Finite Group Databasebeta

← #27272664

Subgroups — #27272664

Order 13568.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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#20283064C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274476C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic, complemented
5#20272540C211yesmaximal, characteristic, complemented
6#20272540C211yesmaximal, characteristic, complemented
7C53×#1272C211yesmaximal, characteristic, complemented
8C2×#13133215C411yescharacteristic
9C2×#13133197C411yescharacteristic
10C2×#13133193C411yescharacteristic
11C2×C42×D53C2211yescharacteristic
12C2×#13133129—21no—
13C2×#13133129—21no—
14C2×#13133127—21no—
15#16810529C2211yescharacteristic
16#16810520—21no—
17#16810520—21no—
18#16810492C411yescharacteristic
19#16810482—21no—
20#16810482—21no—
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#391C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×#88322—21no—
32C22×#88322—21no—
33#13133215—42no—
34#13133197—42no—
35#13133193—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—42no—
49#13133129—24no—
50#13133129—42no—
51#13133127—42no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53C2×C411yescharacteristic
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61C23×D53C22⋊C411yescharacteristic
62C2×Q8×C53—21no—
63C2×Q8×C53—21no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C2×C4×D53—22no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106—41no—
91C2×Dic106—41no—
92C2×C8×C53—22no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C4⋊C4×C53—41no—
99C42×C53—41no—
100#11719921—42no—
101#11719921—42no—
102C4⋊Dic53—42no—
103C4⋊Dic53—41no—
104#11719919—42no—
105#11719919—42no—
106#11719919—42no—
107#11719919—42no—
108#11719919—42no—
109C4×Dic53—22no—
110C4×Dic53—22no—
111C4×Dic53—41no—
112C2×#88322—22no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53C2×C22⋊C411yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53#10012yes—
126Q8×C53—81no—
127Q8×C53—81no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
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#13111yescharacteristic
141C2×C4×C53#13111yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53#9912yes—
160C2×Dic53#10412yes—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—42no—
167Dic106—81no—
168Dic106—81no—
169C8×C53—42no—
170C8×C53—81no—
171#88322—42no—
172#88322—81no—
173#7707—531nomaximal, Sylow
174C22×C53#40811yescharacteristic
175C22×C53C2×#9911yescharacteristic
176C22×C53C2×#10011yescharacteristic
177C22×C53#41911yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53C2×#10411yescharacteristic
180C22×C53#42111yescharacteristic
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—81no—
195C4×C53—81no—
196Dic53—22no—
197Dic53—22no—
198Dic53—41no—
199Dic53—81no—
200Dic53—81no—
201#2857—531no—
202#1817—531no—
203#1749—531no—
204#1272—531no—
205#1272—531no—
206#1272—531no—
207#1272D5311yescharacteristic, complemented
208C2×C53#147111yescharacteristic
209C2×C53#148511yescharacteristic
210C2×C53#149711yescharacteristic
211C2×C53#150811yescharacteristic
212C2×C53#151111yescharacteristic
213C2×C53#153011yescharacteristic
214C2×C53#155811yescharacteristic
215D53—42no—
216C2×C4⋊Q8—531no—
217C2×C22⋊Q8—531no—
218C2×C22⋊Q8—531no—
219C22×C42—531no—
220C2×C22⋊C8—1061no—
221C2×C22⋊C8—1061no—
222C2×#97—1061no—
223#391—531no—
224#391C2×D5311yescharacteristic
225#389—1061no—
226#389—1061no—
227#384—531no—
228#380—1061no—
229#380—1061no—
230#336—531no—
231#336—531no—
232#336—1061no—
233#336C2×D5311yescharacteristic
234#336C2×D5311yescharacteristic
235C53#770711yescharacteristic, complemented, Sylow
236C22×Q8—531no—
237C22×Q8C4×D5311yescharacteristic
238C23×C4—531no—
239C23×C4—531no—
240C23×C4—531no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—1061no—
244C22×C8—1061no—
245C4⋊Q8—2122no—
246C22⋊Q8—2122no—
247C22⋊Q8—2122no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4C4×D5311yescharacteristic
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—531no—
261C2×C42—2122no—
262C2×C42C22×D5311yescharacteristic
263C22⋊C8—1064no—
264C22⋊C8—2122no—
265#97—2122no—
266#95—21no—
267#95—21no—
268#95—1061no—
269#95—1061no—
270C24—531no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1061no—
276C2×Q8—1061no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1062no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4D4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—1062no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C2×C8—2121no—
313C4⋊C4—41no—
314C4⋊C4—2121no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C4⋊C4—2122no—
318C4⋊C4—2122no—
319C4⋊C4—2122no—
320C4⋊C4—2122no—
321C22⋊C4—2122no—
322C22⋊C4—2122no—
323C42—41no—
324C42—1062no—
325C42—1062no—
326C42—2121no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23C22⋊C4×D5311yescharacteristic
335Q8—81no—
336Q8—81no—
337Q8—4241no—
338Q8—4241no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—532no—
352C2×C4—532no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2122no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4#1313322311yescharacteristic
373C2×C4#1313322511yescharacteristic
374C2×C4#1313324611yescharacteristic
375C2×C4#1313324611yescharacteristic
376C2×C4#1313324911yescharacteristic
377C2×C4#1313324911yescharacteristic
378C8—42no—
379C8—81no—
380C8—4241no—
381C8—2122no—
382C22—1062no—
383C22—1062no—
384C22—1062no—
385C22—2121no—
386C22—2121no—
387C22—2121no—
388C22—2121no—
389C22#1681058811yescharacteristic
390C22#99×D5311yescharacteristic
391C22#100×D5311yescharacteristic
392C22#1681063011yescharacteristic
393C22#1681063011yescharacteristic
394C22#104×D5311yescharacteristic
395C22#1681063811yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—1062no—
405C4—1062no—
406C4—4241no—
407C4—4241no—
408C4—2121no—
409C2—2122no—
410C2#2027307311yescharacteristic
411C2#2027313411yescharacteristic
412C2#2027318611yescharacteristic
413C2#2027323311yescharacteristic
414C2#2027324611yescharacteristic
415C2#2027332411yescharacteristic
416C2#2027343911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7707
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #20272541C2semidirect
5. #20272540C2semidirect
6. #20272540C2semidirect
7. C53×#1272C2semidirect
207. #1272D53semidirect
235. C53#7707semidirect

Maximal subgroups

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

subgroupindex
#202830642
#202747822
#202744762
#202725412
#202725402
C53×#12722
#770753