Finite Group Databasebeta

← #27281276

Subgroups — #27281276

Order 13568.

732 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277556C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic, complemented
3#20274659C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic, complemented
5#20272620C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#3007C211yesmaximal, characteristic
8C2×Q8×Dic53—21no—
9C2×#13133144—22no—
10#16810587C2211yescharacteristic
11#16810526—22no—
12#16810526—21no—
13#16810483—21no—
14#16810483C2211yescharacteristic
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810390—22no—
19#16810360—21no—
20#16810360—21no—
21C22×#41×C53C2211yescharacteristic
22C2×#137×C53C412yescomplemented
23C2×C22⋊Q8×C53C2211yescharacteristic
24C53×#452—22no—
25C53×#451—21no—
26C53×#450—21no—
27C53×#417C412yescomplemented
28C53×#407C2211yescharacteristic
29C2×#11719917—21no—
30Q8×Dic53—44no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53D412yes—
38C2×C4×Dic53—22no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41#13133144—44no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48C2×#41×C53—21no—
49C2×#41×C53—21no—
50C2×#41×C53—44no—
51C2×#41×C53C2×C412yes—
52C2×#41×C53D412yes—
53C22×Q8×C53C2×C411yescharacteristic
54C22×Q8×C53D411yescharacteristic
55C22×D4×C53—21no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59#137×C53—44no—
60C2×Q16×C53—22no—
61C2×SD16×C53—22no—
62C2×M16×C53C2×C412yes—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53C2×C411yescharacteristic
69C2×C22⋊C4×C53—21no—
70#103×C53—22no—
71#102×C53—22no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C24×C53—21no—
75C24×C53C2×D411yescharacteristic
76#41×C53—24no—
77#41×C53—24no—
78#41×C53—42no—
79#41×C53—42no—
80#41×C53—44no—
81#41×C53—44no—
82C2×Q8×C53—24no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—42no—
87C2×Q8×C53C22⋊C412yes—
88C2×D4×C53—22no—
89C2×D4×C53—22no—
90C2×D4×C53—21no—
91C2×D4×C53—42no—
92C2×D4×C53—44no—
93C2×D4×C53C22⋊C412yes—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—42no—
100C22×C4×C53—44no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—21no—
108C22×Dic53—22no—
109C22×Dic53—21no—
110C22×Dic53—41no—
111Q16×C53—44no—
112SD16×C53—44no—
113M16×C53—42no—
114C2×C8×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊Dic53—22no—
122C22⋊Dic53—42no—
123C22⋊Dic53—44no—
124C22⋊Dic53—41no—
125C22⋊Dic53—42no—
126C22⋊Dic53—82no—
127C4⋊Dic53—44no—
128C4⋊Dic53—42no—
129C4×Dic53—44no—
130C4×Dic53—42no—
131C4×Dic53—42no—
132#11719917—42no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53C4×D411yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53#12111yescharacteristic
146Q8×C53—24no—
147Q8×C53—22no—
148Q8×C53—42no—
149Q8×C53—81no—
150Q8×C53#9914yes—
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—44no—
154D4×C53—44no—
155D4×C53—42no—
156C2×C4×C53—24no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—44no—
165C2×C4×C53—44no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—81no—
169C2×C4×C53—81no—
170C2×C4×C53C2×C22⋊C411yescharacteristic
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C2×Dic53—22no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—41no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C8×C53—42no—
187#88322—81no—
188#9364—531nomaximal, Sylow
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—82no—
203C22×C53—82no—
204C22×C53#38611yescharacteristic
205C22×C53#45511yescharacteristic
206C22×C53#45611yescharacteristic
207C4×C53—21no—
208C4×C53—24no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—81no—
212C4×C53—81no—
213C4×C53C2×#9912yes—
214Dic53—42no—
215Dic53—82no—
216Dic53—81no—
217Dic53—82no—
218#3007D5311yescharacteristic
219#2341—531no—
220#1996—531no—
221#1789—531no—
222#1508—531no—
223#1305—531no—
224#1297—531no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—41no—
228C2×C53—41no—
229C2×C53—82no—
230C2×C53#188211yescharacteristic
231C2×C53#189811yescharacteristic
232C2×C53#190611yescharacteristic
233C22×#41C2×D5311yescharacteristic
