Finite Group Databasebeta

← #27280543

Subgroups — #27280543

Order 13568.

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

normal subgroups only

This group has 594 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×#16810934C211yesmaximal, characteristic
2#20277586C211yesmaximal, characteristic, complemented
3#20275552C211yesmaximal, characteristic
4#20274802C211yesmaximal, characteristic, complemented
5#20272582C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1519C211yesmaximal, characteristic
8C2×#13133173—21no—
9C22×C4⋊Dic53C2211yescharacteristic
10#16810934—21no—
11#16810934—22no—
12#16810934—21no—
13C2×#13133150C412yescomplemented
14C2×#13133138C412yescomplemented
15C2×#13133132—21no—
16C2×#13133132—21no—
17C2×C22⋊#88322C2211yescharacteristic
18#16810519—21no—
19#16810506—22no—
20#16810484—21no—
21#16810483C2211yescharacteristic
22C4⋊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—
31#13133173—42no—
32#13133173—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C412yes—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719919—22no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53D411yescharacteristic
44C2×C22⋊Dic53D411yescharacteristic
45C2×C4×Dic53—22no—
46C2×C4×Dic53—21no—
47#13133150—22no—
48#13133150—22no—
49#13133138—22no—
50#13133138—22no—
51#13133132—42no—
52#13133132—42no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55#13133124—22no—
56#13133124—22no—
57#13133124—21no—
58#13133124—21no—
59C22×D4×C53—21no—
60C22×D4×C53C2×C411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C22×C8×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53C2×C411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73#95×C53D412yes—
74C24×C53—21no—
75C24×C53C2×D411yescharacteristic
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—42no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—22no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—22no—
99C22×Dic53—21no—
100C22×Dic53—41no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—41no—
104C22×Dic53—41no—
105C2×C8×C53—42no—
106C2×C8×C53—41no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—42no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊Dic53—21no—
115C22⋊Dic53—21no—
116C22⋊Dic53—21no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53D812yes—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—42no—
125C4⋊Dic53C22⋊C412yes—
126C4⋊Dic53C22⋊C412yes—
127#11719919—44no—
128C4×Dic53—44no—
129C4×Dic53—42no—
130C2×#88322—22no—
131C2×#88322—42no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53C4×D411yescharacteristic
140C23×C53#12011yescharacteristic
141C23×C53#12111yescharacteristic
142C23×C53C2×D811yescharacteristic
143C23×C53#13511yescharacteristic
144C23×C53AGL(1,ℤ/8)11yescharacteristic
145C23×C53AGL(1,ℤ/8)11yescharacteristic
146D4×C53—42no—
147D4×C53—81no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12311yescharacteristic
168C2×Dic53—22no—
169C2×Dic53—22no—
170C2×Dic53—22no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C8×C53—81no—
188#88322—42no—
189#9260—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—161no—
204C22×C53—161no—
205C22×C53#38611yescharacteristic
206C22×C53C4×D811yescharacteristic
207C22×C53#44211yescharacteristic
208C22×C53#44711yescharacteristic
209C22×C53#44911yescharacteristic
210C22×C53#45911yescharacteristic
211C22×C53#46311yescharacteristic
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—82no—
216C4×C53—81no—
217C4×C53#41012yes—
218Dic53—42no—
219Dic53—42no—
220Dic53—42no—
221Dic53—81no—
222Dic53—161no—
223C2×#480—531no—
224#2344—531no—
225#1964—531no—
226#1823—531no—
227#1519D5311yescharacteristic
228#1289—531no—
229#1282—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—161no—
237C2×C53#187911yescharacteristic
238C2×C53#189911yescharacteristic
239C2×C53#190511yescharacteristic
240C2×#121C2×D5311yescharacteristic
241C2×#117—1061no—
242C22×C4⋊C4—531no—
243#480—1062no—
244#480—1061no—
245#480—1061no—
246C2×#107—532no—
247C2×#102—532no—
248C2×#99—1061no—
249C2×#99—1061no—
250C2×C22⋊C8—531no—
251C2×C22⋊C8C2×D5311yescharacteristic
252#389—1061no—
253#387—1062no—
254#381—1061no—
255#380—531no—
256#380C2×D5311yescharacteristic
257C4×C4⋊C4—1061no—
258C4×C22⋊C4—531no—
259#340—1062no—
260#327—22no—
261C53#926011yescharacteristic, complemented, Sylow
262C22×D4—21no—
263C22×D4C4×D5311yescharacteristic
264C23×C4—1061no—
265C23×C4C22×D5311yescharacteristic
266C22×C8—21no—
267C22×C8—532no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272#117—2122no—
273#117—2122no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—532no—
276C2×C4⋊C4—532no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1062no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4C4×D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—1061no—
290C2×C42—1062no—
291C2×C42—1061no—
292#107—1062no—
293#107—1062no—
294#102—1062no—
295#102—1062no—
296#99—2122no—
297#99—2122no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—1061no—
301C22⋊C8—1061no—
302#95—1062no—
303#95—1062no—
304#95—1061no—
305#95—1061no—
306#95C22⋊D5312yes—
307C24—21no—
308C24D4×D5311yescharacteristic
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—1062no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1062no—
326C22×C4—1061no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4#1171994511yescharacteristic
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—1062no—
341C2×C8—2122no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4—532no—
346C4⋊C4—532no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—2122no—
352C4⋊C4—2124no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—532no—
358C22⋊C4—1061no—
359C22⋊C4—1061no—
360C22⋊C4—1061no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42—2122no—
364C42—2124no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23#1313317711yescharacteristic
373C23#1313318511yescharacteristic
374C23#1313319011yescharacteristic
375C23D8×D5311yescharacteristic
376C23#1313323811yescharacteristic
377C23#1313324111yescharacteristic
378C23#1313324411yescharacteristic
379D4—42no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—1062no—
398C2×C4—1062no—
399C2×C4—1062no—
400C2×C4—4241no—
401C2×C4—4241no—
402C2×C4—4241no—
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—2122no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2122no—
416C2×C4C22⋊C4×D5311yescharacteristic
417C2×C4C4×C22⋊D5311yescharacteristic
418C2×C4#1313318811yescharacteristic
419C2×C4#1313319911yescharacteristic
420C8—81no—
421C8—2122no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—161no—
436C22—161no—
437C22#1681049911yescharacteristic
438C22#1681070311yescharacteristic
439C22#1681073211yescharacteristic
440C22#1681075511yescharacteristic
441C22#1681076711yescharacteristic
442C22#1681081711yescharacteristic
443C22#1681083711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—82no—
448C4—81no—
449C4—8481no—
450C4—4241no—
451C4—2122no—
452C4—2122no—
453C4—2122no—
454C4#1681059512yes—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—161no—
462C2#2027506011yescharacteristic
463C2#2027517011yescharacteristic
464C2#2027520611yescharacteristic
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#9260
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. #20277586C2semidirect
4. #20274802C2semidirect
5. #20272582C2semidirect
6. #20272561C2semidirect
13. C2×#13133150C4semidirect
14. C2×#13133138C4semidirect
261. C53#9260semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
#202775862
#202755522
#202748022
#202725822
#202725612
C53×#15192
#926053