Finite Group Databasebeta

← #27278860

Subgroups — #27278860

Order 13568.

616 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
1C2×#16810823C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic
3#20275520C211yesmaximal, characteristic
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2355C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—22no—
11#16810823C414yescomplemented
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×#13133137—22no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19Q8⋊#88322—22no—
20C2×C4×Q8×C53C2211yescharacteristic
21C53×#391—21no—
22C53×#387—22no—
23C53×#384C2211yescharacteristic
24C53×#382—21no—
25C4×C4⋊C4×C53—21no—
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214C2×C412yes—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322C2×C412yes—
45#13133137—24no—
46#13133137—44no—
47#13133137—42no—
48#13133137—42no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×Q8×C53D411yescharacteristic
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53C2×C412yes—
55C4×Q8×C53D412yes—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D412yes—
63C2×C42×C53—22no—
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#95×C53—22no—
68#95×C53—22no—
69#95×C53—22no—
70#95×C53—21no—
71C23×D53—41no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—42no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53C22⋊C412yes—
114C4⋊C4×C53SD1612yescomplemented
115C42×C53—22no—
116C42×C53—22no—
117C42×C53—41no—
118C42×C53—41no—
119C42×C53—82no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53#4111yescharacteristic
124#11719924—48no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53#12011yescharacteristic
135C22×D53—42no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—81no—
140C22×D53—81no—
141C22×D53—81no—
142Q8×C53—22no—
143Q8×C53—22no—
144Q8×C53—41no—
145Q8×C53—41no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×C4×C53C2×SD1611yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53#13511yescharacteristic
172C2×C4×C53#13511yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53#13711yescharacteristic
177D212—44no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—82no—
183D212—81no—
184D212—81no—
185D212—81no—
186D212—81no—
187#88322—42no—
188#88322—42no—
189#88322—81no—
190#8978—531nomaximal, Sylow
191C22×C53#38611yescharacteristic
192C22×C53#44811yescharacteristic
193C22×C53#44911yescharacteristic
194C22×C53#44911yescharacteristic
195C22×C53#44911yescharacteristic
196C22×C53#45311yescharacteristic
197C22×C53#45411yescharacteristic
198C2×D53—82no—
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—161no—
202C2×D53—161no—
203C2×D53—161no—
204C2×D53—161no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—42no—
208C4×C53—42no—
209C4×C53—42no—
210C4×C53—82no—
211C4×C53—81no—
212C4×C53#41611yescharacteristic
213C4×C53#41811yescharacteristic
214C4×C53C4×SD1611yescharacteristic
215C4×C53#44011yescharacteristic
216C4×C53#46011yescharacteristic
217C4×C53#46411yescharacteristic
218C4×C53#48211yescharacteristic
219C4×C53#48211yescharacteristic
220C2×#460—531no—
221#2355D5311yescharacteristic, complemented
222#1982—531no—
223#1957—531no—
224C2×Q8⋊C8—531no—
225#1296—531no—
226#1279—531no—
227C2×C53#162011yescharacteristic
228C2×C53#162211yescharacteristic
229C2×C53#162611yescharacteristic
230C2×C53#162811yescharacteristic
231C2×C53#187511yescharacteristic
232C2×C53#188011yescharacteristic
233C2×C53#188311yescharacteristic
234D53—162no—
235C22×SD16—1061no—
236C2×#127—531no—
237C2×C4×Q8C2×D5311yescharacteristic
238#460—534no—
239#460—1062no—
240C2×C4⋊C8—531no—
241C2×C4⋊C8—531no—
242C2×#102—1061no—
243C2×#102—1062no—
244C2×#102—1061no—
245C2×#102—1061no—
246C2×C4×C8—531no—
247#391—21no—
248#387—22no—
249#384C2×D5311yescharacteristic
250#382—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2×D5311yescharacteristic
