Finite Group Databasebeta

← #27277726

Subgroups — #27277726

Order 13568.

466 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 85 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#20277683C211yesmaximal, characteristic
2C4×#13133131C211yesmaximal, characteristic
3#20273051C211yesmaximal, characteristic
4#20272601C211yesmaximal, characteristic, complemented
5#20272600C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#1284C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D53C2211yescharacteristic
9C2×#13133131—21no—
10#16810587C2211yescharacteristic
11#16810523—21no—
12#16810512—21no—
13#16810511—21no—
14#16810487C411yescharacteristic, complemented
15#16810485C411yescharacteristic, complemented
16#16810474—21no—
17#16810474C411yescharacteristic, complemented
18C4×C22⋊Dic53C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810470C411yescharacteristic, complemented
21#16810389—21no—
22#16810348—21no—
23#16810346—21no—
24C53×#407C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C2×#11719917—21no—
30C4×C22⋊D53—21no—
31C4×C22⋊D53—21no—
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919D411yescharacteristic
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53D411yescharacteristic
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921D411yescharacteristic
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53C2×C411yescharacteristic
48#13133131—24no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124C2×C411yescharacteristic
55#13133124C2×C411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C2×M16×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C42×C53—21no—
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53C2×C411yescharacteristic
66#95×C53C2×C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C23×D53C22⋊C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53—81no—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—41no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—41no—
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C22⋊C411yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53—81no—
92M16×C53—81no—
93C2×C8×C53—41no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—81no—
97C42×C53—81no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—22no—
101C22⋊Dic53—42no—
102C22⋊Dic53—41no—
103#11719921—21no—
104#11719921—21no—
105#11719921—42no—
106#11719921—81no—
107#11719919—21no—
108#11719919—21no—
109#11719919—42no—
110#11719919—42no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53—42no—
114C4×Dic53—81no—
115C4×Dic53—81no—
116C4×Dic53—81no—
117#11719917—81no—
118C2×#88322—41no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53#12311yescharacteristic
125C23×C53#12311yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—41no—
130C22×D53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53#11711yescharacteristic
147C2×C4×C53#11711yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×C4×C53#12311yescharacteristic
150C22⋊D53—42no—
151C22⋊D53—42no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—22no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164C2×Dic53—41no—
165C2×Dic53—81no—
166C2×Dic53—81no—
167C2×Dic53#10412yes—
168C4×D53—81no—
169C4×D53—81no—
170C8×C53—81no—
171#88322—81no—
172#8778—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#38111yescharacteristic
184C22×C53C2×#10411yescharacteristic
185C22×C53#42111yescharacteristic
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—81no—
190C2×D53—81no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53—82no—
198Dic53—22no—
199Dic53—42no—
200Dic53—41no—
201Dic53—41no—
202Dic53—81no—
203Dic53—82no—
204#2357—531no—
205C4×#99—531no—
206#1465—531no—
207#1297—531no—
208#1297—531no—
209#1284—531no—
210#1284D5311yescharacteristic, complemented
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53#182611yescharacteristic
216C2×C53#182711yescharacteristic
217C2×C53#182811yescharacteristic
218D53—81no—
219C2×C4×D4—531no—
220C2×#99—1061no—
221#407—531no—
222#407C2×D5311yescharacteristic
223#390—1061no—
224#388—1061no—
225#388—1061no—
226#382—531no—
227#381—531no—
228#381C2×D5311yescharacteristic
229C4×C4⋊C4—531no—
230C4×C4⋊C4—1061no—
231C4×C22⋊C4—531no—
232C4×C22⋊C4—531no—
233C4×C22⋊C4—531no—
234C4×C22⋊C4C2×D5311yescharacteristic
235#342—1061no—
236#324—1061no—
237#323—1061no—
238C53#877811yescharacteristic, complemented, Sylow
239C22×D4—531no—
240C23×C4—531no—
241C23×C4C22×D5311yescharacteristic
242C2×M16—21no—
243C2×M16—1061no—
244C4×D4—1061no—
245C4×D4—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C42—21no—
260C2×C42—531no—
261C2×C42—1061no—
262C2×C42—1061no—
263C2×C42—1061no—
264C2×C42—1061no—
265#99—1064no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270#95—21no—
271#95—21no—
272#95—531no—
273#95—531no—
274#95—1061no—
275#95—1061no—
276#95—1061no—
277#95C4×D5311yescharacteristic
278#95C4×D5311yescharacteristic
279C24—531no—
280C24D4×D5311yescharacteristic
281C2×D4—1062no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—2121no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—81no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—4241no—
297C22×C4—2121no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4#1171994511yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4#1171994711yescharacteristic
303C22×C4#1171994911yescharacteristic
304M16—81no—
305M16—4241no—
306C2×C8—41no—
307C2×C8—2121no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—2122no—
311C4⋊C4—2122no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1062no—
320C22⋊C4—4241no—
321C22⋊C4—2122no—
322C22⋊C4—2122no—
323C22⋊C4—2121no—
324C42—81no—
325C42—1061no—
326C42—1061no—
327C42—4241no—
328C42—4241no—
329C42—4241no—
330C42—2122no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—1061no—
336C23—1061no—
337C23—1061no—
338C23—2121no—
339C23—2121no—
340C23C22⋊C4×D5311yescharacteristic
341C23#1313319911yescharacteristic
342C23#1313320211yescharacteristic
343D4—2122no—
344D4—2122no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—532no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1062no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—4241no—
367C2×C4—4241no—
368C2×C4—4241no—
369C2×C4—4241no—
370C2×C4—2121no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313317411yescharacteristic
379C2×C4#1313317511yescharacteristic
380C2×C4#1313319911yescharacteristic
381C2×C4#1313320211yescharacteristic
382C8—81no—
383C8—4241no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—81no—
393C22—81no—
394C22—4241no—
395C22—4241no—
396C22—2121no—
397C22—2121no—
398C22—2121no—
399C22#1681048511yescharacteristic
400C22#104×D5311yescharacteristic
401C22#1681063811yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—81no—
407C4—81no—
408C4—82no—
409C4—1062no—
410C4—4241no—
411C4—4242no—
412C4—2122no—
413C4—2121no—
414C4—2121no—
415C2—21no—
416C2—21no—
417C2—41no—
418C2—41no—
419C2—4241no—
420C2#2027481311yescharacteristic
421C2#2027481711yescharacteristic
422C2#2027482211yescharacteristic
subgroupG/H
centerC22#16810485
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1284C2
socleC22×C53#381
prime radicalC22×C22⋊D53D4
cosocleC23×C4×C53C23
hypercenter#1284D53
hypocenterC53#8778
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
4. #20272601C2semidirect
5. #20272600C2semidirect
6. #20272567C2semidirect
7. C53×#1284C2semidirect
14. #16810487C4semidirect
15. #16810485C4semidirect
17. #16810474C4semidirect
20. #16810470C4semidirect
210. #1284D53semidirect
238. C53#8778semidirect

Maximal subgroups

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

subgroupindex
#202776832
C4×#131331312
#202730512
#202726012
#202726002
#202725672
C53×#12842
#877853