Finite Group Databasebeta

← #27283223

Subgroups — #27283223

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283016C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic, complemented
3C2×#16810359C212yesmaximal, complemented
4C2×#16810356C212yesmaximal
5C4×#95×C53C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7C2×#13133196C2211yescharacteristic
8C2×#13133191C2211yescharacteristic
9C2×C4⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11#16810529C411yescharacteristic, complemented
12#16810525C411yescharacteristic, complemented
13#16810514—21no—
14#16810514—21no—
15#16810502—21no—
16#16810502—21no—
17#16810489C411yescharacteristic, complemented
18#16810378—21no—
19#16810359—24no—
20#16810356—24no—
21C22×C42×C53—21no—
22C22×C42×C53C2211yescharacteristic
23C2×#95×C53—21no—
24C2×#95×C53C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C22×D212C2×C411yescharacteristic
27C22×C22⋊D53—21no—
28C22×#88322—22no—
29#13133205—42no—
30#13133196—22no—
31#13133196—22no—
32#13133191—22no—
33#13133191—22no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C4⋊#88322—42no—
44C22⋊#88322—22no—
45C22⋊#88322—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53—22no—
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×C42×C53—24no—
55C2×C42×C53—42no—
56C2×C42×C53—42no—
57C2×C42×C53—42no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62#95×C53—24no—
63#95×C53—22no—
64#95×C53D414yes—
65C24×C53C2×D411yescharacteristic
66C23×D53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—24no—
72C22×C4×C53—24no—
73C22×C4×C53—24no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—42no—
82C22×C4×C53—42no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53#4112yes—
91C2×C22⋊D53—42no—
92C2×C22⋊D53—42no—
93C2×C22⋊D53—41no—
94C2×C22⋊D53—41no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C2×D212—22no—
99C2×D212—41no—
100C2×D212—41no—
101C2×Dic106—22no—
102C2×Dic106—41no—
103C2×Dic106—41no—
104C42×C53—22no—
105C42×C53—22no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53—48no—
109C22⋊Dic53—42no—
110#11719921—42no—
111#11719921—42no—
112#11719921—42no—
113#11719921—42no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118#11719919—41no—
119#11719919—41no—
120C2×#88322—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12112yes—
129C23×C53#13512yes—
130C23×C53AGL(1,ℤ/8)11yescharacteristic
131C23×C53#13711yescharacteristic
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C2×C4×C53—24no—
140C2×C4×C53—24no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—28no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—44no—
158C2×C4×C53—44no—
159C2×C4×C53—44no—
160C2×C4×C53—44no—
161C2×C4×C53—44no—
162C2×C4×C53—44no—
163C2×C4×C53—44no—
164C2×C4×C53#9912yes—
165C2×C4×C53#10412yes—
166C2×C4×C53#10412yes—
167C2×C4×C53C4×D412yes—
168C2×C4×C53#12411yescharacteristic
169C2×C4×C53#12711yescharacteristic
170C22⋊D53—82no—
171C22⋊D53—82no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181D212—82no—
182Dic106—82no—
183#88322—82no—
184#9713—531nomaximal, Sylow
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53#39011yescharacteristic
194C22×C53C2×#9911yescharacteristic
195C22×C53#41011yescharacteristic
196C22×C53#41611yescharacteristic
197C22×C53#41711yescharacteristic
198C22×C53C2×#10411yescharacteristic
199C22×C53C2×#10411yescharacteristic
200C22×C53#46312yes—
201C22×C53#46412yes—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—22no—
211C4×C53—21no—
212C4×C53—22no—
213C4×C53—22no—
214C4×C53—22no—
215C4×C53—48no—
216C4×C53—41no—
217Dic53—81no—
218Dic53—81no—
219Dic53—81no—
220#2850—531no—
221#1797—531no—
222C2×#328—532no—
223C2×#327—532no—
224C4×#95D5311yescharacteristic, complemented
225C2×C53—22no—
226C2×C53—22no—
227C2×C53#150011yescharacteristic
228C2×C53#150111yescharacteristic
229C2×C53#150411yescharacteristic
230C2×C53#150911yescharacteristic
231C2×C53#196211yescharacteristic
232C2×C53#196411yescharacteristic
