Finite Group Databasebeta

← #27280550

Subgroups — #27280550

Order 13568.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16810939C211yesmaximal, characteristic
2#20277594C211yesmaximal, characteristic, complemented
3#20275552C211yesmaximal, characteristic
4#20274806C211yesmaximal, characteristic, complemented
5#20272582C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1519C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810939—21no—
10#16810939—22no—
11#16810939—21no—
12C2×#13133148C412yescomplemented
13C2×#13133138C412yescomplemented
14C2×#13133132—21no—
15C2×#13133132—21no—
16C2×C22⋊#88322C2211yescharacteristic
17#16810519—21no—
18#16810508—22no—
19#16810483C2211yescharacteristic
20#16810481—21no—
21C4⋊C4×Dic53—21no—
22C22⋊C4×Dic53—21no—
23C4×C22⋊Dic53C2211yescharacteristic
24#16810386—22no—
25C2×#121×C53C2211yescharacteristic
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C53×#327—22no—
29C23×Dic53—21no—
30C22×#88322C2×C412yes—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C412yes—
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—22no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53D411yescharacteristic
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
45#13133148—22no—
46#13133148—22no—
47#13133138—22no—
48#13133138—22no—
49#13133132—42no—
50#13133132—42no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53#13133124—22no—
54#13133124—22no—
55#13133124—21no—
56#13133124—21no—
57C22×D4×C53—21no—
58C22×D4×C53C2×C411yescharacteristic
59C23×C4×C53C2311yescharacteristic
60C22×C8×C53—21no—
61#121×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71#95×C53D412yes—
72C24×C53—21no—
73C24×C53C2×D411yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—42no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—42no—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C22×Dic53—82no—
100C2×C8×C53—42no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—42no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊Dic53—21no—
110C22⋊Dic53—21no—
111C22⋊Dic53—21no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53SD1612yes—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—42no—
120C4⋊Dic53C22⋊C412yes—
121C4⋊Dic53C22⋊C412yes—
122#11719919—82no—
123C4×Dic53—42no—
124C4×Dic53—82no—
125C2×#88322—22no—
126C2×#88322—42no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—81no—
133C23×C53—81no—
134C23×C53C4×D411yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53#12111yescharacteristic
137C23×C53C2×SD1611yescharacteristic
138C23×C53#13511yescharacteristic
139C23×C53AGL(1,ℤ/8)11yescharacteristic
140C23×C53#13711yescharacteristic
141D4×C53—42no—
142D4×C53—81no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—81no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12311yescharacteristic
163C2×Dic53—22no—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—82no—
174C2×Dic53—82no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C8×C53—81no—
179#88322—42no—
180#9261—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—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—161no—
195C22×C53—161no—
196C22×C53#38611yescharacteristic
197C22×C53C4×SD1611yescharacteristic
198C22×C53#44011yescharacteristic
199C22×C53#44811yescharacteristic
200C22×C53#44911yescharacteristic
201C22×C53#46011yescharacteristic
202C22×C53#46411yescharacteristic
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—82no—
207C4×C53—81no—
208C4×C53#41012yes—
209Dic53—42no—
210Dic53—42no—
211Dic53—42no—
212Dic53—81no—
213Dic53—161no—
214C2×#481—531no—
215#2345—531no—
216#1964—531no—
217#1824—531no—
218#1519D5311yescharacteristic
219#1289—531no—
220#1282—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—161no—
228C2×C53#188011yescharacteristic
229C2×C53#189911yescharacteristic
230C2×C53#190411yescharacteristic
231C2×#121C2×D5311yescharacteristic
232C22×C4⋊C4—531no—
233#481—1062no—
234#481—1061no—
235#481—1061no—
236C2×#106—532no—
237C2×#102—532no—
238C2×#99—1061no—
239C2×#99—1061no—
240C2×C22⋊C8—531no—
241C2×C22⋊C8C2×D5311yescharacteristic
242#389—1061no—
243#387—1062no—
244#380—531no—
245#380—1061no—
246#380C2×D5311yescharacteristic
247C4×C4⋊C4—1061no—
248C4×C22⋊C4—531no—
249C4×C22⋊C4—1061no—
250#340—1062no—
251#327—22no—
252C53#926111yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4C4×D5311yescharacteristic
255C23×C4—1061no—
256C23×C4C22×D5311yescharacteristic
257C22×C8—21no—
258C22×C8—532no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4C4×D5311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—531no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—1061no—
279C2×C42—1062no—
280C2×C42—1061no—
281#106—1062no—
282#106—1062no—
283#102—1062no—
284#102—1062no—
285#99—2122no—
286#99—2122no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1061no—
290C22⋊C8—1061no—
291#95—1062no—
292#95—1062no—
293#95—1061no—
294#95—1061no—
295#95C22⋊D5312yes—
296C24—21no—
297C24D4×D5311yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—4242no—
317C22×C4—2121no—
318C22×C4C2×C4×D5311yescharacteristic
319C22×C4#1171994511yescharacteristic
320C22×C4D4×D5311yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4#1171994711yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—1062no—
327C2×C8—2122no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—532no—
332C4⋊C4—532no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—4242no—
338C4⋊C4—2122no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—532no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C42—4242no—
350C42—2122no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23—81no—
358C23#1313317711yescharacteristic
359C23#1313318511yescharacteristic
360C23#1313319011yescharacteristic
361C23SD16×D5311yescharacteristic
362C23#1313323611yescharacteristic
363C23#1313324111yescharacteristic
364C23#1313324611yescharacteristic
365D4—42no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—1062no—
384C2×C4—1062no—
385C2×C4—1062no—
386C2×C4—4241no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4—4242no—
390C2×C4—4242no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2122no—
396C2×C4—2121no—
397C2×C4—2122no—
398C2×C4C22⋊C4×D5311yescharacteristic
399C2×C4C4×C22⋊D5311yescharacteristic
400C2×C4#1313318811yescharacteristic
401C2×C4#1313319911yescharacteristic
402C8—81no—
403C8—2122no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—161no—
418C22—161no—
419C22#1681049911yescharacteristic
420C22#1681071411yescharacteristic
421C22#1681072011yescharacteristic
422C22#1681076011yescharacteristic
423C22#1681076511yescharacteristic
424C22#1681082211yescharacteristic
425C22#1681084411yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—81no—
431C4—8481no—
432C4—4241no—
433C4—2122no—
434C4—2122no—
435C4—2122no—
436C4#1681059512yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#2027506711yescharacteristic
445C2#2027517011yescharacteristic
446C2#2027519911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1519C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#1519D53
hypocenterC53#9261
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #20277594C2semidirect
4. #20274806C2semidirect
5. #20272582C2semidirect
6. #20272561C2semidirect
12. C2×#13133148C4semidirect
13. C2×#13133138C4semidirect
252. C53#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
#202775942
#202755522
#202748062
#202725822
#202725612
C53×#15192
#926153