Finite Group Databasebeta

← #27272612

Subgroups — #27272612

Order 13568.

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

normal subgroups only

This group has 494 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
1#20283056C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274407C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic, complemented
5#20272539C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1272C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×#13133193C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11C2×#13133129—21no—
12#16810530C411yescharacteristic
13#16810522C411yescharacteristic
14#16810521C2211yescharacteristic
15#16810506—21no—
16#16810505—21no—
17#16810497—21no—
18#16810485—21no—
19#16810480C411yescharacteristic
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24C53×#391C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C2×#11719912—24no—
31C2×#11719912—42no—
32C2×#11719912—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133193—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C42×D53—42no—
47#13133129—42no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C22×Q8×C53C2×C411yescharacteristic
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#95×C53—21no—
59#95×C53—21no—
60C23×D53C22⋊C411yescharacteristic
61C2×Q8×C53—21no—
62C2×Q8×C53—21no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—41no—
94C42×C53—41no—
95#11719921—42no—
96#11719921—42no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—42no—
100C4⋊Dic53—41no—
101#11719919—42no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C2×#88322—22no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113#11719912—48no—
114C23×C53C2×C22⋊C411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53#9912yes—
122Q8×C53—81no—
123Q8×C53—81no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
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#13111yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53#10412yes—
157C4×D53—24no—
158C4×D53—24no—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—42no—
162C4×D53—42no—
163C8×C53—42no—
164C8×C53—81no—
165#88322—42no—
166#88322—81no—
167#7695—531nomaximal, Sylow
168C22×C53#40811yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53#41311yescharacteristic
171C22×C53#41711yescharacteristic
172C22×C53#41911yescharacteristic
173C22×C53C2×#10411yescharacteristic
174C22×C53#42111yescharacteristic
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—81no—
189C4×C53—81no—
190Dic53—22no—
191Dic53—41no—
192Dic53—41no—
193Dic53—81no—
194Dic53—81no—
195#2855—531no—
196#1817—531no—
197#1728—531no—
198#1272—531no—
199#1272D5311yescharacteristic, complemented
200#1271—531no—
201#1271—531no—
202C2×C53#146811yescharacteristic
203C2×C53#148511yescharacteristic
204C2×C53#149511yescharacteristic
205C2×C53#150411yescharacteristic
206C2×C53#151111yescharacteristic
207C2×C53#152911yescharacteristic
208C2×C53#155911yescharacteristic
209D53—42no—
210C22×M16—1061no—
211C2×C22⋊Q8—531no—
212C22×C42—531no—
213C2×C22⋊C8—1061no—
214#391—531no—
215#391C2×D5311yescharacteristic
216#389—531no—
217#389—531no—
218#387—1061no—
219#387—1061no—
220#385—1061no—
221#381—1061no—
222#380—531no—
223#336—531no—
224#336—531no—
225#336—1061no—
226#336—1061no—
227#336C2×D5311yescharacteristic
228#336C2×D5311yescharacteristic
229C53#769511yescharacteristic, complemented, Sylow
230C22×Q8C4×D5311yescharacteristic
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C2×M16—1064no—
235C2×M16—2122no—
236C2×M16—2122no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—1061no—
240C22×C8—1061no—
241C22⋊Q8—2122no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4C4×D5311yescharacteristic
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—2122no—
254C2×C42C22×D5311yescharacteristic
255C22⋊C8—2122no—
256#95—21no—
257#95—21no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263C24—531no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—1062no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4C2×C4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290M16—2128no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—1062no—
298C2×C8—2121no—
299C2×C8—2121no—
300C2×C8—2121no—
301C2×C8—2121no—
302C2×C8—2121no—
303C4⋊C4—41no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—2121no—
307C4⋊C4—2122no—
308C4⋊C4—2122no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—2122no—
312C22⋊C4—2122no—
313C42—41no—
314C42—2121no—
315C42—2121no—
316C42—2121no—
317C23—532no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23C22⋊C4×D5311yescharacteristic
325Q8—81no—
326Q8—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—532no—
340C2×C4—1064no—
341C2×C4—1064no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4#1313322311yescharacteristic
361C2×C4#1313322511yescharacteristic
362C2×C4#1313324411yescharacteristic
363C2×C4#1313324611yescharacteristic
364C2×C4#1313324911yescharacteristic
365C2×C4#1313324911yescharacteristic
366C8—42no—
367C8—81no—
368C8—4241no—
369C8—2122no—
370C22—1062no—
371C22—1062no—
372C22—1062no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22#1681058811yescharacteristic
378C22#99×D5311yescharacteristic
379C22#1681060411yescharacteristic
380C22#1681062311yescharacteristic
381C22#1681063011yescharacteristic
382C22#104×D5311yescharacteristic
383C22#1681063811yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—81no—
392C4—1062no—
393C4—4241no—
394C4—4241no—
395C4—2121no—
396C4—2121no—
397C2—2122no—
398C2#2027306011yescharacteristic
399C2#2027313411yescharacteristic
400C2#2027317911yescharacteristic
401C2#2027321811yescharacteristic
402C2#2027324611yescharacteristic
403C2#2027331911yescharacteristic
404C2#2027344211yescharacteristic
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#7695
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. #20272539C2semidirect
6. #20272537C2semidirect
7. C53×#1272C2semidirect
199. #1272D53semidirect
229. C53#7695semidirect

Maximal subgroups

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

subgroupindex
#202830562
#202747822
#202744072
#202725412
#202725392
#202725372
C53×#12722
#769553