Finite Group Databasebeta

← #27287293

Subgroups — #27287293

Order 13568.

446 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 77 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#20275550C211yesmaximal, characteristic
2#20273243C211yesmaximal, characteristic
3#20273230C211yesmaximal, characteristic
4#20272621C211yesmaximal, characteristic, complemented
5#20272601C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#2665C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×#13133131—21no—
10C2×#13133131—21no—
11#16810587C2211yescharacteristic
12#16810587C2211yescharacteristic
13C4×C22⋊Dic53C2211yescharacteristic
14#16810356—21no—
15#16810356—21no—
16#16810356—21no—
17#16810356—21no—
18C53×#400—21no—
19C53×#396—21no—
20C53×#395—21no—
21C53×#393—21no—
22C53×#391—21no—
23C53×#387—21no—
24C53×#381C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C22×D212Q811yescharacteristic, complemented
28C22×C22⋊D53—21no—
29C2×#11719917—21no—
30C2×#11719917—21no—
31#13133191—21no—
32#13133191—21no—
33#13133191—21no—
34#13133191—21no—
35C2×C4⋊Dic53Q811yescharacteristic, complemented
36C2×C22⋊Dic53—21no—
37C2×#11719921—21no—
38C2×C4×Dic53—21no—
39#13133131—42no—
40#13133131—42no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133124—21no—
46C22×Q8×C53—21no—
47C23×C4×C53C2311yescharacteristic
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65#95×C53D411yescharacteristic
66#95×C53D411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C23×D53—21no—
69C2×Q8×C53—42no—
70C2×Q8×C53—81no—
71C2×Q8×C53—81no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—81no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×Q811yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C2×C22⋊D53—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C22×Dic53—21no—
92C22×Dic53—41no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—81no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—81no—
98C4⋊C4×C53—81no—
99C4⋊C4×C53—81no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—81no—
104C42×C53—81no—
105C22⋊Dic53—22no—
106C22⋊Dic53—41no—
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4×Dic53—81no—
112#11719917—81no—
113#11719917—81no—
114C2×#88322—41no—
115C2×#88322—41no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53#12011yescharacteristic
121C23×C53C22⋊Q811yescharacteristic
122C23×C53#12411yescharacteristic
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—81no—
127C22×D53—81no—
128Q8×C53—84no—
129C2×C4×C53—22no—
130C2×C4×C53—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—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53C22⋊Q811yescharacteristic
154C2×C4×C53C22⋊Q811yescharacteristic
155C2×C4×C53#12411yescharacteristic
156C2×C4×C53#12411yescharacteristic
157C22⋊D53—81no—
158C22⋊D53—81no—
159C22⋊D53—81no—
160C22⋊D53—81no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164C2×Dic53—41no—
165C2×Dic53—81no—
166C2×Dic53—81no—
167D212—81no—
168D212—81no—
169#88322—81no—
170#88322—81no—
171#10382—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#39311yescharacteristic
183C22×C53#45311yescharacteristic
184C22×C53#45411yescharacteristic
185C2×D53—81no—
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—161no—
189C2×D53—161no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200Dic53—82no—
201Dic53—81no—
202#2665D5311yescharacteristic, complemented
203#1964—531no—
204#1510—531no—
205#1507—531no—
206#1305—531no—
207#1297—531no—
208#1284—531no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—41no—
212C2×C53—41no—
213C2×C53#202711yescharacteristic
214C2×C53#203111yescharacteristic
215C2×C53#203211yescharacteristic
216D53—161no—
217C2×#121—531no—
218C2×#99—1061no—
219C2×#99—1061no—
220#407—531no—
221#407—531no—
222#400—21no—
223#396—21no—
224#395—21no—
225#393—21no—
226#391—21no—
227#387—21no—
228#381C2×D5311yescharacteristic
229#380C2×D5311yescharacteristic
230C4×C22⋊C4—531no—
231C4×C22⋊C4C2×D5311yescharacteristic
232#327—1061no—
233#327—1061no—
234#327—1061no—
235#327—1061no—
236C53#1038211yescharacteristic, complemented, Sylow
237C22×Q8—21no—
238C22×D4—531no—
239C22×D4—1061no—
240C23×C4C22×D5311yescharacteristic
241C2×M16—1061no—
242C2×M16—1061no—
243#121—1061no—
244#121—1061no—
245#121—1061no—
246#121—1061no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—531no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4C22⋊D5311yescharacteristic
258C2×C42—21no—
259C2×C42—1061no—
260#99—2122no—
261#99—2122no—
262C22⋊C8—1061no—
263C22⋊C8—1061no—
264C22⋊C8—1061no—
265C22⋊C8—1061no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—1061no—
273#95C22⋊D5311yescharacteristic
274#95C22⋊D5311yescharacteristic
275#95C22⋊D5311yescharacteristic
276#95C22⋊D5311yescharacteristic
277C24—1061no—
278C24C2×C22⋊D5311yescharacteristic
279C2×Q8—42no—
280C2×Q8—81no—
281C2×Q8—81no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—4241no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C2×D4—2121no—
290C2×D4—2121no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—81no—
298C22×C4—1061no—
299C22×C4—2121no—
300C22×C4Q8×D5311yescharacteristic
301C22×C4#1171994911yescharacteristic
302C22×C4#1171994911yescharacteristic
303C22×C4#1171994911yescharacteristic
304C22×C4C2×C22⋊D5311yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306M16—4241no—
307M16—4241no—
308C2×C8—2121no—
309C2×C8—2121no—
310C4⋊C4—42no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—81no—
320C22⋊C4—1062no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—81no—
325C42—4241no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—4241no—
331C23—4241no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23#1313318611yescharacteristic
336C23#1313319311yescharacteristic
337C23#1313320611yescharacteristic
338Q8—84no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—4241no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—4241no—
370C2×C4—4241no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4#1313319311yescharacteristic
376C2×C4#1313319311yescharacteristic
377C2×C4#1313320611yescharacteristic
378C2×C4#1313320611yescharacteristic
379C8—4241no—
380C8—4241no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—8481no—
392C22—8481no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22#1681053411yescharacteristic
397C22#1681079111yescharacteristic
398C22#1681079611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—4242no—
410C4—4241no—
411C2—21no—
412C2—21no—
413C2—41no—
414C2—41no—
415C2—8481no—
416C2#2027586811yescharacteristic
417C2#2027588411yescharacteristic
418C2#2027589111yescharacteristic
subgroupG/H
centerC22#16810534
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2665C2
socleC22×C53#393
prime radicalC2×#13133191C22
hypercenter#2665D53
hypocenterC53#10382
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2665

Complements

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

NH ≅ G/Ntype
4. #20272621C2semidirect
5. #20272601C2semidirect
6. #20272567C2semidirect
7. C53×#2665C2semidirect
27. C22×D212Q8semidirect
35. C2×C4⋊Dic53Q8semidirect
202. #2665D53semidirect
236. C53#10382semidirect

Maximal subgroups

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

subgroupindex
#202755502
#202732432
#202732302
#202726212
#202726012
#202725672
C53×#26652
#1038253