Finite Group Databasebeta

← #27294814

Subgroups — #27294814

Order 13568.

496 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274789C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272739C212yesmaximal, complemented
4#20272736C212yesmaximal, complemented
5C53×#2626C211yesmaximal, characteristic
6#16810587—21no—
7#16810479—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic, complemented
13C22×C22⋊C4×C53C2211yescharacteristic
14C53×#399C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#396—21no—
17C53×#396—21no—
18C53×#393C411yescharacteristic, complemented
19C53×#393C2211yescharacteristic
20C53×#385—21no—
21C53×#380C411yescharacteristic, complemented
22C53×#379—21no—
23C2×#11719917—41no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—41no—
26C2×C22⋊Dic53—41no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29C25×C53D411yescharacteristic
30C22×Q8×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C22⋊Q8×C53—41no—
34C22⋊Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—41no—
47C2×C22⋊C4×C53—41no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—22no—
59C24×C53—41no—
60C24×C53—81no—
61C24×C53—81no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—41no—
66C2×Q8×C53—81no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—81no—
95C22⋊C4×C53—81no—
96C22⋊C4×C53—81no—
97C22⋊C4×C53—81no—
98C22⋊Dic53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—82no—
101C22⋊Dic53—82no—
102#11719917—82no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—84no—
124C23×C53—84no—
125C23×C53#9911yescharacteristic
126C23×C53#9911yescharacteristic
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#13712yes—
129Q8×C53—82no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×Dic53—82no—
155C2×Dic53—82no—
156C2×Dic53—82no—
157#88322—82no—
158#11811—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—84no—
180C22×C53—84no—
181C22×C53C2×#9911yescharacteristic
182C22×C53#41911yescharacteristic
183C22×C53#42111yescharacteristic
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—81no—
192Dic53—162no—
193#2626D5311yescharacteristic
194#1819—531no—
195#1462—531no—
196#1352—532no—
197#1351—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—42no—
201C2×C53—41no—
202C2×C53—41no—
203C2×C53—81no—
204C2×C53#151111yescharacteristic
205C2×C53#212111yescharacteristic
206C2×C53#212711yescharacteristic
207C2×C22⋊Q8Dic5311yescharacteristic, complemented
208C22×C22⋊C4C2×D5311yescharacteristic
209#407—1061no—
210#399Dic5311yescharacteristic, complemented
211#399C2×D5311yescharacteristic
212#396—21no—
213#396—21no—
214#393Dic5311yescharacteristic, complemented
215#393C2×D5311yescharacteristic
216#385—21no—
217#380Dic5311yescharacteristic, complemented
218#379—21no—
219#379—1061no—
220#342—532no—
221#342—1061no—
222#323—532no—
223#323—1061no—
224C53#1181111yescharacteristic, complemented, Sylow
225C25C22⋊D5311yescharacteristic
226C22×Q8C2×Dic5311yescharacteristic
227C23×C4C2×Dic5311yescharacteristic
228C23×C4C22⋊D5311yescharacteristic
229C2×M16—2121no—
230C22⋊Q8—41no—
231C22⋊Q8—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic5311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—2121no—
247C2×C22⋊C4—2121no—
248C2×C22⋊C4C2×Dic5311yescharacteristic
249C2×C22⋊C4C22⋊D5311yescharacteristic
250C2×C22⋊C4C22⋊D5311yescharacteristic
251C2×C22⋊C4C22×D5311yescharacteristic
252C22⋊C8—1062no—
253C22⋊C8—2121no—
254#95—21no—
255#95—21no—
256#95C2×Dic5311yescharacteristic
257#95C2×Dic5311yescharacteristic
258C24—21no—
259C24—21no—
260C24—22no—
261C24—41no—
262C24—81no—
263C24—81no—
264C24C22⋊Dic5311yescharacteristic
265C24C22⋊Dic5311yescharacteristic
266C24C2×C22⋊D5311yescharacteristic
267C2×Q8—41no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—2122no—
280C22×C4C22⋊Dic5311yescharacteristic
281C22×C4C22⋊Dic5311yescharacteristic
282C22×C4C22×Dic5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284M16—4242no—
285C2×C8—2122no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—4242no—
303C22⋊C4—4242no—
304C22⋊C4—2122no—
305C22⋊C4—2122no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—84no—
327C23#1313313211yescharacteristic
328C23#1313313211yescharacteristic
329C23C2×C22⋊Dic5311yescharacteristic
330C23#1313324812yes—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—4242no—
359C8—4242no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×#1313313211yescharacteristic
383C22#1681063211yescharacteristic
384C22#1681064011yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—8482no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2#2027324811yescharacteristic
401C2#2027636111yescharacteristic
402C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2626C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2626D53
hypocenterC53#11811
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
4. #20272736C2semidirect
12. C2×C22⋊Q8×C53C4semidirect
14. C53×#399C4semidirect
18. C53×#393C4semidirect
21. C53×#380C4semidirect
207. C2×C22⋊Q8Dic53semidirect
210. #399Dic53semidirect
214. #393Dic53semidirect
217. #380Dic53semidirect
224. C53#11811semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202730422
#202727392
#202727362
C53×#26262
#1181153