Finite Group Databasebeta

← #27284561

Subgroups — #27284561

Order 13568.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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#20285468C211yesmaximal, characteristic
2C2×#16810936C211yesmaximal, characteristic
3#20275551C211yesmaximal, characteristic
4#20274801C211yesmaximal, characteristic, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2503C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810951—21no—
10#16810941—21no—
11#16810936—22no—
12#16810936—22no—
13#16810854—22no—
14#16810773—21no—
15#16810762—21no—
16C2×#13133150—21no—
17C2×#13133137—21no—
18C2×#13133131—22no—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810483C2211yescharacteristic
22#16810385—22no—
23#16810360—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#400—22no—
27C53×#391—21no—
28C53×#382—21no—
29C53×#381—22no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719924—22no—
34C22×#88322—21no—
35C22×#88322D412yes—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133150—24no—
45#13133148—22no—
46#13133141—22no—
47#13133137—22no—
48#13133137—24no—
49#13133131—44no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—21no—
55C22×Q8×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×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×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53D412yes—
72C2×C42×C53—21no—
73#95×C53—24no—
74#95×C53—24no—
75#95×C53—21no—
76C24×C53C2×D411yescharacteristic
77C23×D53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—42no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C2×C22⋊D53—41no—
97C2×C22⋊D53—41no—
98C2×C22⋊D53—42no—
99C2×C22⋊D53—42no—
100C22×Dic53—21no—
101C22×Dic53—41no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—42no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53—42no—
118C22⋊Dic53—42no—
119#11719924—82no—
120#11719921—41no—
121#11719921—41no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—42no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—42no—
128C2×#88322—42no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53#12011yescharacteristic
132C23×C53#12111yescharacteristic
133C23×C53#12711yescharacteristic
134C23×C53#13512yes—
135C23×C53AGL(1,ℤ/8)12yes—
136C22×D53—41no—
137C22×D53—42no—
138C22×D53—81no—
139C22×D53—81no—
140Q8×C53—42no—
141Q8×C53—81no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—44no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C22⋊D53—82no—
165C22⋊D53—81no—
166C22⋊D53—81no—
167C2×Dic53—41no—
168C2×Dic53—42no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172D212—42no—
173D212—42no—
174#88322—42no—
175#88322—42no—
176#9953—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—42no—
184C22×C53—42no—
185C22×C53#39211yescharacteristic
186C22×C53#44912yes—
187C22×C53#46312yes—
188C22×C53#49511yescharacteristic
189C22×C53#49611yescharacteristic
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C2×D53—162no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—82no—
198C4×C53—81no—
199C4×C53—82no—
200C4×C53—81no—
201C4×C53#45812yes—
202Dic53—81no—
203Dic53—161no—
204#3187—531no—
205C2×#480—531no—
206#2503D5311yescharacteristic, complemented
207#1964—531no—
208#1823—531no—
209#1520—531no—
210#1282—531no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53#200911yescharacteristic
216C2×C53#201211yescharacteristic
217C2×C53#202211yescharacteristic
218D53—161no—
219C2×C22⋊Q8C2×D5311yescharacteristic
220C2×#121—531no—
221C22×C4⋊C4C2×D5311yescharacteristic
222#483—1061no—
223#481—1061no—
224#480—1062no—
225#480—1062no—
226#465—1062no—
227#450—1061no—
228#448—1061no—
229C2×#107—1061no—
230C2×#102—1061no—
231C2×#99—1062no—
232C2×C22⋊C8—531no—
233C2×C22⋊C8—531no—
234#400—22no—
235#391—21no—
236#382—21no—
237#381—22no—
238#380—531no—
239C4×C22⋊C4C2×D5311yescharacteristic
240#340—1062no—
241#328—1062no—
242C53#995311yescharacteristic, complemented, Sylow
243C22×Q8C22⋊D5311yescharacteristic
244C22×D4—531no—
245C22×D4—1061no—
246C23×C4—21no—
247C23×C4C22×D5311yescharacteristic
248C2×SD16—1062no—
249C22×C8—532no—
250C22×C8—1061no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255#121—1061no—
256#121—1061no—
257#121—1061no—
258#121—1061no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4C22⋊D5312yes—
268C2×C4⋊C4C22⋊D5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4C22⋊D5312yes—
273C2×C42—21no—
274#107—1064no—
275#106—1062no—
276#103—1062no—
277#102—1064no—
278#102—1062no—
279#99—2124no—
280C22⋊C8—532no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284#95—24no—
285#95—24no—
286#95—21no—
287#95—1061no—
288C24—1061no—
289C24C2×C22⋊D5311yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—42no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—2122no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2122no—
299C2×D4—2122no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—1061no—
310C22×C4—2121no—
311C22×C4D4×D5311yescharacteristic
312C22×C4D4×D5311yescharacteristic
313C22×C4D4×D5311yescharacteristic
314C22×C4#1171994711yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317SD16—4242no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—2122no—
321C2×C8—2122no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4—2122no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—2122no—
338C22⋊C4—2121no—
339C22⋊C4—2121no—
340C42—42no—
341C23—41no—
342C23—41no—
343C23—4241no—
344C23—4241no—
345C23—2122no—
346C23—2121no—
347C23#1313318611yescharacteristic
348C23#1313318811yescharacteristic
349C23#1313321311yescharacteristic
350C23#1313323712yes—
351C23#1313324212yes—
352Q8—42no—
353Q8—81no—
354D4—4241no—
355D4—4242no—
356D4—4241no—
357D4—2122no—
358D4—2122no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—2121no—
381C2×C4—2122no—
382C2×C4#1313318511yescharacteristic
383C2×C4#1313318511yescharacteristic
384C2×C4#1313318911yescharacteristic
385C2×C4#1313318911yescharacteristic
386C8—2122no—
387C8—2122no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—8482no—
397C22—4241no—
398C22—4241no—
399C22—4241no—
400C22#1681053111yescharacteristic
401C22#1681076812yes—
402C22#1681083512yes—
403C22#1681100511yescharacteristic
404C22#1681101311yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—8481no—
413C4—4241no—
414C4#1681081212yes—
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—8481no—
420C2#2027577511yescharacteristic
421C2#2027579211yescharacteristic
422C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2503C2
socleC22×C53#392
prime radicalC2×#13133191C22
hypercenter#2503D53
hypocenterC53#9953
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
4. #20274801C2semidirect
5. #20273287C2semidirect
6. #20272563C2semidirect
7. C53×#2503C2semidirect
206. #2503D53semidirect
242. C53#9953semidirect

Maximal subgroups

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

subgroupindex
#202854682
C2×#168109362
#202755512
#202748012
#202732872
#202725632
C53×#25032
#995353