Finite Group Databasebeta

← #28299961

Subgroups — #28299961

Order 13568.

464 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290572C211yesmaximal, characteristic
2#20290570C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283056C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×#13133274—21no—
9C2×#13133274—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133193C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810506—21no—
18#16810505—21no—
19#16810497—21no—
20#16810485—21no—
21#16810480C411yescharacteristic
22C53×#391C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—42no—
27#13133274—42no—
28#13133274—42no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133193—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43C42×D53—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54C23×D53C22⋊C411yescharacteristic
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53SD1612yes—
78C2×C4×D53Q1612yes—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C4⋊C4×C53—22no—
86C42×C53—41no—
87#11719921—42no—
88#11719921—42no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C4×Dic53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53#10312yes—
106Q8×C53—82no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#10312yes—
121C2×C4×C53#13112yes—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53#9912yes—
135C2×Dic53C2×SD1611yescharacteristic
136C2×Dic53C2×Q1611yescharacteristic
137C2×Dic53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53#13711yescharacteristic
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—44no—
142C4×D53—42no—
143C4×D53—42no—
144#88324—82no—
145#88324—82no—
146#7715—531nomaximal, Sylow
147C22×C53#40811yescharacteristic
148C22×C53C2×#9911yescharacteristic
149C22×C53#41311yescharacteristic
150C22×C53C2×#10311yescharacteristic
151C22×C53#41711yescharacteristic
152C22×C53#42111yescharacteristic
153C22×C53#42111yescharacteristic
154C2×D53—22no—
155C2×D53—22no—
156C2×D53—22no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—22no—
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—81no—
172Dic53—81no—
173#2855—531no—
174#1797—531no—
175#1797—531no—
176#1272—532no—
177#1271—532no—
178C2×C53#146811yescharacteristic
179C2×C53#148111yescharacteristic
180C2×C53#149511yescharacteristic
181C2×C53#151311yescharacteristic
182C2×C53#152011yescharacteristic
183C2×C53#153511yescharacteristic
184C2×C53#155011yescharacteristic
185D53—42no—
186C2×C22⋊Q8—531no—
187C22×C42—531no—
188C2×C4⋊C8—1061no—
189C2×C4⋊C8—1061no—
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192#391—531no—
193#391#615711yescharacteristic
194#389—531no—
195#389—531no—
196#387—1061no—
197#387—1061no—
198#385—1061no—
199#381—1061no—
200#380—531no—
201#336—532no—
202#336—532no—
203C53#771511yescharacteristic, complemented, Sylow
204C22×Q8C2×#615711yescharacteristic
205C23×C4—531no—
206C23×C4—531no—
207C23×C4—531no—
208C22×C8—1062no—
209C22×C8—1062no—
210C22⋊Q8—2122no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4C2×#615711yescharacteristic
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—531no—
222C2×C42—2122no—
223C2×C42C2×#615711yescharacteristic
224C4⋊C8—2122no—
225C4⋊C8—2122no—
226C22⋊C8—2122no—
227C22⋊C8—2122no—
228#95—21no—
229#95—21no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24—531no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—532no—
245C22×C4—532no—
246C22×C4—1061no—
247C22×C4—1062no—
248C22×C4—1061no—
249C22×C4—1062no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4—2121no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22⋊#615711yescharacteristic
259C22×C4C22×#615711yescharacteristic
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C4⋊C4—22no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2122no—
270C4⋊C4—2122no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C42—41no—
276C42—1062no—
277C42—1062no—
278C42—2121no—
279C23—532no—
280C23—1061no—
281C23—1061no—
282C23—1061no—
283C23—1061no—
284C23—1061no—
285C23—1061no—
286C23C2×C22⋊#615711yescharacteristic
287Q8—82no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—532no—
302C2×C4—531no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2124no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4#1313328012yes—
324C2×C4#1313330412yes—
325C8—4242no—
326C8—4242no—
327C22—1062no—
328C22—1062no—
329C22—1062no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22#1681159711yescharacteristic
335C22D53⋊#9911yescharacteristic
336C22#1681160911yescharacteristic
337C22C2×#1313328011yescharacteristic
338C22#1681161411yescharacteristic
339C22#1681162011yescharacteristic
340C22#1681162211yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—1061no—
347C4—1061no—
348C4—1061no—
349C4—1061no—
350C4—1062no—
351C4—4241no—
352C4—4241no—
353C2—2122no—
354C2#2028998111yescharacteristic
355C2#2029002411yescharacteristic
356C2#2029007111yescharacteristic
357C2#2029013311yescharacteristic
358C2#2029015311yescharacteristic
359C2#2029020111yescharacteristic
360C2#2029023211yescharacteristic
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#7715
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289533C2semidirect
203. C53#7715semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202905702
#202895352
#202895332
#202830562
#771553