Finite Group Databasebeta

← #27295147

Subgroups — #27295147

Order 13568.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279714C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic, complemented
3#20272719C214yesmaximal, complemented
4C53×#1461C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic
6C2×#13133206—21no—
7#16810647—21no—
8C2×#13133136—21no—
9#16810540C412yes—
10#16810540C2212yes—
11#16810525—21no—
12#16810525—21no—
13#16810524—21no—
14C4×C4⋊Dic53C411yescharacteristic
15#16810392C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C2×#11719917—22no—
24#13133214—41no—
25#13133214—41no—
26#13133206—42no—
27#13133206—42no—
28C2×C4⋊Dic53—21no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C4⋊#88322—41no—
36#13133136—42no—
37#13133124C2×C412yes—
38C22×Q8×C53D411yescharacteristic
39C4×Q8×C53—41no—
40C4×Q8×C53—41no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C4⋊C8×C53—41no—
48C22⋊C8×C53—22no—
49C4×C8×C53—41no—
50C23×D53—22no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53D812yes—
54C2×Q8×C53SD1612yes—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—22no—
63C2×D212—42no—
64C2×D212—42no—
65C2×D212—42no—
66C2×D212—42no—
67C2×D212—42no—
68C2×D212—41no—
69C2×D212—41no—
70C2×D212—42no—
71C2×D212—82no—
72C2×C8×C53—42no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82#11719921—44no—
83#11719921—42no—
84#11719921—42no—
85#11719921—82no—
86#11719921—82no—
87C4⋊Dic53—42no—
88C4×Dic53—42no—
89#11719917—42no—
90#11719917—42no—
91C2×#88322—42no—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—42no—
100Q8×C53—42no—
101Q8×C53—42no—
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—22no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#10212yes—
118C2×C4×C53#10212yes—
119C2×C4×C53C2×D811yescharacteristic
120C2×C4×C53C2×SD1611yescharacteristic
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124D212—82no—
125D212—84no—
126D212—84no—
127D212—82no—
128C8×C53—82no—
129#88322—82no—
130#11882—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10211yescharacteristic
135C22×C53#42111yescharacteristic
136C2×D53—82no—
137C2×D53—82no—
138C2×D53—82no—
139C2×D53—82no—
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—82no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53#35512yes—
152Dic53—82no—
153#2548—531no—
154#1801—531no—
155#1461D5311yescharacteristic, complemented
156#1345—534no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53#151911yescharacteristic
160C2×C53#212211yescharacteristic
161C2×C53C2×#35511yescharacteristic
162D53—162no—
163C2×#127—531no—
164C2×#124—1061no—
165C2×C4×Q8C2×D5311yescharacteristic
166#431—21no—
167#423—1061no—
168C2×#101—1061no—
169#394—532no—
170#394—532no—
171#390—1061no—
172#390—1061no—
173#390—1061no—
174C4×C4⋊C4—531no—
175#343—532no—
176#324—21no—
177#324C2×D5312yes—
178C53#1188211yescharacteristic, complemented, Sylow
179C22×Q8D21211yescharacteristic
180C22×D4—532no—
181C22×D4—1061no—
182C22×D4—1061no—
183C2×M16—1062no—
184#127—2121no—
185#127—2121no—
186#124—2122no—
187#124—2122no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4C4×D5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22×D5311yescharacteristic
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—1061no—
199C2×C42C4×D5311yescharacteristic
200C2×C42D21211yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—2122no—
206C22⋊C8—22no—
207C4×C8—41no—
208#95—532no—
209C24—1062no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1171991312yes—
213C2×Q8D42412yes—
214C2×D4—4242no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2122no—
220C2×D4—2121no—
221C2×D4—2121no—
222C2×D4—2122no—
223C22×C4—1062no—
224C22×C4#1171992111yescharacteristic
225C22×C4#1171992111yescharacteristic
226C22×C4#1171992111yescharacteristic
227C22×C4#1171992111yescharacteristic
228C22×C4C2×C4×D5311yescharacteristic
229C22×C4C2×D21211yescharacteristic
230C22×C4C2×C22⋊D5311yescharacteristic
231M16—2122no—
232M16—2122no—
233C2×C8—42no—
234C2×C8—2122no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—2122no—
239C22⋊C4—4242no—
240C22⋊C4—4242no—
241C22⋊C4—2124no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—2122no—
251C23—2122no—
252C23—2122no—
253C23—2122no—
254C23—2122no—
255C23—2122no—
256C23—2122no—
257C23—2122no—
258C23C2×#1171992111yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—4244no—
262D4—4242no—
263D4—4244no—
264D4—4242no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313313111yescharacteristic
282C2×C4#1313313111yescharacteristic
283C2×C4#1313313912yes—
284C2×C4#1313313912yes—
285C2×C4C2×D42411yescharacteristic
286C2×C4C2×#1171991311yescharacteristic
287C8—82no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—4242no—
292C22—4242no—
293C22—4242no—
294C22—4242no—
295C22—4242no—
296C22—4242no—
297C22—4242no—
298C22C2×#1313313111yescharacteristic
299C22C2×#1313313911yescharacteristic
300C22#1681063911yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—4242no—
310C4#1681041912yes—
311C2—21no—
312C2—21no—
313C2—8482no—
314C2#2027328411yescharacteristic
315C2#2027636411yescharacteristic
316C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1461C2
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1461D53
hypocenterC53#11882
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #20274711C2semidirect
3. #20272719C2semidirect
4. C53×#1461C2semidirect
155. #1461D53semidirect
178. C53#11882semidirect

Maximal subgroups

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

subgroupindex
#202797142
#202747112
#202727192
C53×#14612
#1188253