Finite Group Databasebeta

← #27294706

Subgroups — #27294706

Order 13568.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279032C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20272740C214yesmaximal, complemented
4C53×#1466C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic, complemented
6C2×#13133132—21no—
7#16810562C412yescomplemented
8#16810559C2212yes—
9#16810521—21no—
10#16810503—21no—
11#16810495—21no—
12#16810483—21no—
13#16810483C411yescharacteristic, complemented
14C4×C22⋊Dic53—21no—
15#16810390C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#323—21no—
19C53×#323C2212yes—
20C23×Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—41no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53C2×C412yes—
26C2×#11719919—22no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—41no—
31#13133132—24no—
32#13133124—22no—
33#13133124—22no—
34#13133124—41no—
35#13133124C2×C412yes—
36C22×D4×C53D411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47#97×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C2×D411yescharacteristic
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53—82no—
69C2×C8×C53—42no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C42×C53—21no—
74C42×C53—21no—
75C22⋊Dic53—24no—
76C22⋊Dic53—42no—
77C22⋊Dic53—42no—
78C22⋊Dic53—82no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—42no—
83#11719919—82no—
84C4×Dic53—82no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#10412yes—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53#13712yes—
96D4×C53—44no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—81no—
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53#9911yescharacteristic
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—44no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—82no—
121C2×Dic53—84no—
122C2×Dic53—82no—
123C8×C53—82no—
124#11786—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53C2×#9911yescharacteristic
138C22×C53#41911yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53#35212yes—
146Dic53—82no—
147Dic53—84no—
148Dic53—82no—
149#2487—531no—
150C4×#99—531no—
151#1466D5311yescharacteristic
152#1352—534no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53—42no—
156C2×C53—42no—
157C2×C53#150811yescharacteristic
158C2×C53C2×#35211yescharacteristic
159C2×C53#212811yescharacteristic
160C2×C4×D4C2×D5311yescharacteristic
161C22×C4⋊C4—531no—
162#432—21no—
163C2×#99—1061no—
164#400—532no—
165#399—532no—
166#389—1061no—
167#386—1061no—
168#385—1061no—
169#380—531no—
170#380—1061no—
171C4×C22⋊C4—1061no—
172#342—532no—
173#323—21no—
174#323C2×D5312yes—
175C53#1178611yescharacteristic, complemented, Sylow
176C22×D4D21211yescharacteristic
177C23×C4—1061no—
178C23×C4C4×D5311yescharacteristic
179C23×C4D21211yescharacteristic
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—2121no—
187C2×C4⋊C4—2121no—
188C2×C4⋊C4C4×D5311yescharacteristic
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C22⋊C4—1061no—
192C2×C22⋊C4C22⋊D5311yescharacteristic
193C2×C22⋊C4C22×D5311yescharacteristic
194C2×C42—2121no—
195C2×C42C22⋊D5311yescharacteristic
196C4⋊C8—41no—
197#99—1064no—
198C22⋊C8—22no—
199#97—41no—
200#95—532no—
201#95—1062no—
202#95—1062no—
203#95—2121no—
204C24#1171992111yescharacteristic
205C24C2×D21211yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—4242no—
218C22×C4—2122no—
219C22×C4—2122no—
220C22×C4#1171992111yescharacteristic
221C22×C4#1171992111yescharacteristic
222C22×C4#1171992111yescharacteristic
223C22×C4C2×C4×D5311yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225C2×C8—42no—
226C4⋊C4—41no—
227C4⋊C4—4242no—
228C4⋊C4—2122no—
229C4⋊C4—2122no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—1064no—
235C22⋊C4—4242no—
236C22⋊C4—2122no—
237C22⋊C4—2122no—
238C42—21no—
239C42—21no—
240C42—4242no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#1313314312yes—
250C23C2×#1171992111yescharacteristic
251C23#1313325012yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—4242no—
268C2×C4—4244no—
269C2×C4—4242no—
270C2×C4—2124no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4#1313313111yescharacteristic
278C2×C4#1313313111yescharacteristic
279C8—82no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#1313313111yescharacteristic
293C22#1681063311yescharacteristic
294C22C2×#1313314311yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—4242no—
301C4—4244no—
302C4—4242no—
303C4#1681041412yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2027323611yescharacteristic
309C2C2×#1681041411yescharacteristic
310C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1466C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1466D53
hypocenterC53#11786
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
5. C22×C4⋊Dic53C4semidirect
7. #16810562C4semidirect
13. #16810483C4semidirect
175. C53#11786semidirect

Maximal subgroups

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

subgroupindex
#202790322
C4×#131331322
#202727402
C53×#14662
#1178653