Finite Group Databasebeta

← #27294708

Subgroups — #27294708

Order 13568.

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

normal subgroups only

This group has 460 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#20279888C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272735C214yesmaximal, complemented
4C4×#99×C53C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic, complemented
6#16810587—21no—
7#16810535C412yescomplemented
8#16810535C2212yes—
9#16810526—21no—
10#16810488—21no—
11#16810483—21no—
12#16810483—21no—
13C4×C4⋊Dic53C411yescharacteristic, complemented
14#16810349—21no—
15#16810347C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C2×#99×C53—21no—
18C4×C22⋊C4×C53—21no—
19C53×#342C2212yes—
20C22×Dic106C2×C412yes—
21C2×#11719917—41no—
22#13133217—41no—
23#13133217—41no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C4×Dic53—21no—
30C22⋊#88322—22no—
31C22⋊#88322—41no—
32#13133124—22no—
33#13133124—22no—
34#13133124C2×C412yes—
35C22×D4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C4×D4×C53—41no—
39C4×D4×C53—41no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53—41no—
45C2×C42×C53C2×C411yescharacteristic
46#99×C53—24no—
47#95×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×C53—42no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C2×Dic106—42no—
68C2×Dic106—82no—
69C4⋊C4×C53—41no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—42no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—82no—
77C22⋊Dic53—82no—
78C22⋊Dic53—82no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83C4×Dic53—42no—
84#11719917—82no—
85C2×#88322—42no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53#13712yes—
97D4×C53—44no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—44no—
110C2×C4×C53—81no—
111C2×C4×C53—81no—
112C2×C4×C53—82no—
113C2×C4×C53—82no—
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#10412yes—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123Dic106—84no—
124#88322—82no—
125#11787—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53C2×#9911yescharacteristic
139C22×C53#41911yescharacteristic
140C22×C53C2×#10411yescharacteristic
141C4×C53—21no—
142C4×C53—22no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—84no—
147C4×C53#35412yes—
148Dic53—82no—
149Dic53—82no—
150#2563—531no—
151C4×#99D5311yescharacteristic
152#1465—531no—
153#1351—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—42no—
157C2×C53—81no—
158C2×C53#150811yescharacteristic
159C2×C53#212211yescharacteristic
160C2×C53C2×#35411yescharacteristic
161C2×C4⋊Q8—531no—
162C2×C4×D4C2×D5311yescharacteristic
163C2×#99—21no—
164#407—1061no—
165#393—532no—
166#393—532no—
167#390—1061no—
168#382—1061no—
169#380—1061no—
170#380—1061no—
171C4×C4⋊C4—531no—
172C4×C22⋊C4—21no—
173#342C2×D5312yes—
174#324—1061no—
175#323—532no—
176C53#1178711yescharacteristic, complemented, Sylow
177C22×Q8—532no—
178C22×D4C22⋊D5311yescharacteristic
179C23×C4D21211yescharacteristic
180C23×C4C22⋊D5311yescharacteristic
181C2×M16—2121no—
182C4⋊Q8—2121no—
183C4⋊Q8—2121no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4D21211yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4C4×D5311yescharacteristic
194C2×C22⋊C4C22×D5311yescharacteristic
195C2×C42—41no—
196C2×C42—1061no—
197C2×C42C4×D5311yescharacteristic
198#99—24no—
199C22⋊C8—1062no—
200C22⋊C8—2121no—
201#95—41no—
202#95—532no—
203#95—1062no—
204#95—1062no—
205C24#1171992111yescharacteristic
206C24C2×C22⋊D5311yescharacteristic
207C2×Q8—4242no—
208C2×Q8—2122no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4#1171992111yescharacteristic
222C22×C4#1171992111yescharacteristic
223C22×C4#1171992111yescharacteristic
224C22×C4C2×C4×D5311yescharacteristic
225C22×C4C2×D21211yescharacteristic
226M16—4242no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—2122no—
230C4⋊C4—2122no—
231C4⋊C4—2121no—
232C4⋊C4—2121no—
233C22⋊C4—24no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—4242no—
238C22⋊C4—4242no—
239C42—21no—
240C42—21no—
241C42—82no—
242C42—2122no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×#1171992111yescharacteristic
253C23#1313324712yes—
254Q8—4244no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4#1313313111yescharacteristic
279C2×C4#1313313111yescharacteristic
280C2×C4#1313314512yes—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1313313111yescharacteristic
295C22#1681063111yescharacteristic
296C22C2×#1313314511yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—84no—
303C4—4242no—
304C4—4242no—
305C4#1681041712yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2027323411yescharacteristic
311C2#2027636411yescharacteristic
312C2C2×#1681041711yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11787
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20272735C2semidirect
5. C2×#13133217C4semidirect
7. #16810535C4semidirect
13. C4×C4⋊Dic53C4semidirect
176. C53#11787semidirect

Maximal subgroups

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

subgroupindex
#202798882
#202730532
#202727352
C4×#99×C532
#1178753