Finite Group Databasebeta

← #28299982

Subgroups — #28299982

Order 13568.

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

normal subgroups only

This group has 444 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
1#20290606C211yesmaximal, characteristic
2#20290605C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5#20283049C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C22×C4×Dic53C2211yescharacteristic
13#16810530C411yescharacteristic
14#16810530C2211yescharacteristic
15#16810519C411yescharacteristic
16#16810519C2211yescharacteristic
17#16810504—21no—
18#16810504—21no—
19#16810495—21no—
20#16810484—21no—
21#16810483C411yescharacteristic
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—42no—
26C22⋊#88324—42no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—42no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53C2311yescharacteristic
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C22⋊Q8×C53—42no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52C24×C53C22⋊C411yescharacteristic
53C2×Q8×C53—22no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—22no—
86C4⋊Dic53—41no—
87#11719919—42no—
88#11719919—42no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#9912yes—
100C23×C53C2×C22⋊C411yescharacteristic
101Q8×C53—82no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—44no—
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#13112yes—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—44no—
126C2×Dic53—44no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53#13512yes—
134C2×Dic53#13712yes—
135#88324—82no—
136#88324—82no—
137#7721—531nomaximal, Sylow
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53#40811yescharacteristic
146C22×C53C2×#9911yescharacteristic
147C22×C53#41311yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41911yescharacteristic
150C22×C53#42111yescharacteristic
151C22×C53#42111yescharacteristic
152C4×C53—22no—
153C4×C53—41no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—22no—
157Dic53—22no—
158Dic53—41no—
159Dic53—41no—
160Dic53—81no—
161Dic53—81no—
162#2855—531no—
163#1817—531no—
164#1817—531no—
165#1272—532no—
166#1271—532no—
167C2×C53—42no—
168C2×C53#146811yescharacteristic
169C2×C53#148211yescharacteristic
170C2×C53#149711yescharacteristic
171C2×C53#151111yescharacteristic
172C2×C53#151611yescharacteristic
173C2×C53#153511yescharacteristic
174C2×C53#155411yescharacteristic
175C2×C22⋊Q8#615711yescharacteristic
176C22×C42—531no—
177C2×C22⋊C8—1061no—
178C2×C22⋊C8—1061no—
179#391—531no—
180#391—531no—
181#389—531no—
182#389—531no—
183#387—1061no—
184#387—1061no—
185#385—1061no—
186#381—1061no—
187#380—531no—
188#336—532no—
189#336—532no—
190#336—1061no—
191#336—1061no—
192C53#772111yescharacteristic, complemented, Sylow
193C22×Q8C2×#615711yescharacteristic
194C23×C4—531no—
195C23×C4—531no—
196C23×C4C2×#615711yescharacteristic
197C22×C8—1062no—
198C22×C8—1062no—
199C22⋊Q8—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4C2×#615711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1061no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—2122no—
213C22⋊C8—2122no—
214C22⋊C8—2122no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222C24C22⋊#615711yescharacteristic
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4C22⋊#615711yescharacteristic
244C22×C4C22×#615711yescharacteristic
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—1062no—
254C4⋊C4—2121no—
255C4⋊C4—2122no—
256C4⋊C4—2122no—
257C22⋊C4—42no—
258C22⋊C4—2122no—
259C42—2121no—
260C42—2121no—
261C42—2121no—
262C42—2121no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#1313327512yes—
270C23C2×C22⋊#615711yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—532no—
285C2×C4—532no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—2124no—
297C2×C4—2124no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4#1313330412yes—
305C8—4242no—
306C8—4242no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#1681159611yescharacteristic
315C22C2×#1313327511yescharacteristic
316C22#1681161011yescharacteristic
317C22#1681161311yescharacteristic
318C22#1681161611yescharacteristic
319C22#1681162011yescharacteristic
320C22#1681162211yescharacteristic
321C4—22no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—1062no—
326C4—1062no—
327C4—4241no—
328C4—4241no—
329C4—2121no—
330C4—2121no—
331C2—42no—
332C2#2028998011yescharacteristic
333C2#2029002811yescharacteristic
334C2#2029007711yescharacteristic
335C2#2029012511yescharacteristic
336C2#2029014111yescharacteristic
337C2#2029020211yescharacteristic
338C2#2029024111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7721
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289534C2semidirect
192. C53#7721semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202906052
#202895352
#202895342
#202830492
#772153