Finite Group Databasebeta

← #27301577

Subgroups — #27301577

Order 13568.

722 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810607C211yesmaximal, characteristic
2C2×#16810413C212yesmaximal, complemented
3#20274633C211yesmaximal, characteristic
4#20272833C212yesmaximal, complemented
5C53×#2406C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810607—24no—
8C2×#13133136—21no—
9C2×#13133134C2212yes—
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810486—21no—
13#16810413—24no—
14C22×#41×C53C2211yescharacteristic
15C2×#123×C53C411yescharacteristic, complemented
16C2×#123×C53C2211yescharacteristic
17C22×C4⋊C4×C53C411yescharacteristic, complemented
18C53×#395—21no—
19C53×#394—21no—
20C53×#386C411yescharacteristic, complemented
21C53×#386C2211yescharacteristic
22C53×#385—21no—
23C53×#385—21no—
24C53×#381C411yescharacteristic, complemented
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—42no—
28C2×#11719917—41no—
29C22×#88322—41no—
30C2×C22⋊Dic53—22no—
31#13133136—24no—
32#13133134—22no—
33#13133134D414yes—
34#13133132—22no—
35#13133132—22no—
36#13133132—42no—
37#13133124—41no—
38#13133124—41no—
39C2×#41×C53—22no—
40C2×#41×C53—22no—
41C2×#41×C53—41no—
42C2×#41×C53—41no—
43C2×#41×C53—42no—
44C22×Q8×C53D411yescharacteristic
45C22×D4×C53C2×C411yescharacteristic
46C22×D4×C53D411yescharacteristic
47C22×D4×C53C2311yescharacteristic
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53#123×C53—22no—
54#123×C53—22no—
55#123×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—41no—
61C2×C4⋊C4×C53—41no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53C2×C411yescharacteristic
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53C22⋊C411yescharacteristic
73C24×C53C22×C411yescharacteristic
74C24×C53C2×D411yescharacteristic
75#41×C53—44no—
76#41×C53—44no—
77#41×C53—44no—
78#41×C53—44no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53C2×D412yes—
92C2×D4×C53#4112yes—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53—42no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C22⋊C411yescharacteristic
116C22×C4×C53C22⋊C411yescharacteristic
117C22×C4×C53C22⋊C411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×Dic53—42no—
120C4⋊C4×C53—42no—
121C4⋊C4×C53—42no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—42no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—42no—
130C22⋊C4×C53—42no—
131C22⋊C4×C53—41no—
132C22⋊Dic53—42no—
133C22⋊Dic53—42no—
134#11719917—24no—
135#11719917—44no—
136#11719917—42no—
137C2×#88322—44no—
138C2×#88322—42no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53#9912yes—
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53C4×D412yes—
154C23×C53#12112yes—
155Q8×C53—42no—
156Q8×C53—42no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—44no—
160D4×C53—44no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—82no—
194C2×C4×C53#9911yescharacteristic
195C2×C4×C53#9911yescharacteristic
196C2×C4×C53#12011yescharacteristic
197C2×C4×C53#12311yescharacteristic
198C2×Dic53—82no—
199C2×Dic53—82no—
200C2×Dic53—82no—
201#88322—44no—
202#13081—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53#38611yescharacteristic
221C22×C53C2×#9911yescharacteristic
222C22×C53C2×#9911yescharacteristic
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—42no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53—81no—
231C4×C53—81no—
232C4×C53—81no—
233Dic53—162no—
234C2×#413—531no—
235#2406D5311yescharacteristic
236C2×#352—532no—
237#1782—531no—
238#1390—532no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53—42no—
242C2×C53—42no—
243C2×C53—42no—
244C2×C53#189811yescharacteristic
245C2×C53#212011yescharacteristic
246C2×C53#212311yescharacteristic
247C22×#41C2×D5311yescharacteristic
248C22×M16—1061no—
249C2×#123Dic5311yescharacteristic, complemented
250C2×#123C2×D5311yescharacteristic
251C22×C4⋊C4Dic5311yescharacteristic, complemented
252#413—1064no—
253C2×#101—1061no—
254C2×#100—532no—
255C2×#99—532no—
256C2×#99—1061no—
257#395—21no—
258#394—21no—
259#386Dic5311yescharacteristic, complemented
260#386C2×D5311yescharacteristic
261#385—21no—
262#385—21no—
263#381—1061no—
264#381Dic5311yescharacteristic, complemented
265#352—1064no—
266C53#1308111yescharacteristic, complemented, Sylow
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—41no—
270C2×#41—41no—
271C2×#41—42no—
272C22×Q8C22⋊D5311yescharacteristic
273C22×D4C2×Dic5311yescharacteristic
274C22×D4C22⋊D5311yescharacteristic
275C22×D4C22×D5311yescharacteristic
276C23×C4C2×Dic5311yescharacteristic
277C23×C4C2×Dic5311yescharacteristic
278C23×C4C2×Dic5311yescharacteristic
279C23×C4C22⋊D5311yescharacteristic
280C23×C4C22⋊D5311yescharacteristic
281C2×M16—1064no—
282C2×M16—1062no—
283C2×M16—2122no—
284C2×M16—2121no—
285C22×C8—2121no—
286#123—22no—
287#123—22no—
288#123—42no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—1062no—
300C2×C22⋊C4C2×Dic5311yescharacteristic
301C2×C22⋊C4C2×Dic5311yescharacteristic
302#101—1064no—
303#100—534no—
304#100—1062no—
305#99—1062no—
306#99—1062no—
307#99—2122no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—2121no—
313#95—2121no—
314C24C22⋊Dic5311yescharacteristic
315C24C22×Dic5311yescharacteristic
316C24C2×C22⋊D5311yescharacteristic
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4D4×D5312yes—
334C2×D4#1171994712yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—2122no—
358C22×C4C22⋊Dic5311yescharacteristic
359C22×C4C22⋊Dic5311yescharacteristic
360C22×C4C22⋊Dic5311yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362M16—1064no—
363M16—2124no—
364M16—2122no—
365C2×C8—2124no—
366C2×C8—2122no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—2122no—
380C22⋊C4—2122no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#1313313212yes—
394C23C2×C22⋊Dic5311yescharacteristic
395C23D4×Dic5312yes—
396C23#1313318912yes—
397Q8—42no—
398Q8—42no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—4242no—
437C2×C4—4242no—
438C2×C4—4242no—
439C2×C4#1313313211yescharacteristic
440C2×C4#1313313211yescharacteristic
441C2×C4#1313318511yescharacteristic
442C2×C4#1313319811yescharacteristic
443C8—2124no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22#1681050311yescharacteristic
462C22C2×#1313313211yescharacteristic
463C22C2×#1313313211yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—8482no—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2#2027516911yescharacteristic
481C2#2027635611yescharacteristic
482C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2406C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#2406D53
hypocenterC53#13081
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. C2×#16810413C2semidirect
4. #20272833C2semidirect
15. C2×#123×C53C4semidirect
17. C22×C4⋊C4×C53C4semidirect
20. C53×#386C4semidirect
24. C53×#381C4semidirect
249. C2×#123Dic53semidirect
251. C22×C4⋊C4Dic53semidirect
259. #386Dic53semidirect
264. #381Dic53semidirect
266. C53#13081semidirect

Maximal subgroups

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

subgroupindex
C2×#168106072
C2×#168104132
#202746332
#202728332
C53×#24062
#1308153