233C2×C53#198311yescharacteristic
234D53—161no—
235C2×#124—531no—
236C2×C22⋊Q8—531no—
237C2×#121—531no—
238C22×C42—21no—
239C22×C42C2×D5311yescharacteristic
240C2×C4⋊C8—1061no—
241C2×C22⋊C8—532no—
242#391—531no—
243#390—531no—
244#388—1061no—
245#388—1061no—
246#386—1061no—
247#386—1061no—
248#382—531no—
249C2×#95—21no—
250C2×#95C2×D5312yes—
251#336—1061no—
252#328—1064no—
253#327—1064no—
254C53#971311yescharacteristic, complemented, Sylow
255C22×Q8—531no—
256C22×D4—531no—
257C22×D4—1061no—
258C23×C4—22no—
259C23×C4C22⋊D5311yescharacteristic
260C23×C4C22⋊D5311yescharacteristic
261C23×C4C22×D5311yescharacteristic
262C22×C8—1062no—
263#124—2122no—
264C22⋊Q8—1062no—
265C22⋊Q8—1062no—
266#121—1062no—
267#121—1062no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4—1061no—
275C2×C22⋊C4—1061no—
276C2×C22⋊C4—1061no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C4×D5311yescharacteristic
282C2×C42C4×D5311yescharacteristic
283C2×C42D21211yescharacteristic
284C2×C42D21211yescharacteristic
285C4⋊C8—2122no—
286C22⋊C8—1062no—
287C22⋊C8—1062no—
288#95—24no—
289#95—22no—
290#95—1061no—
291#95—1061no—
292#95—1061no—
293#95—1061no—
294#95D21214yes—
295C24—1061no—
296C24C2×C22⋊D5311yescharacteristic
297C2×Q8—1062no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×D4—1062no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2122no—
304C2×D4—2122no—
305C2×D4—2121no—
306C2×D4—2121no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4#1171992111yescharacteristic
327C22×C4#1171992111yescharacteristic
328C22×C4#1171992111yescharacteristic
329C22×C4#1171992111yescharacteristic
330C22×C4C2×C4×D5311yescharacteristic
331C22×C4C2×D21211yescharacteristic
332C22×C4C2×D21212yes—
333C22×C4#1171994512yes—
334C2×C8—2122no—
335C2×C8—2122no—
336C2×C8—2122no—
337C4⋊C4—1062no—
338C4⋊C4—1062no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C22⋊C4—2122no—
342C22⋊C4—2122no—
343C22⋊C4—2122no—
344C22⋊C4—2122no—
345C22⋊C4—2122no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—48no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23—2121no—
364C23C2×#1171992111yescharacteristic
365C23#1313319112yes—
366C23#1313323712yes—
367C23#1313324311yescharacteristic
368C23#1313324811yescharacteristic
369Q8—4242no—
370D4—4242no—
371D4—4242no—
372D4—4242no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4#1313313112yes—
408C2×C4#1313314512yes—
409C2×C4#1313314512yes—
410C2×C4C4×D21212yes—
411C2×C4#1313320511yescharacteristic
412C2×C4#1313321411yescharacteristic
413C8—4242no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—4241no—
423C22—4241no—
424C22—4241no—
425C22—4241no—
426C22—4241no—
427C22—4241no—
428C22—4241no—
429C22#1681052511yescharacteristic
430C22C2×#1313313111yescharacteristic
431C22#1681059611yescharacteristic
432C22#1681061711yescharacteristic
433C22#1681062211yescharacteristic
434C22C2×#1313314511yescharacteristic
435C22C2×#1313314511yescharacteristic
436C22#1681083812yes—
437C22#1681084512yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—48no—
445C4—41no—
446C4—4241no—
447C4—4241no—
448C4—4241no—
449C2—22no—
450C2—22no—
451C2—8481no—
452C2#2027319911yescharacteristic
453C2#2027320411yescharacteristic
454C2#2027321711yescharacteristic
455C2#2027323811yescharacteristic
456C2#2027554311yescharacteristic
457C2#2027555111yescharacteristic
458C2#2027564711yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC4×#95×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC4×#95D53
hypocenterC53#9713
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. #20274695C2semidirect
3. C2×#16810359C2semidirect
5. C4×#95×C53C2semidirect
6. C2×#13133205C4semidirect
11. #16810529C4semidirect
12. #16810525C4semidirect
17. #16810489C4semidirect
224. C4×#95D53semidirect
254. C53#9713semidirect

Maximal subgroups

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

subgroupindex
#202830162
#202746952
C2×#168103592
C2×#168103562
C4×#95×C532
#971353