Finite Group Databasebeta

← #28308920

Subgroups — #28308920

Order 13568.

432 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290421C211yesmaximal, characteristic
2#20289947C211yesmaximal, characteristic
3#20289714C212yesmaximal, complemented
4#20289713C212yesmaximal, complemented
5#20280541C211yesmaximal, characteristic
6C22×#11719940—21no—
7#16811640—21no—
8#16811503—21no—
9#16811503C2212yes—
10#16811469C2212yes—
11#16811466—21no—
12C2×#13133216C411yescharacteristic
13C2×#13133173C2211yescharacteristic
14#16810563C2211yescharacteristic
15#16810561C2211yescharacteristic
16#16810559C411yescharacteristic
17#16810543—21no—
18#16810535—21no—
19#16810518—21no—
20#16810487C411yescharacteristic
21#16810481—21no—
22C53×#399C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—22no—
25C2×#11719940—41no—
26C2×#11719940—42no—
27C22×#88324—41no—
28#13133284—41no—
29C22⋊#88324—22no—
30C4×#88324—41no—
31C22×Dic106—21no—
32C23×Dic53D411yescharacteristic
33#13133216—41no—
34#13133216—41no—
35#13133173—41no—
36#13133173—41no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919D411yescharacteristic
42C2×#11719919C2311yescharacteristic
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53D411yescharacteristic
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124C2×C411yescharacteristic
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53SD1612yes—
71C22×Dic53Q1612yes—
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×Dic106—42no—
75C2×Dic106—81no—
76C2×Dic106—81no—
77#11719940—24no—
78#11719940—44no—
79#11719940—42no—
80C2×#88324—42no—
81C2×#88324—44no—
82C2×#88324—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—81no—
87C22⋊Dic53—41no—
88C22⋊Dic53—81no—
89C4⋊Dic53—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—41no—
94#11719919—42no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10312yes—
105C23×C53C2×C22⋊C411yescharacteristic
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×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53#9911yescharacteristic
133C2×Dic53#9911yescharacteristic
134C2×Dic53#10312yes—
135C2×Dic53#10412yes—
136C2×Dic53C2×SD1611yescharacteristic
137C2×Dic53C2×Q1611yescharacteristic
138Dic106—84no—
139#88324—44no—
140#88324—82no—
141#11723—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53C2×#9911yescharacteristic
150C22×C53C2×#10311yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—21no—
157Dic53—21no—
158Dic53—41no—
159Dic53—41no—
160Dic53—81no—
161Dic53—81no—
162Dic53#32812yes—
163Dic53#32812yes—
164#2628—531no—
165#1742—531no—
166#1458—531no—
167#1353—532no—
168#1353—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53C2×#32811yescharacteristic
173C2×C53#212311yescharacteristic
174C2×C53#213411yescharacteristic
175C22×M16—1061no—
176C2×C4⋊Q8—531no—
177C2×#117—531no—
178#431—1061no—
179#400—531no—
180#400—531no—
181#399—531no—
182#399#615711yescharacteristic
183#395—1061no—
184#393—1061no—
185#389—1061no—
186#382—531no—
187#380—1061no—
188#343—532no—
189#343—1061no—
190#324—532no—
191#323—1061no—
192C53#1172311yescharacteristic, complemented, Sylow
193C22×Q8—1061no—
194C23×C4—531no—
195C2×M16—1062no—
196C2×M16—1064no—
197C2×M16—2122no—
198C2×M16—2121no—
199C22×C8—2121no—
200C4⋊Q8—2121no—
201C4⋊Q8—2121no—
202#117—2121no—
203#117—2121no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C2×#615711yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4C2×#615711yescharacteristic
216C2×C42—531no—
217C2×C42—531no—
218C4⋊C8—2121no—
219C22⋊C8—1062no—
220C4×C8—2121no—
221#95—531no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95C2×#615711yescharacteristic
228C24C22⋊#615711yescharacteristic
229C2×Q8—4241no—
230C2×Q8—4241no—
231C2×Q8—2122no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—531no—
235C22×C4—532no—
236C22×C4—532no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1062no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4C22⋊#615711yescharacteristic
246C22×C4C22×#615711yescharacteristic
247M16—1064no—
248M16—2122no—
249M16—2124no—
250C2×C8—2122no—
251C2×C8—2124no—
252C2×C8—2122no—
253C4⋊C4—41no—
254C4⋊C4—81no—
255C4⋊C4—4241no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2122no—
262C22⋊C4—41no—
263C22⋊C4—81no—
264C22⋊C4—4241no—
265C22⋊C4—2121no—
266C42—1061no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23#1313328012yes—
275C23C2×C22⋊#615711yescharacteristic
276Q8—4244no—
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—531no—
288C2×C4—531no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—1062no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C8—4242no—
310C8—2124no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22D53⋊#9911yescharacteristic
319C22C2×#1313328011yescharacteristic
320C22D53⋊#10411yescharacteristic
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—532no—
326C4—532no—
327C4—1061no—
328C4—1061no—
329C4—4241no—
330C4—4241no—
331C4—2121no—
332C4—2121no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2D53⋊#32811yescharacteristic
337C2#2029097811yescharacteristic
338C2#2029101511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#399C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×#11719919C23
hypercenter#399#6157
hypocenterC53#11723
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #20289714C2semidirect
4. #20289713C2semidirect
192. C53#11723semidirect

Maximal subgroups

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

subgroupindex
#202904212
#202899472
#202897142
#202897132
#202805412
#1172353