Finite Group Databasebeta

← #28309398

Subgroups — #28309398

Order 13568.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#20290614C211yesmaximal, characteristic
2C2×#16811469C211yesmaximal, characteristic
3#20289716C212yesmaximal, complemented
4#20289699C212yesmaximal, complemented
5#20280096C211yesmaximal, characteristic
6#16811593—21no—
7C2×C22⋊#88324—21no—
8#16811503—21no—
9#16811503C2212yes—
10#16811469—21no—
11#16811469C2212yes—
12C22×#11719919C2211yescharacteristic
13#16810554C411yescharacteristic
14#16810554C2211yescharacteristic
15#16810542C411yescharacteristic
16#16810542C2211yescharacteristic
17#16810518—21no—
18#16810518—21no—
19#16810495—21no—
20#16810495—21no—
21C4×C22⋊Dic53C411yescharacteristic
22C2×C22⋊Q8×C53C411yescharacteristic
23C2×#11719940—22no—
24C22×#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—41no—
27C22⋊#88324—41no—
28C22⋊#88324—41no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—41no—
36C2×#11719919—41no—
37C2×#11719919C2×C411yescharacteristic
38C2×#11719919D411yescharacteristic
39C2×#11719919D411yescharacteristic
40C2×#11719919C2311yescharacteristic
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124C2×C411yescharacteristic
48#13133124C2×C411yescharacteristic
49C22×Q8×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C22⋊Q8×C53—41no—
52C22⋊Q8×C53—41no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C2×Q8×C53—41no—
58C2×Q8×C53—81no—
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—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—22no—
73C22×Dic53—41no—
74C22×Dic53—81no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79#11719940—42no—
80#11719940—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—82no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊Dic53—24no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—42no—
94#11719919—42no—
95#11719919—81no—
96#11719919—81no—
97C4×Dic53—81no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53#9911yescharacteristic
103C23×C53#9911yescharacteristic
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
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—81no—
117C2×C4×C53—81no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—81no—
137C2×Dic53#13512yes—
138#88324—82no—
139#88324—82no—
140#11854—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53#35612yes—
151C22×C53C2×#9911yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53#42111yescharacteristic
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—81no—
157C4×C53—81no—
158Dic53—22no—
159Dic53—42no—
160Dic53—42no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164#2583—531no—
165#1821—531no—
166C2×#324—531no—
167#1354—532no—
168#1346—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—22no—
172C2×C53—41no—
173C2×C53#151611yescharacteristic
174C2×C53#212111yescharacteristic
175C2×C53C2×#35611yescharacteristic
176C2×C22⋊Q8#615711yescharacteristic
177C22×C4⋊C4—531no—
178#407—1061no—
179C2×C22⋊C8—1061no—
180#398—531no—
181#398—531no—
182#395—531no—
183#395—531no—
184#389—1061no—
185#389—1061no—
186#385—1061no—
187#385—1061no—
188C4×C22⋊C4—531no—
189#343—532no—
190#343—1061no—
191#324—532no—
192#324—1061no—
193C53#1185411yescharacteristic, complemented, Sylow
194C22×Q8C2×#615711yescharacteristic
195C23×C4—531no—
196C23×C4—531no—
197C23×C4C2×#615711yescharacteristic
198C2×M16—1062no—
199C22×C8—2121no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—2121no—
212C2×C4⋊C4—2121no—
213C2×C4⋊C4C2×#615711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—1061no—
216C2×C42—1061no—
217C22⋊C8—1062no—
218C22⋊C8—2121no—
219C22⋊C8—2121no—
220C22⋊C8—2121no—
221#95—531no—
222#95—531no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227C24C22⋊#615711yescharacteristic
228C2×Q8—41no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—81no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—1062no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—4241no—
247C22×C4—2121no—
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22×#615711yescharacteristic
250M16—2122no—
251M16—2122no—
252C2×C8—4242no—
253C2×C8—2122no—
254C2×C8—2122no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—4241no—
258C4⋊C4—4241no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—1064no—
268C42—4241no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23#1313327511yescharacteristic
274C23#1313327511yescharacteristic
275C23C2×C22⋊#615711yescharacteristic
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—532no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—4241no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2122no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C8—4242no—
310C8—4242no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1681153312yes—
321C22C2×#1313327511yescharacteristic
322C22#1681161311yescharacteristic
323C22#1681162011yescharacteristic
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—1062no—
329C4—2122no—
330C4—2122no—
331C4—2121no—
332C4—2121no—
333C4—2121no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2029014111yescharacteristic
339C2#2029097011yescharacteristic
340C2C2×#1681153311yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#11854
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289716C2semidirect
4. #20289699C2semidirect
193. C53#11854semidirect

Maximal subgroups

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

subgroupindex
#202906142
C2×#168114692
#202897162
#202896992
#202800962
#1185453