Finite Group Databasebeta

← #27294307

Subgroups — #27294307

Order 13568.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274454C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272721C214yesmaximal, complemented
4C53×#2621C211yesmaximal, characteristic
5C22×#11719917—21no—
6#16810676—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C22⋊Q8×C53—21no—
13C2×#117×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398C2212yes—
16C53×#397—21no—
17C53×#397C412yes—
18C53×#391—21no—
19C53×#382C411yescharacteristic
20C2×#11719917—24no—
21C2×#11719917—22no—
22C2×#11719917—41no—
23C2×#11719917—42no—
24C22×#88322—41no—
25C4⋊#88322—41no—
26C22⋊#88322—22no—
27C4×#88322—41no—
28C22×Q8×C53—21no—
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32C22⋊Q8×C53—42no—
33C22⋊Q8×C53—42no—
34#117×C53—41no—
35#117×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—21no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53C2×C412yes—
51C24×C53C22⋊C411yescharacteristic
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—42no—
55C2×Q8×C53—42no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53D812yes—
65C22×C4×C53SD1612yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—44no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83#11719917—24no—
84#11719917—44no—
85#11719917—42no—
86C2×#88322—42no—
87C2×#88322—44no—
88C2×#88322—42no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#10212yes—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—82no—
96Q8×C53—82no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53#10212yes—
120C2×C4×C53#10412yes—
121C2×C4×C53C2×D811yescharacteristic
122C2×C4×C53C2×SD1611yescharacteristic
123#88322—44no—
124#88322—82no—
125#11700—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—81no—
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10211yescharacteristic
135C22×C53C2×#10411yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—82no—
143C4×C53—82no—
144C4×C53#32712yes—
145C4×C53#32712yes—
146#2621D5311yescharacteristic
147#1742—531no—
148#1458—531no—
149#1346—534no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—42no—
153C2×C53C2×#32711yescharacteristic
154C2×C53#212311yescharacteristic
155C2×C53#213511yescharacteristic
156C22×M16—1061no—
157C2×C4⋊Q8Dic5311yescharacteristic
158C2×C22⋊Q8—21no—
159C2×#117C2×D5311yescharacteristic
160#431—1061no—
161#398—21no—
162#398C2×D5312yes—
163#397—21no—
164#397Dic5312yes—
165#391—21no—
166#382Dic5311yescharacteristic
167#343—532no—
168#343—1061no—
169#324—532no—
170#323—1061no—
171C53#1170011yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4C22⋊D5311yescharacteristic
174C2×M16—1064no—
175C2×M16—1062no—
176C2×M16—2121no—
177C2×M16—2122no—
178C22×C8—2121no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×Dic5312yes—
189C2×C4⋊C4C22⋊D5311yescharacteristic
190C2×C4⋊C4C22×D5311yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4C2×Dic5311yescharacteristic
193C2×C22⋊C4C22⋊D5311yescharacteristic
194C2×C42C2×Dic5311yescharacteristic
195C2×C42C22⋊D5311yescharacteristic
196C4⋊C8—2121no—
197C22⋊C8—1062no—
198C4×C8—2121no—
199#95—22no—
200#95—21no—
201#95—22no—
202#95C2×Dic5312yes—
203C24C22⋊Dic5311yescharacteristic
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4#1171992212yes—
214C22×C4#1171992312yes—
215C22×C4C22⋊Dic5311yescharacteristic
216C22×C4C22⋊Dic5311yescharacteristic
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22×Dic5311yescharacteristic
219C22×C4C2×C22⋊D5311yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221M16—1064no—
222M16—2124no—
223M16—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2124no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1313313812yes—
246C23C2×C22⋊Dic5311yescharacteristic
247Q8—82no—
248Q8—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#1313313211yescharacteristic
270C2×C4#1313313211yescharacteristic
271C2×C4#1313313812yes—
272C2×C4#1313314412yes—
273C2×C4C2×#1171992211yescharacteristic
274C2×C4C2×#1171992311yescharacteristic
275C8—4242no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1313313211yescharacteristic
285C22C2×#1313313811yescharacteristic
286C22C2×#1313314411yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1681035712yes—
296C4#1681035712yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1681035711yescharacteristic
301C2#2027636911yescharacteristic
302C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2621C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2621D53
hypocenterC53#11700
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
171. C53#11700semidirect

Maximal subgroups

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

subgroupindex
#202744542
#202730292
#202727212
C53×#26212
#1170053