Finite Group Databasebeta

← #28303901

Subgroups — #28303901

Order 13568.

604 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C2×#16811645C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3C2×Q8⋊#88324C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277253C211yesmaximal, characteristic
6#16811645—24no—
7C2×C4⋊#88324C2212yes—
8C2×#13133274—21no—
9C2×#13133274—21no—
10C2×#13133272—21no—
11C2×C4×#88324C2212yes—
12#16811489—21no—
13Q8⋊#88324—24no—
14C2×#13133193C411yescharacteristic
15C2×Q8×Dic53C2211yescharacteristic
16C2×C42×D53C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810510—21no—
19#16810497—21no—
20#16810480C411yescharacteristic
21C4⋊C4×Dic53C2211yescharacteristic
22#16810474—21no—
23#16810470—21no—
24C53×#391C411yescharacteristic
25C22×#88324—22no—
26C22×#88324—22no—
27C4⋊#88324—22no—
28C4⋊#88324D414yes—
29#13133274—24no—
30#13133274—24no—
31#13133272—24no—
32C4×#88324—22no—
33C4×#88324—22no—
34C22×C4×D53C2×C411yescharacteristic
35C22×C4×D53D411yescharacteristic
36C22×C4×D53D411yescharacteristic
37#13133193—22no—
38#13133193—22no—
39Q8×Dic53—22no—
40Q8×Dic53—22no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53D411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53C2311yescharacteristic
53C42×D53—22no—
54C42×D53—22no—
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58C22×Q8×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53C2×C411yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63C23×D53C22⋊C411yescharacteristic
64C2×Q8×C53—21no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53M1612yes—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—22no—
83C2×C4×D53—21no—
84C2×C4×D53—22no—
85C2×C4×D53—21no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—41no—
89C2×#88324—24no—
90C2×#88324—24no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C2×#88324—42no—
95C4⋊C4×C53—22no—
96C42×C53—22no—
97#11719921—24no—
98#11719921—42no—
99C4⋊Dic53—21no—
100C4⋊Dic53—22no—
101C4⋊Dic53—42no—
102C4⋊Dic53M1612yes—
103#11719919—24no—
104#11719919—42no—
105C4×Dic53—21no—
106C4×Dic53—22no—
107C4×Dic53—21no—
108C4×Dic53—41no—
109C4×Dic53—41no—
110C4×Dic53—42no—
111C4×Dic53C2×D412yes—
112C4×Dic53#4112yes—
113C23×C53C2×C22⋊C411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53#10412yes—
121Q8×C53—24no—
122Q8×C53—42no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C4×D412yes—
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13112yes—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—22no—
144C2×Dic53—21no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53#12011yescharacteristic
151C2×Dic53#12112yes—
152C2×Dic53#12311yescharacteristic
153C2×Dic53#13512yes—
154C2×Dic53#13712yes—
155C4×D53—44no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159C4×D53—42no—
160#88324—44no—
161#88324—44no—
162#8941—531nomaximal, Sylow
163C22×C53#38611yescharacteristic
164C22×C53#40711yescharacteristic
165C22×C53#40811yescharacteristic
166C22×C53#41611yescharacteristic
167C22×C53#41911yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C22×C53#42111yescharacteristic
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—42no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—81no—
181C4×C53#43512yes—
182C4×C53#43612yes—
183Dic53—22no—
184Dic53—41no—
185Dic53—42no—
186Dic53—81no—
187Dic53#45212yes—
188Dic53#48512yes—
189C2×#432—531no—
190C4×C22⋊Q8—531no—
191C4×C22⋊C8—531no—
192C2×Q8⋊C8—532no—
193#1279—532no—
194C2×C53#148811yescharacteristic
195C2×C53#149111yescharacteristic
196C2×C53#149411yescharacteristic
