Finite Group Databasebeta

← #27293334

Subgroups — #27293334

Order 13568.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#20276324C211yesmaximal, characteristic, complemented
3C22⋊C16×D53C211yesmaximal, characteristic, complemented
4#20272656C211yesmaximal, characteristic, complemented
5#20272656C211yesmaximal, characteristic, complemented
6#20272641C211yesmaximal, characteristic, complemented
7C53×#1315C211yesmaximal, characteristic, complemented
8C2×#13133224C411yescharacteristic
9C22×C8×D53C2211yescharacteristic
10#16811130C411yescharacteristic
11C2×#13133156—21no—
12C2×C16×D53—21no—
13#16810697—21no—
14#16810697—21no—
15#16810686—21no—
16#16810686—21no—
17#16810649C2211yescharacteristic
18#16810586C411yescharacteristic
19#16810581C411yescharacteristic
20C22⋊#11719908C2211yescharacteristic
21C22⋊#11719908C2211yescharacteristic
22#16810402—21no—
23#16810402—21no—
24C53×#408C2211yescharacteristic
25C53×C22⋊C16C2211yescharacteristic
26C53×C22⋊C16C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28#13133224—42no—
29C2×#11719917—21no—
30C2×#11719917C2×C411yescharacteristic
31C22×#88322—21no—
32C22×#88322C2×C411yescharacteristic
33C2×C8×D53—42no—
34C2×C8×D53D411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37C2×C8×D53D411yescharacteristic
38#13133198—21no—
39#13133193C811yescharacteristic
40#13133193C811yescharacteristic
41#13133188C811yescharacteristic
42#13133188C811yescharacteristic
43#13133185—21no—
44C4×C22⋊D53—21no—
45C4×C22⋊D53—21no—
46#13133176C2×C411yescharacteristic
47#13133156—42no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C16×D53—42no—
51#13133146—21no—
52#13133146—21no—
53#13133129—21no—
54#13133129—21no—
55#13133127—21no—
56C8×Dic53—21no—
57C2×#41×C53C2×C411yescharacteristic
58C2×M16×C53C2×C411yescharacteristic
59C22×C8×C53C2311yescharacteristic
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C22⋊C8×C53—21no—
63C22⋊C8×C53—21no—
64C23×D53M1611yescharacteristic
65#41×C53—41no—
66#41×C53—41no—
67C2×Q8×C53C2×C811yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53C2×C811yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53C22×C411yescharacteristic
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C22×Dic53M1611yescharacteristic
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53C22⋊C411yescharacteristic
78C2×C4×D53C22⋊C411yescharacteristic
79C2×C4×D53C2×C811yescharacteristic
80C2×C4×D53C2×C811yescharacteristic
81M16×C53—41no—
82C2×C8×C53—21no—
83C2×C8×C53—41no—
84C2×C8×C53C2×D411yescharacteristic
85C2×C8×C53C2×D411yescharacteristic
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719921—21no—
90#11719921—21no—
91C4⋊Dic53C2×C811yescharacteristic
92C4⋊Dic53C2×C811yescharacteristic
93#11719919—21no—
94#11719919—21no—
95C4×Dic53—21no—
96#11719917—42no—
97#11719917—41no—
98C2×#88322—22no—
99C2×#88322—21no—
100C2×#88322—42no—
101C2×#88322C22⋊C411yescharacteristic
102C2×#88322C22⋊C411yescharacteristic
103C8×D53—22no—
104C8×D53—22no—
105C8×D53—41no—
106C8×D53—41no—
107C16×C53—41no—
108C16×C53—41no—
109#11719908—41no—
110#11719908—41no—
111C23×C53—21no—
112C23×C53C2×M1611yescharacteristic
113C22×D53—22no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53C22⋊C811yescharacteristic
117C22×D53C22⋊C811yescharacteristic
118Q8×C53—41no—
119D4×C53—41no—
120D4×C53—42no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53C22×C811yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—41no—
133C2×Dic53C22⋊C811yescharacteristic
134C2×Dic53C22⋊C811yescharacteristic
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—41no—
140C4×D53—41no—
141C8×C53—21no—
142C8×C53—21no—
143C8×C53—41no—
144#88322—21no—
145#88322—41no—
146#88322#9912yes—
147#11470—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C2×C22⋊C811yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—44no—
