Finite Group Databasebeta

← #27294874

Subgroups — #27294874

Order 13568.

464 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280599C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic, complemented
3#20272740C212yesmaximal, complemented
4#20272713C212yesmaximal, complemented
5C53×#1466C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810540C411yescharacteristic, complemented
11#16810538—21no—
12#16810532C2211yescharacteristic
13#16810531—21no—
14#16810514—21no—
15#16810502—21no—
16#16810486C411yescharacteristic, complemented
17#16810483—21no—
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26#13133191—41no—
27#13133191—41no—
28C2×C4⋊Dic53—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38#13133132—42no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56#97×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—42no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—81no—
79C2×C22⋊D53—81no—
80C2×C22⋊D53—81no—
81C2×C22⋊D53—81no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C2×D212—41no—
87C2×D212—81no—
88C2×C8×C53—42no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C42×C53—21no—
93C42×C53—21no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—81no—
97C22⋊Dic53—81no—
98#11719921—42no—
99#11719921—41no—
100#11719921—41no—
101#11719921—81no—
102#11719921—81no—
103C4⋊Dic53—41no—
104C4⋊Dic53—82no—
105#11719919—81no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53AGL(1,ℤ/8)11yescharacteristic
117C23×C53#13711yescharacteristic
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125D4×C53—42no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#9911yescharacteristic
143C22⋊D53—82no—
144C22⋊D53—82no—
145C22⋊D53—84no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—82no—
156C2×Dic53—82no—
157C2×Dic53—82no—
158D212—82no—
159C8×C53—82no—
160#11824—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53C2×#9911yescharacteristic
176C22×C53#41711yescharacteristic
177C22×C53#42111yescharacteristic
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—41no—
191Dic53—81no—
192Dic53—81no—
193Dic53—81no—
194Dic53—162no—
195#2635—531no—
196#1799—531no—
197#1466D5311yescharacteristic, complemented
198#1352—532no—
199#1343—532no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—41no—
203C2×C53—41no—
204C2×C53—81no—
205C2×C53#151311yescharacteristic
206C2×C53#212111yescharacteristic
207C2×C53#212811yescharacteristic
208D53—161no—
209C2×#121—531no—
210C2×C4×D4C2×D5311yescharacteristic
211#432—21no—
212C2×#99—1061no—
213#400—531no—
214#399—531no—
215#394—531no—
216#394—1061no—
217#392—531no—
218#392—1061no—
219#388—1061no—
220#386—1061no—
221#381—531no—
222#380—1061no—
223#342—532no—
224#323—21no—
225#323C2×D5312yes—
226C53#1182411yescharacteristic, complemented, Sylow
227C22×D4—531no—
228C22×D4—1061no—
229C22×D4—1061no—
230C22×D4D21211yescharacteristic
231C23×C4C4×D5311yescharacteristic
232C23×C4D21211yescharacteristic
233#121—2121no—
234#121—2121no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—2121no—
240C2×C4⋊C4C4×D5311yescharacteristic
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4C22⋊D5311yescharacteristic
249C2×C22⋊C4C22×D5311yescharacteristic
250C2×C42C22⋊D5311yescharacteristic
251C4⋊C8—41no—
252#99—2122no—
253C22⋊C8—22no—
254#97—41no—
255#95—531no—
256#95—531no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—2121no—
261C24—1061no—
262C24#1171992111yescharacteristic
263C24C2×D21211yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—4241no—
269C2×D4—4241no—
270C2×D4—4241no—
271C2×D4—4241no—
272C2×D4—4241no—
273C2×D4—2121no—
274C2×D4—2122no—
275C2×D4—2121no—
276C2×D4—2121no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—2122no—
287C22×C4#1171992111yescharacteristic
288C22×C4#1171992111yescharacteristic
289C22×C4#1171992111yescharacteristic
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292C2×C8—42no—
293C4⋊C4—41no—
294C4⋊C4—4242no—
295C4⋊C4—4241no—
296C4⋊C4—2121no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—4241no—
300C22⋊C4—4241no—
301C22⋊C4—4241no—
302C22⋊C4—4241no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C22⋊C4—2122no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42—21no—
309C42—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23—2121no—
323C23—2121no—
324C23—2121no—
325C23—2121no—
326C23C2×#1171992111yescharacteristic
327C23#1313324511yescharacteristic
328C23#1313325011yescharacteristic
329D4—42no—
330D4—42no—
331D4—4242no—
332D4—4242no—
333D4—4242no—
334D4—4244no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—4242no—
350C2×C4—4242no—
351C2×C4—4242no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4#1313313111yescharacteristic
362C2×C4#1313313111yescharacteristic
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—4241no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22C2×#1313313111yescharacteristic
386C22#1681062511yescharacteristic
387C22#1681063911yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—8482no—
395C4—4241no—
396C4—4241no—
397C4—4241no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—81no—
403C2—8481no—
404C2#2027325911yescharacteristic
405C2#2027636211yescharacteristic
406C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1466C2
socleC22×C53C2×#99
prime radical#20280599C2
cosocleC2×C22⋊C4×C53C23
hypercenter#1466D53
hypocenterC53#11824
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #20274703C2semidirect
3. #20272740C2semidirect
4. #20272713C2semidirect
5. C53×#1466C2semidirect
6. C2×#13133191C4semidirect
8. #16810562C4semidirect
10. #16810540C4semidirect
16. #16810486C4semidirect
197. #1466D53semidirect
226. C53#11824semidirect

Maximal subgroups

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

subgroupindex
#202805992
#202747032
#202727402
#202727132
C53×#14662
#1182453