Finite Group Databasebeta

← #28301550

Subgroups — #28301550

Order 13568.

710 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 136 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#20290441C211yesmaximal, characteristic, complemented
2C4×C22⋊#88324C211yesmaximal, characteristic, complemented
3C2×#16811467C212yesmaximal
4#20289553C212yesmaximal, complemented
5C2×#16810513C211yesmaximal, characteristic
6C2×C22⋊#88324C2212yes—
7C2×C22⋊#88324C2212yes—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489—21no—
11#16811489—21no—
12#16811467C418yescomplemented
13C22×#11719921C411yescharacteristic
14C22×#11719921C411yescharacteristic
15C22×#11719921C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810513—21no—
18#16810513—21no—
19#16810513—21no—
20#16810513—21no—
21C2×#13133124C411yescharacteristic
22C2×#13133124C2211yescharacteristic
23C22×C4⋊C4×C53C411yescharacteristic
24C22×#88324—22no—
25C22×#88324—22no—
26C22⋊#88324—22no—
27C22⋊#88324—22no—
28C22⋊#88324C2×C414yes—
29C22⋊#88324C2×C414yes—
30#13133272—42no—
31C4×#88324—42no—
32C24×D53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C23×Dic53C2311yescharacteristic
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—22no—
43C2×#11719921—42no—
44C2×#11719921—42no—
45C2×#11719921C2×C414yes—
46C2×C4×Dic53—42no—
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—42no—
61C24×C53C22×C411yescharacteristic
62C23×D53—21no—
63C23×D53—21no—
64C23×D53—22no—
65C23×D53—22no—
66C23×D53—41no—
67C23×D53—41no—
68C23×D53—41no—
69C23×D53—41no—
70C23×D53C4212yes—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C4212yes—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53C22⋊C411yescharacteristic
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53C22×C412yes—
97C22×Dic53C2×D411yescharacteristic
98C22×Dic53C2×D411yescharacteristic
99C22×Dic53#4112yes—
100C2×#88324—24no—
101C2×#88324—24no—
102C2×#88324—42no—
103C2×#88324—42no—
104C2×#88324—42no—
105C2×#88324—42no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—42no—
108#11719921—24no—
109#11719921—24no—
110#11719921—42no—
111#11719921—42no—
112#11719921—42no—
113#11719921—42no—
114C4×Dic53—22no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C2×C4211yescharacteristic
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53#11711yescharacteristic
125C23×C53#13112yes—
126C23×C53#13112yes—
127C22×D53—22no—
128C22×D53—24no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143C22×D53—82no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—22no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—44no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—82no—
159C2×C4×C53#10012yes—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—44no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—82no—
176C2×Dic53#9912yes—
177C2×Dic53C4×D412yes—
178C2×Dic53C4×D412yes—
179#88324—44no—
180#88324—44no—
181#8155—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53C4×C22⋊C411yescharacteristic
191C22×C53#40512yes—
192C22×C53#40811yescharacteristic
193C22×C53#40811yescharacteristic
194C22×C53C2×#9911yescharacteristic
195C22×C53#41011yescharacteristic
196C22×C53C2×#10011yescharacteristic
197C22×C53#41311yescharacteristic
198C22×C53#43312yes—
199C2×D53—42no—
200C2×D53—42no—
201C2×D53—42no—
202C2×D53—84no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—84no—
208C4×C53—22no—
209C4×C53—41no—
210C4×C53—42no—
211C4×C53—81no—
212Dic53—22no—
213Dic53—22no—
214Dic53—22no—
215Dic53—41no—
216Dic53—81no—
217C2×#388—531no—
218#1749—531no—
219C4×C22⋊C8—531no—
220C2×#323—532no—
221#1280—532no—
222C2×C53—22no—
223C2×C53—22no—
224C2×C53#146811yescharacteristic
225C2×C53#146911yescharacteristic
226C2×C53#147111yescharacteristic
227C2×C53#147311yescharacteristic
228C2×C53#175011yescharacteristic
229C2×C53C4×#9911yescharacteristic
230C2×C53C4×#10011yescharacteristic
231D53—82no—
