Group #24907121 — subgroups

Subgroups — #24907121

Order 10624.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19128080	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128051	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128049	C₂	1	1	yes	maximal, characteristic
4	#19128049	C₂	1	1	yes	maximal, characteristic
5	#19128048	C₂	1	1	yes	maximal, characteristic
6	#19127423	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050580	—	2	2	no	—
9	#14581792	C₂²	1	1	yes	characteristic
10	#14581789	C₂²	1	1	yes	characteristic
11	#14581716	—	2	2	no	—
12	#14581714	—	2	2	no	—
13	#14581712	—	2	2	no	—
14	#14581712	—	2	2	no	—
15	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	#106×C₈₃	—	2	2	no	—
20	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×D₃₃₂	—	2	1	no	—
23	C₂×D₃₃₂	—	2	1	no	—
24	C₂×D₃₃₂	—	2	1	no	—
25	#13050580	—	4	4	no	—
26	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
27	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₈₃	—	2	1	no	—
29	C₄⋊Dic₈₃	—	2	1	no	—
30	C₂×Q₈×C₈₃	—	2	1	no	—
31	C₂×Q₈×C₈₃	—	2	1	no	—
32	C₂×C₈×C₈₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₈₃	—	2	1	no	—
34	C₄⋊C₄×C₈₃	—	2	1	no	—
35	C₄⋊C₄×C₈₃	—	2	1	no	—
36	C₄⋊C₄×C₈₃	—	2	1	no	—
37	C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₂²×D₈₃	—	4	1	no	—
39	Q₈×C₈₃	—	4	2	no	—
40	Q₈×C₈₃	—	4	2	no	—
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	—	4	1	no	—
43	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₈₃	—	4	1	no	—
47	D₃₃₂	—	4	2	no	—
48	D₃₃₂	—	4	2	no	—
49	D₃₃₂	—	4	2	no	—
50	Dic₁₆₆	—	4	2	no	—
51	C₈×C₈₃	—	2	2	no	—
52	C₈₃⋊C₈	—	2	2	no	—
53	C₈₃⋊C₈	—	2	2	no	—
54	C₂²×C₈₃	#127	1	1	yes	characteristic
55	C₂×D₈₃	—	8	1	no	—
56	C₂×D₈₃	—	8	1	no	—
57	C₂×D₈₃	—	8	1	no	—
58	C₄×C₈₃	—	8	1	no	—
59	C₄×C₈₃	—	8	1	no	—
60	C₄×C₈₃	#135	1	2	yes	—
61	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₈₃	#137	1	2	yes	—
63	Dic₈₃	—	8	1	no	—
64	C₂×C₈₃	#495	1	1	yes	characteristic
65	C₂×C₈₃	#496	1	1	yes	characteristic
66	C₂×C₈₃	#497	1	1	yes	characteristic
67	D₈₃	—	16	1	no	—
68	#1720	—	83	1	no	maximal, Sylow
69	C₈₃	#1720	1	1	yes	characteristic, complemented, Sylow
70	#498	D₈₃	1	1	yes	characteristic, complemented
71	#494	—	83	1	no	—
72	#487	—	83	1	no	—
73	#486	—	83	1	no	—
74	#486	—	83	1	no	—
75	#486	—	83	1	no	—
76	#322	—	83	1	no	—
77	C₂×Q₁₆	—	166	2	no	—
78	C₄⋊Q₈	—	83	1	no	—
79	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
81	#127	—	83	1	no	—
82	#106	—	2	2	no	—
83	#103	—	166	2	no	—
84	#102	—	166	2	no	—
85	#102	—	166	2	no	—
86	#102	—	166	2	no	—
87	C₄×C₈	—	83	1	no	—
88	C₄×C₈	—	83	1	no	—
89	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	166	1	no	—
93	C₂×D₄	—	166	1	no	—
94	C₂×D₄	—	166	1	no	—
95	C₂×D₄	—	166	1	no	—
96	Q₁₆	—	332	4	no	—
97	C₂×C₈	—	83	2	no	—
98	C₂×C₈	—	83	2	no	—
99	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	166	1	no	—
105	C₄⋊C₄	—	166	1	no	—
106	C₄²	C₂²×D₈₃	1	1	yes	characteristic
107	C₂³	—	332	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	332	2	no	—
111	D₄	—	332	2	no	—
112	D₄	—	332	2	no	—
113	D₄	—	332	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	332	1	no	—
117	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	166	2	no	—
122	C₈	—	166	2	no	—
123	C₂²	—	664	1	no	—
124	C₂²	—	664	1	no	—
125	C₂²	—	664	1	no	—
126	C₂²	#14581788	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	664	1	no	—
130	C₄	#14581813	1	2	yes	—
131	C₄	#14581819	1	2	yes	—
132	C₄	#14581823	1	2	yes	—
133	C₂	—	1328	1	no	—
134	C₂	#19128082	1	1	yes	characteristic
135	C₂	#19128087	1	1	yes	characteristic
136	C₂	#19128093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128080

C₂

yes

maximal, characteristic, complemented

#19128051

C₂

yes

maximal, characteristic, complemented

#19128049

C₂

yes

maximal, characteristic

#19128049

C₂

yes

maximal, characteristic

#19128048

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, characteristic, complemented

#498×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050580

—

#14581792

C₂²

yes

characteristic

#14581789

C₂²

yes

—

—

—

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

#106×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

#13050580

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#127

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#135

yes

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

#137

yes

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1720

—

maximal, Sylow

C₈₃

#1720

yes

characteristic, complemented, Sylow

#498

D₈₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

C₂×D₈₃

yes

—

—

—

166

—

#102

—

166

—

#102

—

166

—

#102

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄²

C₂²×D₈₃

yes

characteristic

107

C₂³

—

332

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

332

—

111

D₄

—

332

—

112

D₄

—

332

—

113

D₄

—

332

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

332

—

117

C₂×C₄

D₄×D₈₃

yes

characteristic

118

C₂×C₄

D₄×D₈₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

120

C₈

—

121

C₈

—

166

—

122

C₈

—

166

—

123

C₂²

—

664

—

124

C₂²

—

664

—

125

C₂²

—

664

—

126

C₂²

#14581788

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

664

—

130

C₄

#14581813

yes

—

131

C₄

#14581819

yes

—

132

C₄

#14581823

yes

—

133

C₂

—

1328

—

134

C₂

#19128082

yes

characteristic

135

C₂

#19128087

yes

characteristic

136

C₂

#19128093

yes

characteristic

	subgroup	G/H
center	C₂²	#14581788
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#498×C₈₃	C₂
socle	C₂²×C₈₃	#127
prime radical	#14581789	C₂²
hypercenter	#498	D₈₃
hypocenter	C₈₃	#1720
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14581788

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#498×C₈₃

C₂

C₂²×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #19128080	C₂	semidirect
2. #19128051	C₂	semidirect
6. #19127423	C₂	semidirect
7. #498×C₈₃	C₂	semidirect
69. C₈₃	#1720	semidirect
70. #498	D₈₃	semidirect

H ≅ G/N

type

1. #19128080

C₂

semidirect

2. #19128051

C₂

semidirect

6. #19127423

C₂

semidirect

7. #498×C₈₃

C₂

semidirect

69. C₈₃

#1720

semidirect

70. #498

D₈₃

semidirect

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

subgroup	index
#19128080	2
#19128051	2
#19128049	2
#19128048	2
#19127423	2
#498×C₈₃	2
#1720	83

subgroup

index

#498×C₈₃

Subgroups — #24907121

Conjugacy classes of subgroups

Complements

Maximal subgroups