Finite Group Databasebeta

← #27280560

Subgroups — #27280560

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810762C211yesmaximal, characteristic
2#20275551C211yesmaximal, characteristic
3#20274650C211yesmaximal, characteristic, complemented
4#20273282C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7Q8×C22⋊C4×C53C211yesmaximal, characteristic, complemented
8C22×#11719924C412yescomplemented
9C2×#13133191C2211yescharacteristic
10#16810762—21no—
11#16810762—22no—
12#16810762—21no—
13C2×#13133141—22no—
14C2×#13133141C412yescomplemented
15C2×#13133131—21no—
16C2×#13133131—21no—
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19#16810483C2211yescharacteristic
20#16810356—22no—
21C23×Q8×C53C2211yescharacteristic
22C2×C4×Q8×C53—21no—
23C53×#391—21no—
24C53×#391—22no—
25C53×#380—21no—
26C53×#380C2211yescharacteristic
27C4×C22⋊C4×C53—21no—
28C4×C22⋊C4×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C2×#11719924—22no—
32C2×#11719924—22no—
33C2×#11719924—44no—
34C22×#88322—21no—
35C22×#88322C2×C412yes—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133141—22no—
45#13133141—22no—
46#13133141—44no—
47#13133131—42no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C22×Q8×C53—21no—
56C22×Q8×C53—21no—
57C22×Q8×C53—44no—
58C22×Q8×C53C2×C412yes—
59C22×Q8×C53D411yescharacteristic
60C22×Q8×C53D412yes—
61C22×Q8×C53D411yescharacteristic
62C23×C4×C53—21no—
63C23×C4×C53C2311yescharacteristic
64C4×Q8×C53—44no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C42×C53—22no—
71C2×C42×C53—21no—
72#95×C53—22no—
73#95×C53—22no—
74#95×C53—21no—
75#95×C53D412yes—
76C24×C53C2×D411yescharacteristic
77C23×D53—21no—
78C2×Q8×C53—24no—
79C2×Q8×C53—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—24no—
84C2×Q8×C53—24no—
85C2×Q8×C53—42no—
86C2×Q8×C53—42no—
87C2×Q8×C53—42no—
88C2×Q8×C53—44no—
89C2×Q8×C53—44no—
90C2×Q8×C53—44no—
91C2×Q8×C53C22⋊C412yes—
92C2×Q8×C53C22⋊C412yes—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—42no—
101C22×C4×C53—44no—
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C2×C22⋊D53—41no—
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C22×Dic53—21no—
115C22×Dic53—41no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212—42no—
119C4⋊C4×C53—44no—
120C4⋊C4×C53—42no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53SD1612yescomplemented
127C42×C53—44no—
128C42×C53—42no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131#11719924—44no—
132#11719924—44no—
133#11719921—41no—
134#11719921—41no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—42no—
138C2×#88322—22no—
139C2×#88322—42no—
140C2×#88322—42no—
141C2×#88322—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53C4×D411yescharacteristic
145C23×C53#12011yescharacteristic
146C23×C53#12111yescharacteristic
147C23×C53C2×SD1611yescharacteristic
148C23×C53#13511yescharacteristic
149C23×C53AGL(1,ℤ/8)11yescharacteristic
150C23×C53#13711yescharacteristic
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—81no—
155C22×D53—81no—
156Q8×C53—24no—
157Q8×C53—22no—
158Q8×C53—22no—
159Q8×C53—22no—
160Q8×C53—44no—
161Q8×C53—44no—
162Q8×C53—42no—
163Q8×C53#9914yescomplemented
164C2×C4×C53—24no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—44no—
175C2×C4×C53—44no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53C2×C22⋊C411yescharacteristic
186C2×C4×C53C4×D411yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12311yescharacteristic
189C22⋊D53—81no—
190C22⋊D53—81no—
191C22⋊D53—81no—
192C22⋊D53—81no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—81no—
197C2×Dic53—81no—
198C2×Dic53—81no—
199D212—42no—
200D212—81no—
201#88322—42no—
202#88322—81no—
203#9262—531nomaximal, Sylow
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53#38611yescharacteristic
215C22×C53C4×SD1611yescharacteristic
216C22×C53#44011yescharacteristic
217C22×C53#44811yescharacteristic
218C22×C53#44911yescharacteristic
219C22×C53#46011yescharacteristic
220C22×C53#46411yescharacteristic
221C2×D53—81no—
222C2×D53—81no—
223C2×D53—81no—
224C2×D53—161no—
225C2×D53—161no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—24no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—42no—
232C4×C53—81no—
233C4×C53—82no—
234C4×C53C2×#9912yes—
235Dic53—81no—
236Dic53—161no—
237C2×#448—531no—
238Q8×C22⋊C4D5311yescharacteristic, complemented
239#1964—531no—
240#1787—531no—
241#1519—531no—
242#1289—531no—
243#1282—531no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53#188011yescharacteristic
251C2×C53#189811yescharacteristic
