Finite Group Databasebeta

← #27358272

Subgroups — #27358272

Order 13568.

1230 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#16811204C211yesmaximal, characteristic
2#20285517C212yesmaximal, complemented
3#20273717C214yesmaximal, complemented
4#20273683C214yesmaximal, complemented
5#20273230C212yesmaximal, complemented
6#20273045C211yesmaximal, characteristic, complemented
7C53×#2583C211yesmaximal, characteristic, complemented
8C22×#11719945—21no—
9#16811204—24no—
10#16811204—22no—
11#16811204C2214yes—
12C2×#13133197—22no—
13C2×#13133192—22no—
14C2×#13133191—21no—
15C2×#13133191C2212yes—
16#16810956—22no—
17#16810936C2214yescomplemented
18#16810871—24no—
19#16810773—22no—
20#16810762C2214yescomplemented
21#16810587C2212yes—
22#16810356C2218yescomplemented
23#16810349C2214yescomplemented
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#398C2212yescomplemented
27C53×#395C2212yescomplemented
28C53×#389—22no—
29C53×#385—21no—
30C53×#385—21no—
31C4×C22⋊C4×C53C2211yescharacteristic
32C2×#11719945—24no—
33C2×#11719945—24no—
34C2×#11719945—48no—
35C22×Dic106—21no—
36C22×D212—21no—
37C22×D212—21no—
38C22×D212D412yes—
39C22×C22⋊D53—22no—
40C22×C4×D53—22no—
41C2×#11719924—24no—
42C2×#11719917—22no—
43#13133197—28no—
44#13133197—28no—
45#13133192—28no—
46#13133192—28no—
47#13133191—24no—
48#13133191—24no—
49#13133191—22no—
50#13133191—22no—
51#13133191D414yes—
52#13133191C2314yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53D412yes—
55C2×#11719921—22no—
56C2×#11719921—22no—
57#13133150—24no—
58#13133141—24no—
59#13133137—24no—
60C22⋊#88322—22no—
61C22⋊#88322C2314yes—
62C22×Q8×C53C2311yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53D411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53D412yes—
74C2×C4⋊C4×C53C2311yescharacteristic
75C2×C4⋊C4×C53C2312yes—
76C2×C22⋊C4×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C42×C53—21no—
79#95×C53—22no—
80#95×C53—22no—
81#95×C53C2312yes—
82C24×C53C2×D411yescharacteristic
83C23×D53—22no—
84C2×Q8×C53—22no—
85C2×Q8×C53—81no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—24no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—81no—
95C22×C4×C53—81no—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2411yescharacteristic
102C2×C22⋊D53—24no—
103C2×C22⋊D53—24no—
104C2×C22⋊D53—48no—
105C22×Dic53—22no—
106#11719945—416no—
107#11719945—48no—
108#11719945—48no—
109C2×D212—24no—
110C2×D212—24no—
111C2×D212—22no—
112C2×D212—44no—
113C2×D212—82no—
114C2×D212—82no—
115C2×D212C2×D414yes—
116C2×C4×D53—28no—
117C2×C4×D53—48no—
118C2×Dic106—24no—
119C2×Dic106—44no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—42no—
124C4⋊C4×C53—44no—
125C4⋊C4×C53—82no—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C42×C53—81no—
130#11719924—84no—
131#11719921—28no—
132#11719921—24no—
133#11719921—24no—
134C4⋊Dic53—24no—
135C4⋊Dic53—22no—
136C4⋊Dic53C2×D414yes—
137#11719917—82no—
138C2×#88322—24no—
139C23×C53—22no—
140C23×C53—22no—
141C23×C53—41no—
142C23×C53#12011yescharacteristic
143C23×C53#12011yescharacteristic
144C23×C53C22×D411yescharacteristic
145C22×D53—24no—
146C22×D53—44no—
147C22×D53—44no—
148C22×D53—44no—
149Q8×C53—44no—
150C2×C4×C53—24no—
151C2×C4×C53—24no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—81no—
169C2×C4×C53#12012yes—
170C2×C4×C53AGL(1,ℤ/8)14yes—
171C2×C4×C53C22×D412yes—
172C22⋊D53—48no—
173C22⋊D53—48no—
174C2×Dic53—24no—
175C2×Dic53—44no—
176D212—44no—
177D212—48no—
178D212—82no—
179D212—84no—
180C4×D53—416no—
181Dic106—48no—
182#88322—44no—
183#18325—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—44no—
192C22×C53#45612yes—
193C22×C53C2×#12011yescharacteristic
194C22×C53C2×AGL(1,ℤ/8)12yes—
195C2×D53—44no—
196C2×D53—44no—
197C2×D53—44no—
198C2×D53—88no—
199C4×C53—24no—
200C4×C53—22no—
201C4×C53—22no—
202C4×C53—44no—
203C4×C53—42no—
204C4×C53—81no—
205C4×C53—81no—
206Dic53—44no—
207C2×#537—531no—
208#3193—532no—
209#2583D5311yescharacteristic, complemented
210#1610—534no—
211#1604—534no—
212#1507—532no—
213#1463—531no—
214C2×C53—22no—
215C2×C53—22no—
216C2×C53—41no—
217C2×C53#300311yescharacteristic
