Finite Group Databasebeta

← #27278719

Subgroups — #27278719

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283824C211yesmaximal, characteristic, complemented
2C4×#13133243C211yesmaximal, characteristic
3#20273106C211yesmaximal, characteristic, complemented
4#20273098C211yesmaximal, characteristic, complemented
5#20272565C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#1954C211yesmaximal, characteristic, complemented
8C2×#13133243C411yescharacteristic
9C4×#11719945C2211yescharacteristic
10#16810747—21no—
11#16810733—21no—
12#16810723—21no—
13C4×#11719923—21no—
14C4×#11719922—21no—
15#16810688—21no—
16C8×C22⋊D53—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810675—21no—
20#16810663C411yescharacteristic
21C4⋊C8×D53—21no—
22#16810628C411yescharacteristic
23#16810622C411yescharacteristic
24C4×#11719917C2211yescharacteristic
25#16810355—21no—
26#16810352—21no—
27D4⋊#88322—21no—
28D4⋊#88322—21no—
29C2×C4×D4×C53C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C2×C22⋊C8×C53—22no—
32C2×C4×C8×C53C2211yescharacteristic
33C2×#11719945C2×C411yescharacteristic
34#13133243—22no—
35#13133243C814yescomplemented
36C2×#11719923—21no—
37C2×#11719922—21no—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719917C2×C411yescharacteristic
40C2×C8×D53—22no—
41C4×Dic106D411yescharacteristic
42C4×Dic106D411yescharacteristic
43C4×D212D411yescharacteristic
44C4×D212D411yescharacteristic
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47#13133176C2×C411yescharacteristic
48#13133174—21no—
49C42×D53—21no—
50#13133150—21no—
51#13133148—21no—
52C4⋊#88322—21no—
53C4⋊#88322D411yescharacteristic
54C4⋊#88322D411yescharacteristic
55#13133146—22no—
56#13133140—21no—
57#13133138—21no—
58#13133138—21no—
59#13133137—21no—
60C22⋊#88322—21no—
61C22⋊#88322—21no—
62#13133129—22no—
63#13133128—21no—
64C4×#88322—21no—
65C4×#88322—21no—
66C4×#88322—21no—
67C8×Dic53—22no—
68C22×D4×C53C2×C411yescharacteristic
69C23×C4×C53—21no—
70C22×C8×C53—21no—
71C22×C8×C53D412yes—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76C2×C4⋊C4×C53C2×C411yescharacteristic
77C2×C22⋊C4×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53—21no—
81C22⋊C8×C53—44no—
82C4×C8×C53—21no—
83C4×C8×C53—21no—
84C24×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53C2×C812yescomplemented
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×C22⋊D53—21no—
100#11719945—21no—
101#11719945—21no—
102#11719945C2×C812yes—
103#11719945M1612yes—
104C2×D212C22⋊C411yescharacteristic
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×C4×D53—21no—
108C2×Dic106C22⋊C411yescharacteristic
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C2×C8×C53—42no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C22⋊Dic53—21no—
123#11719923—24no—
124#11719922—24no—
125#11719921—21no—
126#11719921—21no—
127C4⋊Dic53C22⋊C411yescharacteristic
128C4⋊Dic53C22⋊C411yescharacteristic
129#11719919—21no—
130#11719919—21no—
131C4×Dic53—21no—
132C4×Dic53—21no—
133C4×Dic53—21no—
134#11719917—21no—
135#11719917—21no—
136#11719917—21no—
137#11719917C2×C812yescomplemented
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—41no—
143C8×D53—82no—
144C23×C53—22no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53C4×D411yescharacteristic
150C22×D53—21no—
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—41no—
154D4×C53—41no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—82no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×C4×C53C22×C811yescharacteristic
170C2×C4×C53C2×M1611yescharacteristic
171C2×C4×C53#13111yescharacteristic
172C2×C4×C53#13111yescharacteristic
173C22⋊D53—22no—
174C22⋊D53—22no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178D212—21no—
179D212C22⋊C812yescomplemented
180C4×D53—22no—
181C4×D53—22no—
182C4×D53—41no—
183C4×D53—41no—
184C4×D53—81no—
185Dic106—21no—
186Dic106C22⋊C812yescomplemented
187C8×C53—42no—
188C8×C53—42no—
189#88322—22no—
190#88322—41no—
191#88322—41no—
192#88322—41no—
193#88322—81no—
194#8954—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—44no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53#38611yescharacteristic
203C22×C53C8×D411yescharacteristic
204C22×C53#43511yescharacteristic
205C2×D53—22no—
206C2×D53—41no—
207C2×D53—41no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—81no—
212C4×C53C2×C22⋊C811yescharacteristic
213C4×C53#40811yescharacteristic
214C4×C53C8×D411yescharacteristic
215C4×C53#43511yescharacteristic
216C4×C53#45311yescharacteristic
217C4×C53#45411yescharacteristic
218Dic53—22no—
219Dic53—41no—
220Dic53—41no—
221#2988—531no—
222C4×AGL(1,ℤ/8)—531no—
223#1954D5311yescharacteristic, complemented
224#1478—531no—
225#1476—531no—
226#1283—531no—
227#1278—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—82no—
