Group #24908248 — subgroups

Subgroups — #24908248

Order 10624.

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

normal subgroups only

This group has 196 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	C₂×#14581728	C₂	1	1	yes	maximal, characteristic
2	#19127744	C₂	1	2	yes	maximal
3	C₂×#14581726	C₂	1	1	yes	maximal, characteristic
4	#19127469	C₂	1	2	yes	maximal
5	#405×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	#14581728	C₄	1	4	yes	complemented
9	#14581726	—	2	1	no	—
10	#14581726	—	2	1	no	—
11	#14581725	—	2	2	no	—
12	#14581723	—	2	2	no	—
13	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#117×C₈₃	C₂²	1	1	yes	characteristic
16	#97×C₈₃	—	2	1	no	—
17	C₄×C₈×C₈₃	—	2	1	no	—
18	C₂²×Dic₈₃	—	2	2	no	—
19	#13050574	—	2	2	no	—
20	#13050574	C₂×C₄	1	4	yes	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	C₄⋊Dic₈₃	—	2	2	no	—
24	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₈₃	—	2	1	no	—
26	M₁₆×C₈₃	Q₈	1	2	yes	—
27	C₂×C₈×C₈₃	—	2	1	no	—
28	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₃	Q₈	1	2	yes	—
32	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₈₃	—	2	1	no	—
35	C₄²×C₈₃	—	2	1	no	—
36	C₂³×C₈₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₈₃	—	2	1	no	—
38	C₂×C₄×C₈₃	—	2	1	no	—
39	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₂×D₄	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	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
45	C₂×Dic₈₃	—	4	2	no	—
46	C₂×Dic₈₃	—	8	2	no	—
47	C₈×C₈₃	—	2	2	no	—
48	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
49	C₈×C₈₃	C₄⋊C₄	1	2	yes	—
50	C₈₃⋊C₈	—	2	4	no	—
51	C₂²×C₈₃	—	2	1	no	—
52	C₂²×C₈₃	—	2	1	no	—
53	C₂²×C₈₃	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₈₃	#121	1	1	yes	characteristic
55	C₂²×C₈₃	#125	1	1	yes	characteristic
56	C₄×C₈₃	—	4	1	no	—
57	C₄×C₈₃	—	4	1	no	—
58	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
60	C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₈₃	—	8	2	no	—
63	C₂×C₈₃	—	2	1	no	—
64	C₂×C₈₃	—	2	1	no	—
65	C₂×C₈₃	#389	1	1	yes	characteristic
66	C₂×C₈₃	#472	1	1	yes	characteristic
67	C₂×C₈₃	#473	1	1	yes	characteristic
68	#1952	—	83	1	no	maximal, Sylow
69	C₈₃	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	83	1	no	—
71	#428	—	83	2	no	—
72	C₂×#107	—	83	1	no	—
73	#405	D₈₃	1	1	yes	characteristic
74	#343	—	83	2	no	—
75	C₂×M₁₆	—	83	2	no	—
76	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
78	#117	C₂×D₈₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	—	83	2	no	—
80	#108	—	83	4	no	—
81	#107	—	166	1	no	—
82	#107	—	166	1	no	—
83	#107	—	166	2	no	—
84	#106	—	166	2	no	—
85	#97	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₂²×C₄	—	166	2	no	—
88	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
89	M₁₆	—	2	1	no	—
90	M₁₆	—	83	4	no	—
91	M₁₆	—	166	2	no	—
92	M₁₆	Dic₁₆₆	1	2	yes	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	166	2	no	—
95	C₂×C₈	Dic₁₆₆	1	2	yes	—
96	C₂×C₈	C₄×D₈₃	1	2	yes	—
97	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
99	C₄⋊C₄	—	166	2	no	—
100	C₄⋊C₄	—	166	2	no	—
101	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
102	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₂³	#13050590	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	332	2	no	—
109	C₂×C₄	—	664	2	no	—
110	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
115	C₂×C₄	#13050590	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	166	4	no	—
118	C₈	Dic₈₃⋊C₄	1	2	yes	—
119	C₈	Dic₈₃⋊C₄	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	C₄×Dic₁₆₆	1	1	yes	characteristic
123	C₂²	#14581766	1	1	yes	characteristic
124	C₂²	#14581784	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	664	2	no	—
128	C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
129	C₄	C₄×D₃₃₂	1	1	yes	characteristic
130	C₄	#14581771	1	1	yes	characteristic
131	C₄	#14581792	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#19127599	1	1	yes	characteristic
135	C₂	#19127971	1	1	yes	characteristic
136	C₂	#19127976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581728

C₂

yes

maximal, characteristic

#19127744

C₂

yes

maximal

C₂×#14581726

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal

#405×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581728

C₄

yes

—

—

—

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

#13050574

—

#13050574

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

Q₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

—

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈×C₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×Q₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

#343

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

#108

—

#107

—

166

—

#107

—

166

—

#107

—

166

—

#106

—

166

—

#97

—

C₄×C₈

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

166

—

M₁₆

Dic₁₆₆

yes

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

Dic₁₆₆

yes

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

166

—

100

C₄⋊C₄

—

166

—

101

C₄⋊C₄

D₃₃₂

yes

characteristic

102

C₄⋊C₄

D₃₃₂

yes

characteristic

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

#13050590

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

664

—

110

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₄

C₂×D₃₃₂

yes

characteristic

113

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

114

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

115

C₂×C₄

#13050590

yes

characteristic

116

C₈

—

117

C₈

—

166

—

118

C₈

Dic₈₃⋊C₄

yes

—

119

C₈

Dic₈₃⋊C₄

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄×Dic₁₆₆

yes

characteristic

123

C₂²

#14581766

yes

characteristic

124

C₂²

#14581784

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

664

—

128

C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

129

C₄

C₄×D₃₃₂

yes

characteristic

130

C₄

#14581771

yes

characteristic

131

C₄

#14581792

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#19127599

yes

characteristic

135

C₂

#19127971

yes

characteristic

136

C₂

#19127976

yes

characteristic

	subgroup	G/H
center	C₂²	#14581766
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#405	D₈₃
hypocenter	C₈₃	#1952
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#14581766

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#405×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #14581728	C₄	semidirect
69. C₈₃	#1952	semidirect

H ≅ G/N

type

8. #14581728

C₄

semidirect

69. C₈₃

#1952

semidirect

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

subgroup	index
C₂×#14581728	2
#19127744	2
C₂×#14581726	2
#19127469	2
#405×C₈₃	2
#1952	83

subgroup

index

#405×C₈₃

Subgroups — #24908248

Conjugacy classes of subgroups

Complements

Maximal subgroups