Finite Group Databasebeta

← #28302963

Subgroups — #28302963

Order 13568.

614 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290570C211yesmaximal, characteristic
2C2×#16811477C212yesmaximal
3#20289891C211yesmaximal, characteristic
4#20289588C212yesmaximal
5C4×#13133193C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133274C2212yes—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4—21no—
10C2×(C2×#6157)⋊C4C2212yes—
11#16811489—21no—
12#16811477—24no—
13C2×#13133193C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810530C411yescharacteristic, complemented
16#16810519C411yescharacteristic, complemented
17#16810504—21no—
18#16810494—21no—
19#16810480C2211yescharacteristic
20C4×#11719919—21no—
21C4×C4⋊Dic53C411yescharacteristic, complemented
22C4×#11719921—21no—
23C2×C4×Q8×C53C411yescharacteristic, complemented
24C23×#6157—22no—
25C22×#88324—22no—
26C4⋊#88324—42no—
27#13133274—22no—
28#13133274—22no—
29(C2×#6157)⋊C4—24no—
30(C2×#6157)⋊C4—22no—
31(C2×#6157)⋊C4—22no—
32(C2×#6157)⋊C4—42no—
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×C4×D53C2311yescharacteristic
36#13133193—21no—
37#13133193—21no—
38#13133193D412yes—
39#13133193Q812yes—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C42×D53—42no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56C22×Q8×C53C2×C411yescharacteristic
57C4×Q8×C53—42no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2×C411yescharacteristic
62C23×D53C2×D411yescharacteristic
63C22×#6157—24no—
64C22×#6157—42no—
65C22×#6157—42no—
66C22×#6157—42no—
67C22×#6157—42no—
68C22×#6157—42no—
69C22×#6157—42no—
70C2×Q8×C53—21no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53C4⋊C412yes—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53C2×D411yescharacteristic
93C2×C4×D53C2×Q811yescharacteristic
94C2×C4×D53#4111yescharacteristic
95C2×C4×D53#4111yescharacteristic
96C2×#88324—42no—
97C2×#88324—42no—
98C2×#88324—42no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—42no—
101C42×C53—22no—
102C42×C53—42no—
103#11719921—22no—
104#11719921—22no—
105#11719921—42no—
106C4⋊Dic53—21no—
107C4⋊Dic53—41no—
108C4⋊Dic53C4⋊C412yes—
109#11719919—22no—
110#11719919—22no—
111#11719919—42no—
112C4×Dic53—22no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—42no—
116C23×C53C2×C22⋊C411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53#12111yescharacteristic
122C22×D53C22⋊Q811yescharacteristic
123C22×D53#13512yes—
124C22×D53#13712yes—
125C2×#6157—44no—
126C2×#6157—44no—
127C2×#6157—44no—
128C2×#6157—82no—
129C2×#6157—82no—
130C2×#6157—82no—
131C2×#6157—82no—
132Q8×C53—42no—
133Q8×C53—42no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#9912yes—
147C2×C4×C53#10412yes—
148C2×C4×C53C2×C4⋊C411yescharacteristic
149C2×C4×C53#11711yescharacteristic
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—22no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53#12111yescharacteristic
165C2×Dic53C22⋊Q811yescharacteristic
166C4×D53—41no—
167C4×D53—41no—
168C4×D53—41no—
169C4×D53—41no—
170C4×D53—44no—
171C4×D53—44no—
172#88324—82no—
173#8574—531nomaximal, Sylow
174C22×C53#38011yescharacteristic
175C22×C53C2×#9911yescharacteristic
176C22×C53#41011yescharacteristic
177C22×C53#41611yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53C2×#10411yescharacteristic
180C22×C53#42111yescharacteristic
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—22no—
188C2×D53—22no—
189C2×D53#46412yes—
190C2×D53#47912yes—
191#6157—84no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—22no—
195C4×C53—22no—
196C4×C53—42no—
197C4×C53—81no—
198Dic53—21no—
199Dic53—21no—
200Dic53—42no—
201Dic53—41no—
202Dic53—41no—
203Dic53—81no—
204C4×C22⋊Q8—531no—
205#1797—531no—
206C2×#328—532no—
207#1434—531no—
208#1294—532no—
209C2×C53#150011yescharacteristic
210C2×C53#150811yescharacteristic
211C2×C53#151511yescharacteristic
212C2×C53#151611yescharacteristic