234C2×#137Dic5312yescomplemented
235C2×C22⋊Q8C2×D5311yescharacteristic
236C2×C4×Q8—1061no—
237#452—22no—
238#451—21no—
239#450—21no—
240C2×#104—1062no—
241#417Dic5312yescomplemented
242#407—531no—
243#407C2×D5311yescharacteristic
244#390—1062no—
245#390—1061no—
246#380—531no—
247#380—1061no—
248C4×C22⋊C4—531no—
249C4×C22⋊C4—531no—
250C4×C22⋊C4—1061no—
251#342—1062no—
252#328—1061no—
253#328—1061no—
254C53#936411yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—21no—
257C2×#41—44no—
258C2×#41C2×Dic5312yes—
259C2×#41C22⋊D5312yes—
260C22×Q8C2×Dic5311yescharacteristic
261C22×Q8C22⋊D5311yescharacteristic
262C22×D4—21no—
263C22×D4C22⋊D5311yescharacteristic
264C23×C4—21no—
265C23×C4C22×D5311yescharacteristic
266#137—44no—
267C2×Q16—22no—
268C2×SD16—22no—
269C2×M16—1061no—
270C2×M16C2×Dic5312yes—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C4×Q8—2124no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—1062no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4C2×Dic5311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—532no—
282C2×C22⋊C4—1061no—
283C2×C22⋊C4—1062no—
284C2×C22⋊C4—1062no—
285C2×C22⋊C4—1061no—
286C2×C42—1062no—
287C2×C42—1061no—
288C2×C42—1061no—
289#104—2124no—
290#103—22no—
291#102—22no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296#95—531no—
297#95—531no—
298#95—1061no—
299#95—1061no—
300C24—21no—
301C24D4×D5311yescharacteristic
302#41—24no—
303#41—24no—
304#41—42no—
305#41—42no—
306#41—44no—
307#41—44no—
308C2×Q8—24no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8C22⋊Dic5312yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4C22⋊Dic5312yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4—1062no—
330C22×C4—2121no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4#1171994711yescharacteristic
334C22×C4C22×Dic5311yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337Q16—44no—
338SD16—44no—
339M16—42no—
340M16—2122no—
341C2×C8—22no—
342C2×C8—2121no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—2124no—
348C4⋊C4—2122no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1062no—
352C22⋊C4—4242no—
353C22⋊C4—2122no—
354C22⋊C4—2124no—
355C22⋊C4—2121no—
356C22⋊C4—2122no—
357C42—2122no—
358C42—2124no—
359C42—2122no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23D4×Dic5311yescharacteristic
370C23#1313318511yescharacteristic
371C23#1313318911yescharacteristic
372Q8—24no—
373Q8—22no—
374Q8—42no—
375Q8—81no—
376Q8#1313313214yes—
377D4—22no—
378D4—22no—
379D4—44no—
380D4—44no—
381D4—42no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—1062no—
397C2×C4—4241no—
398C2×C4—4241no—
399C2×C4—2121no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4—2121no—
403C2×C4—2122no—
404C2×C4—2121no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4C2×C22⋊Dic5311yescharacteristic
409C2×C4D4×Dic5311yescharacteristic
410C2×C4#1313318911yescharacteristic
411C2×C4#1313319811yescharacteristic
412C8—42no—
413C8—4241no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—82no—
428C22—82no—
429C22#1681050311yescharacteristic
430C22#1681079911yescharacteristic
431C22#1681080411yescharacteristic
432C4—21no—
433C4—24no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—4241no—
439C4—4242no—
440C4—4242no—
441C4—2122no—
442C4C2×#1313313212yes—
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—82no—
448C2#2027508611yescharacteristic
449C2#2027516911yescharacteristic
450C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3007C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#3007D53
hypocenterC53#9364
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
2. #20275701C2semidirect
4. #20273235C2semidirect
5. #20272620C2semidirect
6. #20272600C2semidirect
22. C2×#137×C53C4semidirect
27. C53×#417C4semidirect
234. C2×#137Dic53semidirect
241. #417Dic53semidirect
254. C53#9364semidirect

Maximal subgroups

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

subgroupindex
#202775562
#202757012
#202746592
#202732352
#202726202
#202726002
C53×#30072
#936453