48	C₂²×C₆₁	#107	1	1	yes	characteristic
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	#102	1	1	yes	characteristic
53	C₄×C₆₁	#102	1	1	yes	characteristic
54	C₄×C₆₁	#103	1	1	yes	characteristic
55	C₄×C₆₁	#103	1	1	yes	characteristic
56	C₄×C₆₁	#108	1	2	yes	—
57	#1301	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	—	2	1	no	—
59	C₂×C₆₁	—	2	1	no	—
60	C₂×C₆₁	#340	1	1	yes	characteristic
61	C₂×C₆₁	#341	1	1	yes	characteristic
62	C₂×C₆₁	#344	1	1	yes	characteristic
63	#431	—	61	1	no	—
64	#423	—	61	1	no	—
65	C₂×C₄⋊C₈	D₆₁	1	1	yes	characteristic
66	C₆₁	#1301	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	122	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	61	2	no	—
72	C₄⋊C₈	—	122	1	no	—
73	C₄⋊C₈	—	122	1	no	—
74	C₄⋊C₈	Dic₆₁	1	2	yes	—
75	C₂²⋊C₈	—	61	2	no	—
76	C₄×C₈	—	122	1	no	—
77	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
80	M₁₆	—	244	1	no	—
81	M₁₆	—	244	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	122	2	no	—
86	C₂×C₈	—	122	2	no	—
87	C₄²	C₄×D₆₁	1	1	yes	characteristic
88	C₄²	D₂₄₄	1	1	yes	characteristic
89	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
91	C₂³	#11758196	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₆₁	1	1	yes	characteristic
95	C₂×C₄	#11758196	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
97	C₂×C₄	#11758198	1	1	yes	characteristic
98	C₂×C₄	#11758198	1	1	yes	characteristic
99	C₂×C₄	#11758199	1	1	yes	characteristic
100	C₂×C₄	#11758200	1	1	yes	characteristic
101	C₂×C₄	#11758201	1	1	yes	characteristic
102	C₂×C₄	#11758202	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	244	2	no	—
106	C₈	—	244	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#14519684	1	1	yes	characteristic
110	C₂²	#14519708	1	1	yes	characteristic
111	C₂²	#14519710	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#14519697	1	1	yes	characteristic
116	C₄	#14519698	1	1	yes	characteristic
117	C₄	#14519700	1	1	yes	characteristic
118	C₄	#14519701	1	1	yes	characteristic
119	C₄	#14519713	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#16839524	1	1	yes	characteristic
123	C₂	#16839527	1	1	yes	characteristic
124	C₂	#16839534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839815

C₂

yes

maximal, characteristic

#16839786

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

C₂²⋊#92418

C₄

yes

—

C₄×#92418

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

—

—

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

characteristic

C₂×C₄×C₆₁

Q₁₆

yes

characteristic

C₈×C₆₁

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#1301

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

C₂×C₄⋊C₈

D₆₁

yes

characteristic

C₆₁

#1301

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

C₄⋊C₈

Dic₆₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

122

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

M₁₆

—

244

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

C₄×D₆₁

yes

characteristic

C₄²

D₂₄₄

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²⋊D₆₁

yes

characteristic

C₂³

#11758196

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₄

#11758196

yes

characteristic

C₂×C₄

C₄⋊Dic₆₁

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#11758200

yes

characteristic

101

C₂×C₄

#11758201

yes

characteristic

102

C₂×C₄

#11758202

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

104

C₈

—

105

C₈

—

244

—

106

C₈

—

244

—

107

C₂²

—

108

C₂²

—

109

C₂²

#14519684

yes

characteristic

110

C₂²

#14519708

yes

characteristic

111

C₂²

#14519710

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#14519697

yes

characteristic

116

C₄

#14519698

yes

characteristic

117

C₄

#14519700

yes

characteristic

118

C₄

#14519701

yes

characteristic

119

C₄

#14519713

yes

—

120

C₂

—

121

C₂

—

122

C₂

#16839524

yes

characteristic

123

C₂

#16839527

yes

characteristic

124

C₂

#16839534

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11758198
commutator	C₂×C₄×C₆₁	C₄²
Frattini	C₂×C₄²	C₂×D₆₁
Fitting	C₂×C₄⋊C₈×C₆₁	C₂
socle	C₂²×C₆₁	#95
hypercenter	C₂×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#1301
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#11758198

commutator

C₂×C₄×C₆₁

C₄²

Frattini

C₂×C₄²

C₂×D₆₁

Fitting

C₂×C₄⋊C₈×C₆₁

C₂

socle

C₂²×C₆₁

#95

hypercenter

C₂×C₄⋊C₈

D₆₁

hypocenter

C₆₁

#1301

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
66. C₆₁	#1301	semidirect

H ≅ G/N

type

66. C₆₁

#1301

semidirect

Subgroups — #21443716

Conjugacy classes of subgroups

Complements