Finite Group Databasebeta

← #27273071

Subgroups — #27273071

Order 13568.

478 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 81 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#20281674C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic, complemented
3#20276040C211yesmaximal, characteristic, complemented
4#20272539C211yesmaximal, characteristic, complemented
5#20272537C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2013C211yesmaximal, characteristic
8C2×#13133210—21no—
9C2×#13133150—21no—
10C2×#13133148—21no—
11#16810559—21no—
12#16810546—21no—
13#16810537—21no—
14#16810521C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810518—21no—
17#16810516—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C22×SD16×C53—21no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C53×#390C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29#13133210—42no—
30#13133210—42no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133150—42no—
45#13133148—42no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53—21no—
50C22×D4×C53D411yescharacteristic
51C2×SD16×C53—42no—
52C2×SD16×C53—42no—
53C2×SD16×C53—42no—
54C22×C8×C53—21no—
55C4⋊Q8×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62#103×C53—42no—
63C24×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C22×C4×C53—21no—
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83SD16×C53—48no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C42×C53—41no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99C22⋊Dic53—81no—
100C22⋊Dic53—81no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53SD1612yescomplemented
107#11719919—42no—
108#11719919—41no—
109#11719919—41no—
110#11719919—41no—
111#11719919—41no—
112#11719919—81no—
113C4×Dic53—42no—
114C2×#88322—22no—
115C2×#88322—22no—
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
128Q8×C53—42no—
129Q8×C53—42no—
130Q8×C53—81no—
131Q8×C53—81no—
132D4×C53—42no—
133D4×C53—81no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C8×C53—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—82no—
189C4×C53—81no—
190Dic53—42no—
191Dic53—81no—
192Dic53—81no—
193Dic53—81no—
194#2727—531no—
195#2058—531no—
196#2057—531no—
197#2013D5311yescharacteristic
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—21no—
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×#126—1061no—
212C2×#107—1061no—
213C2×#106—1061no—
214C2×#103—21no—
215#399—1061no—
216#396—1061no—
217#394—1061no—
218#390C2×D5311yescharacteristic
219#389—531no—
220#389—531no—
221#389—1061no—
222#388—1061no—
223#340—1061no—
224#340—1061no—
225#336—531no—
226#336—531no—
227#336C2×D5311yescharacteristic
228C53#778011yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×D4C22⋊D5311yescharacteristic
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8—21no—
235C22×C8—1061no—
236C22×C8—1061no—
237C4⋊Q8—42no—
238#126—2122no—
239#126—2122no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4C22⋊D5311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C42—1061no—
258C2×C42C22×D5311yescharacteristic
259#107—2122no—
260#106—2122no—
261#103—42no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4D4×D5311yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285SD16—48no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1062no—
290C2×C8—1062no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—532no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—4241no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C4⋊C4—2122no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—4241no—
317C22⋊C4—4241no—
318C22⋊C4—2122no—
319C22⋊C4—2122no—
320C42—41no—
321C42—2122no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1313318511yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—1062no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4SD16×D5311yescharacteristic
361C2×C4#1313323711yescharacteristic
362C2×C4#1313323811yescharacteristic
363C2×C4#1313324311yescharacteristic
364C2×C4#1313324411yescharacteristic
365C2×C4#1313324611yescharacteristic
366C8—42no—
367C8—2122no—
368C8—2122no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#1681076611yescharacteristic
377C22#1681077211yescharacteristic
378C22#1681078211yescharacteristic
379C22#1681079011yescharacteristic
380C22#1681080411yescharacteristic
381C22#1681080411yescharacteristic
382C22#1681080911yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—82no—
391C4—4241no—
392C4—4241no—
393C4—4241no—
394C4—2122no—
395C2—161no—
396C2#2027365911yescharacteristic
397C2#2027367511yescharacteristic
398C2#2027368211yescharacteristic
399C2#2027383911yescharacteristic
400C2#2027396911yescharacteristic
401C2#2027399711yescharacteristic
402C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2013C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2013D53
hypocenterC53#7780
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #20276045C2semidirect
3. #20276040C2semidirect
4. #20272539C2semidirect
5. #20272537C2semidirect
106. C4⋊Dic53SD16semidirect
228. C53#7780semidirect

Maximal subgroups

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

subgroupindex
#202816742
#202760452
#202760402
#202725392
#202725372
#202725362
C53×#20132
#778053