Finite Group Databasebeta

← #27272500

Subgroups — #27272500

Order 13568.

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

normal subgroups only

This group has 468 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#20283024C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274472C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×#13133208C411yescharacteristic
9C2×C42×D53C2211yescharacteristic
10C2×#13133129—21no—
11C2×#13133129—21no—
12C2×C8×Dic53—21no—
13#16810521C2211yescharacteristic
14#16810509—21no—
15#16810505—21no—
16#16810497—21no—
17#16810494—21no—
18#16810488C411yescharacteristic
19#16810480C411yescharacteristic
20#16810480C411yescharacteristic
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×#88322—21no—
31C22×#88322—21no—
32#13133208—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C42×D53—42no—
44#13133129—24no—
45#13133129—42no—
46C8×Dic53—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#95×C53—21no—
59C23×D53C22⋊C411yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C8×C53—22no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C42×C53—41no—
94#11719921—42no—
95#11719921—42no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—42no—
99C4⋊Dic53—41no—
100#11719919—42no—
101#11719919—41no—
102#11719919—41no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—41no—
106C2×#88322—22no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53#10412yes—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53#13111yescharacteristic
133C2×C4×C53#13111yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53#9912yes—
152C2×Dic53#10112yes—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—42no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159C8×C53—42no—
160C8×C53—81no—
161#88322—42no—
162#88322—81no—
163#7671—531nomaximal, Sylow
164C22×C53#40811yescharacteristic
165C22×C53C2×#9911yescharacteristic
166C22×C53C2×#10111yescharacteristic
167C22×C53#41611yescharacteristic
168C22×C53#41611yescharacteristic
169C22×C53C2×#10411yescharacteristic
170C22×C53#42111yescharacteristic
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—81no—
185C4×C53—81no—
186Dic53—22no—
187Dic53—22no—
188Dic53—41no—
189Dic53—81no—
190Dic53—81no—
191#2851—531no—
192#1817—531no—
193C4×C22⋊C8—531no—
194#1271—531no—
195#1271—531no—
196#1271—531no—
197#1271D5311yescharacteristic, complemented
198C2×C53#147211yescharacteristic
199C2×C53#148211yescharacteristic
200C2×C53#149411yescharacteristic
201C2×C53#150111yescharacteristic
202C2×C53#151611yescharacteristic
203C2×C53#152811yescharacteristic
204C2×C53#154411yescharacteristic
205D53—42no—
206C2×#125—531no—
207C22×C42—531no—
208C2×C22⋊C8—1061no—
209C2×C22⋊C8—1061no—
210C2×C4×C8—1061no—
211#389—531no—
212#389C2×D5311yescharacteristic
213#387—1061no—
214#387—1061no—
215#385—1061no—
216#385—1061no—
217#382—531no—
218#380—531no—
219#380—531no—
220#336—531no—
221#336—531no—
222#336—1061no—
223#336C2×D5311yescharacteristic
224#336C2×D5311yescharacteristic
225C53#767111yescharacteristic, complemented, Sylow
226C23×C4—531no—
227C23×C4—531no—
228C23×C4—531no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—1061no—
232C22×C8—1061no—
233#125—2122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4C4×D5311yescharacteristic
241C2×C4⋊C4C4×D5311yescharacteristic
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—2122no—
248C2×C42C22×D5311yescharacteristic
249C22⋊C8—1064no—
250C22⋊C8—2122no—
251C4×C8—2122no—
252#95—21no—
253#95—1061no—
254#95—1061no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258C24—531no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—1062no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4C2×C4×D5311yescharacteristic
279C22×C4D4×D5311yescharacteristic
280C22×C4D4×D5311yescharacteristic
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—1062no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—2121no—
301C4⋊C4—2122no—
302C4⋊C4—2122no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C22⋊C4—2122no—
306C22⋊C4—2122no—
307C42—41no—
308C42—1062no—
309C42—1062no—
310C42—2121no—
311C23—532no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23C22⋊C4×D5311yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—532no—
332C2×C4—532no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4#1313322311yescharacteristic
353C2×C4#1313322511yescharacteristic
354C2×C4#1313323611yescharacteristic
355C2×C4#1313323811yescharacteristic
356C2×C4#1313323811yescharacteristic
357C2×C4#1313324011yescharacteristic
358C8—42no—
359C8—81no—
360C8—4241no—
361C8—2122no—
362C22—1062no—
363C22—1062no—
364C22—1062no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22#1681058811yescharacteristic
370C22#99×D5311yescharacteristic
371C22#101×D5311yescharacteristic
372C22#1681061611yescharacteristic
373C22#1681061811yescharacteristic
374C22#104×D5311yescharacteristic
375C22#1681063811yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—1062no—
385C4—1062no—
386C4—4241no—
387C4—4241no—
388C4—2121no—
389C2—2122no—
390C2#2027307711yescharacteristic
391C2#2027312011yescharacteristic
392C2#2027317411yescharacteristic
393C2#2027320311yescharacteristic
394C2#2027327011yescharacteristic
395C2#2027331511yescharacteristic
396C2#2027337811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7671
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272539C2semidirect
5. #20272538C2semidirect
6. #20272537C2semidirect
7. C53×#1271C2semidirect
197. #1271D53semidirect
225. C53#7671semidirect

Maximal subgroups

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

subgroupindex
#202830242
#202747822
#202744722
#202725392
#202725382
#202725372
C53×#12712
#767153