Finite Group Databasebeta

← #27295146

Subgroups — #27295146

Order 13568.

476 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274711C211yesmaximal, characteristic, complemented
2#20273039C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2548C211yesmaximal, characteristic
5#16810676—21no—
6#16810647—21no—
7C2×#13133136—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C2×#127×C53C411yescharacteristic
12C2×#124×C53—21no—
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#394C412yes—
15C53×#394C2212yes—
16C53×#390—21no—
17C53×#390—21no—
18C53×#390—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133136—42no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27C22×Q8×C53D411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C412yes—
31#127×C53—41no—
32#127×C53—41no—
33#124×C53—42no—
34#124×C53—42no—
35C4×Q8×C53—41no—
36C4×Q8×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53—21no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49#95×C53C2×C412yes—
50C24×C53—22no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53D812yes—
54C2×Q8×C53SD1612yes—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—42no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—82no—
64C22×C4×C53—22no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—42no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—82no—
80C22⋊C4×C53—82no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—42no—
88#11719917—42no—
89#11719917—42no—
90C2×#88322—42no—
91C2×#88322—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53C2×C22⋊C411yescharacteristic
100Q8×C53—42no—
101Q8×C53—42no—
102D4×C53—82no—
103D4×C53—84no—
104D4×C53—84no—
105D4×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#10212yes—
125C2×C4×C53#10212yes—
126C2×C4×C53C2×D811yescharacteristic
127C2×C4×C53C2×SD1611yescharacteristic
128#88322—82no—
129#88322—82no—
130#11882—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—82no—
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10211yescharacteristic
142C22×C53#42111yescharacteristic
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—82no—
152C4×C53#35512yes—
153#2548D5311yescharacteristic
154#1801—531no—
155#1461—531no—
156#1345—534no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—162no—
160C2×C53#151911yescharacteristic
161C2×C53#212211yescharacteristic
162C2×C53C2×#35511yescharacteristic
163C2×#127Dic5311yescharacteristic
164C2×#124—21no—
165C2×C4×Q8C2×D5311yescharacteristic
166#431—1061no—
167#423—1061no—
168C2×#101—1061no—
169#394Dic5312yes—
170#394C2×D5312yes—
171#390—21no—
172#390—21no—
173#390—21no—
174C4×C4⋊C4Dic5311yescharacteristic
175#343—532no—
176#324—532no—
177#324—1061no—
178C53#1188211yescharacteristic, complemented, Sylow
179C22×Q8C22⋊D5311yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2×Dic5312yes—
183C2×M16—1062no—
184#127—41no—
185#127—41no—
186#124—42no—
187#124—42no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22×D5311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42C2×Dic5311yescharacteristic
200C2×C42C22⋊D5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—2121no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—2122no—
206C22⋊C8—1062no—
207C4×C8—2121no—
208#95C2×Dic5312yes—
209C24—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1171992212yes—
213C2×Q8#1171992312yes—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—82no—
223C22×C4—22no—
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22×Dic5311yescharacteristic
229C22×C4C2×C22⋊D5311yescharacteristic
230C22×C4C2×C22⋊D5311yescharacteristic
231M16—2122no—
232M16—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23C2×C22⋊Dic5311yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—84no—
262D4—82no—
263D4—84no—
264D4—82no—
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—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#1313313211yescharacteristic
282C2×C4#1313313211yescharacteristic
283C2×C4#1313313812yes—
284C2×C4#1313313812yes—
285C2×C4C2×#1171992211yescharacteristic
286C2×C4C2×#1171992311yescharacteristic
287C8—4242no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#1313313211yescharacteristic
299C22C2×#1313313811yescharacteristic
300C22#1681064011yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4#1681042012yes—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#2027328311yescharacteristic
315C2#2027636511yescharacteristic
316C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2548C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2548D53
hypocenterC53#11882
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #20274711C2semidirect
2. #20273039C2semidirect
3. #20272718C2semidirect
178. C53#11882semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202727182
C53×#25482
#1188253