Finite Group Databasebeta

← #27411495

Subgroups — #27411495

Order 13568.

1378 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 275 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#20277094C213yesmaximal
2#20275881C214yesmaximal, complemented
3#20275878C214yesmaximal
4#20274448C213yesmaximal, complemented
5C53×#3458C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×#13133176C2213yes—
8#16810484C2216yes—
9#16810480—26no—
10#16810478C2216yes—
11C4×#11719921—26no—
12#16810390C22112yes—
13C53×#544—26no—
14C53×#543C2214yes—
15C53×#533—26no—
16C2×#117×C53C2213yes—
17C24×D53C2311yescharacteristic
18C23×Dic53C2311yescharacteristic
19C22×C4×D53—23no—
20C22×C4×D53—23no—
21C22×C4×D53—26no—
22C22×C4×D53Q814yescomplemented
23#13133176—212no—
24C2×C4⋊Dic53—23no—
25C2×C22⋊Dic53C2316yes—
26C2×#11719921—212no—
27C2×C4×Dic53—23no—
28#13133124—212no—
29C23×C4×C53C2311yescharacteristic
30C4⋊Q8×C53—212no—
31#125×C53—212no—
32C22⋊Q8×C53—224no—
33C4×Q8×C53—28no—
34#117×C53—26no—
35#117×C53—26no—
36C2×C4⋊C4×C53D416yes—
37C2×C22⋊C4×C53C2316yes—
38C2×C42×C53D416yes—
39C24×C53C2411yescharacteristic
40C23×D53—23no—
41C23×D53—23no—
42C23×D53—44no—
43C23×D53C2×Q812yes—
44C2×Q8×C53—28no—
45C22×C4×C53C2×D4112yes—
46C22×C4×C53C2×D413yes—
47C22×C4×C53C2×D413yes—
48C22×C4×C53C2×Q812yes—
49C22×C4×C53#4116yes—
50C22×Dic53—23no—
51C22×Dic53—23no—
52C22×Dic53—26no—
53C22×Dic53C2×Q812yes—
54C2×C4×D53—212no—
55C2×C4×D53—212no—
56C2×C4×D53—43no—
57C2×C4×D53—41no—
58C2×C4×D53—46no—
59C2×C4×D53—46no—
60C4⋊C4×C53—224no—
61C4⋊C4×C53—212no—
62C22⋊C4×C53—212no—
63C42×C53—26no—
64C42×C53—26no—
65C22⋊Dic53—212no—
66#11719921—424no—
67C4⋊Dic53—46no—
68C4×Dic53—46no—
69C23×C53—42no—
70C23×C53C22×D413yes—
71C23×C53C22×Q811yescharacteristic
72C23×C53C2×#4113yes—
73C22×D53—212no—
74C22×D53—26no—
75C22×D53—44no—
76C22×D53—412no—
77C22×D53—46no—
78C22×D53—41no—
79C22×D53—43no—
80Q8×C53—88no—
81C2×C4×C53—212no—
82C2×C4×C53—412no—
83C2×C4×C53#12013yes—
84C2×C4×C53#12011yescharacteristic
85C2×C4×C53C22⋊Q816yes—
86C2×C4×C53C22⋊Q816yes—
87C2×C4×C53#12416yes—
88C2×C4×C53#12416yes—
89C2×Dic53—26no—
90C2×Dic53—41no—
91C2×Dic53—43no—
92C2×Dic53—412no—
93C2×Dic53—46no—
94C4×D53—412no—
95C4×D53—44no—
96#23467—531nomaximal, Sylow
97C22×C53—26no—
98C22×C53—41no—
99C22×C53—43no—
100C22×C53C2×#12011yescharacteristic
101C22×C53C2×C22⋊Q813yes—
102C22×C53C2×#12413yes—
103C2×D53—212no—
104C2×D53—412no—
105C2×D53—46no—
106C2×D53—42no—
107C2×D53—44no—
108C4×C53—412no—
109C4×C53#39316yes—
110C4×C53#39312yes—
111Dic53—42no—
112Dic53—86no—
113#3458D5311yescharacteristic, complemented
114#2280—533no—
115#2030—534no—
116#2029—534no—
117#1740—533no—
118C2×C53—23no—
119C2×C53—23no—
120C2×C53C2×#39311yescharacteristic
