Finite Group Databasebeta

← #28303268

Subgroups — #28303268

Order 13568.

358 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C4×#13133287C211yesmaximal, characteristic, complemented
2#20290276C211yesmaximal, characteristic, complemented
3#20290180C212yesmaximal
4#20289581C212yesmaximal, complemented
5#20283693C211yesmaximal, characteristic
6#16811645C2212yes—
7C2×#13133287—21no—
8C4×#11719936C2212yes—
9#16811499—21no—
10#16811486—21no—
11#16811484—21no—
12#16811480—22no—
13#16811478—22no—
14#16811155C2211yescharacteristic
15#16810753C411yescharacteristic
16#16810747C411yescharacteristic
17#16810692—21no—
18D4×#88322C411yescharacteristic
19#16810686—21no—
20#16810681—21no—
21#16810679C2211yescharacteristic
22#16810581—21no—
23C4×C8×D53C2211yescharacteristic
24C53×#436C411yescharacteristic
25C2×#11719936—22no—
26#13133287—24no—
27C4⋊#88324—22no—
28#13133274—22no—
29#13133272—22no—
30#13133272—22no—
31C4×#88324—22no—
32C22×#88322—21no—
33C2×C8×D53D411yescharacteristic
34C2×C8×D53D411yescharacteristic
35#13133210—21no—
36#13133208D411yescharacteristic
37#13133204D411yescharacteristic
38#13133199—21no—
39#13133197Q811yescharacteristic
40#13133188Q811yescharacteristic
41C4×Dic106C2×C411yescharacteristic
42C4×C22⋊D53—21no—
43#13133176—21no—
44C42×D53C2311yescharacteristic
45C4⋊#88322C2×C411yescharacteristic
46#13133146—21no—
47#13133146—21no—
48C22⋊#88322—21no—
49#13133129—21no—
50#13133127—21no—
51#13133127—21no—
52C4×#88322C2×C411yescharacteristic
53C8×Dic53—21no—
54C8×Dic53D411yescharacteristic
55C8×Dic53D411yescharacteristic
56C2×M16×C53—21no—
57C4×D4×C53C2×C411yescharacteristic
58C4⋊C8×C53C2×C411yescharacteristic
59C22⋊C8×C53—21no—
60C4×C8×C53C2×C411yescharacteristic
61C2×D4×C53C4⋊C411yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C4×D53C2×D411yescharacteristic
65C2×C4×D53C2×Q811yescharacteristic
66C2×C4×D53#4111yescharacteristic
67C2×Dic106C4⋊C411yescharacteristic
68C2×#88324—42no—
69C2×#88324—42no—
70#11719936—24no—
71#11719936—42no—
72M16×C53—42no—
73C2×C8×C53—21no—
74C2×C8×C53C22⋊C411yescharacteristic
75C2×C8×C53C22⋊C411yescharacteristic
76C4⋊C4×C53C4⋊C411yescharacteristic
77C22⋊C4×C53—21no—
78C42×C53C22×C411yescharacteristic
79C22⋊Dic53—21no—
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82#11719921—21no—
83C4⋊Dic53—21no—
84C4⋊Dic53—21no—
85C4⋊Dic53C4⋊C411yescharacteristic
86#11719919—21no—
87#11719919—21no—
88C4×Dic53—21no—
89C4×Dic53C2×D411yescharacteristic
90C4×Dic53C2×D411yescharacteristic
91C4×Dic53#4111yescharacteristic
92C2×#88322—21no—
93C2×#88322—42no—
94C2×#88322C22⋊C411yescharacteristic
95C2×#88322C22⋊C411yescharacteristic
96C8×D53—22no—
97C8×D53—41no—
98C23×C53—21no—
99C22×D53C22⋊Q811yescharacteristic
100D4×C53—41no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53C2×C4⋊C411yescharacteristic
106C2×C4×C53#11711yescharacteristic
107C22⋊D53—42no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53#12111yescharacteristic
111C2×Dic53#12111yescharacteristic
112C2×Dic53C22⋊Q811yescharacteristic
113C4×D53—21no—
114C4×D53—21no—
115C4×D53—41no—
116C4×D53—41no—
117Dic106—41no—
118#88324—44no—
119#88324—82no—
120C8×C53—21no—
121C8×C53—41no—
122C8×C53#10412yes—
123#88322—21no—
124#88322—21no—
125#88322—41no—
126#8693—531nomaximal, Sylow
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53#38011yescharacteristic
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53C2×#10411yescharacteristic
