Finite Group Databasebeta

← #27281422

Subgroups — #27281422

Order 13568.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 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#20279671C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic, complemented
3#20273284C212yesmaximal, complemented
4#20272595C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×#13133189—21no—
8C2×#13133132—21no—
9#16810553C411yescharacteristic, complemented
10#16810548C411yescharacteristic, complemented
11#16810540C411yescharacteristic, complemented
12#16810532C411yescharacteristic, complemented
13#16810514—21no—
14#16810502—21no—
15#16810500—21no—
16#16810483C2212yes—
17C2×#13133124C2211yescharacteristic
18#16810390—21no—
19#16810358—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432—21no—
23C53×#407—21no—
24C2×C22⋊C8×C53C2212yes—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28C23×Dic53—21no—
29#13133191—22no—
30#13133191—22no—
31#13133189—44no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41#13133132—42no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—22no—
46#13133124C2×C411yescharacteristic
47#13133124C2×C411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C2×M16×C53—22no—
52C22×C8×C53—22no—
53C4×D4×C53—22no—
54C4×D4×C53—22no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C4⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C22⋊C8×C53D414yescomplemented
63#97×C53—22no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C23×D53—21no—
67C2×D4×C53—22no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C2×D4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—81no—
74C22×C4×C53—81no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D412yes—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4112yes—
82C2×C22⋊D53—42no—
83C2×C22⋊D53—42no—
84C2×C22⋊D53—42no—
85C2×C22⋊D53—42no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—81no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—81no—
96C2×D212—22no—
97C2×D212—81no—
98M16×C53—82no—
99C2×C8×C53—24no—
100C2×C8×C53—82no—
101C4⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—22no—
104C42×C53—21no—
105C42×C53—21no—
106C22⋊Dic53—42no—
107C22⋊Dic53—42no—
108C22⋊Dic53—42no—
109#11719921—42no—
110#11719921—42no—
111#11719921—41no—
112#11719921—41no—
113#11719921—81no—
114C4⋊Dic53—22no—
115C4⋊Dic53—82no—
116#11719919—42no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12112yes—
126C23×C53AGL(1,ℤ/8)11yescharacteristic
127C23×C53AGL(1,ℤ/8)12yes—
128C23×C53#13711yescharacteristic
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136D4×C53—42no—
137D4×C53—42no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53C4×D412yes—
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C22⋊D53—84no—
155C22⋊D53—82no—
156C22⋊D53—82no—
157C2×Dic53—22no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—84no—
167C2×Dic53—82no—
168C2×Dic53—82no—
169C2×Dic53—82no—
170D212—82no—
171C8×C53—44no—
172#9386—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—21no—
176C22×C53—22no—
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#38611yescharacteristic
186C22×C53#41711yescharacteristic
187C22×C53#41811yescharacteristic
188C22×C53#46812yes—
189C22×C53#46912yes—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203Dic53—42no—
204Dic53—81no—
205Dic53—81no—
206Dic53—162no—
207#2973D5311yescharacteristic, complemented
208#2543—531no—
209#1799—531no—
210#1519—532no—
211#1294—532no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—81no—
217C2×C53#188411yescharacteristic
218C2×C53#189511yescharacteristic
219C2×C53#191311yescharacteristic
220D53—161no—
221C2×#121—531no—
222C2×#121—1061no—
223C2×C4×D4C2×D5311yescharacteristic
224#435—24no—
225#432—21no—
226C2×#99—1061no—
227#407—21no—
228C2×C22⋊C8C2×D5312yes—
229#398—531no—
230#397—531no—
231#394—531no—
232#392—531no—
233#388—1061no—
234#386—1061no—
235#386—1061no—
236#380—532no—
237C2×#95—531no—
238#342—1061no—
239#327—1064no—
240C53#938611yescharacteristic, complemented, Sylow
241C22×D4—531no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4D21211yescharacteristic
245C23×C4—1061no—
246C23×C4C22⋊D5311yescharacteristic
247C23×C4C22×D5311yescharacteristic
248C2×M16—22no—
249C22×C8—22no—
250#121—1062no—
251#121—1062no—
252#121—2124no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—1062no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4C4×D5311yescharacteristic
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4C4×D5311yescharacteristic
266C2×C22⋊C4D21211yescharacteristic
267C2×C42C22⋊D5311yescharacteristic
268C4⋊C8—22no—
269#99—2122no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8C22⋊D5314yescomplemented
273#97—22no—
274#95—531no—
275#95—531no—
276#95—1061no—
277#95—1061no—
278#95—1061no—
279#95—1062no—
280C24—1061no—
281C24#1171992111yescharacteristic
282C24C2×D21211yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—1062no—
288C2×D4—4241no—
289C2×D4—4241no—
290C2×D4—2122no—
291C2×D4—2122no—
292C2×D4—2122no—
293C2×D4—2122no—
294C2×D4—2121no—
295C2×D4—2121no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—4241no—
304C22×C4—2122no—
305C22×C4—2122no—
306C22×C4—2122no—
307C22×C4#1171992111yescharacteristic
308C22×C4#1171992111yescharacteristic
309C22×C4#1171992111yescharacteristic
310C22×C4C2×C4×D5311yescharacteristic
311C22×C4#1171994512yes—
312C22×C4C2×C22⋊D5312yes—
313C22×C4C2×C22⋊D5311yescharacteristic
314M16—82no—
315C2×C8—24no—
316C2×C8—82no—
317C4⋊C4—22no—
318C4⋊C4—1062no—
319C4⋊C4—4242no—
320C4⋊C4—2122no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—4241no—
324C22⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4—2122no—
327C22⋊C4—2122no—
328C22⋊C4—2122no—
329C22⋊C4—2121no—
330C22⋊C4—2121no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—2121no—
341C23—2121no—
342C23—2121no—
343C23—2121no—
344C23—2121no—
345C23—2121no—
346C23—2121no—
347C23C2×#1171992111yescharacteristic
348C23#1313319112yes—
349C23#1313324512yes—
350C23#1313324511yescharacteristic
351C23#1313325011yescharacteristic
352D4—42no—
353D4—42no—
354D4—4242no—
355D4—4242no—
356D4—4242no—
357D4—4244no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—1062no—
372C2×C4—4242no—
373C2×C4—4242no—
374C2×C4—4242no—
375C2×C4—4244no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4C4×C22⋊D5312yes—
385C2×C4#1313318611yescharacteristic
386C2×C4#1313320011yescharacteristic
387C8—44no—
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—4241no—
401C22—4241no—
402C22—4241no—
403C22—4241no—
404C22—4241no—
405C22—4241no—
406C22—4241no—
407C22#1681050211yescharacteristic
408C22#1681062511yescharacteristic
409C22#1681062911yescharacteristic
410C22#1681086912yes—
411C22#1681087612yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—8482no—
419C4—4241no—
420C4—4241no—
421C4—2122no—
422C2—22no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—8481no—
428C2#2027509811yescharacteristic
429C2#2027515311yescharacteristic
430C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radical#20279671C2
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#9386
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20274703C2semidirect
3. #20273284C2semidirect
4. #20272595C2semidirect
5. C53×#2973C2semidirect
9. #16810553C4semidirect
10. #16810548C4semidirect
11. #16810540C4semidirect
12. #16810532C4semidirect
62. C22⋊C8×C53D4semidirect
207. #2973D53semidirect
240. C53#9386semidirect
272. C22⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796712
#202747032
#202732842
#202725952
C53×#29732
#938653