218C2×C53C2×#45611yescharacteristic
219C2×C53#302611yescharacteristic
220D53—84no—
221C22×#41—1061no—
222#537—534no—
223#537—1064no—
224#537—1062no—
225C2×C22⋊Q8—1062no—
226C2×C22⋊Q8C2×D5311yescharacteristic
227C2×#121—532no—
228C2×#121—1062no—
229C2×#121—1061no—
230C22×C4⋊C4C2×D5311yescharacteristic
231#484—1062no—
232#480—534no—
233#468—1064no—
234#450—1062no—
235#448—534no—
236#407—532no—
237#398C2×D5312yescomplemented
238#395C2×D5312yescomplemented
239#389—22no—
240#385—21no—
241#385—21no—
242C4×C22⋊C4C2×D5311yescharacteristic
243#327—538no—
244#324—534no—
245C53#1832511yescharacteristic, complemented, Sylow
246C2×#41—1064no—
247C2×#41—1064no—
248C2×#41—2128no—
249C22×Q8—1061no—
250C22×Q8C22×D5311yescharacteristic
251C22×D4—532no—
252C22×D4—1061no—
253C22×D4—1061no—
254C22×D4—1062no—
255C23×C4—1062no—
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C2×SD16—1064no—
260C2×M16—1062no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—1068no—
264C22⋊Q8—1068no—
265#121—534no—
266#121—534no—
267#121—1068no—
268#121—1068no—
269#121—1064no—
270#121—1062no—
271#121—1062no—
272#121—1064no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—532no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4C22⋊D5312yes—
281C2×C4⋊C4C22×D5311yescharacteristic
282C2×C4⋊C4C22×D5312yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—1062no—
286C2×C22⋊C4—1062no—
287C2×C42—21no—
288#107—1064no—
289#103—1064no—
290#102—1064no—
291C22⋊C8—534no—
292C22⋊C8—1062no—
293#95—22no—
294#95—22no—
295#95C22×D5312yes—
296C24—1062no—
297C24C2×C22⋊D5311yescharacteristic
298#41—21216no—
299#41—2128no—
300#41—2128no—
301C2×Q8—22no—
302C2×Q8—81no—
303C2×Q8—1064no—
304C2×Q8—2124no—
305C2×D4—534no—
306C2×D4—1064no—
307C2×D4—1064no—
308C2×D4—1064no—
309C2×D4—1064no—
310C2×D4—1062no—
311C2×D4—4242no—
312C2×D4—4242no—
313C2×D4—2124no—
314C2×D4—2128no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—1068no—
326C22×C4—1062no—
327C22×C4—2128no—
328C22×C4D4×D5312yes—
329C22×C4D4×D5312yes—
330C22×C4C2×C22⋊D5312yes—
331C22×C4C2×C22⋊D5312yes—
332C22×C4C2×C22⋊D5311yescharacteristic
333C22×C4C23×D5311yescharacteristic
334SD16—4244no—
335M16—4242no—
336C2×C8—1064no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C4⋊C4—534no—
344C4⋊C4—1064no—
345C4⋊C4—1062no—
346C22⋊C4—24no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—1068no—
350C22⋊C4—1064no—
351C22⋊C4—1064no—
352C42—81no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—1064no—
357C23—2124no—
358C23—2124no—
359C23—2124no—
360C23#1313318511yescharacteristic
361C23#1313318511yescharacteristic
362C23C22×C22⋊D5311yescharacteristic
363Q8—44no—
364Q8—2128no—
365D4—4242no—
366D4—4244no—
367D4—2124no—
368D4—2128no—
369D4—2128no—
370D4—2128no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—1064no—
391C2×C4—21216no—
392C2×C4—2124no—
393C2×C4#1313318512yes—
394C2×C4#1313324214yes—
395C2×C4C2×D4×D5312yes—
396C8—2124no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—4248no—
406C22—2124no—
407C22—2124no—
408C22—2124no—
409C22#1681080512yes—
410C22C2×#1313318511yescharacteristic
411C22C2×#1313324212yes—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—2124no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—4244no—
424C2#2028394111yescharacteristic
425C2C2×#1681080511yescharacteristic
426C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2583C2
socleC22×C53C2×#120
prime radicalC2×#16811204C2
hypercenter#2583D53
hypocenterC53#18325
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
2. #20285517C2semidirect
3. #20273717C2semidirect
4. #20273683C2semidirect
5. #20273230C2semidirect
6. #20273045C2semidirect
7. C53×#2583C2semidirect
17. #16810936C22semidirect
20. #16810762C22semidirect
22. #16810356C22semidirect
23. #16810349C22semidirect
26. C53×#398C22semidirect
27. C53×#395C22semidirect
209. #2583D53semidirect
237. #398C2×D53semidirect
238. #395C2×D53semidirect
245. C53#18325semidirect

Maximal subgroups

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

subgroupindex
C2×#168112042
#202855172
#202737172
#202736832
#202732302
#202730452
C53×#25832
#1832553