Finite Group Databasebeta

← #27532710

Subgroups — #27532710

Order 13568.

688 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275893C216yesmaximal, complemented
2#20275891C212yesmaximal, complemented
3#20275882C213yesmaximal, complemented
4#20274632C213yesmaximal
5C53×#2030C211yesmaximal, characteristic, complemented
6C22×#11719945C2211yescharacteristic
7#16811249C2213yes—
8C2×#13133131C2216yes—
9C2×#13133131C2216yes—
10#16810486C2213yes—
11#16810390C2213yes—
12#16810389C2216yes—
13C53×#543C2211yescharacteristic
14C53×#381C2213yes—
15C53×#342C2213yes—
16C2×#11719945—43no—
17C2×#11719945—43no—
18C22×Dic106Q811yescharacteristic, complemented
19C22×D212Q811yescharacteristic, complemented
20C22×C22⋊D53C2312yes—
21C22×C4×D53Q812yescomplemented
22#13133196—43no—
23#13133194—46no—
24C2×C22⋊Dic53C2316yes—
25#13133131—46no—
26#13133131—46no—
27#13133124—43no—
28C23×C4×C53C2311yescharacteristic
29C22⋊Q8×C53—43no—
30C2×C22⋊C4×C53C2313yes—
31C2×C22⋊C4×C53C2313yes—
32#95×C53—43no—
33C24×C53C2411yescharacteristic
34C23×D53C2×Q812yes—
35C2×Q8×C53—41no—
36C22×C4×C53—23no—
37C22×C4×C53—23no—
38C22×C4×C53—23no—
39C22×C4×C53—23no—
40C22×C4×C53C2×Q811yescharacteristic
41C22×C4×C53C2×Q811yescharacteristic
42C2×C22⋊D53—26no—
43C2×C22⋊D53—26no—
44C22×Dic53—26no—
45C22×Dic53C2×Q812yes—
46#11719945—412no—
47#11719945—44no—
48C2×D212—43no—
49C2×C4×D53—46no—
50C2×Dic106—43no—
51C2×Dic106—43no—
52C4⋊C4×C53—43no—
53C22⋊C4×C53—26no—
54C22⋊C4×C53—26no—
55C22⋊Dic53—26no—
56C22⋊Dic53—26no—
57C4⋊Dic53—43no—
58#11719919—46no—
59C23×C53—81no—
60C23×C53C22×Q811yescharacteristic
61C23×C53H(5,𝔽2)13yes—
62C23×C53H(5,𝔽2)13yes—
63C22×D53—26no—
64C22×D53—82no—
65Q8×C53—161no—
66C2×C4×C53—23no—
67C2×C4×C53—23no—
68C2×C4×C53—43no—
69C2×C4×C53—41no—
70C2×C4×C53—43no—
71C2×C4×C53—43no—
72C2×C4×C53—83no—
73C2×C4×C53—83no—
74C22⋊D53—46no—
75C22⋊D53—42no—
76C2×Dic53—26no—
77C2×Dic53—46no—
78C2×Dic53—86no—
79D212—44no—
80C4×D53—48no—
81Dic106—44no—
82Dic106—163no—
83#32874—531nomaximal, Sylow
84C22×C53—43no—
85C22×C53—43no—
86C22×C53—41no—
87C22×C53#54313yes—
88C22×C53H(3,𝔽4)13yes—
89C22×C53H(3,𝔽4)11yescharacteristic
90C2×D53—42no—
91C2×D53—86no—
92C4×C53—41no—
93C4×C53—41no—
94C4×C53—83no—
95C4×C53—83no—
96Dic53—42no—
97Dic53—86no—
98#2032—536no—
99#2032—532no—
100#2030—533no—
101#2030D5311yescharacteristic, complemented
102#1782—533no—
103C2×C53—23no—
104C2×C53—21no—
105C2×C53—23no—
106C2×C53#284011yescharacteristic
107D53—82no—
108C22×#41—531no—
109#543—533no—
110#543C2×D5311yescharacteristic
111C2×#99—536no—
112C2×#99—536no—
113#381—533no—
114#381C2×D5313yes—
115#342—536no—
116#342—533no—
117#342C2×D5313yes—
118C53#3287411yescharacteristic, complemented, Sylow
119C2×#41—2123no—
120C2×#41—2123no—
121C22×Q8—531no—
122C22×D4—532no—
123C22×D4—531no—
124C23×C4—532no—
125C23×C4C22×D5311yescharacteristic
126C22⋊Q8—43no—
127C22⋊Q8—2126no—
128C22⋊Q8—2123no—
129C2×C22⋊C4—536no—
130C2×C22⋊C4C22×D5313yes—
131C2×C22⋊C4C22×D5313yes—
132#99—2126no—
133#99—2126no—
134#95—43no—
135#95—2123no—
136C24—532no—
137C24C23×D5311yescharacteristic
138#41—21212no—
139#41—2124no—
140C2×Q8—41no—
141C2×Q8—2123no—
142C2×Q8—2123no—
143C2×D4—1066no—
144C2×D4—1066no—
145C2×D4—2123no—
146C22×C4—23no—
147C22×C4—23no—
148C22×C4—23no—
149C22×C4—23no—
150C22×C4—532no—
151C22×C4—1066no—
152C22×C4—2126no—
153C22×C4Q8×D5311yescharacteristic
154C22×C4Q8×D5311yescharacteristic
155C4⋊C4—43no—
156C4⋊C4—2126no—
157C4⋊C4—2123no—
158C22⋊C4—26no—
159C22⋊C4—26no—
160C22⋊C4—1066no—
161C22⋊C4—1066no—
162C23—81no—
163C23—1066no—
164C23—4242no—
165C23C2×Q8×D5311yescharacteristic
166C23#1313326213yes—
167C23#1313326313yes—
168Q8—161no—
169Q8—8483no—
170Q8—2124no—
171D4—2124no—
172D4—2126no—
173D4—2122no—
174C2×C4—23no—
175C2×C4—23no—
176C2×C4—43no—
177C2×C4—41no—
178C2×C4—43no—
179C2×C4—43no—
180C2×C4—83no—
181C2×C4—83no—
182C2×C4—1066no—
183C2×C4—4246no—
184C2×C4—2128no—
185C2×C4—2126no—
186C22—43no—
187C22—43no—
188C22—41no—
189C22—4246no—
190C22—2122no—
191C22#1681124813yes—
192C22#1681136513yes—
193C22#1681136611yescharacteristic
194C4—41no—
195C4—41no—
196C4—83no—
197C4—83no—
198C4—4246no—
199C4—2122no—
200C2—21no—
201C2—23no—
202C2—23no—
203C2—4242no—
204C2#2028296711yescharacteristic
subgroupG/H
centerC2#20282967
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#2030C2
socleC2×C53#2840
prime radicalC22×#11719945C22
hypercenter#2030D53
hypocenterC53#32874
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #2030

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20275893C2semidirect
2. #20275891C2semidirect
3. #20275882C2semidirect
5. C53×#2030C2semidirect
18. C22×Dic106Q8semidirect
19. C22×D212Q8semidirect
21. C22×C4×D53Q8semidirect
101. #2030D53semidirect
118. C53#32874semidirect

Maximal subgroups

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

subgroupindex
#202758932
#202758912
#202758822
#202746322
C53×#20302
#3287453