161C2×D53#35012yes—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53C2×#9911yescharacteristic
165C4×C53#41311yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170#2929—531no—
171#2112—531no—
172C2×C22⋊C16—531no—
173#1323—531no—
174#1323—531no—
175#1315—531no—
176#1315D5311yescharacteristic, complemented
177C2×C53—41no—
178C2×C53—81no—
179C2×C53#145911yescharacteristic
180C2×C53C2×#35011yescharacteristic
181C2×C53#211711yescharacteristic
182D53—22no—
183D53—41no—
184C2×#131—531no—
185C23×C8—531no—
186#525—531no—
187C2×M32—1061no—
188C22×C16—1061no—
189#435—1061no—
190#435—1061no—
191C8×D4—1061no—
192C8×D4—1061no—
193#424—531no—
194#408C2×D5311yescharacteristic
195#407—531no—
196#405—531no—
197C22⋊C16—531no—
198C22⋊C16—531no—
199C22⋊C16—1061no—
200C22⋊C16—1061no—
201C22⋊C16C2×D5311yescharacteristic
202C22⋊C16C2×D5311yescharacteristic
203C53#1147011yescharacteristic, complemented, Sylow
204C2×#41C4×D5311yescharacteristic
205C23×C4—531no—
206#131—2122no—
207C2×M16—531no—
208C2×M16—1061no—
209C2×M16C4×D5311yescharacteristic
210C22×C8—531no—
211C22×C8—531no—
212C22×C8—531no—
213C22×C8—531no—
214C22×C8—531no—
215C22×C8—1061no—
216C22×C8—2122no—
217C22×C8C22×D5311yescharacteristic
218#123—1061no—
219C22⋊Q8—531no—
220C22⋊Q8—531no—
221#121—531no—
222#121—531no—
223#120—1061no—
224C4×D4—1061no—
225C4×D4—1061no—
226#117—531no—
227M32—2122no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—1061no—
231C2×C16—1061no—
232C2×C16—2122no—
233C4⋊C8—1061no—
234C4⋊C8—1061no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—1061no—
238C22⋊C8—1061no—
239#97—1061no—
240C4×C8—1061no—
241C24—531no—
242#41—41no—
243#41—41no—
244C2×Q8C8×D5311yescharacteristic
245C2×D4—21no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4C8×D5311yescharacteristic
249C22×C4—21no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4C2×C4×D5311yescharacteristic
258M16—41no—
259M16—2122no—
260M16—2121no—
261C2×C8—21no—
262C2×C8—41no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—1062no—
266C2×C8—1062no—
267C2×C8—1062no—
268C2×C8—1061no—
269C2×C8—2121no—
270C2×C8—2121no—
271C2×C8—2122no—
272C2×C8D4×D5311yescharacteristic
273C2×C8D4×D5311yescharacteristic
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—1061no—
284C16—41no—
285C16—41no—
286C16—2121no—
287C16—2121no—
288C23—21no—
289C23—531no—
290C23—531no—
291C23—1062no—
292C23—2121no—
293C23—2121no—
294C23M16×D5311yescharacteristic
295Q8—41no—
296D4—41no—
297D4—42no—
298D4—2122no—
299D4—2122no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4C22⋊C4×D5311yescharacteristic
317C2×C4C2×C8×D5311yescharacteristic
318C8—21no—
319C8—21no—
320C8—41no—
321C8—532no—
322C8—1061no—
323C8—2121no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—532no—
331C22—1061no—
332C22—1061no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22—2124no—
337C22C22⋊C8×D5311yescharacteristic
338C4—21no—
339C4—41no—
340C4—1061no—
341C4—1061no—
342C4—2121no—
343C4—2121no—
344C4#99×D5311yescharacteristic
345C4#1681060411yescharacteristic
346C2—41no—
347C2—81no—
348C2—1062no—
349C2—2121no—
350C2#2027303011yescharacteristic
351C2#350×D5311yescharacteristic
352C2#2027634111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1315C2
socleC22×C53C2×C22⋊C8
prime radical#16811130C4
cosocleC22×C8×C53C23
hypercenter#1315D53
hypocenterC53#11470
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #20276324C2semidirect
3. C22⋊C16×D53C2semidirect
4. #20272656C2semidirect
5. #20272656C2semidirect
6. #20272641C2semidirect
7. C53×#1315C2semidirect
176. #1315D53semidirect
203. C53#11470semidirect

Maximal subgroups

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

subgroupindex
#202834122
#202763242
C22⋊C16×D532
#202726562
#202726412
C53×#13152
#1147053