Finite Group Databasebeta

← #28303236

Subgroups — #28303236

Order 13568.

350 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290637C211yesmaximal, characteristic
2#20290265C211yesmaximal, characteristic
3#20290148C212yesmaximal
4#20289570C212yesmaximal
5#20283644C211yesmaximal, characteristic
6#16811642C2212yes—
7#16811567C2212yes—
8#16811497—21no—
9#16811497—21no—
10#16811486—21no—
11#16811485—21no—
12#16811477—22no—
13#16811476—22no—
14#16811177C2211yescharacteristic
15#16810754C411yescharacteristic, complemented
16#16810752—21no—
17#16810750C411yescharacteristic, complemented
18C8×Dic106—21no—
19C8×Dic106C411yescharacteristic, complemented
20#16810648—21no—
21#16810645C2211yescharacteristic
22#16810577—21no—
23C4×C8×D53C2211yescharacteristic
24C8×Q8×C53C411yescharacteristic, complemented
25C2×C4⋊#6157—22no—
26C4⋊#88324—22no—
27#13133274—22no—
28C4×#88324—22no—
29(C2×#6157)⋊C4—22no—
30(C2×#6157)⋊C4—22no—
31C2×#11719912—21no—
32C2×C8×D53D411yescharacteristic
33C2×C8×D53D411yescharacteristic
34#13133216Q811yescharacteristic
35#13133215Q811yescharacteristic
36#13133207—21no—
37#13133197D411yescharacteristic
38#13133195—21no—
39#13133193D411yescharacteristic
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
60C22×#6157—42no—
61C2×Q8×C53C4⋊C411yescharacteristic
62C2×C4×D53C2×D411yescharacteristic
63C2×C4×D53C2×D411yescharacteristic
64C2×C4×D53#4111yescharacteristic
65C2×Dic106—21no—
66C2×Dic106C4⋊C411yescharacteristic
67C2×#88324—42no—
68C4⋊#6157—42no—
69C4⋊#6157—42no—
70C2×C8×C53—21no—
71C2×C8×C53C22⋊C411yescharacteristic
72C2×C8×C53C22⋊C411yescharacteristic
73C4⋊C4×C53—21no—
74C4⋊C4×C53C4⋊C411yescharacteristic
75C42×C53—21no—
76C42×C53C22×C411yescharacteristic
77#11719921—21no—
78#11719921—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53C4⋊C411yescharacteristic
81#11719919—21no—
82#11719919—21no—
83#11719919—21no—
84#11719919—21no—
85#11719919—21no—
86C4×Dic53—21no—
87C4×Dic53C2×D411yescharacteristic
88C4×Dic53C2×Q811yescharacteristic
89C4×Dic53#4111yescharacteristic
90C2×#88322—21no—
91C2×#88322C22⋊C411yescharacteristic
92C2×#88322C22⋊C411yescharacteristic
93#11719912—42no—
94C8×D53—22no—
95C8×D53—22no—
96C22×D53#12111yescharacteristic
97C2×#6157—82no—
98C2×#6157—82no—
99C2×#6157—82no—
100Q8×C53—41no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53C2×C22⋊C411yescharacteristic
104C2×C4×C53C2×C4⋊C411yescharacteristic
105C2×C4×C53#11711yescharacteristic
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53#12111yescharacteristic
109C2×Dic53C22⋊Q811yescharacteristic
110C2×Dic53C22⋊Q811yescharacteristic
111C4×D53—41no—
112C4×D53—41no—
113C4×D53#13512yes—
114C4×D53#13712yes—
115Dic106—42no—
116Dic106—41no—
117#88324—82no—
118C8×C53—21no—
119C8×C53—21no—
120C8×C53—41no—
121#88322—21no—
122#88322—21no—
123#88322—41no—
124#8681—531nomaximal, Sylow
125C22×C53#38011yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53#46412yes—
129#6157—162no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53#41611yescharacteristic
135C4×C53#41911yescharacteristic
136Dic53—21no—
137Dic53—21no—
138Dic53—41no—
139Dic53—41no—
140Dic53#47912yes—
141#2960—531no—
142#1835—531no—
143#1564—531no—
144#1518—532no—
145#1286—532no—
146C2×C53#180511yescharacteristic