232C22×C4⋊C4#615711yescharacteristic
233C22×C22⋊C4—531no—
234C22×C22⋊C4—531no—
235C22×C22⋊C4—531no—
236C22×C42—531no—
237C2×C22⋊C8—532no—
238C2×C22⋊C8—532no—
239C2×#97—1061no—
240C2×C4×C8—1061no—
241#388—1061no—
242#388—1061no—
243#388—1061no—
244#388—1061no—
245C2×#95—531no—
246C2×#95—531no—
247#336—1061no—
248#336—1061no—
249#323—538no—
250C53#815511yescharacteristic, complemented, Sylow
251C25—531no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4—531no—
255C23×C4C2×#615711yescharacteristic
256C23×C4C2×#615711yescharacteristic
257C23×C4C2×#615711yescharacteristic
258C22×C8—1062no—
259C22×C8—1062no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—42no—
263C2×C22⋊C4—534no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C22⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—2122no—
272C2×C22⋊C4—2122no—
273C2×C42—531no—
274C2×C42—531no—
275C2×C42—531no—
276C2×C42—531no—
277C2×C42—2122no—
278C22⋊C8—534no—
279C22⋊C8—534no—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282#97—2122no—
283C4×C8—2122no—
284#95—1061no—
285#95—1061no—
286#95—1061no—
287#95—1061no—
288C24—532no—
289C24—1062no—
290C24—1062no—
291C24—1061no—
292C24—1061no—
293C24—2121no—
294C24—2121no—
295C24—2121no—
296C24—2121no—
297C24C22×#615711yescharacteristic
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—531no—
312C22×C4—532no—
313C22×C4—532no—
314C22×C4—531no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—1061no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4—2121no—
324C22×C4C4×#615712yes—
325C22×C4C22⋊#615711yescharacteristic
326C22×C4C22⋊#615711yescharacteristic
327C2×C8—1064no—
328C2×C8—1064no—
329C2×C8—2122no—
330C2×C8—2122no—
331C2×C8—2122no—
332C2×C8—2122no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C22⋊C4—1064no—
336C22⋊C4—1064no—
337C22⋊C4—2122no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C22⋊C4—2122no—
341C42—1062no—
342C42—1062no—
343C42—2121no—
344C42—2121no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—1064no—
350C23—1062no—
351C23—4242no—
352C23—4242no—
353C23—4242no—
354C23—4242no—
355C23—2122no—
356C23—2122no—
357C23—2122no—
358C23—2122no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23—2122no—
366C23C2×C4×#615711yescharacteristic
367C23C2×C22⋊#615711yescharacteristic
368C23#1313329611yescharacteristic
369C23#1313330412yes—
370C23#1313330512yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—532no—
387C2×C4—532no—
388C2×C4—532no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—4242no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2124no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4#1313327712yes—
406C8—2124no—
407C8—2124no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—4244no—
417C22—4244no—
418C22—4241no—
419C22—4241no—
420C22—4241no—
421C22—4241no—
422C22—2122no—
423C22—2122no—
424C22—2122no—
425C22C4×C22⋊#615711yescharacteristic
426C22#1681158812yes—
427C22#1681159611yescharacteristic
428C22#1681159811yescharacteristic
429C22D53⋊#9911yescharacteristic
430C22#1681160411yescharacteristic
431C22C2×#1313327711yescharacteristic
432C22#1681160911yescharacteristic
433C22#1681164812yes—
434C4—22no—
435C4—42no—
436C4—41no—
437C4—81no—
438C4—1062no—
439C4—1062no—
440C4—1062no—
441C4—4241no—
442C4—2121no—
443C2—22no—
444C2—22no—
445C2—4242no—
446C2#2028998311yescharacteristic
447C2#2028998611yescharacteristic
448C2#2028999111yescharacteristic
449C2#2028999911yescharacteristic
450C2#2029044411yescharacteristic
451C2#2029046011yescharacteristic
452C2C4×#1313327711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8155
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. C4×C22⋊#88324C2semidirect
4. #20289553C2semidirect
12. #16811467C4semidirect
250. C53#8155semidirect

Maximal subgroups

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

subgroupindex
#202904412
C4×C22⋊#883242
C2×#168114672
#202895532
C2×#168105132
#815553