Finite Group Databasebeta

← #27273004

Subgroups — #27273004

Order 13568.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276048C211yesmaximal, characteristic, complemented
2#20275961C211yesmaximal, characteristic, complemented
3#20275851C211yesmaximal, characteristic
4#20272538C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2724C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133148—21no—
10C2×#13133140—21no—
11C2×#13133140—21no—
12C2×#13133138—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C2×#123×C53—21no—
19C53×#398—21no—
20C53×#397—21no—
21C53×#397—21no—
22C53×#390C2211yescharacteristic
23C53×#389C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#385—21no—
26C53×#385—21no—
27C22×Dic106—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133217—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34#13133148—42no—
35#13133140—42no—
36#13133140—42no—
37#13133138—42no—
38C22×D4×C53D411yescharacteristic
39C23×C4×C53—21no—
40#123×C53—42no—
41#123×C53—42no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53SD1612yes—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—42no—
72C22×C4×C53—81no—
73C22×C4×C53—81no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—41no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—81no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C42×C53—41no—
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C2×#88322—22no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C23×C53—22no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53#12011yescharacteristic
125D4×C53—42no—
126D4×C53—81no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
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—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—84no—
149C2×C4×C53C2×SD1611yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×Dic53—42no—
156C2×Dic53—81no—
157C2×Dic53—81no—
158Dic106—42no—
159Dic106—42no—
160Dic106—81no—
161Dic106—81no—
162#88322—42no—
163#88322—81no—
164#88322—81no—
165#7770—531nomaximal, Sylow
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#44911yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45411yescharacteristic
177C22×C53#45511yescharacteristic
178C22×C53#45611yescharacteristic
179C22×C53B(4,𝔽2)11yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—81no—
191Dic53—82no—
192#2724D5311yescharacteristic
193#2058—531no—
194#2043—531no—
195#2023—531no—
196#1271—531no—
197#1271—531no—
198#1270—531no—
199C2×C53—82no—
200C2×C53#159711yescharacteristic
201C2×C53#160111yescharacteristic
202C2×C53#160311yescharacteristic
203C2×C53#163511yescharacteristic
204C2×C53#164011yescharacteristic
205C2×C53#165411yescharacteristic
206C2×C53#166411yescharacteristic
207C2×C4⋊Q8—531no—
208C2×#123—21no—
209C2×#106—1061no—
210C2×#103—1061no—
211C2×#103—1061no—
212C2×#102—1061no—
213#398—21no—
214#397—21no—
215#397—21no—
216#390C2×D5311yescharacteristic
217#389C2×D5311yescharacteristic
218#389C2×D5311yescharacteristic
219#385—21no—
220#385—21no—
221#340—1061no—
222#340—1061no—
223#336—531no—
224#336—531no—
225#336—531no—
226C53#777011yescharacteristic, complemented, Sylow
227C22×Q8—1061no—
228C22×D4C22⋊D5311yescharacteristic
229C23×C4—21no—
230C22×C8—1061no—
231C22×C8—1061no—
232C22×C8—1061no—
233C4⋊Q8—2122no—
234#123—42no—
235#123—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4C22⋊D5311yescharacteristic
242C2×C4⋊C4C22⋊D5311yescharacteristic
243C2×C4⋊C4C22⋊D5311yescharacteristic
244C2×C4⋊C4C22⋊D5311yescharacteristic
245C2×C4⋊C4C22⋊D5311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42C22×D5311yescharacteristic
252#106—2122no—
253#103—2122no—
254#103—2122no—
255#102—2122no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×Q8—1062no—
263C2×Q8—2121no—
264C2×Q8—2121no—
265C2×Q8—2121no—
266C2×Q8—2121no—
267C2×Q8—2121no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4#1171992312yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—2121no—
281C22×C4C2×C22⋊D5311yescharacteristic
282C22×C4C2×C22⋊D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284C2×C8—1062no—
285C2×C8—2121no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—41no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1313318611yescharacteristic
325Q8—4241no—
326Q8—4241no—
327Q8—2122no—
328Q8—2122no—
329D4—42no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—84no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—2122no—
356C2×C4C2×#1171992311yescharacteristic
357C2×C4#1313323711yescharacteristic
358C2×C4#1313323711yescharacteristic
359C2×C4#1313324211yescharacteristic
360C2×C4#1313324311yescharacteristic
361C2×C4#1313324711yescharacteristic
362C8—4241no—
363C8—4241no—
364C8—2122no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1681076811yescharacteristic
373C22#1681077411yescharacteristic
374C22#1681078511yescharacteristic
375C22#1681079611yescharacteristic
376C22#1681080111yescharacteristic
377C22#1681080611yescharacteristic
378C22#1681081011yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—4242no—
391C2—82no—
392C2#2027364411yescharacteristic
393C2#2027366711yescharacteristic
394C2#2027367811yescharacteristic
395C2#2027386611yescharacteristic
396C2#2027390111yescharacteristic
397C2#2027399911yescharacteristic
398C2#2027404911yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2724C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2724D53
hypocenterC53#7770
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #20276048C2semidirect
2. #20275961C2semidirect
4. #20272538C2semidirect
5. #20272538C2semidirect
226. C53#7770semidirect

Maximal subgroups

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

subgroupindex
#202760482
#202759612
#202758512
#202725382
#202725362
C53×#27242
#777053