Finite Group Databasebeta

← #28303248

Subgroups — #28303248

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⋊#334C211yesmaximal, characteristic
3#20290232C212yesmaximal
4#20289584C212yesmaximal
5#20283644C211yesmaximal, characteristic
6#16811628C2212yes—
7D53⋊(C4⋊C8)C2212yes—
8#16811499—21no—
9#16811499—21no—
10#16811485—21no—
11#16811484—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—22no—
110C4×D53—41no—
111C4×D53#13512yes—
112C4×D53#13712yes—
113Dic106—42no—
114Dic106—41no—
115#88324—82no—
116#88324—82no—
117C8×C53—21no—
118C8×C53—21no—
119C8×C53—41no—
120#88322—21no—
121#88322—21no—
122#88322—41no—
123#8687—531nomaximal, Sylow
124C22×C53#38011yescharacteristic
125C2×D53—21no—
126C2×D53—21no—
127C2×D53#47912yes—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53#41611yescharacteristic
133C4×C53#41911yescharacteristic
134Dic53—21no—
135Dic53—41no—
136Dic53—41no—
137Dic53#46412yes—
138Dic53#47112yes—
139#2960—531no—
140#1837—531no—
141C2×#334—531no—
142#1550—532no—
143#1292—532no—
144C2×C53#180511yescharacteristic
145C2×C53#181311yescharacteristic
146C2×C53#181511yescharacteristic
147D53—22no—
148#533—531no—
149#446—531no—
150#446—531no—
151#446—1061no—
152C8×Q8—531no—
153C8×Q8—1061no—
154C8×Q8#615711yescharacteristic
155#424—1061no—
156#423—531no—
157#423—532no—
158C2×C4⋊C8—532no—
159#405—1061no—
160C2×C4×C8—531no—
161#341—1061no—
162#341—1061no—
163#334—1061no—
164#334—1061no—
165#333—1062no—
166#332—1062no—
167C53#868711yescharacteristic, complemented, Sylow
168C2×M16—1061no—
169C2×M16—1062no—
170C22×C8—531no—
171C22×C8—531no—
172C22×C8—1062no—
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×C42—531no—
184C4⋊C8—21no—
185C4⋊C8—531no—
186C4⋊C8—1061no—
187C4⋊C8—1061no—
188C4⋊C8—1062no—
189C4⋊C8—1062no—
190C4⋊C8—1062no—
191C4⋊C8—1062no—
192C4⋊C8—1061no—
193C4⋊C8—1061no—
194C4⋊C8C2×#615711yescharacteristic
195#97—1061no—
196#97—1061no—
197C4×C8—21no—
198C4×C8—531no—
199C4×C8—531no—
200C4×C8—531no—
201C4×C8—1061no—
202C4×C8C2×#615711yescharacteristic
203C2×Q8—531no—
204C2×Q8—1061no—
205C2×Q8C4⋊#615711yescharacteristic
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—531no—
209M16—2122no—
210M16—2122no—
211M16—2122no—
212C2×C8—21no—
213C2×C8—531no—
214C2×C8—531no—
215C2×C8—1062no—
216C2×C8—1062no—
217C2×C8—1061no—
218C2×C8—4242no—
219C2×C8—2122no—
220C2×C8—2122no—
221C2×C8C22⋊#615711yescharacteristic
222C2×C8C22⋊#615711yescharacteristic
223C4⋊C4—21no—
224C4⋊C4—531no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4C4⋊#615711yescharacteristic
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C42—21no—
235C42—531no—
236C42—531no—
237C42—531no—
238C42—1061no—
239C42C22×#615711yescharacteristic
240C23—531no—
241Q8—41no—
242Q8—2121no—
243Q8—2122no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—531no—
247C2×C4—531no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—531no—
251C2×C4—1062no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—2121no—
255C2×C4C2×C22⋊#615711yescharacteristic
256C2×C4C2×C4⋊#615711yescharacteristic
257C2×C4#1313329611yescharacteristic
258C8—21no—
259C8—21no—
260C8—41no—
261C8—1061no—
262C8—1061no—
263C8—4242no—
264C8—4242no—
265C8—2121no—
266C22—532no—
267C22—1061no—
268C22—1061no—
269C22#1681156511yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—532no—
275C4—532no—
276C4—1061no—
277C4—2121no—
278C4—2121no—
279C4#1681161311yescharacteristic
280C4#1681161611yescharacteristic
281C2—1062no—
282C2#2029058611yescharacteristic
283C2#2029059911yescharacteristic
284C2#2029060311yescharacteristic
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#8687
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
167. C53#8687semidirect

Maximal subgroups

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

subgroupindex
#202906402
D53⋊#3342
#202902322
#202895842
#202836442
#868753