Finite Group Databasebeta

← #27276585

Subgroups — #27276585

Order 13568.

1168 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 139 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#20274643C211yesmaximal, characteristic, complemented
2#20274633C211yesmaximal, characteristic, complemented
3#20273244C214yesmaximal, complemented
4C2×C53×#535C211yesmaximal, characteristic
5C2×#13133136—21no—
6C2×#13133134—21no—
7C2×#13133132—21no—
8C2×#13133132—21no—
9#16810587C2212yescomplemented
10#16810486C2212yescomplemented
11#16810357—28no—
12C22×#41×C53C2211yescharacteristic
13C23×D4×C53C411yescharacteristic, complemented
14C53×#535—28no—
15C2×#127×C53—21no—
16C2×#124×C53—21no—
17C2×#121×C53C412yescomplemented
18C2×#121×C53C2212yes—
19C2×#120×C53—21no—
20C2×#120×C53—21no—
21C2×#117×C53C411yescharacteristic, complemented
22C2×#11719917—22no—
23C2×C22⋊Dic53—22no—
24#13133136—42no—
25#13133134—42no—
26#13133132—42no—
27#13133132—42no—
28C22⋊#88322—24no—
29#13133124—24no—
30C25×C53—21no—
31C2×#41×C53—24no—
32C2×#41×C53—42no—
33C2×#41×C53—42no—
34C22×Q8×C53D411yescharacteristic
35C22×D4×C53—22no—
36C22×D4×C53—22no—
37C22×D4×C53—24no—
38C22×D4×C53—21no—
39C22×D4×C53—42no—
40C22×D4×C53—42no—
41C22×D4×C53C2×C412yes—
42C22×D4×C53D411yescharacteristic
43C22×D4×C53D411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48#127×C53—28no—
49#124×C53—28no—
50#121×C53—24no—
51#121×C53—24no—
52#121×C53—24no—
53#121×C53D418yes—
54#120×C53—28no—
55#120×C53—28no—
56#117×C53—24no—
57C2×C4⋊C4×C53C2×C412yes—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53—22no—
61C2×C42×C53—21no—
62C24×C53—24no—
63C24×C53—22no—
64C24×C53—42no—
65C24×C53—42no—
66C24×C53—84no—
67C24×C53C22⋊C411yescharacteristic
68C24×C53C22⋊C411yescharacteristic
69C24×C53C2×D411yescharacteristic
70#41×C53—416no—
71C2×Q8×C53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—41no—
74C2×D4×C53—24no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—28no—
80C2×D4×C53—28no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—416no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—84no—
93C2×D4×C53—84no—
94C2×D4×C53C22⋊C414yes—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C22⋊C411yescharacteristic
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D414yes—
109C22×Dic53—42no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53C22⋊C414yes—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—28no—
114C22⋊C4×C53—28no—
115C42×C53—24no—
116C22⋊Dic53—42no—
117C22⋊Dic53—42no—
118#11719917—42no—
119#11719917—42no—
120C2×#88322—42no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—24no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—48no—
132C23×C53—44no—
133C23×C53—44no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—88no—
138C23×C53—88no—
139C23×C53—88no—
140C23×C53#9912yes—
141C23×C53C2×C22⋊C411yescharacteristic
142C23×C53#12012yes—
143Q8×C53—44no—
144D4×C53—44no—
145D4×C53—44no—
146D4×C53—44no—
147D4×C53—44no—
148D4×C53—48no—
149D4×C53—44no—
150D4×C53—88no—
151D4×C53—88no—
152C2×C4×C53—28no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—24no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—44no—
164C2×C4×C53—44no—
165C2×C4×C53—44no—
166C2×C4×C53#9912yes—
167C2×C4×C53C2×C22⋊C412yes—
168C2×C4×C53#12012yes—
169C2×Dic53—84no—
170C2×Dic53—82no—
171#88322—82no—
172#8516—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—24no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—44no—
182C22×C53—44no—
183C22×C53—41no—
184C22×C53—44no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—84no—
190C22×C53—84no—
191C22×C53—816no—
192C22×C53#37911yescharacteristic
193C22×C53C2×#9911yescharacteristic
194C22×C53C2×#9911yescharacteristic
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—22no—
198C4×C53—44no—
199C4×C53—41no—
200Dic53—162no—
201C2×#535D5311yescharacteristic
202#1785—531no—
