Finite Group Databasebeta

← #28309382

Subgroups — #28309382

Order 13568.

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

normal subgroups only

This group has 396 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#20289714C212yesmaximal, complemented
4#20289699C212yesmaximal, complemented
5#20278998C211yesmaximal, characteristic
6#16811593—21no—
7C2×C22⋊#88324—21no—
8#16811503—21no—
9#16811503C2212yes—
10#16811469—21no—
11#16811469C2212yes—
12C22×#11719919C2211yescharacteristic
13#16810561C411yescharacteristic
14#16810561C2211yescharacteristic
15#16810554C411yescharacteristic
16#16810554C2211yescharacteristic
17#16810507—21no—
18#16810507—21no—
19#16810495—21no—
20#16810486C411yescharacteristic
21#16810484—21no—
22C2×#117×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—41no—
34C2×#11719919—41no—
35C2×#11719919C2×C411yescharacteristic
36C2×#11719919D411yescharacteristic
37C2×#11719919D411yescharacteristic
38C2×#11719919C2311yescharacteristic
39C2×C22⋊Dic53—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124C2×C411yescharacteristic
48#13133124C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50#117×C53—41no—
51#117×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C2×C42×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53—42no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77#11719940—42no—
78#11719940—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—82no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C42×C53—42no—
87C22⋊Dic53—42no—
88#11719919—41no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—42no—
93#11719919—42no—
94C23×C53—22no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53#9911yescharacteristic
99C23×C53#9911yescharacteristic
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—81no—
131C2×Dic53#13512yes—
132#88324—82no—
133#88324—82no—
134#11850—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53#35212yes—
145C22×C53C2×#9911yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53#42111yescharacteristic
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—81no—
151C4×C53—81no—
152Dic53—22no—
153Dic53—41no—
154Dic53—41no—
155Dic53—41no—
156Dic53—81no—
157Dic53—81no—
158#2484—531no—
159#1821—531no—
160C2×#324—531no—
161#1353—532no—
162#1346—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—22no—
166C2×C53—41no—
167C2×C53#151611yescharacteristic
168C2×C53C2×#35211yescharacteristic
169C2×C53#213311yescharacteristic
170C2×#117#615711yescharacteristic
171C22×C4⋊C4—531no—
172#407—1061no—
173C2×C22⋊C8—1061no—
174#400—531no—
175#400—531no—
176#398—531no—
177#398—531no—
178#387—1061no—
179#387—1061no—
180#385—1061no—
181#381—531no—
182#381—1061no—
183#343—532no—
184#343—1061no—
185#324—532no—
186#324—1061no—
187C53#1185011yescharacteristic, complemented, Sylow
188C23×C4—531no—
189C23×C4—531no—
190C23×C4C2×#615711yescharacteristic
191C2×M16—1062no—
192C22×C8—2121no—
193#117—41no—
194#117—41no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—2121no—
202C2×C4⋊C4—2121no—
203C2×C4⋊C4C2×#615711yescharacteristic
204C2×C4⋊C4C2×#615711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—1061no—
207C2×C42—21no—
208C22⋊C8—1062no—
209C22⋊C8—2121no—
210C22⋊C8—2121no—
211C22⋊C8—2121no—
212#95—531no—
213#95—531no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221C24C22⋊#615711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—1062no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—2121no—
239C22×C4—2122no—
240C22×C4C22⋊#615711yescharacteristic
241C22×C4C22×#615711yescharacteristic
242M16—2122no—
243M16—2122no—
244C2×C8—4242no—
245C2×C8—2122no—
246C2×C8—2122no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—2122no—
258C42—42no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23#1313327511yescharacteristic
264C23#1313327511yescharacteristic
265C23C2×C22⋊#615711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—532no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—4241no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2122no—
295C2×C4—2121no—
296C2×C4—2121no—
297C8—4242no—
298C8—4242no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1681152512yes—
309C22C2×#1313327511yescharacteristic
310C22#1681161311yescharacteristic
311C22#1681162011yescharacteristic
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—1062no—
317C4—4241no—
318C4—4241no—
319C4—2121no—
320C4—2121no—
321C4—2121no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2029014111yescharacteristic
327C2C2×#1681152511yescharacteristic
328C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×#117#6157
hypocenterC53#11850
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20289714C2semidirect
4. #20289699C2semidirect
187. C53#11850semidirect

Maximal subgroups

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

subgroupindex
#202906142
C2×#168114692
#202897142
#202896992
#202789982
#1185053