Finite Group Databasebeta

← #27281878

Subgroups — #27281878

Order 13568.

556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 109 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#20283741C211yesmaximal, characteristic
2#20279026C211yesmaximal, characteristic
3#20274805C211yesmaximal, characteristic, complemented
4#20274700C211yesmaximal, characteristic
5#20272583C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1824C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D53C2211yescharacteristic
9#16810694—22no—
10#16810691—22no—
11#16810677—21no—
12C2×#13133131—21no—
13#16810585—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810561C412yescomplemented
16#16810558C412yescomplemented
17#16810519—21no—
18#16810502—21no—
19#16810496—21no—
20#16810483C2211yescharacteristic
21#16810483C2211yescharacteristic
22#16810480—21no—
23#16810389—21no—
24#16810385—22no—
25C22×C4⋊C4×C53C2211yescharacteristic
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C53×#340—22no—
29C22×C22⋊D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C2×#11719912—22no—
32C22×#88322D412yes—
33C4×C22⋊D53—21no—
34C4×C22⋊D53—21no—
35C4×C22⋊D53—21no—
36C4×C22⋊D53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×#11719919—22no—
40C2×#11719919C2×C411yescharacteristic
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53D411yescharacteristic
44C2×#11719921—22no—
45C2×#11719921C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47#13133146—22no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133129—22no—
52#13133127—22no—
53#13133124—22no—
54#13133124—22no—
55#13133124—21no—
56#13133124C2×C412yes—
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C412yes—
67C2×C22⋊C4×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70#95×C53—21no—
71C24×C53C2×D411yescharacteristic
72C23×D53C22⋊C411yescharacteristic
73C22×C4×C53—22no—
74C22×C4×C53—42no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—82no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C2×C22⋊D53—21no—
86C2×C22⋊D53—21no—
87C2×C22⋊D53—21no—
88C2×C22⋊D53—21no—
89C2×C22⋊D53—21no—
90C2×C22⋊D53—21no—
91C22×Dic53—22no—
92C22×Dic53—41no—
93C22×Dic53C22⋊C411yescharacteristic
94C22×Dic53C22⋊C411yescharacteristic
95C22×Dic53C22⋊C411yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—42no—
99C2×C8×C53—22no—
100C2×C8×C53—42no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—42no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊Dic53—21no—
109C22⋊Dic53—21no—
110C22⋊Dic53—41no—
111C22⋊Dic53—41no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114#11719921—21no—
115#11719921—21no—
116#11719921—82no—
117C4⋊Dic53—42no—
118C4⋊Dic53—42no—
119#11719919—21no—
120#11719919—21no—
121#11719919—82no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C2×#88322—22no—
125C2×#88322—42no—
126#11719912—82no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53C2×C22⋊C411yescharacteristic
130C23×C53#12111yescharacteristic
131C23×C53#12111yescharacteristic
132C23×C53AGL(1,ℤ/8)11yescharacteristic
133C23×C53#13711yescharacteristic
134C23×C53#13711yescharacteristic
135C23×C53#13711yescharacteristic
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—41no—
140C22×D53—41no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C22⋊D53—41no—
160C22⋊D53—41no—
161C22⋊D53—41no—
162C22⋊D53—41no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C4×D53—42no—
177C4×D53—81no—
178C8×C53—42no—
179#88322—42no—
180#9473—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53#38611yescharacteristic
192C22×C53#41711yescharacteristic
193C22×C53#41911yescharacteristic
194C22×C53#46811yescharacteristic
195C22×C53#46811yescharacteristic
196C22×C53#47011yescharacteristic
197C22×C53#47011yescharacteristic
198C2×D53—42no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—81no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—82no—
207C4×C53—161no—
208C4×C53#45812yes—
209Dic53—41no—
210Dic53—41no—
211Dic53—41no—
212Dic53—82no—
213Dic53—161no—
214#2973—531no—
215#2487—531no—
216#1824—531no—
217#1824D5311yescharacteristic, complemented
218#1799—531no—
219#1289—531no—
