Finite Group Databasebeta

← #27299243

Subgroups — #27299243

Order 13568.

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 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#20283412C211yesmaximal, characteristic
2#20276354C212yesmaximal
3#20274517C212yesmaximal, complemented
4#20272838C211yesmaximal, characteristic, complemented
5C53×#1391C211yesmaximal, characteristic, complemented
6C2×#13133224C2211yescharacteristic
7C22×C8×D53C411yescharacteristic, complemented
8#16811130C2211yescharacteristic
9#16810697—22no—
10#16810686—22no—
11#16810649C411yescharacteristic, complemented
12#16810607C2212yes—
13#16810586C411yescharacteristic, complemented
14#16810581C411yescharacteristic, complemented
15#16810579—22no—
16#16810412C414yescomplemented
17#16810409C414yescomplemented
18#16810393—22no—
19C53×#410C2212yes—
20C53×#408C2211yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22#13133224—41no—
23#13133224—41no—
24C2×#11719917—22no—
25C2×#11719917C2×C411yescharacteristic
26C2×#11719917D412yes—
27C22×#88322C2×C411yescharacteristic
28C22×#88322D412yes—
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—41no—
32C2×C8×D53—41no—
33#13133198C2×C412yes—
34#13133193—21no—
35#13133188—21no—
36#13133185C2×C412yes—
37C4×C22⋊D53—22no—
38#13133176C2×C411yescharacteristic
39#13133146—22no—
40#13133136—22no—
41#13133134C2×C414yes—
42#13133129—22no—
43#13133128—42no—
44#13133127—21no—
45C4×#88322—42no—
46C8×Dic53—21no—
47C2×#41×C53C2311yescharacteristic
48C2×M16×C53C2×C411yescharacteristic
49C22×C8×C53C2×C411yescharacteristic
50#117×C53—22no—
51#99×C53—22no—
52#99×C53C2×C414yes—
53C22⋊C8×C53—22no—
54C23×D53C4211yescharacteristic
55#41×C53—41no—
56#41×C53—41no—
57C2×Q8×C53#4111yescharacteristic
58C2×D4×C53C22×C412yes—
59C2×D4×C53#4111yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C2×C22⋊D53—22no—
63C22×Dic53C4211yescharacteristic
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53—41no—
67C2×C4×D53—41no—
68M16×C53—41no—
69C2×C8×C53—21no—
70C2×C8×C53—21no—
71C2×C8×C53—41no—
72C4⋊C4×C53—42no—
73C22⋊C4×C53—24no—
74C42×C53—42no—
75C22⋊Dic53—22no—
76C22⋊Dic53C4212yes—
77#11719921—22no—
78C4⋊Dic53—21no—
79#11719919—22no—
80C4×Dic53—21no—
81#11719917—24no—
82#11719917—41no—
83#11719917—42no—
84#11719917—42no—
85C2×#88322—22no—
86C2×#88322—22no—
87C2×#88322—21no—
88C2×#88322—21no—
89C2×#88322—42no—
90C2×#88322C22⋊C411yescharacteristic
91C2×#88322C22⋊C412yes—
92C2×#88322C22⋊C411yescharacteristic
93C8×D53—42no—
94C8×D53—41no—
95C8×D53—41no—
96C23×C53C2×C4211yescharacteristic
97C23×C53C4×D412yes—
98C22×D53—21no—
99C22×D53—42no—
100C22×D53—41no—
101Q8×C53—41no—
102D4×C53—42no—
103D4×C53—41no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—44no—
109C2×C4×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53#11711yescharacteristic
111C2×C4×C53C4×D412yes—
112C22⋊D53—44no—
113C2×Dic53—21no—
114C2×Dic53—22no—
115C2×Dic53—41no—
116C4×D53—42no—
117C4×D53—41no—
118C4×D53—41no—
119C8×C53—41no—
120C8×C53—41no—
121#88322—21no—
122#88322—22no—
123#88322—44no—
124#88322#9911yescharacteristic
125#88322#9911yescharacteristic
126#12623—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53C4×C22⋊C411yescharacteristic
132C2×D53—22no—
133C2×D53—41no—
134C2×D53—42no—
135C2×D53—41no—
136C2×D53—42no—
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—84no—
140C4×C53C2×#9911yescharacteristic
141C4×C53#41011yescharacteristic
142Dic53—42no—
143Dic53—41no—
