Finite Group Databasebeta

← #27280142

Subgroups — #27280142

Order 13568.

612 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 105 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#20280522C211yesmaximal, characteristic
2#20277312C211yesmaximal, characteristic
3#20273287C211yesmaximal, characteristic, complemented
4C2×#16810346C211yesmaximal, characteristic
5#20272583C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1520C211yesmaximal, characteristic, complemented
8C22×#11719921C2211yescharacteristic
9C2×C22⋊#88322C2211yescharacteristic
10#16810558C412yescomplemented
11#16810545—22no—
12#16810534C412yescomplemented
13#16810515—22no—
14#16810513—22no—
15#16810495—21no—
16#16810483C2211yescharacteristic
17#16810483C2211yescharacteristic
18#16810478—21no—
19#16810478—21no—
20C22⋊C4×Dic53—21no—
21#16810360—22no—
22#16810346—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#328—22no—
27C24×D53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—22no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×#11719921—22no—
36C2×#11719921—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921C2×C412yes—
44C2×#11719921D412yes—
45C2×C4×Dic53—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—22no—
50#13133124C2×C412yes—
51#13133124D412yes—
52C22×Q8×C53C2×C411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C22×C8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66#95×C53D412yes—
67C24×C53C2×D411yescharacteristic
68C23×D53—21no—
69C23×D53—21no—
70C23×D53—21no—
71C23×D53—21no—
72C23×D53—21no—
73C23×D53—21no—
74C23×D53—81no—
75C23×D53—81no—
76C23×D53C22⋊C412yes—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—81no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—81no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—22no—
93C22×Dic53—22no—
94C22×Dic53—81no—
95C22×Dic53C22⋊C412yes—
96C2×C8×C53—41no—
97C2×C8×C53—81no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—81no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊Dic53—41no—
108C22⋊Dic53—41no—
109C22⋊Dic53—41no—
110C22⋊Dic53—41no—
111#11719921—22no—
112#11719921—22no—
113#11719921—22no—
114#11719921—22no—
115#11719921—82no—
116#11719921—82no—
117#11719921—82no—
118#11719921—82no—
119C4⋊Dic53—81no—
120C4⋊Dic53—81no—
121#11719919—82no—
122C4×Dic53—82no—
123C2×#88322—41no—
124C2×#88322—81no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12011yescharacteristic
129C23×C53#12311yescharacteristic
130C23×C53#13511yescharacteristic
131C23×C53#13511yescharacteristic
132C23×C53#13711yescharacteristic
133C23×C53#13711yescharacteristic
134C22×D53—22no—
135C22×D53—22no—
136C22×D53—22no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—82no—
148C22×D53—82no—
149C22×D53—82no—
150C22×D53—82no—
151Q8×C53—81no—
152Q8×C53—81no—
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—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×Dic53—22no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—81no—
180C8×C53—81no—
181#88322—81no—
182#9201—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53#38611yescharacteristic
194C22×C53#41611yescharacteristic
195C22×C53#41911yescharacteristic
196C22×C53#45211yescharacteristic
197C22×C53#45211yescharacteristic
198C22×C53#48511yescharacteristic
199C22×C53#48511yescharacteristic
200C2×D53—44no—
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—42no—
204C2×D53—82no—
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—82no—
208C2×D53—82no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—82no—
214C4×C53—81no—
215C4×C53—81no—
216Dic53—42no—
217Dic53—82no—
218Dic53—82no—
219#2626—531no—
220#2312—531no—
221#1520—531no—
222#1520D5311yescharacteristic, complemented
223C2×#323—531no—
224#1289—531no—
225#1289—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53#187311yescharacteristic
233C2×C53#191111yescharacteristic
234C2×C53#191311yescharacteristic
235D53—82no—
236C2×C22⋊Q8C2×D5311yescharacteristic
237C22×C22⋊C4—531no—
238C2×C22⋊C8—531no—
239C2×C22⋊C8C2×D5311yescharacteristic
240#399—532no—
241#396—1062no—
242#393—532no—
