Finite Group Databasebeta

← #27281134

Subgroups — #27281134

Order 13568.

520 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#20285517C211yesmaximal, characteristic
2#20275550C211yesmaximal, characteristic
3#20273243C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic, complemented
5#20272621C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#2378C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810956C412yescomplemented
10#16810936—21no—
11#16810871—22no—
12#16810773C412yescomplemented
13#16810762—21no—
14C2×#13133131—21no—
15C2×#13133131—21no—
16#16810587C2211yescharacteristic
17#16810587C2211yescharacteristic
18C4×C22⋊Dic53C2211yescharacteristic
19#16810360—22no—
20#16810356—21no—
21#16810356—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#391—21no—
24C53×#389—21no—
25C53×#388—21no—
26C53×#388—21no—
27C53×#387—21no—
28C53×#387—21no—
29C53×#381C2211yescharacteristic
30C53×#380C2211yescharacteristic
31C22×D212C2×C411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719924—22no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C22⋊Dic53—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44#13133150—22no—
45#13133141—22no—
46#13133137—22no—
47#13133131—42no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322C2×C412yes—
53#13133124—21no—
54C22×Q8×C53D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53C2×C412yes—
59C22⋊Q8×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C42×C53—21no—
70C2×C42×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53D411yescharacteristic
78#95×C53D411yescharacteristic
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—81no—
83C2×Q8×C53C22⋊C412yes—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—81no—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×C22⋊D53—41no—
98C2×C22⋊D53—41no—
99C2×C22⋊D53—41no—
100C2×C22⋊D53—41no—
101C2×C22⋊D53—41no—
102C2×C22⋊D53—41no—
103C22×Dic53—21no—
104C22×Dic53—41no—
105C2×D212—21no—
106C2×D212—21no—
107C2×D212—81no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—81no—
112C4⋊C4×C53—81no—
113C4⋊C4×C53—81no—
114C4⋊C4×C53C22⋊C412yes—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—81no—
120C42×C53—81no—
121C42×C53—81no—
122C22⋊Dic53—22no—
123C22⋊Dic53—41no—
124#11719924—82no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4×Dic53—81no—
130#11719917—81no—
131#11719917—81no—
132C2×#88322—22no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53C4×D411yescharacteristic
139C23×C53#12011yescharacteristic
140C23×C53#12111yescharacteristic
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—81no—
145C22×D53—81no—
146Q8×C53—42no—
147Q8×C53—81no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
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—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53C2×C22⋊C411yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12311yescharacteristic
176C22⋊D53—81no—
177C22⋊D53—81no—
178C22⋊D53—81no—
179C22⋊D53—81no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—42no—
183C2×Dic53—41no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186D212—42no—
187D212—81no—
188#88322—42no—
189#88322—81no—
190#9344—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53#38611yescharacteristic
202C22×C53#45311yescharacteristic
203C22×C53#45411yescharacteristic
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—161no—
208C2×D53—161no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—81no—
219Dic53—82no—
220Dic53—81no—
221#3193—531no—
222#2378D5311yescharacteristic, complemented
223#1964—531no—
224#1510—531no—
225#1508—531no—
226#1305—531no—
227#1284—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—41no—
231C2×C53—41no—
232C2×C53#188311yescharacteristic
233C2×C53#190511yescharacteristic
234C2×C53#191311yescharacteristic
235D53—161no—
236C2×C22⋊Q8C2×D5311yescharacteristic
237C2×#121—531no—
238#484—532no—
239#480—1061no—
240#468—1062no—
241#450—532no—
242#448—1061no—
243C2×#99—1061no—
244C2×#99—1061no—
245#407—531no—
246#407—531no—
247#391—21no—
248#389—21no—
249#388—21no—
250#388—21no—
251#387—21no—
252#387—21no—
253#381C2×D5311yescharacteristic
254#380C2×D5311yescharacteristic
255C4×C22⋊C4—531no—
256#328—1062no—
257#327—1061no—
258#327—1061no—
259C53#934411yescharacteristic, complemented, Sylow
260C22×Q8C22⋊D5311yescharacteristic
261C22×D4—531no—
262C22×D4—1061no—
263C23×C4C22×D5311yescharacteristic
264C2×SD16—1062no—
265C2×M16—1061no—
266C2×M16—1061no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8C4×D5312yes—
270C22⋊Q8D21212yes—
271#121—1061no—
272#121—1061no—
273#121—1061no—
274#121—1061no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4C22⋊D5311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—1061no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4C22⋊D5311yescharacteristic
286C2×C22⋊C4C22⋊D5311yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—1061no—
290#107—1062no—
291#103—1062no—
292#102—1062no—
293#99—2122no—
294#99—2122no—
295C22⋊C8—532no—
296C22⋊C8—1061no—
297C22⋊C8—1061no—
298C22⋊C8—1061no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—1061no—
306#95C22⋊D5311yescharacteristic
307#95C22⋊D5311yescharacteristic
308C24—1061no—
309C24C2×C22⋊D5311yescharacteristic
310C2×Q8—21no—
311C2×Q8—81no—
312C2×Q8#1171992112yes—
313C2×D4—1061no—
314C2×D4—1061no—
315C2×D4—4241no—
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—21no—
327C22×C4—21no—
328C22×C4—81no—
329C22×C4—1061no—
330C22×C4—2121no—
331C22×C4C2×C4×D5311yescharacteristic
332C22×C4C2×D21211yescharacteristic
333C22×C4#1171994511yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—4242no—
338M16—4241no—
339M16—4241no—
340C2×C8—1062no—
341C2×C8—2121no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4#1171992112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—81no—
356C22⋊C4—1062no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C42—81no—
361C42—81no—
362C42—4241no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—4241no—
368C23—4241no—
369C23—2121no—
370C23—2121no—
371C23—2121no—
372C23C4×C22⋊D5311yescharacteristic
373C23#1313318611yescharacteristic
374C23#1313319111yescharacteristic
375Q8—42no—
376Q8—81no—
377D4—4241no—
378D4—4241no—
379D4—4241no—
380D4—4241no—
381D4—4241no—
382D4—2122no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
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—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—2122no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C2×#1171992111yescharacteristic
414C2×C4C4×C22⋊D5311yescharacteristic
415C2×C4#1313319111yescharacteristic
416C2×C4#1313320011yescharacteristic
417C8—4241no—
418C8—2122no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—81no—
428C22—81no—
429C22—8481no—
430C22—8481no—
431C22—4241no—
432C22—4241no—
433C22—4241no—
434C22#1681050211yescharacteristic
435C22#1681079111yescharacteristic
436C22#1681079611yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—4242no—
448C4—4241no—
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—8481no—
454C2#2027509111yescharacteristic
455C2#2027520911yescharacteristic
456C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2378C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2378D53
hypocenterC53#9344
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2378

Complements

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

NH ≅ G/Ntype
4. #20273235C2semidirect
5. #20272621C2semidirect
6. #20272567C2semidirect
7. C53×#2378C2semidirect
9. #16810956C4semidirect
12. #16810773C4semidirect
222. #2378D53semidirect
259. C53#9344semidirect

Maximal subgroups

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

subgroupindex
#202855172
#202755502
#202732432
#202732352
#202726212
#202725672
C53×#23782
#934453