220#1289—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53#189011yescharacteristic
228C2×C53#189411yescharacteristic
229C2×C53#191011yescharacteristic
230D53—81no—
231C2×C4×D4—531no—
232C22×C4⋊C4C2×D5311yescharacteristic
233#435—1062no—
234#435—1062no—
235#432—1061no—
236C2×#99—1061no—
237#407—1061no—
238C2×C22⋊C8—531no—
239C2×C22⋊C8C2×D5311yescharacteristic
240#400—532no—
241#399—532no—
242#389—1061no—
243#386—1061no—
244#385—1061no—
245#380—531no—
246#380—531no—
247#380—1061no—
248#380C2×D5311yescharacteristic
249#342—1061no—
250#340—22no—
251#340—1062no—
252C53#947311yescharacteristic, complemented, Sylow
253C22×D4—531no—
254C23×C4—21no—
255C23×C4—531no—
256C23×C4C22×D5311yescharacteristic
257C2×M16—1062no—
258C22×C8—532no—
259C22×C8C22⋊D5312yes—
260C4×D4—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263C4×D4—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4C4×D5312yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—531no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—1062no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C42—531no—
282C4⋊C8—1062no—
283#99—2122no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—1062no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289#97—1062no—
290#95—21no—
291#95—532no—
292#95—1062no—
293#95—1062no—
294#95—1061no—
295C24—531no—
296C24D4×D5311yescharacteristic
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—531no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1062no—
315C22×C4—2122no—
316C22×C4—2121no—
317C22×C4C2×C4×D5311yescharacteristic
318C22×C4#1171994511yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4D4×D5311yescharacteristic
321C22×C4#1171994711yescharacteristic
322C22×C4C2×C22⋊D5311yescharacteristic
323M16—4242no—
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—1062no—
327C2×C8—2122no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—4242no—
336C4⋊C4—2122no—
337C4⋊C4—2122no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—1061no—
341C22⋊C4—1061no—
342C22⋊C4—1061no—
343C22⋊C4—1061no—
344C22⋊C4—4242no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C42—1061no—
350C42—1061no—
351C23—41no—
352C23—41no—
353C23—1061no—
354C23—1061no—
355C23—1061no—
356C23—2121no—
357C23—2121no—
358C23C22⋊C4×D5311yescharacteristic
359C23#1313318811yescharacteristic
360C23#1313319011yescharacteristic
361C23#1313324411yescharacteristic
362C23#1313324611yescharacteristic
363C23#1313324911yescharacteristic
364C23#1313324911yescharacteristic
365D4—2121no—
366D4—2121no—
367D4—2121no—
368D4—2121no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—4241no—
389C2×C4—4241no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4#1313317711yescharacteristic
399C2×C4C4×C22⋊D5311yescharacteristic
400C2×C4#1313318511yescharacteristic
401C2×C4#1313319911yescharacteristic
402C8—42no—
403C8—2122no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—4241no—
415C22—2122no—
416C22—2121no—
417C22—2121no—
418C22—2121no—
419C22#1681049911yescharacteristic
420C22#1681062311yescharacteristic
421C22#1681063011yescharacteristic
422C22#1681086611yescharacteristic
423C22#1681087211yescharacteristic
424C22#1681087811yescharacteristic
425C22#1681088011yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—161no—
431C4—8481no—
432C4—4242no—
433C4—2121no—
434C4—2121no—
435C4—2121no—
436C4#1681081212yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—4241no—
444C2#2027512511yescharacteristic
445C2#2027514511yescharacteristic
446C2#2027523911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1824C2
socleC22×C53#386
prime radicalC22×C22⋊D53D4
cosocleC23×C4×C53C23
hypercenter#1824D53
hypocenterC53#9473
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #20274805C2semidirect
5. #20272583C2semidirect
6. #20272582C2semidirect
7. C53×#1824C2semidirect
15. #16810561C4semidirect
16. #16810558C4semidirect
217. #1824D53semidirect
252. C53#9473semidirect

Maximal subgroups

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

subgroupindex
#202837412
#202790262
#202748052
#202747002
#202725832
#202725822
C53×#18242
#947353