Finite Group Databasebeta

← #27288488

Subgroups — #27288488

Order 13568.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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#20285487C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic
3#20275699C211yesmaximal, characteristic
4#20275138C211yesmaximal, characteristic
5#20274893C211yesmaximal, characteristic
6#20272582C211yesmaximal, characteristic
7C53×#1892C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—21no—
10#16810956—21no—
11#16810862—21no—
12#16810862—21no—
13#16810779—21no—
14#16810778—21no—
15C2×#13133146—21no—
16C2×#13133142—21no—
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810390—21no—
21#16810390—21no—
22#16810389—21no—
23#16810389—21no—
24#16810386—21no—
25#16810377—21no—
26C23×C8×C53C2211yescharacteristic
27C2×C22⋊Q8×C53C2211yescharacteristic
28C2×#103×C53—21no—
29C53×#380C2211yescharacteristic
30C53×#340—21no—
31C22×Dic106D411yescharacteristic
32C2×#11719925—21no—
33C2×#11719925—21no—
34C22×#88322D411yescharacteristic
35C22×#88322D411yescharacteristic
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39#13133196—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D411yescharacteristic
46C2×C22⋊Dic53D411yescharacteristic
47C2×C4×Dic53—21no—
48#13133150—21no—
49#13133150—21no—
50#13133146—42no—
51#13133142—42no—
52#13133141—21no—
53#13133141—21no—
54#13133140—21no—
55#13133140—21no—
56C22⋊#88322—21no—
57C22⋊#88322—21no—
58#13133124—21no—
59#13133124—21no—
60C22×Q8×C53D411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C22×C8×C53—21no—
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65C22×C8×C53—42no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75#103×C53—42no—
76#95×C53—21no—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—41no—
98C2×Dic106—21no—
99C2×Dic106—21no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊Dic53—21no—
120C22⋊Dic53—21no—
121C22⋊Dic53—21no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53Q1612yescomplemented
127#11719925—81no—
128#11719925—81no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133#11719919—41no—
134#11719919—41no—
135C4×Dic53—41no—
136C4×Dic53—41no—
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—41no—
140C2×#88322—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53#12111yescharacteristic
144C23×C53#12111yescharacteristic
145C23×C53#12111yescharacteristic
146C23×C53C2×Q1611yescharacteristic
147C23×C53#13511yescharacteristic
148C23×C53#13711yescharacteristic
149C23×C53#13711yescharacteristic
150Q8×C53—42no—
151Q8×C53—41no—
152Q8×C53—41no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12311yescharacteristic
171C2×C4×C53#12311yescharacteristic
172C2×C4×C53#12411yescharacteristic
173C2×Dic53—22no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186Dic106—42no—
187Dic106—41no—
188Dic106—41no—
189C8×C53—42no—
190#88322—41no—
191#88322—41no—
192#10566—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#39411yescharacteristic
204C22×C53#46211yescharacteristic
205C22×C53#46211yescharacteristic
206C22×C53#46411yescharacteristic
207C22×C53#46411yescharacteristic
208C22×C53#46711yescharacteristic
209C22×C53#47011yescharacteristic
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53—161no—
216C4×C53B(4,𝔽2)11yescharacteristic
217C4×C53#45811yescharacteristic
218Dic53—42no—
219Dic53—81no—
220Dic53—81no—
221Dic53—81no—
222Dic53—161no—
223#3189—531no—
224#1996—531no—
225#1996—531no—
226#1892—531no—
227#1892D5311yescharacteristic
228#1850—531no—
229#1289—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53#204411yescharacteristic
237C2×C53#205311yescharacteristic
238C2×C53#205611yescharacteristic
239C23×C8C2×D5311yescharacteristic
240C2×C22⋊Q8—531no—
241C2×C22⋊Q8C2×D5311yescharacteristic
242#484—1061no—
243#484—1061no—
244#467—1061no—
245#467—1061no—
246#451—1061no—
247#451—1061no—
248C2×C4⋊C8—1061no—
249C2×#103—21no—
250C2×#103—1061no—
251C2×C22⋊C8—531no—
252#380—531no—
253#380C2×D5311yescharacteristic
254C4×C22⋊C4—531no—
255#342—1061no—
256#342—1061no—
257#342—1061no—
258#342—1061no—
259#340—21no—
260#340—1061no—
261#336—1061no—
262C53#1056611yescharacteristic, complemented, Sylow
263C22×Q8—531no—
264C22×Q8C22⋊D5311yescharacteristic
265C23×C4C22×D5311yescharacteristic
266C2×Q16—1061no—
267C2×Q16—1061no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8—531no—
273C22×C8—531no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—1061no—
279C22⋊Q8—1061no—
280C22⋊Q8—1061no—
281C22⋊Q8—1061no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—531no—
285C2×C4⋊C4—1061no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4C22⋊D5311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C42—1061no—
295#107—1061no—
296#107—1061no—
297C4⋊C8—2122no—
298#103—42no—
299#103—1061no—
300#103—1061no—
301#103—1061no—
302#103—1061no—
303#103—2122no—
304C22⋊C8—1061no—
305C22⋊C8—1061no—
306#95—21no—
307#95—1061no—
308#95—1061no—
309C24D4×D5311yescharacteristic
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—1061no—
315C2×Q8—1061no—
316C2×Q8—2121no—
317C2×Q8—2121no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4—2121no—
328C22×C4#1171994511yescharacteristic
329C22×C4#1171994511yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334Q16—4241no—
335Q16—4241no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—1061no—
344C2×C8—1061no—
345C2×C8—2121no—
346C2×C8—2121no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4—2121no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—532no—
364C22⋊C4—1061no—
365C22⋊C4—1061no—
366C22⋊C4—1061no—
367C22⋊C4—2121no—
368C22⋊C4—2121no—
369C22⋊C4—2121no—
370C22⋊C4—2121no—
371C42—2121no—
372C42—2121no—
373C23—41no—
374C23—41no—
375C23#1313318911yescharacteristic
376C23#1313319011yescharacteristic
377C23#1313319011yescharacteristic
378C23Q16×D5311yescharacteristic
379C23#1313323811yescharacteristic
380C23#1313324611yescharacteristic
381C23#1313324911yescharacteristic
382Q8—42no—
383Q8—41no—
384Q8—41no—
385Q8—2122no—
386Q8—2121no—
387Q8—2121no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—1062no—
405C2×C4—4241no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4#1313318511yescharacteristic
418C2×C4#1313319911yescharacteristic
419C2×C4#1313320011yescharacteristic
420C2×C4#1313320311yescharacteristic
421C8—42no—
422C8—2121no—
423C8—2121no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#1681053811yescharacteristic
435C22#1681083111yescharacteristic
436C22#1681083211yescharacteristic
437C22#1681084311yescharacteristic
438C22#1681084811yescharacteristic
439C22#1681086311yescharacteristic
440C22#1681088011yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—81no—
445C4—81no—
446C4—161no—
447C4—8481no—
448C4—4241no—
449C4—4241no—
450C4—4241no—
451C4—2122no—
452C4#1681080911yescharacteristic
453C4#1681081211yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#2027596411yescharacteristic
461C2#2027601911yescharacteristic
462C2#2027603611yescharacteristic
subgroupG/H
centerC22#16810538
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1892C2
socleC22×C53#394
prime radicalC24×C53C2×D4
hypercenter#1892D53
hypocenterC53#10566
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
126. C22⋊Dic53Q16semidirect
262. C53#10566semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202757012
#202756992
#202751382
#202748932
#202725822
C53×#18922
#1056653