Finite Group Databasebeta

← #28303252

Subgroups — #28303252

Order 13568.

346 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290640C211yesmaximal, characteristic
2D53⋊#335C211yesmaximal, characteristic
3#20290232C212yesmaximal
4#20289584C212yesmaximal
5#20283644C211yesmaximal, characteristic
6#16811628C2212yes—
7D53⋊(C4⋊C8)C2212yes—
8#16811499—21no—
9#16811499—21no—
10#16811487—21no—
11#16811486—21no—
12#16811483—22no—
13#16811482—22no—
14#16811177C2211yescharacteristic
15#16810754C411yescharacteristic
16#16810752—21no—
17#16810750C411yescharacteristic
18C8×Dic106—21no—
19C8×Dic106C411yescharacteristic
20#16810648—21no—
21#16810645C2211yescharacteristic
22#16810577—21no—
23C4×C8×D53C2211yescharacteristic
24C8×Q8×C53C411yescharacteristic
25C2×#11719936—22no—
26C2×D53⋊C8—22no—
27#13133284—22no—
28#13133284—22no—
29#13133284—22no—
30C4⋊#88324—22no—
31C2×#11719912—21no—
32C2×C8×D53D411yescharacteristic
33C2×C8×D53D411yescharacteristic
34#13133216D411yescharacteristic
35#13133215D411yescharacteristic
36#13133207—21no—
37#13133197Q811yescharacteristic
38#13133195—21no—
39#13133193Q811yescharacteristic
40C4×Dic106—21no—
41C4×Dic106C2×C411yescharacteristic
42#13133174—21no—
43C42×D53C2311yescharacteristic
44C4⋊#88322—21no—
45C4⋊#88322C2×C411yescharacteristic
46#13133146—21no—
47#13133146—21no—
48#13133146—21no—
49#13133128—21no—
50#13133127—21no—
51C4×#88322C2×C411yescharacteristic
52C8×Dic53—21no—
53C8×Dic53D411yescharacteristic
54C8×Dic53D411yescharacteristic
55C4×Q8×C53C2×C411yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53C2×C411yescharacteristic
58C4×C8×C53—21no—
59C4×C8×C53C2×C411yescharacteristic
60C2×Q8×C53C4⋊C411yescharacteristic
61C2×C4×D53C2×D411yescharacteristic
62C2×C4×D53C2×Q811yescharacteristic
63C2×C4×D53#4111yescharacteristic
64C2×Dic106—21no—
65C2×Dic106C4⋊C411yescharacteristic
66C2×#88324—42no—
67C2×#88324—42no—
68#11719936—42no—
69#11719936—42no—
70D53⋊C8—82no—
71C2×C8×C53—21no—
72C2×C8×C53C22⋊C411yescharacteristic
73C2×C8×C53C22⋊C411yescharacteristic
74C4⋊C4×C53—21no—
75C4⋊C4×C53C4⋊C411yescharacteristic
76C42×C53—21no—
77C42×C53C22×C411yescharacteristic
78#11719921—21no—
79#11719921—21no—
80C4⋊Dic53—21no—
81C4⋊Dic53C4⋊C411yescharacteristic
82#11719919—21no—
83#11719919—21no—
84#11719919—21no—
85#11719919—21no—
86#11719919—21no—
87C4×Dic53—21no—
88C4×Dic53C2×D411yescharacteristic
89C4×Dic53C2×D411yescharacteristic
90C4×Dic53#4111yescharacteristic
91C2×#88322—21no—
92C2×#88322C22⋊C411yescharacteristic
93C2×#88322C22⋊C411yescharacteristic
94#11719912—42no—
95C8×D53—22no—
96C8×D53—22no—
97C22×D53C22⋊Q811yescharacteristic
98Q8×C53—41no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53C2×C4⋊C411yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53#12111yescharacteristic
107C2×Dic53#12111yescharacteristic
108C2×Dic53C22⋊Q811yescharacteristic
109C4×D53—21no—
110C4×D53—21no—
111C4×D53—41no—
112C4×D53#13512yes—
113C4×D53#13712yes—
114Dic106—41no—
115Dic106—41no—
116Dic106—41no—
117#88324—82no—
118#88324—82no—
119C8×C53—21no—
120C8×C53—21no—
121C8×C53—41no—
122#88322—21no—
123#88322—21no—
124#88322—41no—
125#8689—531nomaximal, Sylow
126C22×C53#38011yescharacteristic
127C2×D53—21no—
128C2×D53—21no—
129C2×D53#47912yes—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53#41611yescharacteristic
135C4×C53#41911yescharacteristic
