Finite Group Databasebeta

← #28308921

Subgroups — #28308921

Order 13568.

432 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 88 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#20290420C211yesmaximal, characteristic
2#20289948C211yesmaximal, characteristic
3#20289714C214yesmaximal, complemented
4#20280537C211yesmaximal, characteristic
5C2×D53⋊M16—21no—
6#16811639—21no—
7#16811503—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811467—21no—
11C2×#13133174C2211yescharacteristic
12#16810561C2212yes—
13#16810558C412yes—
14#16810542—21no—
15#16810534—21no—
16#16810519—21no—
17#16810489C411yescharacteristic
18#16810480—21no—
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×#11719936—41no—
21D53⋊M16—24no—
22D53⋊M16—42no—
23C2×#11719940—22no—
24C2×D53⋊C8—41no—
25C4⋊#88324—41no—
26C22⋊#88324—22no—
27C4×#88324—41no—
28C22×C4×D53D411yescharacteristic
29#13133174—41no—
30#13133174—41no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—22no—
33C2×#11719919D411yescharacteristic
34C2×#11719919C2311yescharacteristic
35C2×#11719921—22no—
36C2×#11719921C2×C411yescharacteristic
37C2×#11719921D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42#13133124C2×C412yes—
43C22×Q8×C53—21no—
44C4⋊Q8×C53—41no—
45C4⋊Q8×C53—41no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C42×C53C2×C411yescharacteristic
49C23×D53C22⋊C411yescharacteristic
50C2×Q8×C53—42no—
51C2×Q8×C53—82no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×C4×D53—21no—
61C2×C4×D53—21no—
62C2×C4×D53—22no—
63C2×C4×D53SD1612yes—
64C2×C4×D53Q1612yes—
65#11719940—24no—
66#11719940—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69#11719936—44no—
70D53⋊C8—44no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76#11719921—41no—
77#11719921—41no—
78#11719921—82no—
79C4⋊Dic53—42no—
80#11719919—41no—
81#11719919—41no—
82#11719919—82no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C23×C53C2×C22⋊C411yescharacteristic
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—41no—
89C22×D53—41no—
90C22×D53#10312yes—
91Q8×C53—84no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53#9911yescharacteristic
100C2×C4×C53#9911yescharacteristic
101C2×C4×C53#10312yes—
102C2×C4×C53#10412yes—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53C2×SD1611yescharacteristic
113C2×Dic53C2×Q1611yescharacteristic
114C4×D53—22no—
115C4×D53—22no—
116C4×D53—22no—
117C4×D53—22no—
118#88324—44no—
119#88324—82no—
120#11723—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10311yescharacteristic
125C22×C53C2×#10411yescharacteristic
126C2×D53—22no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—81no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—82no—
134C4×C53#32812yes—
135C4×C53#32812yes—
136Dic53—21no—
137Dic53—21no—
138Dic53—41no—
139Dic53—41no—
140Dic53—82no—
141#2628—531no—
142#1742—531no—
143#1458—531no—
144#1353—534no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53C2×#32811yescharacteristic
148C2×C53#212311yescharacteristic
149C2×C53#213411yescharacteristic
150D53—42no—
151C22×M16—1061no—
152C2×C4⋊Q8#615711yescharacteristic
153C2×#117—531no—
154#431—1061no—
155#400—532no—
156#399—532no—
157#395—1061no—
158#393—1061no—
159#389—1061no—
160#382—531no—
161#380—1061no—
162#343—532no—
163#343—1061no—
164#324—532no—
165#323—1061no—
166C53#1172311yescharacteristic, complemented, Sylow
167C22×Q8—21no—
168C23×C4—531no—
169C2×M16—1062no—
170C2×M16—1064no—
171C2×M16—2122no—
172C2×M16—2121no—
173C22×C8—2121no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—2121no—
177#117—2121no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—1062no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4C2×#615712yes—
184C2×C22⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—1062no—
187C2×C42—531no—
188C2×C42C2×#615711yescharacteristic
189C4⋊C8—2121no—
190C22⋊C8—1062no—
191C4×C8—2121no—
192#95—532no—
193#95—1062no—
194#95—1062no—
195#95—1061no—
196C24—531no—
197C2×Q8—42no—
198C2×Q8—82no—
199C22×C4—22no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—1062no—
208C22×C4—1062no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212M16—1064no—
213M16—2122no—
214M16—2124no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2124no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—4242no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2122no—
225C22⋊C4—4242no—
226C22⋊C4—2121no—
227C22⋊C4—2121no—
228C42—21no—
229C42—21no—
230C42—1061no—
231C42—1061no—
232C23—532no—
233C23—1061no—
234C23—1061no—
235C23—2121no—
236C23—2121no—
237C23C2×C22⋊#615711yescharacteristic
238Q8—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—531no—
247C2×C4—531no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4#1313327511yescharacteristic
262C2×C4#1313327511yescharacteristic
263C2×C4#1313328012yes—
264C2×C4#1313328212yes—
265C8—4242no—
266C8—2124no—
267C22—21no—
268C22—21no—
269C22—1062no—
270C22—4241no—
271C22—2122no—
272C22—2121no—
273C22—2121no—
274C22C2×#1313327511yescharacteristic
275C22C2×#1313328011yescharacteristic
276C22C2×#1313328211yescharacteristic
277C4—41no—
278C4—41no—
279C4—82no—
280C4—1061no—
281C4—1061no—
282C4—4242no—
283C4—2121no—
284C4—2121no—
285C4#1681147712yes—
286C4#1681147712yes—
287C2—21no—
288C2—21no—
289C2—2122no—
290C2C2×#1681147711yescharacteristic
291C2#2029097711yescharacteristic
292C2#2029101411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11723
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289714C2semidirect
166. C53#11723semidirect

Maximal subgroups

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

subgroupindex
#202904202
#202899482
#202897142
#202805372
#1172353