147C2×C53#181211yescharacteristic
148C2×C53#181411yescharacteristic
149D53—22no—
150#533—531no—
151#446—531no—
152#446—531no—
153#446—1061no—
154C8×Q8—531no—
155C8×Q8—1061no—
156C8×Q8#615711yescharacteristic, complemented
157#431—532no—
158#424—1061no—
159#423—531no—
160#405—1061no—
161C2×C4×C8—531no—
162#382—532no—
163#340—1061no—
164#340—1061no—
165#335—1061no—
166#334—1061no—
167#328—1062no—
168#328—1062no—
169C53#868111yescharacteristic, complemented, Sylow
170C2×M16—1061no—
171C22×C8—531no—
172C22×C8—531no—
173C4⋊Q8—531no—
174C4⋊Q8—531no—
175#125—1061no—
176C22⋊Q8—531no—
177C22⋊Q8—531no—
178C22⋊Q8—1061no—
179C4×Q8—531no—
180C4×Q8—1061no—
181C4×Q8C2×#615711yescharacteristic
182#117—1061no—
183C2×C4⋊C4—1062no—
184C2×C42—531no—
185C4⋊C8—21no—
186C4⋊C8—531no—
187C4⋊C8—1061no—
188C4⋊C8—1061no—
189C4⋊C8—1062no—
190C4⋊C8—1061no—
191C4⋊C8—1061no—
192C4⋊C8C2×#615711yescharacteristic
193C22⋊C8—1062no—
194#97—1061no—
195#97—1061no—
196C4×C8—21no—
197C4×C8—531no—
198C4×C8—531no—
199C4×C8—531no—
200C4×C8—1062no—
201C4×C8—1061no—
202C4×C8C2×#615711yescharacteristic
203#95—1062no—
204#95—1062no—
205C2×Q8—531no—
206C2×Q8—1061no—
207C2×Q8C4⋊#615711yescharacteristic
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—2122no—
212M16—2122no—
213C2×C8—21no—
214C2×C8—531no—
215C2×C8—531no—
216C2×C8—1062no—
217C2×C8—1062no—
218C2×C8—1061no—
219C2×C8—2122no—
220C2×C8C22⋊#615711yescharacteristic
221C2×C8C22⋊#615711yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—531no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—2122no—
231C4⋊C4—2122no—
232C4⋊C4C4⋊#615711yescharacteristic
233C22⋊C4—1061no—
234C22⋊C4—1061no—
235C42—21no—
236C42—531no—
237C42—531no—
238C42—531no—
239C42—1061no—
240C42C22×#615711yescharacteristic
241C23—531no—
242Q8—41no—
243Q8—2121no—
244Q8—2122no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—531no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—531no—
251C2×C4—531no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—4242no—
255C2×C4—4242no—
256C2×C4—4242no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4C2×C22⋊#615711yescharacteristic
260C2×C4C2×C4⋊#615711yescharacteristic
261C2×C4#1313329611yescharacteristic
262C8—21no—
263C8—21no—
264C8—41no—
265C8—1061no—
266C8—1061no—
267C8—4242no—
268C8—2121no—
269C22—532no—
270C22—1061no—
271C22—1061no—
272C22#1681156511yescharacteristic
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—532no—
278C4—8482no—
279C4—1061no—
280C4—1061no—
281C4—2121no—
282C4—2121no—
283C4#1681161311yescharacteristic
284C4#1681161611yescharacteristic
285C2—1062no—
286C2#2029058611yescharacteristic
287C2#2029059711yescharacteristic
288C2#2029060211yescharacteristic
subgroupG/H
centerC22#16811565
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC8×Q8×C53C4
socleC22×C53#380
prime radicalC22×D53#121
cosocleC42×D53C23
hypercenterC8×Q8#6157
hypocenterC53#8681
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
15. #16810754C4semidirect
17. #16810750C4semidirect
19. C8×Dic106C4semidirect
24. C8×Q8×C53C4semidirect
156. C8×Q8#6157semidirect
169. C53#8681semidirect

Maximal subgroups

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

subgroupindex
#202906372
#202902652
#202901482
#202895702
#202836442
#868153