Group #24906920 — subgroups

Subgroups — #24906920

Order 10624.

144 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19128051	C₂	1	1	yes	maximal, characteristic
2	#19127997	C₂	1	1	yes	maximal, characteristic
3	#19127983	C₂	1	1	yes	maximal, characteristic
4	#19127825	C₂	1	1	yes	maximal, characteristic
5	#19127453	C₂	1	1	yes	maximal, characteristic
6	#19127449	C₂	1	1	yes	maximal, characteristic
7	#332×C₈₃	C₂	1	1	yes	maximal, characteristic
8	#14581792	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#14581726	—	2	1	no	—
11	#14581723	—	2	1	no	—
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₈₃⋊C₈	—	2	1	no	—
14	#14581717	—	2	1	no	—
15	#14581716	—	2	1	no	—
16	#14581716	—	2	1	no	—
17	#14581715	—	2	1	no	—
18	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₈₃	—	2	1	no	—
20	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	—	2	1	no	—
24	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₈₃	—	2	1	no	—
29	C₄⋊Dic₈₃	—	2	1	no	—
30	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
31	Dic₈₃⋊C₄	—	2	1	no	—
32	C₄×Dic₈₃	—	2	1	no	—
33	C₂×Q₈×C₈₃	—	2	1	no	—
34	C₂×C₈×C₈₃	—	2	1	no	—
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₄⋊C₄×C₈₃	—	2	1	no	—
37	C₄⋊C₄×C₈₃	—	2	1	no	—
38	C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	Q₈×C₈₃	—	4	1	no	—
40	Q₈×C₈₃	—	4	1	no	—
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
45	C₂×Dic₈₃	—	2	1	no	—
46	C₂×Dic₈₃	—	2	1	no	—
47	C₂×Dic₈₃	—	4	1	no	—
48	Dic₁₆₆	—	2	1	no	—
49	Dic₁₆₆	—	8	1	no	—
50	Dic₁₆₆	Q₁₆	1	2	yes	—
51	C₈×C₈₃	—	4	1	no	—
52	C₈×C₈₃	—	4	1	no	—
53	C₈₃⋊C₈	—	2	1	no	—
54	C₈₃⋊C₈	—	2	1	no	—
55	C₈₃⋊C₈	—	4	1	no	—
56	C₂²×C₈₃	#121	1	1	yes	characteristic
57	C₄×C₈₃	—	2	1	no	—
58	C₄×C₈₃	—	8	1	no	—
59	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₈₃	#135	1	1	yes	characteristic
61	C₄×C₈₃	#135	1	1	yes	characteristic
62	C₄×C₈₃	#137	1	1	yes	characteristic
63	Dic₈₃	—	2	2	no	—
64	Dic₈₃	—	4	1	no	—
65	Dic₈₃	—	8	1	no	—
66	C₂×C₈₃	#464	1	1	yes	characteristic
67	C₂×C₈₃	#467	1	1	yes	characteristic
68	C₂×C₈₃	#471	1	1	yes	characteristic
69	#1684	—	83	1	no	maximal, Sylow
70	C₈₃	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	83	1	no	—
72	#477	—	83	1	no	—
73	#475	—	83	1	no	—
74	C₈×Q₈	—	83	1	no	—
75	#335	—	83	1	no	—
76	#333	—	83	1	no	—
77	#332	D₈₃	1	1	yes	characteristic
78	C₄⋊Q₈	—	83	1	no	—
79	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
80	C₄×Q₈	—	83	1	no	—
81	#107	—	166	1	no	—
82	#106	—	166	1	no	—
83	C₄⋊C₈	—	83	1	no	—
84	C₄⋊C₈	—	166	1	no	—
85	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
87	#103	—	166	1	no	—
88	#103	—	166	1	no	—
89	#103	—	166	1	no	—
90	#103	—	166	1	no	—
91	C₄×C₈	—	83	1	no	—
92	C₄×C₈	—	166	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	83	1	no	—
95	C₂×Q₈	—	166	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	83	1	no	—
99	C₂×C₈	—	83	1	no	—
100	C₂×C₈	—	166	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	83	1	no	—
104	C₄⋊C₄	—	166	1	no	—
105	C₄⋊C₄	—	166	1	no	—
106	C₄⋊C₄	—	166	1	no	—
107	C₄²	—	166	1	no	—
108	C₄²	C₂²×D₈₃	1	1	yes	characteristic
109	Q₈	—	4	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	83	2	no	—
112	Q₈	—	664	1	no	—
113	Q₈	—	166	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	332	1	no	—
116	C₂×C₄	—	166	1	no	—
117	C₂×C₄	—	166	1	no	—
118	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
119	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
120	C₂×C₄	#13050590	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	332	1	no	—
124	C₈	—	166	1	no	—
125	C₈	—	166	1	no	—
126	C₂²	#14581765	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	332	1	no	—
130	C₄	—	664	1	no	—
131	C₄	—	166	2	no	—
132	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
133	C₄	#14581811	1	1	yes	characteristic
134	C₄	#14581813	1	1	yes	characteristic
135	C₄	#14581821	1	1	yes	characteristic
136	C₂	#19127922	1	1	yes	characteristic
137	C₂	#19127941	1	1	yes	characteristic
138	C₂	#19127967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128051

C₂

yes

maximal, characteristic

#19127997

C₂

yes

maximal, characteristic

#19127983

C₂

yes

maximal, characteristic

#19127825

C₂

yes

maximal, characteristic

#19127453

C₂

yes

maximal, characteristic

#19127449

C₂

yes

maximal, characteristic

#332×C₈₃

C₂

yes

maximal, characteristic

#14581792

C₂²

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581726

—

#14581723

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

—

—

—

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

Q₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#121

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#487

—

#477

—

#475

—

C₈×Q₈

—

#335

—

#333

—

#332

D₈₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

#107

—

166

—

#106

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₈

C₂×D₈₃

yes

characteristic

#103

—

166

—

#103

—

166

—

#103

—

166

—

#103

—

166

—

C₄×C₈

—

C₄×C₈

—

166

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

166

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₄²

—

166

—

108

C₄²

C₂²×D₈₃

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

664

—

113

Q₈

—

166

—

114

C₂×C₄

—

115

C₂×C₄

—

332

—

116

C₂×C₄

—

166

—

117

C₂×C₄

—

166

—

118

C₂×C₄

D₄×D₈₃

yes

characteristic

119

C₂×C₄

D₄×D₈₃

yes

characteristic

120

C₂×C₄

#13050590

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

332

—

124

C₈

—

166

—

125

C₈

—

166

—

126

C₂²

#14581765

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

332

—

130

C₄

—

664

—

131

C₄

—

166

—

132

C₄

Q₁₆×D₈₃

yes

characteristic

133

C₄

#14581811

yes

characteristic

134

C₄

#14581813

yes

characteristic

135

C₄

#14581821

yes

characteristic

136

C₂

#19127922

yes

characteristic

137

C₂

#19127941

yes

characteristic

138

C₂

#19127967

yes

characteristic

	subgroup	G/H
center	C₂²	#14581765
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#332×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#121
hypercenter	#332	D₈₃
hypocenter	C₈₃	#1684
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14581765

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#332×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
70. C₈₃	#1684	semidirect

H ≅ G/N

type

70. C₈₃

#1684

semidirect

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

subgroup	index
#19128051	2
#19127997	2
#19127983	2
#19127825	2
#19127453	2
#19127449	2
#332×C₈₃	2
#1684	83

subgroup

index

#332×C₈₃

Subgroups — #24906920

Conjugacy classes of subgroups

Complements

Maximal subgroups