138C4×C53#42111yescharacteristic
139Dic53—21no—
140Dic53—21no—
141Dic53—21no—
142Dic53—41no—
143Dic53—41no—
144#2966—531no—
145C4×#108—531no—
146#1568—531no—
147#1529—532no—
148#1291—532no—
149C2×C53—81no—
150C2×C53#180611yescharacteristic
151C2×C53#181311yescharacteristic
152C2×C53#181411yescharacteristic
153D53—41no—
154#529—531no—
155#446—531no—
156C8×Q8—531no—
157#436#615711yescharacteristic
158#435—1061no—
159C8×D4—531no—
160C8×D4—1061no—
161#433—1061no—
162#432—531no—
163#432—532no—
164C2×#108—1061no—
165#405—1061no—
166C4×M16—532no—
167C2×C4×C8—531no—
168#341—1061no—
169#335—1061no—
170#334—1061no—
171#330—1062no—
172#329—1062no—
173C53#869311yescharacteristic, complemented, Sylow
174C2×M16—21no—
175C2×M16—1062no—
176C22×C8—531no—
177C22×C8—531no—
178C22×C8—1061no—
179#126—1061no—
180#125—531no—
181#124—531no—
182#123—1061no—
183C22⋊Q8—531no—
184#121—531no—
185C4×Q8—531no—
186C4×D4—1061no—
187C4×D4C2×#615711yescharacteristic
188#117—1061no—
189C2×C42—531no—
190#108—1064no—
191C4⋊C8—531no—
192C4⋊C8—1062no—
193C4⋊C8—1061no—
194C4⋊C8—1061no—
195C4⋊C8C2×#615711yescharacteristic
196C22⋊C8—21no—
197C22⋊C8—1061no—
198C22⋊C8—1061no—
199C22⋊C8—1062no—
200#97—1061no—
201#97—1062no—
202#97—1062no—
203#97—1061no—
204C4×C8—531no—
205C4×C8—531no—
206C4×C8—531no—
207C4×C8—1062no—
208C4×C8—1061no—
209C4×C8C2×#615711yescharacteristic
210C2×Q8—531no—
211C2×D4—1061no—
212C2×D4C4⋊#615711yescharacteristic
213C22×C4—21no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217M16—42no—
218M16—1064no—
219M16—2122no—
220C2×C8—21no—
221C2×C8—531no—
222C2×C8—531no—
223C2×C8—1062no—
224C2×C8—1061no—
225C2×C8—2121no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8C22⋊#615711yescharacteristic
230C2×C8C22⋊#615711yescharacteristic
231C4⋊C4—531no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4—1061no—
235C4⋊C4—1061no—
236C4⋊C4C4⋊#615711yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—1061no—
239C22⋊C4—1061no—
240C22⋊C4—1061no—
241C22⋊C4—1061no—
242C42—531no—
243C42—531no—
244C42—531no—
245C42—1061no—
246C42C22×#615711yescharacteristic
247C23—21no—
248C23—531no—
249Q8—2121no—
250D4—41no—
251D4—2122no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—531no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4C2×C22⋊#615711yescharacteristic
265C2×C4C2×C4⋊#615711yescharacteristic
266C2×C4#1313329611yescharacteristic
267C8—21no—
268C8—41no—
269C8—1061no—
270C8—1061no—
271C8—4242no—
272C8—2124no—
273C8—2121no—
274C8#1313328112yes—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—1061no—
279C22—1061no—
280C22—1061no—
281C22#1681156511yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—1061no—
286C4—1061no—
287C4—1061no—
288C4—2121no—
289C4—2121no—
290C4C2×#1313328111yescharacteristic
291C4#1681162011yescharacteristic
292C2—81no—
293C2—2121no—
294C2#2029058711yescharacteristic
295C2#2029060011yescharacteristic
296C2#2029060111yescharacteristic
subgroupG/H
centerC22#16811565
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#436C4
socleC22×C53#380
prime radical#13133188Q8
cosocleC42×D53C23
hypercenter#436#6157
hypocenterC53#8693
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C4×#13133287C2semidirect
2. #20290276C2semidirect
4. #20289581C2semidirect
173. C53#8693semidirect

Maximal subgroups

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

subgroupindex
C4×#131332872
#202902762
#202901802
#202895812
#202836932
#869353