Finite Group Databasebeta

← #27295626

Subgroups — #27295626

Order 13568.

360 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 79 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#20285174C211yesmaximal, characteristic
2#20276527C211yesmaximal, characteristic, complemented
3#20272755C212yesmaximal, complemented
4#20272753C212yesmaximal, complemented
5C53×#2165C211yesmaximal, characteristic, complemented
6C2×#13133250—21no—
7C2×#13133250—21no—
8#16811173C411yescharacteristic
9C2×#13133217C411yescharacteristic
10#16811017—21no—
11#16811017—21no—
12#16811003C411yescharacteristic
13#16811003C2211yescharacteristic
14#16810996C411yescharacteristic
15#16810996C2211yescharacteristic
16#16810444C2212yes—
17#16810405—21no—
18#16810401—21no—
19C4×M16×C53C2211yescharacteristic
20C53×#364C2212yes—
21C53×#349—21no—
22C53×#347—21no—
23C2×#11719945—21no—
24C22×Dic106—21no—
25#13133250—44no—
26#13133250—44no—
27C2×Dic212—21no—
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×#11719913—21no—
32C2×#11719913—21no—
33C2×#11719913—21no—
34C2×#11719913—21no—
35#13133217—21no—
36#13133217—41no—
37#13133217C2×C411yescharacteristic
38#13133217C2×C411yescharacteristic
39#13133217C2×C411yescharacteristic
40#13133214C2×C411yescharacteristic
41#13133205—21no—
42#13133191—41no—
43C4×D212—41no—
44C2×C4⋊Dic53—41no—
45#13133157—22no—
46C2×M16×C53D411yescharacteristic
47C2×M16×C53D411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49M32×C53—22no—
50#97×C53D411yescharacteristic
51#97×C53D411yescharacteristic
52C4×C8×C53C2×C411yescharacteristic
53C4×C8×C53C2311yescharacteristic
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C2×C22⋊D53—41no—
57C22×Dic53—41no—
58#11719945—42no—
59#11719945—42no—
60C2×D212—21no—
61C2×D212—41no—
62C2×C4×D53—41no—
63C2×Dic106—21no—
64C2×Dic106—21no—
65C2×Dic106—21no—
66C2×Dic106—42no—
67C2×Dic106—42no—
68C2×Dic106—41no—
69M16×C53—22no—
70M16×C53—22no—
71C2×C8×C53C22⋊C411yescharacteristic
72C2×C8×C53C22⋊C411yescharacteristic
73C2×C8×C53C2×D411yescharacteristic
74C2×C8×C53C2×D411yescharacteristic
75C42×C53—21no—
76C42×C53C22⋊C411yescharacteristic
77C42×C53C22×C411yescharacteristic
78#11719921—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83Dic212—42no—
84Dic212—42no—
85Dic212—42no—
86Dic212—42no—
87#11719913—42no—
88#11719913—42no—
89#11719913—42no—
90#11719913—42no—
91C16×C53—42no—
92#11719908—42no—
93C23×C53#10011yescharacteristic
94C22×D53—41no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53#10011yescharacteristic
101C2×C4×C53C2×C22⋊C411yescharacteristic
102C22⋊D53—82no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—82no—
107D212—41no—
108D212—41no—
109D212—82no—
110C4×D53—82no—
111Dic106—41no—
112Dic106—41no—
113Dic106—41no—
114Dic106—41no—
115Dic106—41no—
116Dic106—41no—
117Dic106—82no—
118C8×C53—21no—
119C8×C53—21no—
120C8×C53AGL(1,ℤ/8)12yes—
121C8×C53AGL(1,ℤ/8)12yes—
122#11993—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—41no—
125C22×C53C2×#10011yescharacteristic
126C2×D53—82no—
127C2×D53—81no—
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53#41811yescharacteristic
132C4×C53#41811yescharacteristic
133Dic53—81no—
134Dic53—81no—
135Dic53—81no—
136#3152—531no—
137#2165—531no—
138#2165D5311yescharacteristic, complemented