197C2×C53#149711yescharacteristic
198C2×C53#187011yescharacteristic
199C2×C53#187111yescharacteristic
200C2×C53#187311yescharacteristic
201D53—42no—
202C2×C22⋊Q8—531no—
203C2×C4×Q8—531no—
204C22×C42—531no—
205#432—1064no—
206C2×C4⋊C8—532no—
207C2×C22⋊C8—1061no—
208C2×C22⋊C8—1061no—
209C2×#97—1061no—
210C2×C4×C8—532no—
211#391#615711yescharacteristic
212#389—531no—
213#387—1061no—
214#385—1061no—
215#380—531no—
216C4×C4⋊C4—531no—
217C4×C4⋊C4—1061no—
218C4×C22⋊C4—1061no—
219#336—1061no—
220Q8⋊C8—1064no—
221C53#894111yescharacteristic, complemented, Sylow
222C22×Q8C2×#615711yescharacteristic
223C23×C4—531no—
224C23×C4—531no—
225C23×C4—531no—
226C22×C8—1062no—
227C22×C8—1062no—
228C22⋊Q8—1062no—
229C22⋊Q8—1062no—
230C4×Q8—1062no—
231C4×Q8—1062no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4C2×#615711yescharacteristic
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—1062no—
244C2×C42—1062no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42C2×#615711yescharacteristic
248C4⋊C8—534no—
249C4⋊C8—1062no—
250C22⋊C8—1064no—
251C22⋊C8—1064no—
252#97—1064no—
253C4×C8—1062no—
254C4×C8—1062no—
255#95—21no—
256#95—21no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260C24—531no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8#1171993612yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1061no—
277C22×C4—1062no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—2121no—
283C22×C4C22⋊#615711yescharacteristic
284C22×C4C22⋊#615711yescharacteristic
285C22×C4C22×#615711yescharacteristic
286C2×C8—1064no—
287C2×C8—1064no—
288C2×C8—2122no—
289C2×C8—2122no—
290C2×C8—2122no—
291C2×C8—2122no—
292C4⋊C4—22no—
293C4⋊C4—532no—
294C4⋊C4—1064no—
295C4⋊C4—1061no—
296C4⋊C4—1062no—
297C4⋊C4—2122no—
298C4⋊C4—2122no—
299C22⋊C4—1064no—
300C22⋊C4—2122no—
301C42—22no—
302C42—532no—
303C42—532no—
304C42—1061no—
305C42—1061no—
306C42—1062no—
307C42—2121no—
308C42—2121no—
309C42—2122no—
310C23—532no—
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23C2×C22⋊#615711yescharacteristic
318Q8—24no—
319Q8—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—532no—
335C2×C4—532no—
336C2×C4—532no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1062no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2124no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4D4×#615712yes—
354C2×C4D53⋊M1611yescharacteristic
355C2×C4C2×#1171993611yescharacteristic
356C2×C4#1313330412yes—
357C8—2124no—
358C8—2124no—
359C22—1062no—
360C22—1062no—
361C22—1062no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22#1681157111yescharacteristic
367C22#1681159411yescharacteristic
368C22#1681159711yescharacteristic
369C22#1681161311yescharacteristic
370C22#1681161611yescharacteristic
371C22D53⋊#10411yescharacteristic
372C22#1681162211yescharacteristic
373C4—42no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—532no—
378C4—532no—
379C4—1062no—
380C4—4241no—
381C4—2122no—
382C4—2121no—
383C4#1681165212yes—
384C4#1681165612yes—
385C2—2122no—
386C2#2029004911yescharacteristic
387C2#2029005811yescharacteristic
388C2#2029006711yescharacteristic
389C2#2029007811yescharacteristic
390C2#2029069311yescharacteristic
391C2#2029069711yescharacteristic
392C2#2029070111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#8941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#88324C2semidirect
4. #20289551C2semidirect
221. C53#8941semidirect

Maximal subgroups

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

subgroupindex
C2×#168116452
C4×#131332742
C2×Q8⋊#883242
#202895512
#202772532
#894153