213C2×C53#179911yescharacteristic
214C2×C53#180511yescharacteristic
215C2×C53#180611yescharacteristic
216D53—22no—
217D53—22no—
218C2×C22⋊Q8—531no—
219C2×C4×Q8#615711yescharacteristic, complemented
220C22×C42—531no—
221C2×C4⋊C8—1061no—
222C2×C22⋊C8—532no—
223#391—531no—
224#389—531no—
225#387—1061no—
226#385—1061no—
227#380—531no—
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230C4×C22⋊C4—1061no—
231C2×#95—532no—
232C2×#95—1061no—
233C2×#95—1061no—
234#336—1061no—
235#328—1064no—
236C53#857411yescharacteristic, complemented, Sylow
237C22×Q8C2×#615711yescharacteristic
238C23×C4—531no—
239C23×C4—531no—
240C23×C4—531no—
241C23×C4—1062no—
242C22×C8—1062no—
243C22⋊Q8—532no—
244C22⋊Q8—532no—
245C22⋊Q8—1061no—
246C22⋊Q8—1061no—
247C4×Q8—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4C2×#615711yescharacteristic
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C42—21no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—1061no—
261C2×C42—2122no—
262C2×C42C2×#615711yescharacteristic
263C4⋊C8—2122no—
264C22⋊C8—1062no—
265C22⋊C8—1062no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269#95—1061no—
270#95—1062no—
271#95—1062no—
272#95—1064no—
273#95—1061no—
274#95—2122no—
275C24—531no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊#615712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1064no—
295C22×C4—1061no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2121no—
300C22×C4—2122no—
301C22×C4—2122no—
302C22×C4—2122no—
303C22×C4—2122no—
304C22×C4—2122no—
305C22×C4—2122no—
306C22×C4C22⋊#615711yescharacteristic
307C22×C4C22⋊#615711yescharacteristic
308C22×C4C22×#615711yescharacteristic
309C2×C8—2122no—
310C2×C8—2122no—
311C2×C8—2122no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—532no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—1061no—
318C4⋊C4—2121no—
319C4⋊C4—2122no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—2122no—
323C42—22no—
324C42—42no—
325C42—1062no—
326C42—2121no—
327C42—2121no—
328C42—2122no—
329C23—531no—
330C23—531no—
331C23—532no—
332C23—532no—
333C23—1061no—
334C23—1061no—
335C23—1061no—
336C23—1061no—
337C23C2×C22⋊#615711yescharacteristic
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1062no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—4242no—
364C2×C4—4242no—
365C2×C4—4242no—
366C2×C4—4242no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2124no—
372C2×C4—2124no—
373C2×C4—2124no—
374C2×C4—2124no—
375C2×C4—2124no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4#1313327512yes—
382C2×C4#1313328212yes—
383C2×C4C2×C4⋊#615711yescharacteristic
384C2×C4#1313329611yescharacteristic
385C8—4242no—
386C22—532no—
387C22—532no—
388C22—1061no—
389C22—1061no—
390C22—1061no—
391C22—1061no—
392C22—1062no—
393C22—1062no—
394C22—1062no—
395C22—1062no—
396C22#1681156511yescharacteristic
397C22C2×#1313327511yescharacteristic
398C22#1681160211yescharacteristic
399C22#1681161311yescharacteristic
400C22#1681161611yescharacteristic
401C22C2×#1313328211yescharacteristic
402C22#1681162011yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—42no—
408C4—81no—
409C4—1061no—
410C4—1061no—
411C4—4244no—
412C4—4241no—
413C4—2122no—
414C4—2121no—
415C4—2121no—
416C2—1062no—
417C2—1062no—
418C2#2029008811yescharacteristic
419C2#2029011511yescharacteristic
420C2#2029013711yescharacteristic
421C2#2029014111yescharacteristic
422C2#2029057511yescharacteristic
423C2#2029058611yescharacteristic
424C2#2029058811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#8574
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
15. #16810530C4semidirect
16. #16810519C4semidirect
21. C4×C4⋊Dic53C4semidirect
23. C2×C4×Q8×C53C4semidirect
219. C2×C4×Q8#6157semidirect
236. C53#8574semidirect

Maximal subgroups

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

subgroupindex
#202905702
C2×#168114772
#202898912
#202895882
C4×#131331932
#857453