139#1359—532no—
140#1358—532no—
141C2×C53—21no—
142C2×C53—41no—
143C2×C53#155511yescharacteristic
144D53—161no—
145C2×#137—1061no—
146C2×#137—1061no—
147#532—531no—
148C2×C4⋊Q8—531no—
149#497—1061no—
150#497—1061no—
151#494—531no—
152#494—531no—
153#492—531no—
154#492—531no—
155C4×M16C2×D5311yescharacteristic
156#364—532no—
157#364C2×D5312yes—
158#349—21no—
159#349—1061no—
160#347—21no—
161#347—1061no—
162C53#1199311yescharacteristic, complemented, Sylow
163C2×#41—1061no—
164C22×Q8—1061no—
165#137—2124no—
166#137—2124no—
167C2×Q16—1061no—
168C2×Q16—1061no—
169C2×Q16—1061no—
170C2×Q16—1061no—
171C2×SD16—1061no—
172C2×SD16—1061no—
173C2×SD16—1061no—
174C2×SD16—1061no—
175C2×M16D21211yescharacteristic
176C2×M16C22⋊D5311yescharacteristic
177C4⋊Q8—531no—
178C4⋊Q8—531no—
179C4⋊Q8—531no—
180C4⋊Q8—1061no—
181C4⋊Q8—2121no—
182#127—531no—
183#124—1061no—
184#121—2121no—
185C4×D4—2121no—
186C2×C4⋊C4—2121no—
187C2×C42C4×D5311yescharacteristic
188M32—22no—
189M32—1062no—
190#97D21211yescharacteristic
191#97C22⋊D5311yescharacteristic
192C4×C8C4×D5311yescharacteristic
193C4×C8C22×D5311yescharacteristic
194#41—2122no—
195#41—2122no—
196C2×Q8—1061no—
197C2×Q8—1061no—
198C2×Q8—1061no—
199C2×Q8—2122no—
200C2×Q8—2122no—
201C2×Q8—2121no—
202C2×D4—1061no—
203C2×D4—2121no—
204C2×D4—2121no—
205C22×C4—21no—
206C22×C4—2121no—
207C22×C4—2121no—
208C22×C4#1171992111yescharacteristic
209Q16—2122no—
210Q16—2122no—
211Q16—2122no—
212Q16—2122no—
213SD16—2122no—
214SD16—2122no—
215SD16—2122no—
216SD16—2122no—
217M16—22no—
218M16—22no—
219C2×C8#1171992111yescharacteristic
220C2×C8#1171992111yescharacteristic
221C2×C8C2×D21211yescharacteristic
222C2×C8C2×C22⋊D5311yescharacteristic
223C4⋊C4—2121no—
224C4⋊C4—2121no—
225C4⋊C4—2121no—
226C4⋊C4—2122no—
227C22⋊C4—2122no—
228C42—21no—
229C42#1171992111yescharacteristic
230C42C2×C4×D5311yescharacteristic
231C16—42no—
232C16—2122no—
233C23—2121no—
234C23#1313313311yescharacteristic
235Q8—4242no—
236Q8—2121no—
237Q8—2121no—
238Q8—2121no—
239Q8—2121no—
240Q8—2121no—
241Q8—2121no—
242D4—4242no—
243D4—4242no—
244D4—2121no—
245D4—2121no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—4242no—
252C2×C4—4242no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4#1313313311yescharacteristic
257C2×C4C2×#1171992111yescharacteristic
258C8—21no—
259C8—21no—
260C8#1313324212yes—
261C8#1313324512yes—
262C22—21no—
263C22—41no—
264C22—4242no—
265C22—4241no—
266C22C2×#1313313311yescharacteristic
267C4—22no—
268C4—21no—
269C4—41no—
270C4—4241no—
271C4—4241no—
272C4—4241no—
273C4#1681062711yescharacteristic
274C4#1681062911yescharacteristic
275C2—21no—
276C2—41no—
277C2—8481no—
278C2#2027342811yescharacteristic
subgroupG/H
centerC2#20273428
commutatorC42×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#2165C2
socleC2×C53#1555
prime radical#16811173C4
cosocleC4×C8×C53C23
hypercenter#2165D53
hypocenterC53#11993
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #20276527C2semidirect
3. #20272755C2semidirect
4. #20272753C2semidirect
5. C53×#2165C2semidirect
138. #2165D53semidirect
162. C53#11993semidirect

Maximal subgroups

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

subgroupindex
#202851742
#202765272
#202727552
#202727532
C53×#21652
#1199353