254Q8⋊C8—1062no—
255C53#897811yescharacteristic, complemented, Sylow
256C22×Q8C22⋊D5311yescharacteristic
257C22×D4—1061no—
258C22×D4—1061no—
259C22×D4—1061no—
260C2×SD16—1064no—
261C2×SD16—2122no—
262C2×SD16—2122no—
263C22×C8—1061no—
264C22×C8—1061no—
265C22×C8—1061no—
266#127—532no—
267#127—1061no—
268#127—1061no—
269#127—1061no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D5312yes—
273C4×Q8D21212yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C4⋊C4C22⋊D5311yescharacteristic
279C2×C4⋊C4C22⋊D5311yescharacteristic
280C2×C4⋊C4C22⋊D5312yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C22×D5311yescharacteristic
285C4⋊C8—532no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289#102—1064no—
290#102—2122no—
291#102—2124no—
292#102—2122no—
293C4×C8—1061no—
294C4×C8—1061no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—2121no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8#1171992112yes—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—2122no—
310C2×D4—2122no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×C22⋊D5311yescharacteristic
328C22×C4C2×C22⋊D5311yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330SD16—2128no—
331C2×C8—1062no—
332C2×C8—1062no—
333C2×C8—2121no—
334C2×C8—2121no—
335C2×C8—2121no—
336C2×C8—2121no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4#1171992112yes—
352C4⋊C4#1171992412yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×C4×D5311yescharacteristic
359C42C2×D21211yescharacteristic
360C42#1171994511yescharacteristic
361C42#1171994511yescharacteristic
362C23—4241no—
363C23—4241no—
364C23—4241no—
365C23—4241no—
366C23—4241no—
367C23—4241no—
368C23—2122no—
369C23#1313318611yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—4242no—
375D4—4241no—
376D4—4241no—
377D4—4241no—
378D4—4241no—
379D4—2124no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×#1171992111yescharacteristic
402C2×C4C4×C22⋊D5311yescharacteristic
403C2×C4C4×C22⋊D5311yescharacteristic
404C2×C4#1313319111yescharacteristic
405C2×C4#1313319111yescharacteristic
406C2×C4#1313320011yescharacteristic
407C2×C4C2×#1171992411yescharacteristic
408C2×C4#1313323711yescharacteristic
409C2×C4#1313323711yescharacteristic
410C2×C4#1313323711yescharacteristic
411C2×C4#1313324211yescharacteristic
412C2×C4#1313324211yescharacteristic
413C2×C4#1313324211yescharacteristic
414C2×C4#1313324811yescharacteristic
415C8—4241no—
416C8—2122no—
417C8—2122no—
418C22—8481no—
419C22—8481no—
420C22—8481no—
421C22—8481no—
422C22—4242no—
423C22—4242no—
424C22—4242no—
425C22#1681050211yescharacteristic
426C22#1681076211yescharacteristic
427C22#1681076811yescharacteristic
428C22#1681076811yescharacteristic
429C22#1681076811yescharacteristic
430C22#1681079111yescharacteristic
431C22#1681079611yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4#1681061711yescharacteristic
440C4#1681062711yescharacteristic
441C4C4×#1171992411yescharacteristic
442C4#1681072111yescharacteristic
443C4#1681082311yescharacteristic
444C4#1681084511yescharacteristic
445C4#1681094711yescharacteristic
446C4#1681094711yescharacteristic
447C2—8482no—
448C2#2027377411yescharacteristic
449C2#2027378711yescharacteristic
450C2#2027381111yescharacteristic
451C2#2027382511yescharacteristic
452C2#2027503711yescharacteristic
453C2#2027507011yescharacteristic
454C2#2027509111yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2355C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#2355D53
hypocenterC53#8978
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#88322C2semidirect
5. #20272598C2semidirect
6. #20272556C2semidirect
7. C53×#2355C2semidirect
11. #16810823C4semidirect
114. C4⋊C4×C53SD16semidirect
221. #2355D53semidirect
255. C53#8978semidirect
352. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
#202756422
#202755202
C2×Q8⋊#883222
#202725982
#202725562
C53×#23552
#897853