Finite Group Databasebeta

← #27293336

Subgroups — #27293336

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#20283408C211yesmaximal, characteristic
2#20276324C211yesmaximal, characteristic, complemented
3C22⋊C16×D53C211yesmaximal, characteristic, complemented
4#20272657C211yesmaximal, characteristic, complemented
5#20272640C211yesmaximal, characteristic, complemented
6#20272640C211yesmaximal, characteristic, complemented
7C53×#1323C211yesmaximal, characteristic, complemented
8C2×#13133225C411yescharacteristic
9C22×C8×D53C2211yescharacteristic
10#16811135C411yescharacteristic
11C2×#13133156—21no—
12C2×C16×D53—21no—
13#16810693—21no—
14#16810693—21no—
15#16810688—21no—
16#16810688—21no—
17#16810589C2211yescharacteristic
18#16810586C411yescharacteristic
19#16810579C411yescharacteristic
20C22⋊#11719908C2211yescharacteristic
21C22⋊#11719908C2211yescharacteristic
22#16810402—21no—
23#16810402—21no—
24C53×#424C2211yescharacteristic
25C53×C22⋊C16C2211yescharacteristic
26C53×C22⋊C16C2211yescharacteristic
27C2×#11719945C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29#13133225—42no—
30C2×#11719917C2×C411yescharacteristic
31C2×#11719912—21no—
32C22×#88322C2×C411yescharacteristic
33C2×C8×D53—21no—
34C2×C8×D53—42no—
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37C2×C8×D53D411yescharacteristic
38C2×C8×D53D411yescharacteristic
39#13133201—21no—
40#13133197C811yescharacteristic
41#13133197C811yescharacteristic
42#13133192C811yescharacteristic
43#13133192C811yescharacteristic
44#13133187—21no—
45C4×D212—21no—
46C4×D212—21no—
47#13133156—42no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C16×D53—42no—
51#13133129—21no—
52#13133129—21no—
53#13133129—21no—
54#13133129—21no—
55#13133128—21no—
56C4×#88322—21no—
57C2×M16×C53C2×C411yescharacteristic
58C22×C8×C53C2311yescharacteristic
59#117×C53C2×C411yescharacteristic
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C23×D53M1611yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C2×C22⋊D53—21no—
67C22×Dic53M1611yescharacteristic
68#11719945—41no—
69#11719945—41no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C2×C811yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53C22⋊C411yescharacteristic
77C2×C4×D53C22⋊C411yescharacteristic
78C2×C4×D53C2×C811yescharacteristic
79C2×C4×D53C2×C811yescharacteristic
80C2×Dic106C2×C811yescharacteristic
81M16×C53—41no—
82C2×C8×C53—21no—
83C2×C8×C53—41no—
84C2×C8×C53C2×D411yescharacteristic
85C2×C8×C53C2×D411yescharacteristic
86C4⋊C4×C53C2×C811yescharacteristic
87C4⋊C4×C53C2×C811yescharacteristic
88C22⋊C4×C53—21no—
89C42×C53—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96#11719917—41no—
97C2×#88322—22no—
98C2×#88322—21no—
99C2×#88322C22⋊C411yescharacteristic
100C2×#88322C22⋊C411yescharacteristic
101#11719912—42no—
102C8×D53—22no—
103C8×D53—22no—
104C8×D53—41no—
105C8×D53—41no—
106C8×D53—42no—
107C16×C53—41no—
108C16×C53—41no—
109#11719908—41no—
110#11719908—41no—
111C23×C53C2×M1611yescharacteristic
112C22×D53—22no—
113C22×D53—21no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53C22⋊C811yescharacteristic
117C22×D53C22⋊C811yescharacteristic
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53C22×C811yescharacteristic
124C22⋊D53—42no—
125C2×Dic53—21no—
126C2×Dic53—41no—
127C2×Dic53C22⋊C811yescharacteristic
128C2×Dic53C22⋊C811yescharacteristic
129D212—41no—
130D212—42no—
131D212—42no—
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—41no—
137C4×D53—41no—
138C4×D53—41no—
139C4×D53—41no—
140Dic106—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×C53C2×C22⋊C811yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—44no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53#35012yes—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#9911yescharacteristic
166C4×C53#41311yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170#2929—531no—
171#2112—531no—
172C2×C22⋊C16—531no—
173#1323—531no—
174#1323D5311yescharacteristic, complemented
175#1315—531no—
176#1315—531no—
177C2×C53—41no—
178C2×C53#145911yescharacteristic
179C2×C53C2×#35011yescharacteristic
180C2×C53#211711yescharacteristic
181D53—22no—
182D53—41no—
183D53—81no—
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#424C2×D5311yescharacteristic
194#408—531no—
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×#41—531no—
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#117C4×D5311yescharacteristic
227M32—2122no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—1061no—
231C2×C16—1061no—
232C2×C16—2122no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C22⋊C8—1061no—
236C22⋊C8—1061no—
237C22⋊C8—1061no—
238C22⋊C8—1061no—
239#97—1061no—
240C4×C8—1061no—
241C24—531no—
242#41—2121no—
243#41—2121no—
244C2×Q8—531no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—1061no—
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—1061no—
275C4⋊C4—1061no—
276C4⋊C4C8×D5311yescharacteristic
277C4⋊C4C8×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—21no—
284C16—41no—
285C16—41no—
286C16—2121no—
287C16—2121no—
288C23—531no—
289C23—531no—
290C23—1062no—
291C23—1061no—
292C23—2121no—
293C23—2121no—
294C23M16×D5311yescharacteristic
295Q8—2121no—
296D4—2121no—
297D4—2122no—
298D4—2122no—
299D4—2122no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—2121no—
312C2×C4—2121no—
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—532no—
328C22—1061no—
329C22—1061no—
330C22—1061no—
331C22—1061no—
332C22—1061no—
333C22—2124no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C22⋊C8×D5311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—1061no—
342C4—1061no—
343C4—2121no—
344C4#99×D5311yescharacteristic
345C4#1681060611yescharacteristic
346C2—41no—
347C2—1062no—
348C2—4241no—
349C2—2121no—
350C2#2027303211yescharacteristic
351C2#350×D5311yescharacteristic
352C2#2027634111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1323C2
socleC22×C53C2×C22⋊C8
prime radical#16811135C4
cosocleC22×C8×C53C23
hypercenter#1323D53
hypocenterC53#11470
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #20276324C2semidirect
3. C22⋊C16×D53C2semidirect
4. #20272657C2semidirect
5. #20272640C2semidirect
6. #20272640C2semidirect
7. C53×#1323C2semidirect
174. #1323D53semidirect
203. C53#11470semidirect

Maximal subgroups

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

subgroupindex
#202834082
#202763242
C22⋊C16×D532
#202726572
#202726402
C53×#13232
#1147053