243#388—1062no—
244#388—1062no—
245#385—1061no—
246#380—531no—
247#380—531no—
248#380C2×D5311yescharacteristic
249#379—1061no—
250#379—1061no—
251C4×C22⋊C4—1061no—
252#328—22no—
253#328—1062no—
254#323—1062no—
255C53#920111yescharacteristic, complemented, Sylow
256C25—531no—
257C22×Q8C4×D5311yescharacteristic
258C23×C4—531no—
259C23×C4C22×D5311yescharacteristic
260C22×C8—21no—
261C22×C8—1061no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4C4×D5311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—532no—
275C2×C22⋊C4—532no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4—1061no—
278C2×C22⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1062no—
282C2×C22⋊C4—1062no—
283C2×C22⋊C4—1062no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4—1061no—
286C2×C42—1062no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1061no—
290C22⋊C8—1061no—
291#95—532no—
292#95—532no—
293#95—1062no—
294#95—1062no—
295#95C22⋊D5312yes—
296C24—532no—
297C24—1061no—
298C24—1061no—
299C24—1061no—
300C24—1061no—
301C24—1061no—
302C24—1061no—
303C24—4241no—
304C24—4241no—
305C24D4×D5311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—81no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—81no—
313C22×C4—532no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—4241no—
319C22×C4C2×C4×D5311yescharacteristic
320C22×C4#1171994511yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4D4×D5311yescharacteristic
323C22×C4#1171994711yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C2×C8—41no—
326C2×C8—81no—
327C2×C8—4241no—
328C2×C8—2121no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—4242no—
335C4⋊C4—4241no—
336C4⋊C4—4241no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—1062no—
342C22⋊C4—1062no—
343C22⋊C4—1062no—
344C22⋊C4—1062no—
345C22⋊C4—4242no—
346C22⋊C4—4242no—
347C22⋊C4—4242no—
348C22⋊C4—4242no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C22⋊C4—2121no—
352C22⋊C4—2121no—
353C42—4242no—
354C23—41no—
355C23—41no—
356C23—1062no—
357C23—1062no—
358C23—1062no—
359C23—4241no—
360C23—4241no—
361C23—4242no—
362C23—4242no—
363C23—4242no—
364C23—4242no—
365C23—2122no—
366C23—2122no—
367C23—2122no—
368C23—2122no—
369C23—2121no—
370C23—2121no—
371C23—2121no—
372C23—2121no—
373C23C22⋊C4×D5311yescharacteristic
374C23#1313318511yescharacteristic
375C23#1313319911yescharacteristic
376C23#1313323811yescharacteristic
377C23#1313324011yescharacteristic
378C23#1313324611yescharacteristic
379C23#1313324911yescharacteristic
380Q8—81no—
381Q8—81no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—1062no—
396C2×C4—4241no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4#1313317711yescharacteristic
406C2×C4C4×C22⋊D5311yescharacteristic
407C2×C4#1313318811yescharacteristic
408C2×C4#1313319011yescharacteristic
409C8—81no—
410C8—4241no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—4242no—
422C22—4242no—
423C22—4242no—
424C22—4242no—
425C22—4242no—
426C22—2124no—
427C22—2122no—
428C22—2122no—
429C22—2122no—
430C22#1681049911yescharacteristic
431C22#1681061811yescharacteristic
432C22#1681063011yescharacteristic
433C22#1681078211yescharacteristic
434C22#1681078411yescharacteristic
435C22#1681096211yescharacteristic
436C22#1681096311yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—4242no—
445C4—4242no—
446C4—2122no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—4242no—
454C2#2027502411yescharacteristic
455C2#2027524411yescharacteristic
456C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1520C2
socleC22×C53#386
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1520D53
hypocenterC53#9201
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
3. #20273287C2semidirect
5. #20272583C2semidirect
6. #20272582C2semidirect
7. C53×#1520C2semidirect
10. #16810558C4semidirect
12. #16810534C4semidirect
222. #1520D53semidirect
255. C53#9201semidirect

Maximal subgroups

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

subgroupindex
#202805222
#202773122
#202732872
C2×#168103462
#202725832
#202725822
C53×#15202
#920153