Finite Group Databasebeta

← #27281136

Subgroups — #27281136

Order 13568.

520 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277881C211yesmaximal, characteristic, complemented
2#20275552C211yesmaximal, characteristic
3#20273244C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic, complemented
5#20272622C211yesmaximal, characteristic, complemented
6#20272569C211yesmaximal, characteristic, complemented
7C53×#3193C211yesmaximal, characteristic
8C2×#13133132—21no—
9C2×#13133132—21no—
10#16810587C2211yescharacteristic
11#16810530—21no—
12#16810522—21no—
13#16810516—21no—
14#16810516—21no—
15#16810509—21no—
16#16810509—21no—
17#16810486C2211yescharacteristic
18#16810483C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810360—22no—
21#16810357—21no—
22#16810357—21no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C53×#484C412yescomplemented
26C53×#480—21no—
27C53×#468—22no—
28C53×#450C412yescomplemented
29C53×#448—21no—
30C53×#407C2211yescharacteristic
31C2×#11719917—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53D411yescharacteristic
38C2×C22⋊Dic53D411yescharacteristic
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42#13133132—42no—
43#13133132—42no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124D411yescharacteristic
54#13133124D411yescharacteristic
55C22×Q8×C53D411yescharacteristic
56C22×D4×C53—21no—
57C22×D4×C53C2×C411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C2×SD16×C53—22no—
60C2×M16×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53C2×C412yes—
64C22⋊Q8×C53D412yes—
65#121×C53—21no—
66#121×C53—21no—
67#121×C53—21no—
68#121×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74#107×C53—22no—
75#103×C53—22no—
76#102×C53—22no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53C2×C412yes—
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×Q8×C53—21no—
82C2×Q8×C53—81no—
83C2×Q8×C53C22⋊C412yes—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—81no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—81no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—41no—
108SD16×C53—82no—
109M16×C53—81no—
110C2×C8×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53C22⋊C412yes—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊Dic53—22no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—81no—
126C4⋊Dic53—81no—
127C4⋊Dic53—81no—
128C4⋊Dic53—81no—
129C4×Dic53—81no—
130C4×Dic53—81no—
131C4×Dic53—81no—
132#11719917—81no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53C4×D411yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53#12111yescharacteristic
146Q8×C53—42no—
147Q8×C53—81no—
148D4×C53—42no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152D4×C53—81no—
153D4×C53—81no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
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×C53C2×C22⋊C411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12311yescharacteristic
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—42no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C8×C53—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—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—161no—
205C22×C53—161no—
206C22×C53#38611yescharacteristic
207C22×C53#45311yescharacteristic
208C22×C53#45411yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53—81no—
216Dic53—81no—
217Dic53—81no—
218Dic53—81no—
219Dic53—81no—
220Dic53—82no—
221#3193D5311yescharacteristic
222#2378—531no—
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—161no—
233C2×C53#188311yescharacteristic
234C2×C53#190511yescharacteristic
235C2×C53#191311yescharacteristic
236C2×C22⋊Q8C2×D5311yescharacteristic
237C2×#121C2×D5311yescharacteristic
238#484Dic5312yescomplemented
239#480—21no—
240#468—22no—
241#450Dic5312yescomplemented
242#448—21no—
243C2×#99—1061no—
244C2×#99—1061no—
245#407—531no—
246#407C2×D5311yescharacteristic
247#391—1061no—
248#389—1061no—
249#388—1061no—
250#388—1061no—
251#387—1061no—
252#387—1061no—
253#381—531no—
254#380—531no—
255C4×C22⋊C4—531no—
256#328—1062no—
257#327—1061no—
258#327—1061no—
259C53#934411yescharacteristic, complemented, Sylow
260C22×Q8C22⋊D5311yescharacteristic
261C22×D4—21no—
262C22×D4C2×Dic5311yescharacteristic
263C23×C4C22×D5311yescharacteristic
264C2×SD16—22no—
265C2×M16—21no—
266C2×M16—1061no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8C2×Dic5312yes—
270C22⋊Q8C22⋊D5312yes—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4C2×Dic5311yescharacteristic
280C2×C4⋊C4C22⋊D5311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—1061no—
286C2×C22⋊C4—1061no—
287C2×C42—1061no—
288C2×C42—1061no—
289C2×C42—1061no—
290#107—22no—
291#103—22no—
292#102—22no—
293#99—2122no—
294#99—2122no—
295C22⋊C8—21no—
296C22⋊C8—1061no—
297C22⋊C8—1061no—
298C22⋊C8C2×Dic5312yes—
299#95—531no—
300#95—531no—
301#95—1061no—
302#95—1061no—
303#95—1061no—
304#95—1061no—
305#95—1061no—
306#95—1061no—
307#95—1061no—
308C24—21no—
309C24D4×D5311yescharacteristic
310C2×Q8—21no—
311C2×Q8—81no—
312C2×Q8C22⋊Dic5312yes—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—81no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—81no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—2121no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4#1171994711yescharacteristic
334C22×C4C22×Dic5311yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—82no—
338M16—81no—
339M16—4241no—
340C2×C8—22no—
341C2×C8—2121no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—4241no—
348C4⋊C4—4241no—
349C4⋊C4—4241no—
350C4⋊C4C22⋊Dic5312yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—1062no—
356C22⋊C4—4241no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C42—4241no—
361C42—4241no—
362C42—4241no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23D4×Dic5311yescharacteristic
373C23#1313318511yescharacteristic
374C23#1313318911yescharacteristic
375Q8—42no—
376Q8—81no—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
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—4241no—
399C2×C4—4241no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2122no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C2×C22⋊Dic5311yescharacteristic
414C2×C4D4×Dic5311yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313319811yescharacteristic
417C8—42no—
418C8—4241no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—161no—
433C22—161no—
434C22#1681050311yescharacteristic
435C22#1681078911yescharacteristic
436C22#1681079411yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—4241no—
445C4—4241no—
446C4—4241no—
447C4—4241no—
448C4—4242no—
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—161no—
454C2#2027509311yescharacteristic
455C2#2027521111yescharacteristic
456C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3193C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#3193D53
hypocenterC53#9344
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
1. #20277881C2semidirect
4. #20273235C2semidirect
5. #20272622C2semidirect
6. #20272569C2semidirect
25. C53×#484C4semidirect
28. C53×#450C4semidirect
238. #484Dic53semidirect
241. #450Dic53semidirect
259. C53#9344semidirect

Maximal subgroups

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

subgroupindex
#202778812
#202755522
#202732442
#202732352
#202726222
#202725692
C53×#31932
#934453