Finite Group Databasebeta

← #28309164

Subgroups — #28309164

Order 13568.

444 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13133277C211yesmaximal, characteristic
2#20289970C211yesmaximal, characteristic
3#20289713C214yesmaximal, complemented
4#20279033C211yesmaximal, characteristic
5C2×#13133277—21no—
6#16811594—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811466—21no—
11C2×C4×C22⋊D53C2211yescharacteristic
12C22×C4⋊Dic53C411yescharacteristic
13#16810563C2212yes—
14#16810560C412yes—
15#16810522—21no—
16#16810500—21no—
17#16810498—21no—
18#16810480—21no—
19C53×#380C411yescharacteristic
20C2×#11719936—41no—
21C2×#11719940—22no—
22#13133277—24no—
23#13133274—41no—
24C22⋊#88324—22no—
25#13133272—41no—
26C4×#88324—41no—
27C22×C22⋊D53D411yescharacteristic
28C23×Dic53—21no—
29C22×C4×D53D411yescharacteristic
30C4×C22⋊D53—41no—
31C4×C22⋊D53—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—41no—
35C2×C4⋊Dic53—41no—
36C2×C4⋊Dic53C2×C412yes—
37C2×#11719919C2311yescharacteristic
38C2×C22⋊Dic53D411yescharacteristic
39C2×#11719921—22no—
40C2×#11719921C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42#13133124—22no—
43#13133124—22no—
44C23×C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C23×D53C22⋊C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C2×C22⋊D53—22no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—41no—
59C22×Dic53—22no—
60C22×Dic53—42no—
61C22×Dic53—82no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×C4×D53—21no—
66C2×C4×D53—21no—
67C2×C4×D53—42no—
68#11719940—24no—
69#11719940—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72#11719936—82no—
73C4⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C22⋊Dic53—41no—
76#11719921—41no—
77#11719921—82no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53—42no—
81C4⋊Dic53—82no—
82#11719919—41no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53#10412yes—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—41no—
95C22×D53—41no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C22⋊D53—44no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—84no—
117C2×Dic53#13712yes—
118C4×D53—42no—
119C4×D53—81no—
120#88324—44no—
121#88324—82no—
122#11790—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53C2×#9911yescharacteristic
131C22×C53#41911yescharacteristic
132C22×C53C2×#10411yescharacteristic
133C2×D53—42no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—81no—
138C4×C53—21no—
139C4×C53—41no—
140C4×C53—82no—
141C4×C53#35212yes—
142Dic53—22no—
143Dic53—41no—
144Dic53—41no—
145Dic53—82no—
146#2487—531no—
147C4×#100—531no—
148#1465—531no—
149#1353—534no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—42no—
153C2×C53#150811yescharacteristic
154C2×C53C2×#35211yescharacteristic
155C2×C53#213411yescharacteristic
156D53—81no—
157C2×C4×D4—531no—
158C22×C4⋊C4—531no—
159C2×#100—1061no—
160#407—1061no—
161C4×M16—1061no—
162#400—532no—
163#399—532no—
164#389—1061no—
165#386—1061no—
166#385—1061no—
167#380—1061no—
168#380#615711yescharacteristic
169#343—532no—
170#324—532no—
171#323—1061no—
172C53#1179011yescharacteristic, complemented, Sylow
173C22×D4—531no—
174C23×C4—531no—
175C23×C4—1061no—
176C23×C4C2×#615711yescharacteristic
177C2×M16—1062no—
178C2×M16—2121no—
179C4×D4—2121no—
180C4×D4—2121no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—2121no—
187C2×C4⋊C4—2121no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—1062no—
192C2×C42—531no—
193#100—1064no—
194C22⋊C8—1062no—
195C22⋊C8—2121no—
196#97—2121no—
197C4×C8—2121no—
198#95—1062no—
199#95—1062no—
200#95C2×#615712yes—
201C24—531no—
202C24C22⋊#615711yescharacteristic
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—2121no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—1062no—
216C22×C4—4242no—
217C22×C4—2122no—
218C22×C4—2122no—
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22×#615711yescharacteristic
221M16—1064no—
222M16—4242no—
223M16—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C4⋊C4—42no—
227C4⋊C4—4242no—
228C4⋊C4—2121no—
229C4⋊C4—2122no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C22⋊C4—42no—
233C22⋊C4—4242no—
234C22⋊C4—2121no—
235C22⋊C4—2121no—
236C42—1061no—
237C42—1061no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—1061no—
243C23—1061no—
244C23—1061no—
245C23—2121no—
246C23—2121no—
247C23#1313328112yes—
248C23C2×C22⋊#615711yescharacteristic
249D4—2124no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—532no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—4241no—
264C2×C4—4244no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4#1313327611yescharacteristic
272C2×C4#1313327611yescharacteristic
273C8—4242no—
274C8—2124no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—4241no—
283C22—2122no—
284C22—2121no—
285C22—2121no—
286C22—2121no—
287C22C2×#1313327611yescharacteristic
288C22#1681161611yescharacteristic
289C22C2×#1313328111yescharacteristic
290C4—21no—
291C4—41no—
292C4—82no—
293C4—1062no—
294C4—4242no—
295C4—2121no—
296C4—2121no—
297C4#1681152612yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—4241no—
302C2#2029011411yescharacteristic
303C2C2×#1681152611yescharacteristic
304C2#2029101611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719919C23
hypercenter#380#6157
hypocenterC53#11790
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289713C2semidirect
172. C53#11790semidirect

Maximal subgroups

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

subgroupindex
C4×#131332772
#202899702
#202897132
#202790332
#1179053