Group #24907503 — subgroups

Subgroups — #24907503

Order 10624.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#13050574	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581727	C₂	1	2	yes	maximal, complemented
3	#19127665	C₂	1	1	yes	maximal, characteristic
4	#19127465	C₂	1	2	yes	maximal, complemented
5	#407×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	—	2	4	no	—
7	C₂×#13050574	C₄	1	4	yes	—
8	C₂×#13050574	C₂²	1	2	yes	—
9	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	#14581727	—	2	4	no	—
12	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₈₃	—	2	2	no	—
16	#13050574	—	2	8	no	—
17	#13050574	—	2	2	no	—
18	#13050574	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	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	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
24	C₂⁴×C₈₃	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	—	2	2	no	—
26	C₂²×C₄×C₈₃	—	2	2	no	—
27	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₈₃	D₄	1	2	yes	—
29	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₈₃	—	2	2	no	—
32	M₁₆×C₈₃	—	2	2	no	—
33	C₂×C₈×C₈₃	—	2	2	no	—
34	C₂³×C₈₃	—	2	2	no	—
35	C₂³×C₈₃	—	4	2	no	—
36	C₂³×C₈₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₈₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	—	2	2	no	—
39	C₂×C₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	2	4	no	—
42	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	2	yes	—
47	C₈×C₈₃	—	4	2	no	—
48	C₈₃⋊C₈	—	2	2	no	—
49	C₈₃⋊C₈	—	4	2	no	—
50	C₈₃⋊C₈	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	—	2	2	no	—
53	C₂²×C₈₃	—	2	4	no	—
54	C₂²×C₈₃	—	4	4	no	—
55	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₈₃	—	2	2	no	—
58	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₈₃	#117	1	1	yes	characteristic
60	C₄×C₈₃	#121	1	2	yes	—
61	C₂×C₈₃	—	2	2	no	—
62	C₂×C₈₃	—	4	2	no	—
63	C₂×C₈₃	#380	1	1	yes	characteristic
64	C₂×C₈₃	#430	1	2	yes	—
65	#1811	—	83	1	no	maximal, Sylow
66	C₈₃	#1811	1	1	yes	characteristic, complemented, Sylow
67	C₂²×M₁₆	—	83	1	no	—
68	C₂×#108	—	83	2	no	—
69	#407	—	83	1	no	—
70	#407	D₈₃	1	1	yes	characteristic
71	#341	—	83	2	no	—
72	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
73	C₂×M₁₆	—	83	4	no	—
74	C₂×M₁₆	—	83	2	no	—
75	C₂×M₁₆	—	166	4	no	—
76	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
77	C₂²×C₈	—	83	1	no	—
78	C₂²×C₈	—	83	1	no	—
79	#108	—	166	4	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	166	2	no	—
82	C₂⁴	Dic₁₆₆	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₈₃	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	166	2	no	—
92	M₁₆	—	166	2	no	—
93	M₁₆	—	166	8	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	83	2	no	—
96	C₂×C₈	—	83	4	no	—
97	C₂×C₈	—	83	2	no	—
98	C₂×C₈	—	166	2	no	—
99	C₂×C₈	—	166	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	Dic₈₃⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₁₆₆	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
110	C₂×C₄	D₄×D₈₃	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₄	#13050589	1	2	yes	—
113	C₈	—	4	2	no	—
114	C₈	—	332	2	no	—
115	C₈	—	83	4	no	—
116	C₈	—	166	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	4	4	no	—
121	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
122	C₂²	#14581769	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
125	C₄	#14581748	1	1	yes	characteristic
126	C₄	#14581764	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	#19127559	1	1	yes	characteristic
130	C₂	#19127749	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13050574

C₂

yes

maximal, characteristic

C₂×#14581727

C₂

yes

maximal, complemented

#19127665

C₂

yes

maximal, characteristic

#19127465

C₂

yes

maximal, complemented

#407×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂×#13050574

C₄

yes

—

C₂×#13050574

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581727

—

C₂²⋊(C₈₃⋊C₈)

—

C₂³×C₄×C₈₃

C₂²

yes

characteristic

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂²⋊C₈×C₈₃

—

#13050574

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂⁴×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₄⋊C₄

yes

—

C₂³×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₂²⋊C₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#380

yes

characteristic

C₂×C₈₃

#430

yes

—

#1811

—

maximal, Sylow

C₈₃

#1811

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#108

—

#407

—

#407

D₈₃

yes

characteristic

#341

—

C₂³×C₄

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

166

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

C₂⁴

Dic₁₆₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

100

C₂³

—

101

C₂³

—

102

C₂³

Dic₈₃⋊C₄

yes

—

103

C₂³

C₂×Dic₁₆₆

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

Dic₈₃⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

110

C₂×C₄

D₄×D₈₃

yes

—

111

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₄

#13050589

yes

—

113

C₈

—

114

C₈

—

332

—

115

C₈

—

116

C₈

—

166

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

122

C₂²

#14581769

yes

—

123

C₄

—

124

C₄

C₂²⋊C₄×D₈₃

yes

characteristic

125

C₄

#14581748

yes

characteristic

126

C₄

#14581764

yes

—

127

C₂

—

128

C₂

—

129

C₂

#19127559

yes

characteristic

130

C₂

#19127749

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#407×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₃	Q₈
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#407	D₈₃
hypocenter	C₈₃	#1811
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#407×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂⁴×C₈₃

Q₈

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#14581727	C₂	semidirect
4. #19127465	C₂	semidirect
66. C₈₃	#1811	semidirect

H ≅ G/N

type

2. C₂×#14581727

C₂

semidirect

4. #19127465

C₂

semidirect

66. C₈₃

#1811

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13050574	2
C₂×#14581727	2
#19127665	2
#19127465	2
#407×C₈₃	2
#1811	83

subgroup

index

#407×C₈₃

Subgroups — #24907503

Conjugacy classes of subgroups

Complements

Maximal subgroups