Finite Group Databasebeta

← #27280157

Subgroups — #27280157

Order 13568.

622 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 113 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#20280459C211yesmaximal, characteristic
2#20277332C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic
4C2×#16810348C211yesmaximal, characteristic, complemented
5#20272583C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1519C211yesmaximal, characteristic
8C2×#13133198—21no—
9C2×#13133173—21no—
10C22×#11719919C2211yescharacteristic
11C2×C22⋊#88322C2211yescharacteristic
12#16810554C412yescomplemented
13#16810546—22no—
14#16810537C412yescomplemented
15#16810520—22no—
16#16810518—22no—
17#16810503—21no—
18#16810483C2211yescharacteristic
19#16810483C2211yescharacteristic
20#16810481—21no—
21#16810357—22no—
22#16810348—22no—
23C2×#121×C53C2211yescharacteristic
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#327—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×#88322—21no—
30#13133198—42no—
31#13133198—42no—
32#13133173—42no—
33#13133173—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—22no—
43C2×#11719919C2×C412yes—
44C2×#11719919D412yes—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47C2×C22⋊Dic53—22no—
48C2×C22⋊Dic53—22no—
49C2×C4×Dic53—22no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52#13133124—22no—
53#13133124—22no—
54#13133124C2×C412yes—
55#13133124D412yes—
56C22×D4×C53—21no—
57C22×D4×C53C2×C411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70#95×C53D412yes—
71C24×C53—21no—
72C24×C53C2×D411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—81no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—81no—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53—81no—
100C22×Dic53C22⋊C412yes—
101C22×Dic53C22⋊C412yes—
102C22×Dic53D812yes—
103C22×Dic53SD1612yes—
104C2×C8×C53—41no—
105C2×C8×C53—81no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—81no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116C22⋊Dic53—41no—
117C22⋊Dic53—44no—
118C22⋊Dic53—82no—
119C4⋊Dic53—81no—
120C4⋊Dic53—81no—
121#11719919—22no—
122#11719919—22no—
123#11719919—22no—
124#11719919—22no—
125#11719919—44no—
126#11719919—44no—
127#11719919—82no—
128C4×Dic53—44no—
129C2×#88322—41no—
130C2×#88322—81no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53#12011yescharacteristic
140C23×C53#12311yescharacteristic
141C23×C53C2×D811yescharacteristic
142C23×C53C2×SD1611yescharacteristic
143C23×C53AGL(1,ℤ/8)11yescharacteristic
144C23×C53#13711yescharacteristic
145D4×C53—81no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×Dic53—22no—
168C2×Dic53—22no—
169C2×Dic53—22no—
170C2×Dic53—22no—
171C2×Dic53—22no—
172C2×Dic53—22no—
173C2×Dic53—22no—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182C2×Dic53—81no—
183C2×Dic53#10214yes—
184C8×C53—81no—
185#88322—81no—
186#9204—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—82no—
198C22×C53—82no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53#38611yescharacteristic
203C22×C53C2×#10211yescharacteristic
204C22×C53#41711yescharacteristic
205C22×C53#44711yescharacteristic
206C22×C53#44811yescharacteristic
207C22×C53#48011yescharacteristic
208C22×C53#48111yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—82no—
214C4×C53—81no—
215Dic53—24no—
216Dic53—42no—
217Dic53—82no—
218Dic53—82no—
219#2618—531no—
220#2316—531no—
221#1519—531no—
222#1519D5311yescharacteristic
223C2×#324—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—161no—
233C2×C53#187411yescharacteristic
234C2×C53#191211yescharacteristic
235C2×C53#191311yescharacteristic
236C2×#123—1061no—
237C2×#121C2×D5311yescharacteristic
238C2×#117—1061no—
239C22×C4⋊C4—531no—
240C2×C22⋊C8—531no—
241C2×C22⋊C8C2×D5311yescharacteristic
242#398—532no—
243#396—1062no—
244#394—532no—
245#389—1062no—
246#389—1062no—
247#386—1061no—
248#380—531no—
249#380—531no—
250#380—1061no—
251#380C2×D5311yescharacteristic
252#327—22no—
253#327—1062no—
254#324—1062no—
255C53#920411yescharacteristic, complemented, Sylow
256C22×D4—21no—
257C22×D4C4×D5311yescharacteristic
258C23×C4—531no—
259C23×C4—531no—
260C23×C4C22×D5311yescharacteristic
261C22×C8—21no—
262C22×C8—1061no—
263#123—2122no—
264#123—2122no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#117—2122no—
270#117—2122no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—532no—
273C2×C4⋊C4—532no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1062no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1062no—
282C2×C4⋊C4—1062no—
283C2×C4⋊C4C4×D5311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1062no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4—1061no—
290C2×C42—1062no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295#95—532no—
296#95—532no—
297#95—1062no—
298#95—1062no—
299#95C22⋊D5312yes—
300C24—21no—
301C24D4×D5311yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—81no—
314C22×C4—532no—
315C22×C4—532no—
316C22×C4—532no—
317C22×C4—532no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1062no—
325C22×C4—1062no—
326C22×C4—4241no—
327C22×C4C2×C4×D5311yescharacteristic
328C22×C4#1171994511yescharacteristic
329C22×C4D4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4#1171994711yescharacteristic
332C22×C4C2×C22⋊D5311yescharacteristic
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—4241no—
336C2×C8—2121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—81no—
340C4⋊C4—1062no—
341C4⋊C4—1062no—
342C4⋊C4—1062no—
343C4⋊C4—1062no—
344C4⋊C4—4242no—
345C4⋊C4—4241no—
346C4⋊C4—4241no—
347C4⋊C4—2124no—
348C4⋊C4—2124no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—4242no—
354C22⋊C4—2124no—
355C22⋊C4—2121no—
356C22⋊C4—2121no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C42—2124no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C22⋊C4×D5311yescharacteristic
368C23#1313318511yescharacteristic
369C23#1313319911yescharacteristic
370C23D8×D5311yescharacteristic
371C23SD16×D5311yescharacteristic
372C23#1313324111yescharacteristic
373C23#1313324611yescharacteristic
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—534no—
393C2×C4—1062no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—1062no—
397C2×C4—1062no—
398C2×C4—1062no—
399C2×C4—1062no—
400C2×C4—4241no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4—2122no—
406C2×C4—2122no—
407C2×C4—2122no—
408C2×C4—2122no—
409C2×C4#1313317711yescharacteristic
410C2×C4C4×C22⋊D5311yescharacteristic
411C2×C4#1313318811yescharacteristic
412C2×C4#1313319011yescharacteristic
413C8—81no—
414C8—4241no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#1681049911yescharacteristic
431C22#102×D5311yescharacteristic
432C22#1681062111yescharacteristic
433C22#1681075511yescharacteristic
434C22#1681076011yescharacteristic
435C22#1681093511yescharacteristic
436C22#1681094011yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—1064no—
444C4—4242no—
445C4—4242no—
446C4—2122no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—161no—
454C2#2027502911yescharacteristic
455C2#2027524911yescharacteristic
456C2#2027525411yescharacteristic
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#9204
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. #20277332C2semidirect
4. C2×#16810348C2semidirect
5. #20272583C2semidirect
6. #20272582C2semidirect
12. #16810554C4semidirect
14. #16810537C4semidirect
255. C53#9204semidirect

Maximal subgroups

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

subgroupindex
#202804592
#202773322
#202732832
C2×#168103482
#202725832
#202725822
C53×#15192
#920453