Finite Group Databasebeta

← #27281297

Subgroups — #27281297

Order 13568.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277600C211yesmaximal, characteristic, complemented
2#20275552C211yesmaximal, characteristic
3#20274814C211yesmaximal, characteristic, complemented
4#20273231C211yesmaximal, characteristic
5#20272620C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#3094C211yesmaximal, characteristic
8C2×#13133132—22no—
9#16810587C2211yescharacteristic
10#16810522—21no—
11#16810516—22no—
12#16810489—21no—
13#16810483C2211yescharacteristic
14C4⋊C4×Dic53—21no—
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810469—21no—
18#16810382—22no—
19#16810357—21no—
20#16810357—21no—
21C2×#121×C53C2211yescharacteristic
22C2×#117×C53C2211yescharacteristic
23C53×#482—22no—
24C53×#481—21no—
25C53×#480—21no—
26C53×#428C412yescomplemented
27C53×#418C412yescomplemented
28C53×#407C2211yescharacteristic
29C2×#11719917—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53D412yes—
35C2×C4×Dic53—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38#13133132—44no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41#13133124—22no—
42#13133124—22no—
43#13133124—21no—
44#13133124—21no—
45#13133124D411yescharacteristic
46#13133124D411yescharacteristic
47C22×D4×C53—21no—
48C22×D4×C53C2×C411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C2×M16×C53C2×C412yes—
51#121×C53—21no—
52#121×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55#117×C53—21no—
56#117×C53—21no—
57#117×C53C2×C412yes—
58#117×C53D412yes—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—21no—
65#107×C53—22no—
66#106×C53—22no—
67#102×C53—22no—
68#102×C53—22no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C24×C53—21no—
72C24×C53C2×D411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—42no—
76C2×D4×C53—42no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—42no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—21no—
93C22×Dic53—41no—
94M16×C53—42no—
95C2×C8×C53—22no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53C22⋊C412yes—
101C4⋊C4×C53C22⋊C412yes—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C42×C53—22no—
107C42×C53—22no—
108C22⋊Dic53—22no—
109C22⋊Dic53—42no—
110C22⋊Dic53—41no—
111C22⋊Dic53—42no—
112C4⋊Dic53—42no—
113C4⋊Dic53—82no—
114C4×Dic53—44no—
115C4×Dic53—42no—
116C4×Dic53—42no—
117#11719917—42no—
118C2×#88322—41no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—42no—
124C23×C53—41no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53C4×D411yescharacteristic
128C23×C53#12011yescharacteristic
129C23×C53#12111yescharacteristic
130D4×C53—42no—
131D4×C53—81no—
132D4×C53—82no—
133D4×C53—81no—
134D4×C53—81no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—41no—
161C2×Dic53—42no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C8×C53—42no—
165#88322—81no—
166#9367—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—162no—
181C22×C53#38611yescharacteristic
182C22×C53#45311yescharacteristic
183C22×C53#45411yescharacteristic
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53—81no—
190C4×C53#41012yes—
191Dic53—42no—
192Dic53—82no—
193Dic53—81no—
194Dic53—82no—
195#3094D5311yescharacteristic
196#2346—531no—
197#1964—531no—
198#1826—531no—
199#1507—531no—
200#1305—531no—
201#1297—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—41no—
205C2×C53—41no—
206C2×C53—161no—
207C2×C53#188311yescharacteristic
208C2×C53#189911yescharacteristic
209C2×C53#190611yescharacteristic
210C2×#121C2×D5311yescharacteristic
211C2×#117C2×D5311yescharacteristic
212#482—22no—
213#481—21no—
214#480—21no—
215#428Dic5312yescomplemented
216#418Dic5312yescomplemented
217C2×#99—1062no—
218#407—531no—
219#407C2×D5311yescharacteristic
220#389—1061no—
221#388—1062no—
222#382—1061no—
223#380—531no—
224C4×C4⋊C4—1061no—
225C4×C22⋊C4—531no—
226C4×C22⋊C4—531no—
227#376—1061no—
228#339—1062no—
229#327—1061no—
230#327—1061no—
231C53#936711yescharacteristic, complemented, Sylow
232C22×D4—21no—
233C22×D4C2×Dic5311yescharacteristic
234C23×C4C22×D5311yescharacteristic
235C2×M16—1061no—
236C2×M16C2×Dic5312yes—
237#121—21no—
238#121—21no—
239#121—21no—
240#121—21no—
241#117—21no—
242#117—21no—
243#117C2×Dic5312yes—
244#117C22⋊D5312yes—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4C2×Dic5311yescharacteristic
248C2×C4⋊C4C22⋊D5311yescharacteristic
249C2×C4⋊C4C22⋊D5311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—532no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C42—21no—
256C2×C42—1062no—
257C2×C42—1061no—
258C2×C42—1061no—
259#107—22no—
260#106—22no—
261#102—22no—
262#102—22no—
263#99—2124no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—1061no—
267C22⋊C8—1061no—
268#95—531no—
269#95—531no—
270#95—1062no—
271#95—1062no—
272#95—1061no—
273#95—1061no—
274C24—21no—
275C24D4×D5311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1062no—
290C22×C4—2121no—
291C22×C4D4×D5311yescharacteristic
292C22×C4#1171994711yescharacteristic
293C22×C4#1171994711yescharacteristic
294C22×C4C22×Dic5311yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297M16—42no—
298M16—2122no—
299C2×C8—22no—
300C2×C8—2121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—4242no—
306C4⋊C4—2122no—
307C4⋊C4C22⋊Dic5312yes—
308C4⋊C4C22⋊Dic5312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—1062no—
314C22⋊C4—2122no—
315C22⋊C4—2121no—
316C22⋊C4—2122no—
317C42—22no—
318C42—22no—
319C42—2122no—
320C42—2122no—
321C42—2124no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—81no—
329C23—81no—
330C23D4×Dic5311yescharacteristic
331C23#1313318511yescharacteristic
332C23#1313318911yescharacteristic
333D4—42no—
334D4—81no—
335D4—82no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—1062no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—2122no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4—2121no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4C2×C22⋊Dic5311yescharacteristic
364C2×C4D4×Dic5311yescharacteristic
365C2×C4#1313318911yescharacteristic
366C2×C4#1313319811yescharacteristic
367C8—42no—
368C8—4241no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—162no—
383C22#1681050311yescharacteristic
384C22#1681078911yescharacteristic
385C22#1681079411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—4241no—
393C4—4242no—
394C4—4242no—
395C4—2122no—
396C4#1681059712yes—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2#2027509311yescharacteristic
403C2#2027517411yescharacteristic
404C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3094C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#3094D53
hypocenterC53#9367
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
1. #20277600C2semidirect
3. #20274814C2semidirect
5. #20272620C2semidirect
6. #20272600C2semidirect
26. C53×#428C4semidirect
27. C53×#418C4semidirect
215. #428Dic53semidirect
216. #418Dic53semidirect
231. C53#9367semidirect

Maximal subgroups

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

subgroupindex
#202776002
#202755522
#202748142
#202732312
#202726202
#202726002
C53×#30942
#936753