Finite Group Databasebeta

← #27273002

Subgroups — #27273002

Order 13568.

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

normal subgroups only

This group has 460 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#20281645C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic, complemented
3#20275958C211yesmaximal, characteristic, complemented
4#20272539C211yesmaximal, characteristic, complemented
5#20272539C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2023C211yesmaximal, characteristic
8C2×#13133198—21no—
9C2×#13133148—21no—
10C2×#13133141—21no—
11C2×#13133141—21no—
12#16810553—21no—
13#16810548—21no—
14#16810548—21no—
15#16810521C2211yescharacteristic
16#16810521C2211yescharacteristic
17#16810498—21no—
18#16810498—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#102×C53—21no—
24C53×#390C2211yescharacteristic
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C23×Dic53—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30#13133198—42no—
31#13133198—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42#13133148—42no—
43#13133141—42no—
44#13133141—42no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×Q8×C53—21no—
51C22×D4×C53D411yescharacteristic
52C22×C8×C53—21no—
53C4⋊Q8×C53—42no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#102×C53—42no—
61C24×C53—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—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53SD1612yes—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—42no—
82C22×Dic53—81no—
83C22×Dic53—81no—
84C2×C8×C53—41no—
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—42no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53—81no—
107#11719919—41no—
108#11719919—41no—
109#11719919—41no—
110#11719919—41no—
111C2×#88322—22no—
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
125Q8×C53—42no—
126Q8×C53—42no—
127Q8×C53—81no—
128Q8×C53—81no—
129D4×C53—42no—
130D4×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—41no—
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—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—84no—
162C8×C53—81no—
163#88322—42no—
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—82no—
187C4×C53—81no—
188Dic53—81no—
189Dic53—81no—
190Dic53—81no—
191Dic53—81no—
192#2724—531no—
193#2058—531no—
194#2043—531no—
195#2023D5311yescharacteristic
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⋊Q8C2×D5311yescharacteristic
208C2×#123—1061no—
209C2×#106—1061no—
210C2×#103—1061no—
211C2×#103—1061no—
212C2×#102—21no—
213#398—1061no—
214#397—1061no—
215#397—1061no—
216#390C2×D5311yescharacteristic
217#389—531no—
218#389—531no—
219#385—1061no—
220#385—1061no—
221#340—21no—
222#340—1061no—
223#336—531no—
224#336—531no—
225#336C2×D5311yescharacteristic
226C53#777011yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×D4C22⋊D5311yescharacteristic
229C23×C4—1061no—
230C22×C8—21no—
231C22×C8—1061no—
232C22×C8—1061no—
233C4⋊Q8—42no—
234#123—2122no—
235#123—2122no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4C22⋊D5311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—2122no—
253#103—2122no—
254#103—2122no—
255#102—42no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4#1171992412yes—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—4241no—
279C22×C4—4241no—
280C22×C4—2122no—
281C22×C4D4×D5311yescharacteristic
282C22×C4D4×D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
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—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—4241no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—2122no—
314C22⋊C4—2122no—
315C22⋊C4—2122no—
316C42—41no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1313318511yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
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—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—4244no—
344C2×C4—2121no—
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×C4—2121no—
355C2×C4—2121no—
356C2×C4C2×#1171992411yescharacteristic
357C2×C4#1313323611yescharacteristic
358C2×C4#1313323811yescharacteristic
359C2×C4#1313324111yescharacteristic
360C2×C4#1313324211yescharacteristic
361C2×C4#1313324611yescharacteristic
362C8—81no—
363C8—4241no—
364C8—2122no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1681076511yescharacteristic
373C22#1681077311yescharacteristic
374C22#1681078211yescharacteristic
375C22#1681079411yescharacteristic
376C22#1681080011yescharacteristic
377C22#1681080411yescharacteristic
378C22#1681080911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—81no—
387C4—4241no—
388C4—4241no—
389C4—4241no—
390C4—4241no—
391C2—82no—
392C2#2027364111yescharacteristic
393C2#2027366611yescharacteristic
394C2#2027367511yescharacteristic
395C2#2027386411yescharacteristic
396C2#2027389811yescharacteristic
397C2#2027399811yescharacteristic
398C2#2027404811yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2023C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2023D53
hypocenterC53#7770
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
2. #20276045C2semidirect
3. #20275958C2semidirect
4. #20272539C2semidirect
5. #20272539C2semidirect
226. C53#7770semidirect

Maximal subgroups

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

subgroupindex
#202816452
#202760452
#202759582
#202725392
#202725362
C53×#20232
#777053