Finite Group Databasebeta

← #28308915

Subgroups — #28308915

Order 13568.

452 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 88 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×#16811503C211yesmaximal, characteristic
2C2×#16811469C211yesmaximal, characteristic
3#20289715C212yesmaximal, complemented
4#20289713C212yesmaximal, complemented
5#20280531C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C22⋊#88324—21no—
8#16811503—21no—
9#16811503C2212yes—
10#16811469—21no—
11#16811469C2212yes—
12C2×#13133196C411yescharacteristic
13C22×#11719919C2211yescharacteristic
14#16810563C411yescharacteristic
15#16810563C2211yescharacteristic
16#16810553—21no—
17#16810553—21no—
18#16810544C411yescharacteristic
19#16810544C2211yescharacteristic
20#16810498—21no—
21#16810484—21no—
22C53×#380C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—22no—
25C2×#11719940—41no—
26C2×#11719940—42no—
27C22×#88324—41no—
28C22×#88324—41no—
29C22⋊#88324—22no—
30C22⋊#88324—41no—
31C22×Dic106C2×C411yescharacteristic
32C23×Dic53D411yescharacteristic
33C23×Dic53D411yescharacteristic
34#13133196—41no—
35#13133196—41no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—41no—
41C2×#11719919—41no—
42C2×#11719919C2×C411yescharacteristic
43C2×#11719919D411yescharacteristic
44C2×#11719919D411yescharacteristic
45C2×#11719919C2311yescharacteristic
46C2×C22⋊Dic53—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C23×C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—81no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—81no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53SD1612yes—
77C22×Dic53Q1612yes—
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C2×Dic106—41no—
81C2×Dic106—81no—
82#11719940—24no—
83#11719940—44no—
84#11719940—42no—
85C2×#88324—44no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—82no—
89C4⋊C4×C53—81no—
90C22⋊C4×C53—42no—
91C22⋊Dic53—41no—
92C22⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4⋊Dic53—81no—
95C4⋊Dic53—81no—
96#11719919—41no—
97#11719919—41no—
98#11719919—41no—
99#11719919—41no—
100#11719919—42no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#9911yescharacteristic
104C23×C53#9911yescharacteristic
105C23×C53#10312yes—
106C23×C53#10412yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—42no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—81no—
139C2×Dic53#10312yes—
140C2×Dic53C2×SD1611yescharacteristic
141C2×Dic53C2×Q1611yescharacteristic
142Dic106—82no—
143#88324—44no—
144#88324—82no—
145#11722—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53#32812yes—
155C22×C53#32812yes—
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10311yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—81no—
162C4×C53—81no—
163Dic53—21no—
164Dic53—22no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—81no—
169Dic53—81no—
170#2627—531no—
171C2×#343—531no—
172C2×#324—531no—
173#1354—532no—
174#1353—532no—
175C2×C53—21no—
176C2×C53—22no—
177C2×C53—22no—
178C2×C53—21no—
179C2×C53C2×#32811yescharacteristic
180C2×C53#212111yescharacteristic
181C2×C53#213311yescharacteristic
182C22×M16—1061no—
183C2×C22⋊Q8—531no—
184C22×C4⋊C4—531no—
185C2×C22⋊C8—1061no—
186#400—531no—
187#400—531no—
188#398—1061no—
189#398—1061no—
190#395—531no—
191#395—531no—
192#385—1061no—
193#381—1061no—
194#380#615711yescharacteristic
195#343—532no—
196#343—1061no—
197#324—532no—
198#324—1061no—
199C53#1172211yescharacteristic, complemented, Sylow
200C22×Q8—531no—
201C23×C4—531no—
202C23×C4—531no—
203C23×C4C2×#615711yescharacteristic
204C2×M16—1062no—
205C2×M16—1064no—
206C2×M16—2122no—
207C2×M16—2121no—
208C22×C8—2121no—
209C22×C8—2121no—
210C22⋊Q8—2121no—
211C22⋊Q8—2121no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—2121no—
222C2×C4⋊C4—2121no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1061no—
225C22⋊C8—1062no—
226C22⋊C8—2121no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95C2×#615711yescharacteristic
234#95C2×#615711yescharacteristic
235C24C22⋊#615711yescharacteristic
236C2×Q8—4241no—
237C2×Q8—2121no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—532no—
244C22×C4—532no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1062no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—4241no—
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22×#615711yescharacteristic
259M16—1064no—
260M16—2122no—
261M16—2124no—
262C2×C8—4242no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2124no—
266C4⋊C4—81no—
267C4⋊C4—4241no—
268C4⋊C4—4241no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2122no—
275C22⋊C4—42no—
276C22⋊C4—2121no—
277C22⋊C4—2121no—
278C23—41no—
279C23—41no—
280C23#1313327611yescharacteristic
281C23#1313327611yescharacteristic
282C23#1313328012yes—
283C23#1313328112yes—
284C23C2×C22⋊#615711yescharacteristic
285Q8—4242no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—532no—
299C2×C4—531no—
300C2×C4—531no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1062no—
306C2×C4—1062no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—4241no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2122no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2122no—
318C2×C4—2121no—
319C2×C4—2121no—
320C8—4242no—
321C8—2124no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1681147612yes—
331C22#1681147612yes—
332C22C2×#1313327611yescharacteristic
333C22C2×#1313328011yescharacteristic
334C22C2×#1313328111yescharacteristic
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—1061no—
340C4—1062no—
341C4—1061no—
342C4—4241no—
343C4—4241no—
344C4—2121no—
345C4—2121no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1681147611yescharacteristic
351C2#2029097211yescharacteristic
352C2#2029101211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×#11719919C23
hypercenter#380#6157
hypocenterC53#11722
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289715C2semidirect
4. #20289713C2semidirect
199. C53#11722semidirect

Maximal subgroups

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

subgroupindex
C2×#168115032
C2×#168114692
#202897152
#202897132
#202805312
#1172253