203#1782—531no—
204#1510—534no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—42no—
208C2×C53—42no—
209C2×C53—81no—
210C2×C53—84no—
211C2×C53#178111yescharacteristic
212C2×C53#179512yes—
213C22×#41C2×D5311yescharacteristic
214C23×D4Dic5311yescharacteristic, complemented
215#535—28no—
216C2×#127—21no—
217C2×#124—21no—
218C2×#121Dic5312yescomplemented
219C2×#121C2×D5312yes—
220C2×#120—21no—
221C2×#120—21no—
222C2×#117Dic5311yescharacteristic, complemented
223C2×#101—1061no—
224C2×#100—1061no—
225C2×#99—1061no—
226C2×#99—1061no—
227#407—532no—
228#381—532no—
229#327—1068no—
230C53#851611yescharacteristic, complemented, Sylow
231C25—21no—
232C2×#41—24no—
233C2×#41—42no—
234C2×#41—42no—
235C22×Q8C22⋊D5311yescharacteristic
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—42no—
241C22×D4—42no—
242C22×D4C2×Dic5312yes—
243C22×D4C22⋊D5311yescharacteristic
244C22×D4C22⋊D5311yescharacteristic
245C22×D4C22⋊D5311yescharacteristic
246C23×C4C2×Dic5311yescharacteristic
247C23×C4C2×Dic5311yescharacteristic
248C23×C4C22×D5311yescharacteristic
249C2×M16—1062no—
250#127—28no—
251#124—28no—
252#121—24no—
253#121—24no—
254#121—24no—
255#121C22⋊D5318yes—
256#120—28no—
257#120—28no—
258#117—24no—
259C2×C4⋊C4C2×Dic5312yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—1062no—
264C2×C42—21no—
265#101—2122no—
266#100—2122no—
267#99—2122no—
268#99—2122no—
269C22⋊C8—1064no—
270#95—1064no—
271C24—24no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24—84no—
276C24C22⋊Dic5311yescharacteristic
277C24C22⋊Dic5311yescharacteristic
278C24C2×C22⋊D5311yescharacteristic
279#41—416no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—416no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4C22⋊Dic5314yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—2122no—
314C22×C4C22⋊Dic5311yescharacteristic
315C22×C4C22⋊Dic5311yescharacteristic
316C22×C4C22×Dic5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318C22×C4C2×C22⋊D5314yes—
319M16—2122no—
320M16—2122no—
321C2×C8—2122no—
322C4⋊C4—22no—
323C4⋊C4C22⋊Dic5314yes—
324C22⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—2122no—
328C22⋊C4—2122no—
329C42—24no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—24no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—42no—
339C23—42no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—88no—
347C23—88no—
348C23—88no—
349C23#1313313212yes—
350C23C2×C22⋊Dic5311yescharacteristic
351C23#1313318612yes—
352Q8—44no—
353D4—44no—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—48no—
358D4—44no—
359D4—88no—
360D4—88no—
361C2×C4—28no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—4242no—
376C2×C4—4244no—
377C2×C4#1313313212yes—
378C2×C4C2×C22⋊Dic5312yes—
379C2×C4#1313318612yes—
380C8—4242no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—44no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—84no—
398C22—84no—
399C22—816no—
400C22#1681047911yescharacteristic
401C22C2×#1313313211yescharacteristic
402C22C2×#1313313211yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—44no—
408C4—8482no—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—42no—
413C2—84no—
414C2—81no—
415C2#2027463011yescharacteristic
416C2#2027468512yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#535C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenterC2×#535D53
hypocenterC53#8516
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. #20274643C2semidirect
2. #20274633C2semidirect
3. #20273244C2semidirect
9. #16810587C22semidirect
10. #16810486C22semidirect
13. C23×D4×C53C4semidirect
17. C2×#121×C53C4semidirect
21. C2×#117×C53C4semidirect
214. C23×D4Dic53semidirect
218. C2×#121Dic53semidirect
222. C2×#117Dic53semidirect
230. C53#8516semidirect

Maximal subgroups

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

subgroupindex
#202746432
#202746332
#202732442
C2×C53×#5352
#851653