Finite Group Databasebeta

← #27295006

Subgroups — #27295006

Order 13568.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#20279002C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272743C212yesmaximal, complemented
4#20272722C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7#16810587—21no—
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810553C411yescharacteristic
11#16810553C2211yescharacteristic
12#16810504—21no—
13#16810504—21no—
14#16810496—21no—
15#16810486—21no—
16#16810486C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C2×#11719917—22no—
24#13133176—41no—
25#13133176—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C4×Dic53—21no—
33C22⋊#88322—41no—
34C22⋊#88322—41no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×C8×C53—42no—
78C2×C8×C53—82no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C22⋊Dic53—42no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719919—42no—
89#11719919—42no—
90C4×Dic53—42no—
91#11719917—42no—
92#11719917—42no—
93C2×#88322—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—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—81no—
120C2×C4×C53#13512yes—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C8×C53—82no—
133#88322—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—22no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—81no—
155Dic53—81no—
156Dic53—81no—
157Dic53—81no—
158#2484—531no—
159#1821—531no—
160C2×#324D5311yescharacteristic
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—531no—
171C22×C4⋊C4C2×D5311yescharacteristic
172#407—1061no—
173C2×C22⋊C8—21no—
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—21no—
186#324C2×D5312yes—
187C53#1185011yescharacteristic, complemented, Sylow
188C23×C4C4×D5311yescharacteristic
189C23×C4D21211yescharacteristic
190C23×C4C22⋊D5311yescharacteristic
191C2×M16—1062no—
192C22×C8—41no—
193#117—2121no—
194#117—2121no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4C4×D5311yescharacteristic
202C2×C4⋊C4D21211yescharacteristic
203C2×C4⋊C4C22⋊D5311yescharacteristic
204C2×C4⋊C4C22×D5311yescharacteristic
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C42—1061no—
208C22⋊C8—22no—
209C22⋊C8—41no—
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—
221C24#1171992111yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238C22×C4#1171992111yescharacteristic
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—2122no—
243M16—2122no—
244C2×C8—42no—
245C2×C8—82no—
246C2×C8—2122no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C22⋊C4—2122no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C42—2122no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1313313111yescharacteristic
264C23#1313313111yescharacteristic
265C23C2×#1171992111yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313323912yes—
297C8—82no—
298C8—4242no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1681041212yes—
309C22C2×#1313313111yescharacteristic
310C22#1681062011yescharacteristic
311C22#1681063911yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—4241no—
319C4—4241no—
320C4—4241no—
321C4—4241no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2027327211yescharacteristic
327C2C2×#1681041211yescharacteristic
328C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11850
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20272743C2semidirect
4. #20272722C2semidirect
187. C53#11850semidirect

Maximal subgroups

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

subgroupindex
#202790022
#202747952
#202727432
#202727222
C2×C53×#3242
#1185053