Group #24906231 — subgroups

Subgroups — #24906231

Order 10624.

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

normal subgroups only

This group has 166 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	#19127725	C₂	1	1	yes	maximal, characteristic
2	C₄×#13050574	C₂	1	1	yes	maximal, characteristic
3	#19127453	C₂	1	2	yes	maximal, complemented
4	#19127451	C₂	1	2	yes	maximal, complemented
5	#432×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	—	2	1	no	—
7	C₂×#13050574	C₄	1	2	yes	—
8	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
10	#14581703	C₄	1	1	yes	characteristic
11	#14581703	C₄	1	1	yes	characteristic
12	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₈₃	—	2	1	no	—
17	#97×C₈₃	—	2	1	no	—
18	#13050574	—	2	4	no	—
19	#13050574	—	4	2	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₈₃	—	2	1	no	—
27	C₂×C₈×C₈₃	—	2	1	no	—
28	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₄²×C₈₃	Q₈	1	1	yes	characteristic
31	C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	—	2	2	no	—
34	C₂×C₄×C₈₃	—	2	1	no	—
35	C₂×C₄×C₈₃	—	2	1	no	—
36	C₂×C₄×C₈₃	—	4	1	no	—
37	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₈₃	—	4	1	no	—
44	C₈×C₈₃	—	4	1	no	—
45	C₈₃⋊C₈	—	2	2	no	—
46	C₈₃⋊C₈	—	4	2	no	—
47	C₈₃⋊C₈	M₁₆	1	4	yes	complemented
48	C₂²×C₈₃	—	2	1	no	—
49	C₂²×C₈₃	—	2	1	no	—
50	C₂²×C₈₃	—	2	1	no	—
51	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₈₃	—	2	1	no	—
53	C₄×C₈₃	—	2	1	no	—
54	C₄×C₈₃	—	2	1	no	—
55	C₄×C₈₃	—	2	1	no	—
56	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
57	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₄×C₈₃	#137	1	1	yes	characteristic
59	C₂×C₈₃	—	4	1	no	—
60	C₂×C₈₃	#423	1	1	yes	characteristic
61	C₂×C₈₃	#428	1	1	yes	characteristic
62	C₂×C₈₃	#430	1	1	yes	characteristic
63	#1569	—	83	1	no	maximal, Sylow
64	C₈₃	#1569	1	1	yes	characteristic, complemented, Sylow
65	#432	D₈₃	1	1	yes	characteristic
66	#423	—	83	1	no	—
67	C₄×M₁₆	—	83	1	no	—
68	#335	—	83	2	no	—
69	#334	—	83	2	no	—
70	C₂×M₁₆	—	83	2	no	—
71	C₂×M₁₆	—	166	1	no	—
72	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
73	C₄⋊C₈	—	83	2	no	—
74	C₄⋊C₈	—	166	1	no	—
75	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
76	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
77	C₂²⋊C₈	—	2	1	no	—
78	#97	—	2	1	no	—
79	#97	—	83	1	no	—
80	#97	—	83	1	no	—
81	C₄×C₈	—	83	2	no	—
82	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
85	M₁₆	—	332	2	no	—
86	M₁₆	—	166	4	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	83	2	no	—
90	C₂×C₈	—	83	2	no	—
91	C₂×C₈	—	166	2	no	—
92	C₄²	Dic₁₆₆	1	1	yes	characteristic
93	C₄²	C₄×D₈₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
95	C₄²	C₂²×D₈₃	1	1	yes	characteristic
96	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
102	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
103	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
107	C₈	—	4	1	no	—
108	C₈	—	4	1	no	—
109	C₈	—	332	2	no	—
110	C₈	—	83	4	no	—
111	C₈	—	166	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	M₁₆×D₈₃	1	2	yes	—
121	C₄	#14581817	1	1	yes	characteristic
122	C₄	#14581822	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	#19127724	1	1	yes	characteristic
125	C₂	#19127744	1	1	yes	characteristic
126	C₂	#19127749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127725

C₂

yes

maximal, characteristic

C₄×#13050574

C₂

yes

maximal, characteristic

#19127453

C₂

yes

maximal, complemented

#19127451

C₂

yes

maximal, complemented

#432×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂×#13050574

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581703

C₄

yes

characteristic

#14581703

C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

M₁₆

yes

complemented

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#432

D₈₃

yes

characteristic

#423

—

C₄×M₁₆

—

#335

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

332

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₂²⋊D₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

102

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

103

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

104

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

106

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

332

—

110

C₈

—

111

C₈

—

166

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

M₁₆×D₈₃

yes

—

121

C₄

#14581817

yes

characteristic

122

C₄

#14581822

yes

characteristic

123

C₂

—

124

C₂

#19127724

yes

characteristic

125

C₂

#19127744

yes

characteristic

126

C₂

#19127749

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#432×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#432	D₈₃
hypocenter	C₈₃	#1569
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#432×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #19127453	C₂	semidirect
4. #19127451	C₂	semidirect
47. C₈₃⋊C₈	M₁₆	semidirect
64. C₈₃	#1569	semidirect

H ≅ G/N

type

3. #19127453

C₂

semidirect

4. #19127451

C₂

semidirect

47. C₈₃⋊C₈

M₁₆

semidirect

64. C₈₃

#1569

semidirect

Subgroups — #24906231

Conjugacy classes of subgroups

Complements