Finite Group Databasebeta

← #27273074

Subgroups — #27273074

Order 13568.

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

normal subgroups only

This group has 478 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#20276043C211yesmaximal, characteristic, complemented
3#20275797C211yesmaximal, characteristic
4#20272538C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2727C211yesmaximal, characteristic
8C22×#11719923—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133150—21no—
11C2×#13133148—21no—
12C2×#13133140—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C2×#126×C53—21no—
19C53×#399—21no—
20C53×#396—21no—
21C53×#394—21no—
22C53×#390C2211yescharacteristic
23C53×#389—21no—
24C53×#389C2211yescharacteristic
25C53×#389C2211yescharacteristic
26C53×#388—21no—
27C22×Dic106—21no—
28C2×#11719923—42no—
29C2×#11719923—42no—
30C2×#11719923—42no—
31C22×#88322—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133217—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37#13133150—42no—
38#13133148—42no—
39#13133140—42no—
40C22×D4×C53D411yescharacteristic
41#126×C53—42no—
42#126×C53—42no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—41no—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
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—81no—
95C4⋊C4×C53SD1612yescomplemented
96C22⋊C4×C53—42no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—81no—
103C22⋊C4×C53—81no—
104C42×C53—41no—
105C42×C53—42no—
106#11719923—48no—
107C4⋊Dic53—22no—
108C4⋊Dic53—22no—
109C4⋊Dic53—41no—
110C4⋊Dic53—41no—
111C2×#88322—22no—
112C2×#88322—22no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53#12011yescharacteristic
128D4×C53—42no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
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—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×SD1611yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×Dic53—42no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160Dic106—42no—
161Dic106—42no—
162Dic106—81no—
163Dic106—81no—
164#88322—42no—
165#88322—42no—
166#88322—42no—
167#7780—531nomaximal, Sylow
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#44911yescharacteristic
176C22×C53#45011yescharacteristic
177C22×C53#45211yescharacteristic
178C22×C53#45311yescharacteristic
179C22×C53#45611yescharacteristic
180C22×C53#45611yescharacteristic
181C22×C53B(4,𝔽2)11yescharacteristic
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—42no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—81no—
192C4×C53—81no—
193Dic53—82no—
194#2727D5311yescharacteristic
195#2058—531no—
196#2057—531no—
197#2013—531no—
198#1271—531no—
199#1271—531no—
200#1270—531no—
201C2×C53—161no—
202C2×C53#160011yescharacteristic
203C2×C53#160311yescharacteristic
204C2×C53#160411yescharacteristic
205C2×C53#163111yescharacteristic
206C2×C53#165011yescharacteristic
207C2×C53#165411yescharacteristic
208C2×C53#166011yescharacteristic
209C22×SD16—1061no—
210C2×C4⋊Q8—531no—
211C2×#126—21no—
212C2×#107—1061no—
213C2×#106—1061no—
214C2×#103—1061no—
215#399—21no—
216#396—21no—
217#394—21no—
218#390C2×D5311yescharacteristic
219#389—21no—
220#389C2×D5311yescharacteristic
221#389C2×D5311yescharacteristic
222#388—21no—
223#340—1061no—
224#340—1061no—
225#336—531no—
226#336—531no—
227#336—531no—
228C53#778011yescharacteristic, complemented, Sylow
229C22×Q8—1061no—
230C22×D4C22⋊D5311yescharacteristic
231C2×SD16—2122no—
232C2×SD16—2122no—
233C2×SD16—2122no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8—1061no—
237C4⋊Q8—2122no—
238#126—42no—
239#126—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4C22⋊D5311yescharacteristic
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C4⋊C4C22⋊D5311yescharacteristic
249C2×C4⋊C4C22⋊D5311yescharacteristic
250C2×C4⋊C4C22⋊D5311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C22×D5311yescharacteristic
259#107—2122no—
260#106—2122no—
261#103—2122no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—1062no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×Q8—2121no—
270C2×Q8—2121no—
271C2×Q8—2121no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2121no—
282C22×C4C2×C22⋊D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285SD16—2128no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—1062no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C4⋊C4#1171992312yescomplemented
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C42—41no—
321C42—42no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1313318611yescharacteristic
330Q8—4241no—
331Q8—4241no—
332Q8—2122no—
333Q8—2122no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
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—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—2122no—
360C2×C4C2×#1171992311yescharacteristic
361C2×C4#1313323711yescharacteristic
362C2×C4#1313323711yescharacteristic
363C2×C4#1313324211yescharacteristic
364C2×C4#1313324311yescharacteristic
365C2×C4#1313324711yescharacteristic
366C8—2122no—
367C8—2122no—
368C8—2122no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#1681076811yescharacteristic
377C22#1681077411yescharacteristic
378C22#1681078511yescharacteristic
379C22#1681079111yescharacteristic
380C22#1681080611yescharacteristic
381C22#1681080611yescharacteristic
382C22#1681081011yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—4242no—
395C2—161no—
396C2#2027366111yescharacteristic
397C2#2027367811yescharacteristic
398C2#2027368411yescharacteristic
399C2#2027384211yescharacteristic
400C2#2027397111yescharacteristic
401C2#2027399911yescharacteristic
402C2#2027402611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2727C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2727D53
hypocenterC53#7780
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #20276048C2semidirect
2. #20276043C2semidirect
4. #20272538C2semidirect
5. #20272538C2semidirect
95. C4⋊C4×C53SD16semidirect
228. C53#7780semidirect
311. C4⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202760482
#202760432
#202757972
#202725382
#202725362
C53×#27272
#778053