Group #24907141 — subgroups

Subgroups — #24907141

Order 10624.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19128104	C₂	1	1	yes	maximal, characteristic
2	#19128099	C₂	1	1	yes	maximal, characteristic
3	#19128080	C₂	1	1	yes	maximal, characteristic
4	#19128053	C₂	1	1	yes	maximal, characteristic
5	#19128051	C₂	1	1	yes	maximal, characteristic
6	#19127422	C₂	1	1	yes	maximal, characteristic
7	#500×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050580	—	2	2	no	—
9	#14581792	C₂²	1	1	yes	characteristic
10	#14581792	C₂²	1	1	yes	characteristic
11	#14581725	—	2	2	no	—
12	#14581724	—	2	2	no	—
13	#14581716	—	2	2	no	—
14	#14581715	—	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	#107×C₈₃	—	2	2	no	—
20	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₆₆	—	2	1	no	—
22	C₂×Dic₁₆₆	—	2	1	no	—
23	#13050580	—	4	4	no	—
24	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
25	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₈₃	—	2	1	no	—
27	C₄⋊Dic₈₃	—	2	1	no	—
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	Q₈×C₈₃	—	4	2	no	—
39	Q₈×C₈₃	—	4	2	no	—
40	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₈₃	—	4	1	no	—
46	C₂×Dic₈₃	—	4	1	no	—
47	Dic₁₆₆	—	4	2	no	—
48	Dic₁₆₆	—	4	2	no	—
49	C₈×C₈₃	Q₁₆	1	4	yes	complemented
50	C₈₃⋊C₈	—	2	2	no	—
51	C₈₃⋊C₈	—	2	2	no	—
52	C₂²×C₈₃	#127	1	1	yes	characteristic
53	C₄×C₈₃	—	8	1	no	—
54	C₄×C₈₃	—	8	1	no	—
55	C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
56	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₈₃	#137	1	2	yes	—
58	Dic₈₃	—	8	1	no	—
59	Dic₈₃	—	8	1	no	—
60	C₂×C₈₃	#494	1	1	yes	characteristic
61	C₂×C₈₃	#496	1	1	yes	characteristic
62	C₂×C₈₃	#497	1	1	yes	characteristic
63	#1723	—	83	1	no	maximal, Sylow
64	C₈₃	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	83	1	no	—
66	#500	D₈₃	1	1	yes	characteristic
67	#498	—	83	1	no	—
68	#494	—	83	1	no	—
69	#487	—	83	1	no	—
70	#487	—	83	1	no	—
71	#322	—	83	1	no	—
72	C₂×Q₁₆	—	166	2	no	—
73	C₄⋊Q₈	—	83	1	no	—
74	C₄⋊Q₈	—	83	1	no	—
75	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	166	2	no	—
79	#106	—	166	2	no	—
80	#103	—	166	2	no	—
81	#103	—	166	2	no	—
82	C₄×C₈	—	83	1	no	—
83	C₄×C₈	—	83	1	no	—
84	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	166	1	no	—
88	C₂×Q₈	—	166	1	no	—
89	Q₁₆	—	332	4	no	—
90	C₂×C₈	—	83	2	no	—
91	C₂×C₈	—	83	2	no	—
92	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	166	1	no	—
98	C₄⋊C₄	—	166	1	no	—
99	C₄⋊C₄	—	166	1	no	—
100	C₄⋊C₄	—	166	1	no	—
101	C₄²	C₂²×D₈₃	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	332	2	no	—
105	Q₈	—	332	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	332	1	no	—
109	C₂×C₄	—	332	1	no	—
110	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
113	C₈	—	166	2	no	—
114	C₈	—	166	2	no	—
115	C₈	#13050580	1	4	yes	complemented
116	C₂²	#14581788	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	664	1	no	—
120	C₄	—	664	1	no	—
121	C₄	C₂×#13050580	1	2	yes	—
122	C₄	#14581816	1	2	yes	—
123	C₄	#14581824	1	2	yes	—
124	C₂	#19128080	1	1	yes	characteristic
125	C₂	#19128089	1	1	yes	characteristic
126	C₂	#19128094	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128104

C₂

yes

maximal, characteristic

#19128099

C₂

yes

maximal, characteristic

#19128080

C₂

yes

maximal, characteristic

#19128053

C₂

yes

maximal, characteristic

#19128051

C₂

yes

maximal, characteristic

#19127422

C₂

yes

maximal, characteristic

#500×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050580

—

#14581792

C₂²

yes

characteristic

#14581792

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

#107×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

#13050580

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

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

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

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

Q₁₆

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#127

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

#137

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#500

—

#500

D₈₃

yes

—

—

—

—

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

#107

—

#107

—

166

—

#106

—

166

—

#103

—

166

—

#103

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

100

C₄⋊C₄

—

166

—

101

C₄²

C₂²×D₈₃

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

332

—

105

Q₈

—

332

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

332

—

110

C₂×C₄

D₄×D₈₃

yes

characteristic

111

C₂×C₄

D₄×D₈₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

113

C₈

—

166

—

114

C₈

—

166

—

115

C₈

#13050580

yes

complemented

116

C₂²

#14581788

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

664

—

120

C₄

—

664

—

121

C₄

C₂×#13050580

yes

—

122

C₄

#14581816

yes

—

123

C₄

#14581824

yes

—

124

C₂

#19128080

yes

characteristic

125

C₂

#19128089

yes

characteristic

126

C₂

#19128094

yes

characteristic

	subgroup	G/H
center	C₂²	#14581788
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#500×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#127
hypercenter	#500	D₈₃
hypocenter	C₈₃	#1723
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14581788

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#500×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
49. C₈×C₈₃	Q₁₆	semidirect
64. C₈₃	#1723	semidirect
115. C₈	#13050580	semidirect

H ≅ G/N

type

49. C₈×C₈₃

Q₁₆

semidirect

64. C₈₃

#1723

semidirect

115. C₈

#13050580

semidirect

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

subgroup	index
#19128104	2
#19128099	2
#19128080	2
#19128053	2
#19128051	2
#19127422	2
#500×C₈₃	2
#1723	83

subgroup

index

#500×C₈₃

Subgroups — #24907141

Conjugacy classes of subgroups

Complements

Maximal subgroups