231C2×C53#186911yescharacteristic
232C2×C53#188111yescharacteristic
233C2×C53#188311yescharacteristic
234D53—42no—
235C2×AGL(1,ℤ/8)—531no—
236C4×#41—531no—
237C2×C4×D4C2×D5311yescharacteristic
238C8×Q8—1061no—
239#442—1061no—
240#440—1061no—
241C4×SD16—1061no—
242C4×D8—1061no—
243C8×D4—1061no—
244C8×D4—1062no—
245#431—531no—
246#431—531no—
247#431—1061no—
248#428—531no—
249C2×C4⋊C8—1061no—
250C2×C4⋊C8C2×D5311yescharacteristic
251#418—531no—
252#417—531no—
253C2×C22⋊C8—22no—
254C4×M16—531no—
255C2×C4×C8C2×D5311yescharacteristic
256Q8⋊C8—1061no—
257D4⋊C8—1061no—
258D4⋊C8—1061no—
259D4⋊C8—1061no—
260C53#895411yescharacteristic, complemented, Sylow
261C2×#41—531no—
262C22×D4C4×D5311yescharacteristic
263C23×C4—21no—
264AGL(1,ℤ/8)—534no—
265AGL(1,ℤ/8)—1062no—
266C2×SD16—1061no—
267C2×D8—1061no—
268C2×M16—531no—
269C2×M16—531no—
270C22×C8—21no—
271C22×C8—1062no—
272C22×C8C22⋊D5312yes—
273C4×Q8—531no—
274C4×Q8—531no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—531no—
280C4×D4—531no—
281C4×D4—1061no—
282C4×D4—1061no—
283#117—531no—
284#117—1061no—
285C2×C4⋊C4C4×D5311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—1061no—
288C2×C42C22×D5311yescharacteristic
289#107—1061no—
290#106—1061no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—531no—
294C4⋊C8—531no—
295C4⋊C8—1062no—
296C4⋊C8—1061no—
297#103—1061no—
298#102—1061no—
299#102—1061no—
300#102—1061no—
301C22⋊C8—44no—
302C22⋊C8—1062no—
303C22⋊C8—1061no—
304C22⋊C8—1061no—
305#97—1061no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—1062no—
309C4×C8—1061no—
310C4×C8—1061no—
311C4×C8—1061no—
312C24—21no—
313#41—532no—
314#41—532no—
315#41—1061no—
316#41—1061no—
317C2×Q8—531no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—531no—
324C2×D4—1061no—
325C2×D4C8×D5312yescomplemented
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4C2×C4×D5311yescharacteristic
335C22×C4#1171994511yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—1064no—
338D8—1064no—
339M16—532no—
340M16—1061no—
341M16—1061no—
342M16—1061no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1061no—
352C2×C8—4242no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—531no—
357C4⋊C4—531no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—1061no—
363C22⋊C4—1061no—
364C22⋊C4—1061no—
365C42—1061no—
366C42—1061no—
367C42—1061no—
368C42D4×D5311yescharacteristic
369C42D4×D5311yescharacteristic
370C42D4×D5311yescharacteristic
371C42#1171994711yescharacteristic
372C23—22no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—1061no—
378C23C4×C22⋊D5311yescharacteristic
379Q8—532no—
380Q8—1061no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385D4—532no—
386D4—1062no—
387D4—1062no—
388D4—1061no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—1061no—
398C2×C4—1062no—
399C2×C4—1062no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—4241no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4C22⋊C4×D5311yescharacteristic
406C2×C4#1313317711yescharacteristic
407C2×C4#1313318511yescharacteristic
408C2×C4#1313318811yescharacteristic
409C2×C4#1313319011yescharacteristic
410C2×C4#1313319911yescharacteristic
411C2×C4C2×C8×D5311yescharacteristic
412C2×C4M16×D5311yescharacteristic
413C2×C4#1313322311yescharacteristic
414C2×C4#1313322511yescharacteristic
415C8—42no—
416C8—42no—
417C8—1062no—
418C8—4241no—
419C8—2121no—
420C8—2121no—
421C8—2121no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—42no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—1062no—
430C22—2121no—
431C22—2121no—
432C22#1681049911yescharacteristic
433C22C8×C22⋊D5311yescharacteristic
434C22#1681069211yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—81no—
439C4—1062no—
440C4—2121no—
441C4—2121no—
442C4C22⋊C8×D5311yescharacteristic
443C4#1681058811yescharacteristic
444C4#1681068611yescharacteristic
445C4#1681069711yescharacteristic
446C4#1681079011yescharacteristic
447C4#1681079511yescharacteristic
448C2—21no—
449C2—21no—
450C2—82no—
451C2—2122no—
452C2#2027500411yescharacteristic
453C2#2027508011yescharacteristic
454C2#2027509411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1954C2
socleC22×C53#386
prime radicalC2×#13133243C4
cosocleC2×C42×C53C23
hypercenter#1954D53
hypocenterC53#8954
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. #20283824C2semidirect
3. #20273106C2semidirect
4. #20273098C2semidirect
5. #20272565C2semidirect
6. #20272552C2semidirect
7. C53×#1954C2semidirect
35. #13133243C8semidirect
90. C2×D4×C53C2×C8semidirect
137. #11719917C2×C8semidirect
179. D212C22⋊C8semidirect
186. Dic106C22⋊C8semidirect
223. #1954D53semidirect
260. C53#8954semidirect
325. C2×D4C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202838242
C4×#131332432
#202731062
#202730982
#202725652
#202725522
C53×#19542
#895453