Finite Group Databasebeta

← #27280385

Subgroups — #27280385

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284715C211yesmaximal, characteristic
2#20275543C211yesmaximal, characteristic, complemented
3#20273209C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C53×#2293C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810936—21no—
10#16810903C412yescomplemented
11#16810818C412yescomplemented
12#16810757—21no—
13C4×#11719922—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17C2×#13133145—21no—
18C2×#13133145—21no—
19#16810356—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53C2211yescharacteristic
22C2×#117×C53—21no—
23C53×#390—21no—
24C53×#389—21no—
25C53×#388—22no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C4×C22⋊C4×C53—21no—
28C43×C53C2211yescharacteristic
29C2×#11719945—21no—
30C2×#11719922—22no—
31C2×#11719917—21no—
32#13133217C2×C411yescharacteristic
33#13133214C2×C411yescharacteristic
34#13133205—21no—
35#13133191—21no—
36#13133191—21no—
37C4×D212—21no—
38C4×D212—21no—
39#13133150—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yes—
44#13133145—42no—
45#13133145—42no—
46#13133138—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53C2×C412yes—
57C4×D4×C53D412yes—
58#117×C53—42no—
59#117×C53—42no—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53—21no—
69C2×C42×C53D411yescharacteristic
70C2×C42×C53D411yescharacteristic
71C2×C42×C53C2311yescharacteristic
72#95×C53—22no—
73#95×C53D412yes—
74C24×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—42no—
77C2×D4×C53C22⋊C412yes—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C2×C22⋊D53—41no—
92#11719945—41no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×Dic106—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—44no—
104C4⋊C4×C53C22⋊C412yes—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—44no—
108C22⋊C4×C53—44no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—42no—
113C42×C53—42no—
114C42×C53—41no—
115C42×C53—41no—
116C42×C53D812yescomplemented
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53#4111yescharacteristic
121#11719922—44no—
122#11719921—41no—
123#11719921—41no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719917—41no—
127#11719917—41no—
128C2×#88322—22no—
129C2×#88322—41no—
130C2×#88322—41no—
131C23×C53—22no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53#12011yescharacteristic
137C22×D53—41no—
138D4×C53—22no—
139D4×C53—22no—
140D4×C53—42no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×C4×C53C2×D811yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C22⋊D53—81no—
170C22⋊D53—81no—
171C2×Dic53—41no—
172D212—42no—
173D212—41no—
174D212—41no—
175D212—81no—
176C4×D53—81no—
177C4×D53—81no—
178Dic106—41no—
179Dic106—41no—
180#88322—42no—
181#88322—81no—
182#88322—81no—
183#9237—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53#38611yescharacteristic
192C22×C53#44711yescharacteristic
193C22×C53#44911yescharacteristic
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—42no—
201C4×C53—42no—
202C4×C53—42no—
203C4×C53—82no—
204C4×C53#42112yes—
205C4×C53C4×D811yescharacteristic
206C4×C53#44211yescharacteristic
207C4×C53#45911yescharacteristic
208C4×C53#46311yescharacteristic
209Dic53—81no—
210#3100—531no—
211#2293D5311yescharacteristic, complemented
212#1962—531no—
213#1502—531no—
214#1486—531no—
215#1303—531no—
216#1276—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—82no—
220C2×C53#187911yescharacteristic
221C2×C53#189711yescharacteristic
222C2×C53#190611yescharacteristic
223D53—161no—
224#532—531no—
225C2×C4×D4C2×D5311yescharacteristic
226C2×#117—21no—
227#480—1061no—
228#474—532no—
229#459—532no—
230#447—1061no—
231C4×D8—1062no—
232#431—531no—
233#431—531no—
234#423—531no—
235C2×#104—1061no—
236C2×#104—1061no—
237#390—21no—
238#389—21no—
239#388—22no—
240C4×C4⋊C4C2×D5311yescharacteristic
241C4×C22⋊C4—21no—
242C43C2×D5311yescharacteristic
243#327—1062no—
244D4⋊C8—1062no—
245C53#923711yescharacteristic, complemented, Sylow
246C2×#41—1061no—
247C22×D4C22⋊D5311yescharacteristic
248C23×C4—21no—
249C2×D8—1062no—
250C2×M16—1061no—
251C4⋊Q8—531no—
252#127—531no—
253#124—1061no—
254#121—1061no—
255#121—1061no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—1061no—
259C4×D4—1061no—
260C4×D4C4×D5312yes—
261C4×D4D21212yes—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42C22⋊D5311yescharacteristic
274C2×C42C22⋊D5311yescharacteristic
275C2×C42C22×D5311yescharacteristic
276#107—1062no—
277C4⋊C8—532no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280C4⋊C8—1061no—
281#104—2122no—
282#104—2122no—
283#102—1062no—
284#102—1062no—
285C22⋊C8—1061no—
286C22⋊C8—1061no—
287C4×C8—1061no—
288C4×C8—1061no—
289#95—22no—
290#95C22⋊D5312yes—
291C24—21no—
292#41—2121no—
293#41—2121no—
294#41—2121no—
295#41—2121no—
296C2×Q8—1061no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—2121no—
303C2×D4#1171992112yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—2121no—
315C22×C4C2×C22⋊D5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—2124no—
319M16—2121no—
320M16—2121no—
321C2×C8—1062no—
322C2×C8—2121no—
323C2×C8—2121no—
324C4⋊C4—21no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—1061no—
328C4⋊C4—1061no—
329C4⋊C4#1171992112yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42#1171992212yescomplemented
344C42C2×C4×D5311yescharacteristic
345C42C2×D21211yescharacteristic
346C42#1171994511yescharacteristic
347C42#1171994511yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—2121no—
354C23#1313318611yescharacteristic
355Q8—2121no—
356Q8—2121no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—2122no—
364D4—2121no—
365D4—2121no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—2121no—
387C2×C4C2×#1171992111yescharacteristic
388C2×C4C4×C22⋊D5311yescharacteristic
389C2×C4C4×C22⋊D5311yescharacteristic
390C2×C4#1313319111yescharacteristic
391C2×C4#1313319111yescharacteristic
392C2×C4#1313320011yescharacteristic
393C2×C4C2×#1171992211yescharacteristic
394C2×C4#1313323711yescharacteristic
395C2×C4#1313324211yescharacteristic
396C2×C4#1313324311yescharacteristic
397C8—4241no—
398C8—4241no—
399C8—2122no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—4241no—
408C22—4241no—
409C22—4241no—
410C22#1681050211yescharacteristic
411C22#1681075711yescharacteristic
412C22#1681076811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—4241no—
421C4#1681064112yes—
422C4C4×#1171992211yescharacteristic
423C4#1681073311yescharacteristic
424C4#1681081811yescharacteristic
425C4#1681083811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—8481no—
430C2#2027506311yescharacteristic
431C2#2027516311yescharacteristic
432C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2293C2
socleC22×C53#386
prime radical#20284715C2
cosocleC2×C42×C53C23
hypercenter#2293D53
hypocenterC53#9237
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #20275543C2semidirect
3. #20273209C2semidirect
4. #20273140C2semidirect
5. #20272618C2semidirect
6. #20272548C2semidirect
7. C53×#2293C2semidirect
10. #16810903C4semidirect
11. #16810818C4semidirect
116. C42×C53D8semidirect
211. #2293D53semidirect
245. C53#9237semidirect
343. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847152
#202755432
#202732092
#202731402
#202726182
#202725482
C53×#22932
#923753