Finite Group Databasebeta

← #27294416

Subgroups — #27294416

Order 13568.

648 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13133132C211yesmaximal, characteristic, complemented
2#20273036C211yesmaximal, characteristic, complemented
3#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2388C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133132—21no—
8C4×C22⋊Dic53—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C22×#41×C53C411yescharacteristic, complemented
13C2×C4×D4×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#395C411yescharacteristic, complemented
16C53×#394—21no—
17C53×#394C411yescharacteristic, complemented
18C53×#393C2211yescharacteristic
19C53×#392—21no—
20C53×#392C2211yescharacteristic
21C4×C22⋊C4×C53—21no—
22C4×C22⋊C4×C53C411yescharacteristic, complemented
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—41no—
25C4⋊#88322—41no—
26#13133132—24no—
27C22⋊#88322—22no—
28C4×#88322—41no—
29#13133124—41no—
30C2×#41×C53—44no—
31C2×#41×C53—41no—
32C2×#41×C53—41no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53—21no—
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C4×D4×C53—44no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53D411yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53C2×C411yescharacteristic
58#95×C53C2×C411yescharacteristic
59C24×C53—21no—
60C24×C53C22⋊C411yescharacteristic
61C24×C53C2×D411yescharacteristic
62#41×C53—44no—
63#41×C53—44no—
64#41×C53—44no—
65#41×C53—44no—
66C2×Q8×C53—42no—
67C2×Q8×C53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—41no—
71C2×D4×C53—42no—
72C2×D4×C53—42no—
73C2×D4×C53—41no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—42no—
77C2×D4×C53—42no—
78C2×D4×C53—42no—
79C22×C4×C53—24no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×Dic53—42no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—42no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C42×C53—21no—
108C42×C53—21no—
109C42×C53—42no—
110C22⋊Dic53—24no—
111C22⋊Dic53—42no—
112C4×Dic53—82no—
113C2×#88322—42no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53#13512yes—
132Q8×C53—42no—
133Q8×C53—42no—
134D4×C53—44no—
135D4×C53—42no—
136D4×C53—48no—
137D4×C53—48no—
138D4×C53—42no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—44no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—44no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53#9911yescharacteristic
164C2×C4×C53#9911yescharacteristic
165C2×C4×C53#10412yes—
166C2×Dic53—44no—
167C2×Dic53—82no—
168C2×Dic53—82no—
169#88322—82no—
170#11725—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53C2×#9911yescharacteristic
191C22×C53#41611yescharacteristic
192C22×C53C2×#10411yescharacteristic
193C4×C53—21no—
194C4×C53—22no—
195C4×C53—42no—
196C4×C53—42no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—81no—
201C4×C53AΣL(1,𝔽16)12yes—
202Dic53—84no—
203#2388D5311yescharacteristic
204C4×#99—531no—
205#1460—531no—
206#1351—532no—
207#1343—532no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—42no—
211C2×C53—81no—
212C2×C53—82no—
213C2×C53#150111yescharacteristic
214C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
215C2×C53#212611yescharacteristic
216C22×#41Dic5311yescharacteristic, complemented
217C2×C4×D4—21no—
218C2×C4×D4C2×D5311yescharacteristic
219#431—1061no—
220C2×#99—1061no—
221#395Dic5311yescharacteristic, complemented
222#394—21no—
223#394Dic5311yescharacteristic, complemented
224#393C2×D5311yescharacteristic
225#392—21no—
226#392C2×D5311yescharacteristic
227C4×C22⋊C4—21no—
228C4×C22⋊C4—1061no—
229C4×C22⋊C4Dic5311yescharacteristic, complemented
230#342—532no—
231#323—532no—
232#323—1061no—
233C53#1172511yescharacteristic, complemented, Sylow
234C2×#41—44no—
235C2×#41—41no—
236C2×#41—41no—
237C22×Q8C2×Dic5311yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×Dic5311yescharacteristic
241C22×D4C22⋊D5311yescharacteristic
242C23×C4—21no—
243C23×C4C2×Dic5311yescharacteristic
244C23×C4C22⋊D5311yescharacteristic
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—44no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C2×Dic5311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4C22×D5311yescharacteristic
258C2×C42—21no—
259C2×C42—2121no—
260C2×C42C22⋊D5311yescharacteristic
261C4⋊C8—2121no—
262#99—1064no—
263C22⋊C8—1062no—
264C4×C8—2121no—
265#95—21no—
266#95—21no—
267#95—2121no—
268#95C2×Dic5311yescharacteristic
269#95C2×Dic5311yescharacteristic
270C24—21no—
271C24C22⋊Dic5311yescharacteristic
272C24C2×C22⋊D5311yescharacteristic
273#41—44no—
274#41—44no—
275#41—44no—
276#41—44no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—2122no—
304C22×C4C22⋊Dic5311yescharacteristic
305C22×C4C22⋊Dic5311yescharacteristic
306C22×C4C22⋊Dic5311yescharacteristic
307C22×C4C22×Dic5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C2×C8—2122no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—1064no—
320C22⋊C4—2122no—
321C42—21no—
322C42—21no—
323C42—42no—
324C42—4242no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊Dic5311yescharacteristic
342C23#1313323712yes—
343Q8—42no—
344Q8—42no—
345D4—44no—
346D4—42no—
347D4—48no—
348D4—48no—
349D4—42no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—4242no—
375C2×C4—4242no—
376C2×C4—2124no—
377C2×C4#1313313211yescharacteristic
378C2×C4#1313313211yescharacteristic
379C2×C4#1313314412yes—
380C8—4242no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×#1313313211yescharacteristic
401C22#1681061911yescharacteristic
402C22C2×#1313314411yescharacteristic
403C4—22no—
404C4—21no—
405C4—42no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—4244no—
412C4#1681041012yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—82no—
417C2—81no—
418C2#2027320611yescharacteristic
419C2C2×#1681041011yescharacteristic
420C2#2027638011yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2388C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2388D53
hypocenterC53#11725
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. C4×#13133132C2semidirect
2. #20273036C2semidirect
3. #20272736C2semidirect
4. #20272712C2semidirect
12. C22×#41×C53C4semidirect
15. C53×#395C4semidirect
17. C53×#394C4semidirect
22. C4×C22⋊C4×C53C4semidirect
216. C22×#41Dic53semidirect
221. #395Dic53semidirect
223. #394Dic53semidirect
229. C4×C22⋊C4Dic53semidirect
233. C53#11725semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730362
#202727362
#202727122
C53×#23882
#1172553