252C2×C53#190511yescharacteristic
253D53—161no—
254C23×Q8C2×D5311yescharacteristic
255C22×SD16—532no—
256C2×#121—531no—
257C2×C4×Q8—21no—
258#448—1062no—
259#448—1061no—
260#448—1061no—
261C2×#103—532no—
262C2×#103—1062no—
263C2×#99—1061no—
264C2×#99—1061no—
265C2×C22⋊C8—531no—
266C2×C22⋊C8—531no—
267#391—21no—
268#391—22no—
269#380—21no—
270#380—531no—
271#380C2×D5311yescharacteristic
272C4×C22⋊C4—21no—
273C4×C22⋊C4C2×D5311yescharacteristic
274#327—1062no—
275C53#926211yescharacteristic, complemented, Sylow
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—44no—
280C22×Q8C4×D5312yes—
281C22×Q8D21212yes—
282C22×Q8C22⋊D5311yescharacteristic
283C22×Q8C22⋊D5311yescharacteristic
284C22×D4—531no—
285C22×D4—1061no—
286C23×C4—21no—
287C23×C4C22×D5311yescharacteristic
288C2×SD16—1062no—
289C2×SD16—1062no—
290C2×SD16—2124no—
291C22×C8—532no—
292C22×C8—1061no—
293#121—1061no—
294#121—1061no—
295#121—1061no—
296#121—1061no—
297C4×Q8—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—1061no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1061no—
304C2×C22⋊C4—1061no—
305C2×C22⋊C4C22⋊D5311yescharacteristic
306C2×C22⋊C4C22⋊D5311yescharacteristic
307C2×C42—22no—
308C2×C42—21no—
309#103—1062no—
310#103—1062no—
311#103—2124no—
312#99—2122no—
313#99—2122no—
314C22⋊C8—1061no—
315C22⋊C8—1061no—
316C22⋊C8—1061no—
317C22⋊C8—1061no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—1061no—
322#95C22⋊D5312yes—
323C24—1061no—
324C24C2×C22⋊D5311yescharacteristic
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8#1171992112yes—
339C2×Q8#1171992112yes—
340C2×D4—1061no—
341C2×D4—1061no—
342C2×D4—2122no—
343C2×D4—2121no—
344C2×D4—2121no—
345C2×D4—2121no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—1061no—
359C22×C4—2121no—
360C22×C4C2×C4×D5311yescharacteristic
361C22×C4C2×D21211yescharacteristic
362C22×C4#1171994511yescharacteristic
363C22×C4#1171994511yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366SD16—2124no—
367SD16—2124no—
368C2×C8—1062no—
369C2×C8—2122no—
370C2×C8—2121no—
371C2×C8—2122no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—2122no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—2121no—
383C22⋊C4—2121no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C22⋊C4#1171992412yescomplemented
387C42—44no—
388C42—42no—
389C23—41no—
390C23—41no—
391C23—4241no—
392C23—4241no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23C4×C22⋊D5311yescharacteristic
397C23#1313318611yescharacteristic
398C23#1313319111yescharacteristic
399C23C2×#1171992411yescharacteristic
400C23#1313323711yescharacteristic
401C23#1313324211yescharacteristic
402C23#1313324811yescharacteristic
403Q8—24no—
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—44no—
409Q8—42no—
410Q8#1313313114yescomplemented
411D4—4241no—
412D4—4241no—
413D4—4241no—
414D4—4241no—
415D4—4241no—
416D4—2122no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—4241no—
439C2×C4—4241no—
440C2×C4—4241no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4C2×#1171992111yescharacteristic
445C2×C4C4×C22⋊D5311yescharacteristic
446C2×C4#1313319111yescharacteristic
447C2×C4#1313320011yescharacteristic
448C8—4241no—
449C8—2122no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—8481no—
461C22—8481no—
462C22—4241no—
463C22—4241no—
464C22—4241no—
465C22#1681050211yescharacteristic
466C22C4×#1171992411yescharacteristic
467C22#1681072111yescharacteristic
468C22#1681076211yescharacteristic
469C22#1681076811yescharacteristic
470C22#1681082311yescharacteristic
471C22#1681084511yescharacteristic
472C4—21no—
473C4—24no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4—8481no—
481C4—4241no—
482C4C2×#1313313112yes—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—8481no—
490C2#2027507011yescharacteristic
491C2#2027516811yescharacteristic
492C2#2027520911yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingQ8×C22⋊C4×C53C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterQ8×C22⋊C4D53
hypocenterC53#9262
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20274650C2semidirect
5. #20272582C2semidirect
6. #20272563C2semidirect
7. Q8×C22⋊C4×C53C2semidirect
8. C22×#11719924C4semidirect
14. C2×#13133141C4semidirect
126. C22⋊C4×C53SD16semidirect
163. Q8×C53#99semidirect
238. Q8×C22⋊C4D53semidirect
275. C53#9262semidirect
386. C22⋊C4#11719924semidirect
410. Q8#13133131semidirect

Maximal subgroups

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

subgroupindex
C2×#168107622
#202755512
#202746502
#202732822
#202725822
#202725632
Q8×C22⋊C4×C532
#926253