Finite Group Databasebeta

← #27281423

Subgroups — #27281423

Order 13568.

618 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 119 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#20274703C211yesmaximal, characteristic, complemented
3#20273283C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C53×#2543C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133132—21no—
9#16810587—21no—
10C2×C22⋊#88322C2212yes—
11#16810483C2212yes—
12#16810390—21no—
13#16810357—24no—
14C2×#121×C53—21no—
15C2×#121×C53C2211yescharacteristic
16C2×C4×D4×C53C2211yescharacteristic
17C53×#398C411yescharacteristic, complemented
18C53×#397C411yescharacteristic, complemented
19C53×#394C411yescharacteristic, complemented
20C53×#392C411yescharacteristic, complemented
21C53×#388—21no—
22C53×#386—21no—
23C53×#386—21no—
24C2×#95×C53C2211yescharacteristic
25C2×#11719917—22no—
26C22×#88322—22no—
27C2×C4⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C4⋊#88322—22no—
30#13133132—42no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322D414yescomplemented
34#13133128—22no—
35#13133124—22no—
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—21no—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43#121×C53—22no—
44#121×C53—22no—
45#121×C53—44no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63#95×C53C2×C411yescharacteristic
64C24×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—22no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—81no—
79C2×D4×C53—81no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—81no—
88C22×C4×C53—81no—
89C22×C4×C53—81no—
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4112yes—
97C22×Dic53—42no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—42no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—81no—
109C42×C53—21no—
110C42×C53—21no—
111C22⋊Dic53—42no—
112C22⋊Dic53—42no—
113C4⋊Dic53—82no—
114#11719917—82no—
115C2×#88322—24no—
116C2×#88322—82no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53C2×C22⋊C411yescharacteristic
132C23×C53#12112yes—
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53AGL(1,ℤ/8)12yes—
135C23×C53#13711yescharacteristic
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—82no—
139D4×C53—84no—
140D4×C53—82no—
141D4×C53—82no—
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—81no—
164C2×C4×C53—84no—
165C2×C4×C53C4×D412yes—
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12311yescharacteristic
168C2×Dic53—82no—
169C2×Dic53—82no—
170C2×Dic53—82no—
171#88322—44no—
172#9386—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—21no—
177C22×C53—42no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#38611yescharacteristic
193C22×C53#41711yescharacteristic
194C22×C53#41811yescharacteristic
195C22×C53#46812yes—
196C22×C53#46912yes—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—22no—
200C4×C53—42no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—81no—
205C4×C53—81no—
206Dic53—162no—
207#2973—531no—
208#2543D5311yescharacteristic
209#1799—531no—
210#1519—532no—
211#1294—532no—
212C2×C53—22no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—81no—
217C2×C53—161no—
218C2×C53#188411yescharacteristic
219C2×C53#189511yescharacteristic
220C2×C53#191311yescharacteristic
221C2×#121—21no—
222C2×#121C2×D5311yescharacteristic
223C2×C4×D4C2×D5311yescharacteristic
224#435—1064no—
225#432—1061no—
226C2×#99—1061no—
227#407—1061no—
228C2×C22⋊C8—532no—
229#398Dic5311yescharacteristic, complemented
230#397Dic5311yescharacteristic, complemented
231#394Dic5311yescharacteristic, complemented
232#392Dic5311yescharacteristic, complemented
233#388—21no—
234#386—21no—
235#386—21no—
236#380—532no—
237C2×#95C2×D5311yescharacteristic
238#342—1061no—
239#327—1064no—
240C53#938611yescharacteristic, complemented, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×Dic5311yescharacteristic
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—21no—
246C23×C4C22⋊D5311yescharacteristic
247C23×C4C22×D5311yescharacteristic
248C2×M16—1062no—
249C22×C8—1062no—
250#121—22no—
251#121—22no—
252#121—44no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1062no—
257C2×C4⋊C4C2×Dic5311yescharacteristic
258C2×C4⋊C4C2×Dic5311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4C2×Dic5311yescharacteristic
266C2×C22⋊C4C22⋊D5311yescharacteristic
267C2×C42C22⋊D5311yescharacteristic
268C4⋊C8—1062no—
269#99—2122no—
270C22⋊C8—534no—
271C22⋊C8—1062no—
272C22⋊C8—1062no—
273#97—1062no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—1062no—
278#95C2×Dic5311yescharacteristic
279#95C2×Dic5311yescharacteristic
280C24—21no—
281C24C22⋊Dic5311yescharacteristic
282C24C2×C22⋊D5311yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—2122no—
307C22×C4C22⋊Dic5311yescharacteristic
308C22×C4C22⋊Dic5311yescharacteristic
309C22×C4C22⋊Dic5311yescharacteristic
310C22×C4D4×D5312yes—
311C22×C4#1171994712yes—
312C22×C4C22×Dic5311yescharacteristic
313C22×C4C2×C22⋊D5311yescharacteristic
314M16—4242no—
315C2×C8—1064no—
316C2×C8—4242no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—4242no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—81no—
329C22⋊C4—2122no—
330C22⋊C4—2122no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C2×C22⋊Dic5311yescharacteristic
348C23#1313318912yes—
349C23#1313324211yescharacteristic
350C23#1313324312yes—
351C23#1313324711yescharacteristic
352D4—42no—
353D4—42no—
354D4—82no—
355D4—84no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—4242no—
382C2×C4—4242no—
383C2×C4—4242no—
384C2×C4D4×Dic5312yes—
385C2×C4#1313318511yescharacteristic
386C2×C4#1313319811yescharacteristic
387C8—2124no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22#1681050311yescharacteristic
408C22#1681062411yescharacteristic
409C22#1681062811yescharacteristic
410C22#1681086712yes—
411C22#1681087412yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—8482no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—161no—
428C2#2027509911yescharacteristic
429C2#2027515411yescharacteristic
430C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2543C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2543D53
hypocenterC53#9386
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. #20274703C2semidirect
3. #20273283C2semidirect
4. #20272594C2semidirect
17. C53×#398C4semidirect
18. C53×#397C4semidirect
19. C53×#394C4semidirect
20. C53×#392C4semidirect
33. C22⋊#88322D4semidirect
229. #398Dic53semidirect
230. #397Dic53semidirect
231. #394Dic53semidirect
232. #392Dic53semidirect
240. C53#9386semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202747032
#202732832
#202725942
C53×#25432
#938653