121C2×C53#302512yes—
122D53—44no—
123C24×C4—531no—
124#544—26no—
125#543C2×D5314yes—
126#533—26no—
127C2×#117—533no—
128C2×#117C2×D5313yes—
129#381—536no—
130#380—1066no—
131#379—536no—
132C4×C22⋊C4—1066no—
133#342—5312no—
134C53#2346711yescharacteristic, complemented, Sylow
135C25—531no—
136C23×C4—531no—
137C23×C4—534no—
138C23×C4—1066no—
139C23×C4—1063no—
140C23×C4—1063no—
141C23×C4C22×D5311yescharacteristic
142C4⋊Q8—212no—
143#125—212no—
144C22⋊Q8—224no—
145C4×Q8—28no—
146#117—26no—
147#117—26no—
148#117—10612no—
149C2×C4⋊C4—1063no—
150C2×C4⋊C4C22⋊D5316yes—
151C2×C22⋊C4—536no—
152C2×C22⋊C4—10612no—
153C2×C22⋊C4C22×D5316yes—
154C2×C42—1063no—
155C2×C42C22⋊D5316yes—
156#95—10612no—
157C24—532no—
158C24—1063no—
159C24—1063no—
160C24—2124no—
161C24C23×D5311yescharacteristic
162C2×Q8—28no—
163C22×C4—532no—
164C22×C4—1063no—
165C22×C4—1063no—
166C22×C4—10612no—
167C22×C4—10612no—
168C22×C4—1066no—
169C22×C4—2123no—
170C22×C4—2121no—
171C22×C4—2126no—
172C22×C4—2126no—
173C22×C4Q8×D5312yes—
174C22×C4#1171994916yes—
175C22×C4C2×C22⋊D53112yes—
176C22×C4C2×C22⋊D5313yes—
177C22×C4C2×C22⋊D5313yes—
178C4⋊C4—224no—
179C4⋊C4—212no—
180C4⋊C4—2126no—
181C22⋊C4—212no—
182C22⋊C4—10612no—
183C22⋊C4—21224no—
184C42—26no—
185C42—26no—
186C42—2126no—
187C23—42no—
188C23—10612no—
189C23—1066no—
190C23—2126no—
191C23—2124no—
192C23—21212no—
193C23—2121no—
194C23—2123no—
195C23C22×C22⋊D5313yes—
196C23C2×Q8×D5311yescharacteristic
197C23C2×#1171994913yes—
198Q8—88no—
199C2×C4—212no—
200C2×C4—412no—
201C2×C4—1066no—
202C2×C4—21212no—
203C2×C4—2126no—
204C2×C4—2121no—
205C2×C4—2123no—
206C2×C4—21212no—
207C2×C4—2124no—
208C2×C4#1313318613yes—
209C2×C4#1313318611yescharacteristic
210C2×C4#1313319316yes—
211C2×C4#1313319316yes—
212C2×C4#1313320616yes—
213C2×C4#1313320616yes—
214C22—26no—
215C22—41no—
216C22—43no—
217C22—10612no—
218C22—21212no—
219C22—2126no—
220C22—2122no—
221C22—2124no—
222C22C2×#1313318611yescharacteristic
223C22C2×#1313319313yes—
224C22C2×#1313320613yes—
225C4—412no—
226C4—4246no—
227C4—2122no—
228C4#1681053416yes—
229C4#1681053412yes—
230C2—23no—
231C2—23no—
232C2—2124no—
233C2C2×#1681053411yescharacteristic
234C2#2028410712yes—
subgroupG/H
centerC2×C4#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#3458C2
socleC22×C53C2×#120
prime radicalC24×D53C23
hypercenter#3458D53
hypocenterC53#23467
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #20275881C2semidirect
4. #20274448C2semidirect
5. C53×#3458C2semidirect
22. C22×C4×D53Q8semidirect
113. #3458D53semidirect
134. C53#23467semidirect

Maximal subgroups

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

subgroupindex
#202770942
#202758812
#202758782
#202744482
C53×#34582
#2346753