Finite Group Databasebeta

← #28299548

Subgroups — #28299548

Order 13568.

588 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290607C211yesmaximal, characteristic, complemented
2#20290572C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5C4×#13133188C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133274—21no—
8C2×#13133274—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133188C411yescharacteristic
13C2×C4×C22⋊D53—21no—
14C2×C42×D53C2211yescharacteristic
15#16810526C2211yescharacteristic
16#16810519C2211yescharacteristic
17#16810511—21no—
18#16810499—21no—
19#16810480C411yescharacteristic
20C4×C4⋊Dic53C411yescharacteristic
21C4×C22⋊Dic53—21no—
22C2×C4×D4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133274—42no—
27#13133274—42no—
28C22×C22⋊D53—21no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133188—22no—
33#13133188C814yescomplemented
34C4×C22⋊D53—44no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×#11719921—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46C42×D53—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53C2×C411yescharacteristic
51C23×C4×C53—21no—
52C4×D4×C53—42no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C2×C42×C53C2×C411yescharacteristic
56C24×C53—21no—
57C23×D53C22⋊C411yescharacteristic
58C2×D4×C53—21no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53C2×C812yes—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C2×C22⋊D53—24no—
70C2×C22⋊D53—42no—
71C2×C22⋊D53—42no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53C2×C812yes—
87C2×C4×D53M1612yes—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C42×C53—22no—
97C22⋊Dic53—24no—
98C22⋊Dic53—41no—
99C22⋊Dic53—42no—
100C22⋊Dic53—41no—
101#11719921—42no—
102C4⋊Dic53—21no—
103C4⋊Dic53—41no—
104C4⋊Dic53C2×C812yes—
105#11719919—42no—
106C4×Dic53—22no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109C4×Dic53—42no—
110C23×C53—22no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53C2×C22⋊C411yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53C22⋊C812yes—
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—21no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—84no—
138C2×C4×C53#9912yes—
139C2×C4×C53C22×C811yescharacteristic
140C2×C4×C53C2×M1611yescharacteristic
141C22⋊D53—48no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53C22⋊C812yes—
155C2×Dic53AGL(1,ℤ/8)11yescharacteristic
156C2×Dic53AGL(1,ℤ/8)12yes—
157C2×Dic53#13711yescharacteristic
158C4×D53—42no—
159C4×D53—42no—
160C4×D53—44no—
161C4×D53—42no—
162C4×D53—42no—
163#88324—82no—
164#88324—82no—
165#7599—531nomaximal, Sylow
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53C2×C22⋊C811yescharacteristic
174C22×C53C2×#9911yescharacteristic
175C22×C53#41311yescharacteristic
176C22×C53#41711yescharacteristic
177C22×C53#41811yescharacteristic
178C22×C53#42111yescharacteristic
179C22×C53#42111yescharacteristic
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—22no—
190C4×C53—41no—
191C4×C53—81no—
192Dic53—22no—
193Dic53—22no—
194Dic53—21no—
195Dic53—21no—
196Dic53—42no—
197Dic53—81no—
198C4×#121—531no—
199#1817—531no—
200#1797—531no—
201#1271—532no—
202#1270—532no—
203C2×C53—82no—
204C2×C53#145911yescharacteristic
205C2×C53#147711yescharacteristic
206C2×C53#147811yescharacteristic
207C2×C53#151011yescharacteristic
208C2×C53#151311yescharacteristic
209C2×C53#153511yescharacteristic
210C2×C53#155711yescharacteristic
211D53—42no—
212C2×#121—531no—
213C2×C4×D4—1061no—
214C2×C4×D4#615711yescharacteristic
215C22×C42—531no—
216C2×C4⋊C8—1061no—
217C2×C22⋊C8—1061no—
218C2×C22⋊C8—1061no—
219#390—531no—
220#389—531no—
221#388—1061no—
222#386—1061no—
223#380—531no—
224C4×C4⋊C4—531no—
225C4×C22⋊C4—1061no—
226#336—532no—
227#336—532no—
228#336—1061no—
229C53#759911yescharacteristic, complemented, Sylow
230C22×D4—1061no—
231C22×D4C2×#615711yescharacteristic
232C23×C4—21no—
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C22×C8—1062no—
237C22×C8—1062no—
238#121—534no—
239#121—1062no—
240C4×D4—42no—
241C4×D4—2124no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4C2×#615711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—1061no—
255C2×C42—2122no—
256C2×C42C2×#615711yescharacteristic
257C4⋊C8—2122no—
258C22⋊C8—2122no—
259C22⋊C8—2122no—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263C24—21no—
264C24—531no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—1064no—
269C2×D4—2122no—
270C2×D4—2122no—
271C2×D4D53⋊C812yes—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1062no—
286C22×C4—1061no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4C22⋊#615711yescharacteristic
293C22×C4C22⋊#615711yescharacteristic
294C22×C4C22×#615711yescharacteristic
295C2×C8—2122no—
296C2×C8—2122no—
297C2×C8—2122no—
298C2×C8—2122no—
299C2×C8—2122no—
300C2×C8—2122no—
301C4⋊C4—41no—
302C4⋊C4—532no—
303C4⋊C4—1061no—
304C4⋊C4—2121no—
305C4⋊C4—2122no—
306C22⋊C4—42no—
307C22⋊C4—1064no—
308C22⋊C4—2122no—
309C22⋊C4—2122no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C42—22no—
313C42—1062no—
314C42—2121no—
315C42—2121no—
316C42—2122no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—532no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23C2×C22⋊#615711yescharacteristic
332D4—42no—
333D4—42no—
334D4—2128no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4—532no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—532no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1062no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2124no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4#1313327512yes—
368C2×C4C2×D53⋊C811yescharacteristic
369C2×C4D53⋊M1611yescharacteristic
370C8—4242no—
371C8—4242no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1062no—
380C22—1062no—
381C22—1062no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22D53⋊(C22⋊C8)11yescharacteristic
387C22C2×#1313327511yescharacteristic
388C22#1681160911yescharacteristic
389C22#1681161411yescharacteristic
390C22#1681161511yescharacteristic
391C22#1681162011yescharacteristic
392C22#1681162111yescharacteristic
393C4—22no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—81no—
398C4—1062no—
399C4—1061no—
400C4—1061no—
401C4—1062no—
402C4—4241no—
403C4—2122no—
404C2—82no—
405C2—2122no—
406C2#2028995011yescharacteristic
407C2#2029001111yescharacteristic
408C2#2029001511yescharacteristic
409C2#2029012011yescharacteristic
410C2#2029013011yescharacteristic
411C2#2029019911yescharacteristic
412C2#2029024611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133188C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7599
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290607C2semidirect
2. #20290572C2semidirect
3. #20289534C2semidirect
4. #20289531C2semidirect
33. #13133188C8semidirect
229. C53#7599semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202905722
#202895342
#202895312
C4×#131331882
#759953