Finite Group Databasebeta

← #28308425

Subgroups — #28308425

Order 13568.

374 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289978C211yesmaximal, characteristic
2#20289958C211yesmaximal, characteristic
3#20289658C212yesmaximal
4#20289657C212yesmaximal
5#20279925C211yesmaximal, characteristic
6#16811643—21no—
7#16811639—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C2×#13133216C411yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14#16810544—21no—
15#16810544—21no—
16#16810530C411yescharacteristic
17#16810520—21no—
18#16810520—21no—
19#16810493C2211yescharacteristic
20#16810492C2211yescharacteristic
21C4×#11719919C411yescharacteristic
22C53×#391C411yescharacteristic
23#13133284—41no—
24C4⋊#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—41no—
28C4×#88324—41no—
29C22×Dic106—21no—
30#13133216—41no—
31#13133216—41no—
32Q8×Dic53—41no—
33Q8×Dic53—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53C2311yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×Dic106—42no—
71C2×Dic106—81no—
72C2×Dic106—81no—
73C2×#88324—42no—
74C2×#88324—42no—
75C4⋊C4×C53—41no—
76C42×C53—22no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81C4⋊Dic53—81no—
82C4⋊Dic53—81no—
83#11719919—24no—
84#11719919—41no—
85#11719919—41no—
86#11719919—81no—
87#11719919—81no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—22no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—44no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#10112yes—
111C2×C4×C53#13112yes—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53#9911yescharacteristic
126C2×Dic53#9911yescharacteristic
127Dic106—84no—
128#88324—82no—
129#88324—82no—
130#11574—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#40811yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10111yescharacteristic
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—42no—
139C4×C53—41no—
140C4×C53—81no—
141Dic53—21no—
142Dic53—42no—
143Dic53—41no—
144Dic53—41no—
145Dic53—41no—
146Dic53—81no—
147Dic53#35612yes—
148#2567—531no—
149#1467—531no—
150#1461—531no—
151#1325—532no—
152#1325—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53#147211yescharacteristic
156C2×C53#212811yescharacteristic
157C2×C53C2×#35611yescharacteristic
158C2×C4⋊Q8—531no—
159C2×C4×Q8—531no—
160#432—1061no—
161#431—1061no—
162#395—1061no—
163#395—1061no—
164#391—531no—
165#391#615711yescharacteristic
166#389—1061no—
167#389—1061no—
168#384—531no—
169#384—531no—
170C4×C4⋊C4—531no—
171#324—532no—
172#324—532no—
173#324—1061no—
174#324—1061no—
175C53#1157411yescharacteristic, complemented, Sylow
176C22×Q8—1061no—
177C22×Q8C2×#615711yescharacteristic
178C4⋊Q8—2121no—
179C4⋊Q8—2121no—
180C4×Q8—2121no—
181C4×Q8—2121no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C42—531no—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42C2×#615711yescharacteristic
198C4⋊C8—2121no—
199C4⋊C8—2121no—
200C22⋊C8—1062no—
201C22⋊C8—1062no—
202#97—2121no—
203C4×C8—2121no—
204#95—21no—
205#95—21no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—4241no—
213C2×Q8—4241no—
214C2×Q8—2122no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—1064no—
230C4⋊C4—4241no—
231C4⋊C4—4241no—
232C4⋊C4—4241no—
233C4⋊C4—4241no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C4⋊C4—2121no—
240C42—22no—
241C42—1061no—
242C42—1061no—
243C42—1061no—
244C42—1061no—
245C42—1061no—
246C42—1061no—
247C42—1062no—
248C23C2×C22⋊#615711yescharacteristic
249Q8—44no—
250Q8—4244no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—531no—
265C2×C4—531no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313327812yes—
280C2×C4#1313330412yes—
281C8—4242no—
282C8—4242no—
283C22—21no—
284C22—21no—
285C22#1681159611yescharacteristic
286C22D53⋊#9911yescharacteristic
287C22C2×#1313327811yescharacteristic
288C4—22no—
289C4—22no—
290C4—42no—
291C4—41no—
292C4—81no—
293C4—532no—
294C4—1061no—
295C4—4241no—
296C4—2122no—
297C4—2121no—
298C4—2121no—
299C4—2121no—
300C2—21no—
301C2—21no—
302C2#2028999411yescharacteristic
303C2#2029099511yescharacteristic
304C2D53⋊#35611yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11574
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
175. C53#11574semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202899582
#202896582
#202896572
#202799252
#1157453