144#2929—531no—
145#2120—532no—
146#1757—532no—
147#1392—531no—
148#1391D5311yescharacteristic, complemented
149C2×C53—21no—
150C2×C53—42no—
151C2×C53—41no—
152C2×C53C4×#9911yescharacteristic
153D53—42no—
154C2×#131—531no—
155C23×C8—531no—
156#525—531no—
157#435—1062no—
158C8×D4—1062no—
159#424—531no—
160#413—532no—
161#410C2×D5312yes—
162#408C2×D5311yescharacteristic
163#407—531no—
164#405—531no—
165#405—1062no—
166#352—534no—
167AΣL(1,𝔽16)—534no—
168#344—1062no—
169C53#1262311yescharacteristic, complemented, Sylow
170C2×#41C22×D5311yescharacteristic
171C23×C4—531no—
172#131—2121no—
173#131—2121no—
174C2×M16—531no—
175C2×M16—532no—
176C2×M16—1062no—
177C2×M16C4×D5311yescharacteristic
178C22×C8—531no—
179C22×C8—532no—
180C22×C8—1061no—
181C22×C8—1061no—
182C22×C8—2121no—
183C22×C8—2121no—
184C22×C8C4×D5311yescharacteristic
185#123—532no—
186C22⋊Q8—1061no—
187#121—1061no—
188#120—532no—
189C4×D4—1062no—
190#117—22no—
191#117—531no—
192C4⋊C8—1062no—
193#101—1062no—
194#100—534no—
195#99—22no—
196#99C4×D5314yes—
197C22⋊C8—22no—
198C22⋊C8—1062no—
199#97—1061no—
200#97—2122no—
201C4×C8—1061no—
202C4×C8—2122no—
203C24—531no—
204#41—41no—
205#41—41no—
206C2×Q8#1171994911yescharacteristic
207C2×D4—1062no—
208C2×D4C2×C4×D5312yes—
209C2×D4#1171994711yescharacteristic
210C22×C4—531no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4—2121no—
214C22×C4—2121no—
215C22×C4C2×C4×D5311yescharacteristic
216C22×C4D4×D5312yes—
217M16—41no—
218M16—1064no—
219M16—2122no—
220M16—2121no—
221M16—2122no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—531no—
226C2×C8—531no—
227C2×C8—532no—
228C2×C8—1062no—
229C2×C8—1062no—
230C2×C8—1061no—
231C2×C8—1061no—
232C2×C8—2121no—
233C2×C8—2121no—
234C2×C8—2122no—
235C2×C8—2122no—
236C4⋊C4—42no—
237C4⋊C4—1061no—
238C4⋊C4—1062no—
239C22⋊C4—24no—
240C22⋊C4—532no—
241C22⋊C4—1062no—
242C22⋊C4—1062no—
243C42—42no—
244C42—1061no—
245C23—1061no—
246C23—2121no—
247C23—2122no—
248C23C42×D5311yescharacteristic
249C23#1313317712yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—2124no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—1062no—
260C2×C4—1061no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4C22⋊C4×D5311yescharacteristic
266C2×C4#1313317511yescharacteristic
267C2×C4#1313317912yes—
268C8—41no—
269C8—41no—
270C8—531no—
271C8—531no—
272C8—1061no—
273C8—1062no—
274C8—2124no—
275C22—22no—
276C22—21no—
277C22—42no—
278C22—41no—
279C22—1062no—
280C22—2121no—
281C22—2122no—
282C22—2122no—
283C22—2121no—
284C22#1681047011yescharacteristic
285C4—22no—
286C4—21no—
287C4—84no—
288C4—2122no—
289C4—2121no—
290C4#99×D5311yescharacteristic
291C4#1681059511yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—2122no—
296C2#2027450211yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC23×C53C2×C42
FrattiniC2×#41C22×D53
FittingC53×#1391C2
socleC2×C53C4×#99
prime radical#16811130C22
cosocleC2×#41×C53C23
hypercenter#1391D53
hypocenterC53#12623
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #20274517C2semidirect
4. #20272838C2semidirect
5. C53×#1391C2semidirect
7. C22×C8×D53C4semidirect
11. #16810649C4semidirect
13. #16810586C4semidirect
14. #16810581C4semidirect
16. #16810412C4semidirect
17. #16810409C4semidirect
148. #1391D53semidirect
169. C53#12623semidirect

Maximal subgroups

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

subgroupindex
#202834122
#202763542
#202745172
#202728382
C53×#13912
#1262353