Finite Group Databasebeta

← #27273044

Subgroups — #27273044

Order 13568.

446 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 83 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#20281676C211yesmaximal, characteristic
2#20275958C211yesmaximal, characteristic, complemented
3#20275848C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272535C211yesmaximal, characteristic
7C53×#2058C211yesmaximal, characteristic, complemented
8C2×#13133212—21no—
9C2×#13133141—21no—
10C2×#13133141—21no—
11C2×#13133137—21no—
12#16810560—21no—
13#16810545—21no—
14#16810540—21no—
15#16810525C2211yescharacteristic
16#16810522—21no—
17#16810521C2211yescharacteristic
18#16810515—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#106×C53—21no—
24C53×#389C2211yescharacteristic
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×D212D411yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30#13133212—42no—
31#13133212—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45#13133141—42no—
46#13133141—42no—
47#13133137—42no—
48#13133124—21no—
49#13133124—21no—
50C22×Q8×C53—21no—
51C22×C8×C53—21no—
52C4⋊Q8×C53—42no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#106×C53—42no—
60#95×C53—21no—
61C23×D53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—41no—
80C2×D212—41no—
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53SD1612yes—
93C42×C53—41no—
94#11719921—42no—
95#11719921—42no—
96#11719921—41no—
97#11719921—41no—
98#11719921—41no—
99#11719921—41no—
100#11719921—81no—
101#11719921—81no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—42no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4⋊Dic53—81no—
108#11719919—41no—
109#11719919—41no—
110C4×Dic53—42no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C23×C53#12011yescharacteristic
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125Q8×C53—42no—
126Q8×C53—42no—
127Q8×C53—81no—
128Q8×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159D212—42no—
160D212—81no—
161C8×C53—42no—
162#88322—81no—
163#88322—81no—
164#7776—531nomaximal, Sylow
165C22×C53#44911yescharacteristic
166C22×C53#45011yescharacteristic
167C22×C53#45211yescharacteristic
168C22×C53#45411yescharacteristic
169C22×C53#45511yescharacteristic
170C22×C53#45611yescharacteristic
171C22×C53B(4,𝔽2)11yescharacteristic
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—42no—
186C4×C53—82no—
187C4×C53—81no—
188Dic53—81no—
189Dic53—81no—
190Dic53—81no—
191#2727—531no—
192#2058D5311yescharacteristic, complemented
193#2043—531no—
194#2023—531no—
195#1271—531no—
196#1271—531no—
197#1270—531no—
198C2×C53#159711yescharacteristic
199C2×C53#160411yescharacteristic
200C2×C53#160611yescharacteristic
201C2×C53#163511yescharacteristic
202C2×C53#164211yescharacteristic
203C2×C53#165211yescharacteristic
204C2×C53#166411yescharacteristic
205D53—161no—
206C2×C4⋊Q8C2×D5311yescharacteristic
207C2×#126—1061no—
208C2×#106—21no—
209C2×#103—1061no—
210C2×#103—1061no—
211C2×#102—1061no—
212#399—1061no—
213#396—1061no—
214#394—1061no—
215#390—531no—
216#389—531no—
217#389—1061no—
218#389C2×D5311yescharacteristic
219#388—1061no—
220#340—21no—
221#340—1061no—
222#336—531no—
223#336—531no—
224#336C2×D5311yescharacteristic
225C53#777611yescharacteristic, complemented, Sylow
226C22×Q8—21no—
227C22×D4—531no—
228C22×C8—21no—
229C22×C8—1061no—
230C22×C8—1061no—
231C4⋊Q8—42no—
232#126—2122no—
233#126—2122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4C22⋊D5311yescharacteristic
244C2×C4⋊C4C22⋊D5311yescharacteristic
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C42—1061no—
252C2×C42C22×D5311yescharacteristic
253#106—42no—
254#103—2122no—
255#103—2122no—
256#102—2122no—
257#95—21no—
258#95—1061no—
259#95—1061no—
260C24—1061no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4—2121no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4D4×D5311yescharacteristic
278C22×C4D4×D5311yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—2121no—
284C2×C8—2121no—
285C2×C8—2121no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—4241no—
300C4⋊C4—2121no—
301C4⋊C4—2121no—
302C4⋊C4—2122no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4#1171992412yes—
306C22⋊C4—4241no—
307C22⋊C4—4241no—
308C22⋊C4—2122no—
309C22⋊C4—2122no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C22⋊C4—2121no—
314C42—41no—
315C42—2122no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23—2121no—
323C23#1313318511yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328D4—4241no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4C2×#1171992411yescharacteristic
355C2×C4#1313323811yescharacteristic
356C2×C4#1313323811yescharacteristic
357C2×C4#1313324211yescharacteristic
358C2×C4#1313324411yescharacteristic
359C2×C4#1313324611yescharacteristic
360C8—42no—
361C8—4241no—
362C8—4241no—
363C22—4241no—
364C22—4241no—
365C22—4241no—
366C22—4241no—
367C22—4241no—
368C22—4241no—
369C22—4241no—
370C22#1681076711yescharacteristic
371C22#1681077311yescharacteristic
372C22#1681078211yescharacteristic
373C22#1681079411yescharacteristic
374C22#1681079911yescharacteristic
375C22#1681080511yescharacteristic
376C22#1681080911yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4—4241no—
387C4—4241no—
388C4—4241no—
389C2—8481no—
390C2#2027364311yescharacteristic
391C2#2027368311yescharacteristic
392C2#2027369211yescharacteristic
393C2#2027386411yescharacteristic
394C2#2027391411yescharacteristic
395C2#2027398211yescharacteristic
396C2#2027404711yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2058C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#2058D53
hypocenterC53#7776
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
2. #20275958C2semidirect
4. #20272539C2semidirect
5. #20272538C2semidirect
7. C53×#2058C2semidirect
192. #2058D53semidirect
225. C53#7776semidirect

Maximal subgroups

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

subgroupindex
#202816762
#202759582
#202758482
#202725392
#202725382
#202725352
C53×#20582
#777653