Finite Group Databasebeta

← #27294978

Subgroups — #27294978

Order 13568.

506 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 83 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#20279873C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic
3#20272735C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C53×#1819C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C2×C22⋊#88322—21no—
8#16810548C411yescharacteristic, complemented
9#16810548C2211yescharacteristic
10#16810535C411yescharacteristic, complemented
11#16810535C2211yescharacteristic
12#16810517—21no—
13#16810517—21no—
14#16810486—21no—
15#16810479—21no—
16C4×C22⋊Dic53C411yescharacteristic, complemented
17#16810347—21no—
18#16810347C2212yes—
19C22×C22⋊C4×C53C2211yescharacteristic
20C53×#379—21no—
21C53×#342—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C25×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53—41no—
50C2×C22⋊C4×C53—41no—
51C2×C22⋊C4×C53—41no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53C2311yescharacteristic
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—22no—
59C24×C53—41no—
60C24×C53—42no—
61C24×C53—81no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×Dic106—41no—
81C2×Dic106—81no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—82no—
89C22⋊C4×C53—82no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C22⋊Dic53—42no—
95C22⋊Dic53—81no—
96C22⋊Dic53—81no—
97C4⋊Dic53—41no—
98#11719919—42no—
99C4×Dic53—81no—
100C2×#88322—42no—
101C2×#88322—82no—
102C23×C53—22no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—84no—
122C23×C53—84no—
123C23×C53#9911yescharacteristic
124C23×C53#9911yescharacteristic
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53#13512yes—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—82no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154Dic106—82no—
155#88322—82no—
156#11845—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—81no—
175C22×C53—84no—
176C22×C53—81no—
177C22×C53—84no—
178C22×C53#35412yes—
179C22×C53C2×#9911yescharacteristic
180C22×C53#41611yescharacteristic
181C22×C53#42111yescharacteristic
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—162no—
187Dic53—81no—
188Dic53—81no—
189Dic53—81no—
190Dic53—81no—
191#2561—531no—
192#1819D5311yescharacteristic
193C2×#323—531no—
194#1351—532no—
195#1344—532no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—22no—
199C2×C53—42no—
200C2×C53—41no—
201C2×C53—81no—
202C2×C53#151611yescharacteristic
203C2×C53#212111yescharacteristic
204C2×C53C2×#35411yescharacteristic
205C2×C22⋊Q8—531no—
206C22×C22⋊C4C2×D5311yescharacteristic
207C2×C22⋊C8—1061no—
208#397—531no—
209#397—531no—
210#393—531no—
211#393—531no—
212#388—1061no—
213#388—1061no—
214#381—1061no—
215#379—21no—
216#379—1061no—
217C4×C22⋊C4—531no—
218#342—21no—
219#342C2×D5312yes—
220#323—532no—
221#323—1061no—
222C53#1184511yescharacteristic, complemented, Sylow
223C25C22⋊D5311yescharacteristic
224C22×Q8—531no—
225C23×C4C4×D5311yescharacteristic
226C23×C4D21211yescharacteristic
227C22×C8—2121no—
228C22⋊Q8—2121no—
229C22⋊Q8—2121no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—1061no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4C4×D5311yescharacteristic
244C2×C22⋊C4D21211yescharacteristic
245C2×C22⋊C4C22⋊D5311yescharacteristic
246C2×C22⋊C4C22×D5311yescharacteristic
247C2×C42—1061no—
248C22⋊C8—1062no—
249C22⋊C8—2121no—
250#95—531no—
251#95—531no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255#95—1061no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—81no—
262C24#1171992111yescharacteristic
263C24#1171992111yescharacteristic
264C24C2×C22⋊D5311yescharacteristic
265C2×Q8—4241no—
266C2×Q8—2121no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4#1171992111yescharacteristic
279C22×C4#1171992111yescharacteristic
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4C2×D21211yescharacteristic
282C2×C8—4242no—
283C2×C8—2122no—
284C4⋊C4—2121no—
285C4⋊C4—2122no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—4241no—
295C22⋊C4—4241no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C22⋊C4—2121no—
299C22⋊C4—2121no—
300C22⋊C4—2122no—
301C42—4241no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23—84no—
323C23#1313313111yescharacteristic
324C23#1313313111yescharacteristic
325C23C2×#1171992111yescharacteristic
326C23#1313323712yes—
327Q8—4242no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C8—4242no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22#1681041712yes—
378C22C2×#1313313111yescharacteristic
379C22#1681061711yescharacteristic
380C22#1681064111yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—162no—
386C4—4241no—
387C4—4241no—
388C4—4241no—
389C4—4241no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2#2027326911yescharacteristic
397C2#2027635911yescharacteristic
398C2C2×#1681041711yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1819C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1819D53
hypocenterC53#11845
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20272735C2semidirect
4. #20272714C2semidirect
6. C2×#13133196C4semidirect
8. #16810548C4semidirect
10. #16810535C4semidirect
16. C4×C22⋊Dic53C4semidirect
222. C53#11845semidirect

Maximal subgroups

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

subgroupindex
#202798732
C2×#168103472
#202727352
#202727142
C53×#18192
#1184553