Finite Group Databasebeta

← #27276765

Subgroups — #27276765

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic, complemented
2#20274695C211yesmaximal, characteristic, complemented
3C2×#16810347C212yesmaximal
4#20272558C212yesmaximal, complemented
5C2×C53×#388C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×C22⋊#88322C2212yes—
10#16810378—21no—
11#16810378—21no—
12#16810347—24no—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×C22⋊C4×C53C411yescharacteristic
15C22×C22⋊C4×C53C411yescharacteristic
16C22×C22⋊C4×C53C2211yescharacteristic
17C22×C42×C53C2211yescharacteristic
18C53×#388—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#388—21no—
22C2×#95×C53C411yescharacteristic
23C2×#95×C53C2211yescharacteristic
24C22×#88322—22no—
25C22×#88322—22no—
26C4⋊#88322—42no—
27C4⋊#88322—42no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C25×C53C2×C411yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—42no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—42no—
51C2×C22⋊C4×C53—42no—
52C2×C22⋊C4×C53D412yes—
53C2×C22⋊C4×C53Q812yes—
54C2×C42×C53—42no—
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53—21no—
64C24×C53—21no—
65C24×C53—22no—
66C24×C53—22no—
67C24×C53—41no—
68C24×C53—41no—
69C24×C53—41no—
70C24×C53—41no—
71C24×C53C4⋊C412yes—
72C24×C53C22×C411yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C4⋊C412yes—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×Q811yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C4⋊C4×C53—42no—
105C4⋊C4×C53—42no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—42no—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53—41no—
117C42×C53—41no—
118C2×#88322—42no—
119C2×#88322—42no—
120C2×#88322—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C2×#88322—42no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—22no—
129C23×C53—24no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53—82no—
142C23×C53—82no—
143C23×C53—82no—
144C23×C53—82no—
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53C2×C4⋊C411yescharacteristic
147C23×C53#11711yescharacteristic
148C23×C53#13112yes—
149C23×C53#13112yes—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—22no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—44no—
170C2×C4×C53—44no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—82no—
182C2×C4×C53—82no—
183C2×C4×C53#9912yes—
184C2×C4×C53#10012yes—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53C22⋊Q811yescharacteristic
188C2×C4×C53C22⋊Q811yescharacteristic
189#88322—82no—
190#88322—82no—
191#8561—531nomaximal, Sylow
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—42no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53—84no—
204C22×C53—84no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#38011yescharacteristic
210C22×C53#40811yescharacteristic
211C22×C53#40811yescharacteristic
212C22×C53C2×#9911yescharacteristic
213C22×C53#41011yescharacteristic
214C22×C53C2×#10011yescharacteristic
215C22×C53#41311yescharacteristic
216C22×C53#42412yes—
217C22×C53#43312yes—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—42no—
225C4×C53—41no—
226C4×C53—41no—
227C4×C53—81no—
228C4×C53—81no—
229C2×#388D5311yescharacteristic
230#1797—531no—
231#1797—531no—
232C2×#323—532no—
233#1280—532no—
234C2×C53—22no—
235C2×C53—22no—
236C2×C53—82no—
237C2×C53#146811yescharacteristic
238C2×C53#146911yescharacteristic
239C2×C53#147111yescharacteristic
240C2×C53#147311yescharacteristic
241C2×C53#179811yescharacteristic
242C2×C53#179911yescharacteristic
243C2×C53#180011yescharacteristic
244C22×C4⋊C4Dic5311yescharacteristic
245C22×C22⋊C4Dic5311yescharacteristic
246C22×C22⋊C4Dic5311yescharacteristic
247C22×C22⋊C4C2×D5311yescharacteristic
248C22×C42C2×D5311yescharacteristic
249C2×C4⋊C8—1061no—
250C2×C4⋊C8—1061no—
251C2×C22⋊C8—532no—
252C2×C22⋊C8—532no—
253#388—21no—
254#388—21no—
255#388—21no—
256#388—21no—
257C2×#95Dic5311yescharacteristic
258C2×#95C2×D5311yescharacteristic
259#336—1061no—
260#336—1061no—
261#323—1064no—
262C53#856111yescharacteristic, complemented, Sylow
263C25C2×Dic5311yescharacteristic
264C23×C4C2×Dic5311yescharacteristic
265C23×C4C2×Dic5311yescharacteristic
266C23×C4C2×Dic5311yescharacteristic
267C23×C4C2×Dic5311yescharacteristic
268C23×C4C2×Dic5311yescharacteristic
269C23×C4C22×D5311yescharacteristic
270C22×C8—1062no—
271C22×C8—1062no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—42no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4Dic10612yes—
286C2×C22⋊C4D21212yes—
287C2×C42—42no—
288C2×C42C22⋊D5311yescharacteristic
289C2×C42C22⋊D5311yescharacteristic
290C2×C42C22⋊D5311yescharacteristic
291C2×C42C22⋊D5311yescharacteristic
292C4⋊C8—2122no—
293C4⋊C8—2122no—
294C22⋊C8—1062no—
295C22⋊C8—1062no—
296C22⋊C8—1062no—
297C22⋊C8—1062no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24C4⋊Dic5312yes—
311C24C22×Dic5311yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4C4⋊Dic5312yes—
333C22×C4C22⋊Dic5311yescharacteristic
334C22×C4C22⋊Dic5311yescharacteristic
335C22×C4C22⋊Dic5311yescharacteristic
336C22×C4C22⋊Dic5311yescharacteristic
337C22×C4C2×Dic10611yescharacteristic
338C22×C4C2×D21211yescharacteristic
339C22×C4#1171994511yescharacteristic
340C22×C4#1171994511yescharacteristic
341C22×C4C2×C22⋊D5311yescharacteristic
342C22×C4C2×C22⋊D5311yescharacteristic
343C2×C8—2122no—
344C2×C8—2122no—
345C2×C8—2122no—
346C2×C8—2122no—
347C2×C8—2122no—
348C2×C8—2122no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—41no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23C2×C22⋊Dic5311yescharacteristic
385C23C2×C4⋊Dic5311yescharacteristic
386C23#1313317611yescharacteristic
387C23#1313322412yes—
388C23#1313322412yes—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#1313313212yes—
423C2×C4#1313313412yes—
424C2×C4#1313319111yescharacteristic
425C2×C4#1313319111yescharacteristic
426C2×C4#1313319611yescharacteristic
427C2×C4#1313319611yescharacteristic
428C8—4242no—
429C8—4242no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—84no—
442C22—84no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#1681048311yescharacteristic
448C22#1681059011yescharacteristic
449C22#1681059011yescharacteristic
450C22C2×#1313313211yescharacteristic
451C22#1681059711yescharacteristic
452C22C2×#1313313411yescharacteristic
453C22#1681060711yescharacteristic
454C22#1681065012yes—
455C22#1681068412yes—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—42no—
463C4—41no—
464C4—41no—
465C4—81no—
466C4—81no—
467C2—22no—
468C2—22no—
469C2—82no—
470C2#2027306311yescharacteristic
471C2#2027306711yescharacteristic
472C2#2027307511yescharacteristic
473C2#2027308411yescharacteristic
474C2#2027469911yescharacteristic
475C2#2027470311yescharacteristic
476C2#2027470711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#388C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#388D53
hypocenterC53#8561
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. #20274695C2semidirect
2. #20274695C2semidirect
4. #20272558C2semidirect
262. C53#8561semidirect

Maximal subgroups

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

subgroupindex
#202746952
C2×#168103472
#202725582
C2×C53×#3882
#856153