Finite Group Databasebeta

← #27279062

Subgroups — #27279062

Order 13568.

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

normal subgroups only

This group has 784 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
1#20283744C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal, complemented
3C4×#13133132C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2388C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133144—24no—
9C2×#13133132—21no—
10#16810587C2212yes—
11C2×C22⋊#88322—21no—
12C4×C22⋊Dic53C2212yes—
13#16810390—21no—
14C22×#41×C53C2211yescharacteristic
15C2×C4×D4×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#395C411yescharacteristic, complemented
18C53×#394—21no—
19C53×#394C411yescharacteristic, complemented
20C53×#393C411yescharacteristic, complemented
21C53×#392—21no—
22C53×#392C411yescharacteristic, complemented
23C4×C22⋊C4×C53—21no—
24C4×C22⋊C4×C53C2211yescharacteristic
25C2×#11719917D414yescomplemented
26C22×#88322—22no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29C4⋊#88322—22no—
30#13133144—48no—
31#13133132—24no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133128—22no—
36#13133124—22no—
37C2×#41×C53—22no—
38C2×#41×C53—22no—
39C2×#41×C53—44no—
40C22×Q8×C53C2×C411yescharacteristic
41C22×D4×C53—21no—
42C22×D4×C53—21no—
43C22×D4×C53C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53—21no—
46C23×C4×C53D411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48C4×D4×C53—22no—
49C4×D4×C53—22no—
50C4×D4×C53—44no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53D411yescharacteristic
62#95×C53—21no—
63#95×C53—21no—
64#95×C53C2×C411yescharacteristic
65#95×C53C2×C411yescharacteristic
66C24×C53—21no—
67C24×C53C22⋊C411yescharacteristic
68C24×C53C2×D411yescharacteristic
69#41×C53—28no—
70#41×C53—44no—
71#41×C53—44no—
72#41×C53—44no—
73C2×Q8×C53—22no—
74C2×Q8×C53—42no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—41no—
79C2×D4×C53—42no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C22×C4×C53—24no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22⋊C411yescharacteristic
99C22×C4×C53C22⋊C411yescharacteristic
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4112yes—
104C22×Dic53—42no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—42no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—42no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—42no—
117C22⋊Dic53—24no—
118C22⋊Dic53—42no—
119C4×Dic53—44no—
120#11719917—44no—
121C2×#88322—24no—
122C2×#88322—82no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—22no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53#10412yescomplemented
139C23×C53C2×C22⋊C411yescharacteristic
140C23×C53#12112yes—
141Q8×C53—24no—
142Q8×C53—42no—
143D4×C53—24no—
144D4×C53—42no—
145D4×C53—44no—
146D4×C53—48no—
147D4×C53—48no—
148C2×C4×C53—22no—
149C2×C4×C53—24no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—44no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—44no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—81no—
172C2×C4×C53C4×D412yes—
173C2×C4×C53#12011yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C2×Dic53—44no—
176C2×Dic53—82no—
177C2×Dic53—82no—
178#88322—44no—
179#9009—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—42no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53#38611yescharacteristic
200C22×C53C2×#10411yescharacteristic
201C22×C53#42111yescharacteristic
202C4×C53—21no—
203C4×C53—22no—
204C4×C53—42no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—42no—
208C4×C53—41no—
209C4×C53—81no—
210C4×C53B(4,𝔽2)12yes—
211Dic53—84no—
212#2973—531no—
213#2388D5311yescharacteristic
214#1789—532no—
215C4×#99—531no—
216#1297—532no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53—42no—
221C2×C53—82no—
222C2×C53#187111yescharacteristic
223C2×C53#188411yescharacteristic
224C2×C53#188911yescharacteristic
225C22×#41C2×D5311yescharacteristic
226C2×C4×D4—21no—
227C2×C4×D4C2×D5311yescharacteristic
228#435—1064no—
229#432—1061no—
230C2×#104—1064no—
231C2×#99—1061no—
232#407—532no—
233C2×C22⋊C8—1061no—
234#395Dic5311yescharacteristic, complemented
235#394—21no—
236#394Dic5311yescharacteristic, complemented
237#393Dic5311yescharacteristic, complemented
238#392—21no—
239#392Dic5311yescharacteristic, complemented
240C4×C22⋊C4—21no—
241C4×C22⋊C4—532no—
242C4×C22⋊C4C2×D5311yescharacteristic
243#342—1061no—
244C53#900911yescharacteristic, complemented, Sylow
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—44no—
248C22×Q8C2×Dic5311yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×Dic5311yescharacteristic
252C22×D4C22⋊D5311yescharacteristic
253C23×C4—21no—
254C23×C4C22⋊D5311yescharacteristic
255C23×C4C22×D5311yescharacteristic
256C2×M16—534no—
257C22×C8—1062no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×Dic5311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1062no—
269C2×C22⋊C4C2×Dic5311yescharacteristic
270C2×C22⋊C4C22⋊D5311yescharacteristic
271C2×C42—21no—
272C2×C42—1062no—
273C2×C42C22⋊D5311yescharacteristic
274C4⋊C8—1062no—
275#104—2128no—
276#99—1064no—
277C22⋊C8—1062no—
278C22⋊C8—1062no—
279C22⋊C8—1062no—
280#97—1062no—
281#95—21no—
282#95—21no—
283#95—1062no—
284#95C2×Dic5311yescharacteristic
285#95C2×Dic5311yescharacteristic
286C24—21no—
287C24C22⋊Dic5311yescharacteristic
288C24C2×C22⋊D5311yescharacteristic
289#41—28no—
290#41—44no—
291#41—44no—
292#41—44no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—2122no—
318C22×C4C22⋊Dic5311yescharacteristic
319C22×C4C22⋊Dic5311yescharacteristic
320C22×C4C22⋊Dic5311yescharacteristic
321C22×C4D4×D5312yes—
322C22×C4#1171994712yes—
323C22×C4C22×Dic5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325M16—2124no—
326C2×C8—1064no—
327C2×C8—4242no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—1064no—
338C22⋊C4—2122no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—2124no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#1313314412yescomplemented
359C23C2×C22⋊Dic5311yescharacteristic
360C23#1313318912yes—
361Q8—24no—
362Q8—42no—
363D4—24no—
364D4—42no—
365D4—44no—
366D4—48no—
367D4—48no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—4242no—
393C2×C4—4242no—
394C2×C4—2124no—
395C2×C4D4×Dic5312yes—
396C2×C4#1313318511yescharacteristic
397C2×C4#1313319811yescharacteristic
398C8—2124no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#1681050311yescharacteristic
419C22C2×#1313314411yescharacteristic
420C22#1681064011yescharacteristic
421C4—21no—
422C4—22no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—81no—
429C4—4244no—
430C4#1681080912yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2—82no—
436C2#2027501811yescharacteristic
437C2#2027509911yescharacteristic
438C2#2027512411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2388C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2388D53
hypocenterC53#9009
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. #20274659C2semidirect
3. C4×#13133132C2semidirect
4. #20272601C2semidirect
17. C53×#395C4semidirect
19. C53×#394C4semidirect
20. C53×#393C4semidirect
22. C53×#392C4semidirect
25. C2×#11719917D4semidirect
138. C23×C53#104semidirect
234. #395Dic53semidirect
236. #394Dic53semidirect
237. #393Dic53semidirect
239. #392Dic53semidirect
244. C53#9009semidirect
358. C23#13133144semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202746592
C4×#131331322
#202726012
C53×#23882
#900953