136Dic53—21no—
137Dic53—41no—
138Dic53—41no—
139Dic53#46412yes—
140Dic53#47211yescharacteristic
141Dic53#47311yescharacteristic
142#2960—531no—
143#1837—531no—
144C2×#335—531no—
145#1550—532no—
146#1292—532no—
147C2×C53#180511yescharacteristic
148C2×C53#181311yescharacteristic
149C2×C53#181611yescharacteristic
150D53—22no—
151#533—531no—
152#446—531no—
153#446—531no—
154#446—1061no—
155C8×Q8—531no—
156C8×Q8—1061no—
157C8×Q8#615711yescharacteristic
158#424—1061no—
159#423—531no—
160#423—532no—
161C2×C4⋊C8—532no—
162#405—1061no—
163C2×C4×C8—531no—
164#341—1061no—
165#341—1061no—
166#335—1061no—
167#335—1061no—
168#333—1062no—
169#332—1062no—
170C53#868911yescharacteristic, complemented, Sylow
171C2×M16—1061no—
172C2×M16—1062no—
173C22×C8—531no—
174C22×C8—531no—
175C22×C8—1062no—
176C4⋊Q8—531no—
177C4⋊Q8—531no—
178#125—1061no—
179C22⋊Q8—531no—
180C22⋊Q8—531no—
181C22⋊Q8—1061no—
182C4×Q8—531no—
183C4×Q8—1061no—
184C4×Q8C2×#615711yescharacteristic
185#117—1061no—
186C2×C42—531no—
187C4⋊C8—21no—
188C4⋊C8—531no—
189C4⋊C8—1061no—
190C4⋊C8—1061no—
191C4⋊C8—1062no—
192C4⋊C8—1062no—
193C4⋊C8—1062no—
194C4⋊C8—1062no—
195C4⋊C8—1061no—
196C4⋊C8—1061no—
197C4⋊C8C2×#615711yescharacteristic
198#97—1061no—
199#97—1061no—
200C4×C8—21no—
201C4×C8—531no—
202C4×C8—531no—
203C4×C8—531no—
204C4×C8—1061no—
205C4×C8C2×#615711yescharacteristic
206C2×Q8—531no—
207C2×Q8—1061no—
208C2×Q8C4⋊#615711yescharacteristic
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212M16—2122no—
213M16—2122no—
214M16—2122no—
215C2×C8—21no—
216C2×C8—531no—
217C2×C8—531no—
218C2×C8—1062no—
219C2×C8—1062no—
220C2×C8—1061no—
221C2×C8—4242no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8C22⋊#615711yescharacteristic
225C2×C8C22⋊#615711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—531no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4C4⋊#615711yescharacteristic
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C42—21no—
238C42—531no—
239C42—531no—
240C42—531no—
241C42—1061no—
242C42C22×#615711yescharacteristic
243C23—531no—
244Q8—41no—
245Q8—2121no—
246Q8—2121no—
247Q8—2121no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—531no—
251C2×C4—531no—
252C2×C4—532no—
253C2×C4—532no—
254C2×C4—531no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—2121no—
260C2×C4C2×C22⋊#615711yescharacteristic
261C2×C4C2×C4⋊#615711yescharacteristic
262C2×C4#1313329611yescharacteristic
263C8—21no—
264C8—21no—
265C8—41no—
266C8—1061no—
267C8—1061no—
268C8—4242no—
269C8—4242no—
270C8—2121no—
271C22—532no—
272C22—1061no—
273C22—1061no—
274C22#1681156511yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—532no—
280C4—531no—
281C4—531no—
282C4—1061no—
283C4—2121no—
284C4—2121no—
285C4#1681161311yescharacteristic
286C4#1681161611yescharacteristic
287C2—1062no—
288C2#2029058611yescharacteristic
289C2#2029059911yescharacteristic
290C2#2029060411yescharacteristic
subgroupG/H
centerC22#16811565
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC8×Q8×C53C4
socleC22×C53#380
prime radicalC22×D53C22⋊Q8
cosocleC42×D53C23
hypercenterC8×Q8#6157
hypocenterC53#8689
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
170. C53#8689semidirect

Maximal subgroups

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

subgroupindex
#202906402
D53⋊#3352
#202902322
#202895842
#202836442
#868953