Finite Group Databasebeta

← #28299661

Subgroups — #28299661

Order 13568.

730 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290378C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289531C212yesmaximal, complemented
4#20277206C211yesmaximal, characteristic
5C22×#11719940—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133176C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810526C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810503—22no—
15C4×C22⋊Dic53—22no—
16C22×#41×C53C411yescharacteristic
17C2×#11719940—28no—
18C2×#11719940—44no—
19C2×#11719940—44no—
20C22×#88324—24no—
21C23×Dic53D412yes—
22#13133176—42no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53—42no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32C2×C4×Dic53C2311yescharacteristic
33#13133124—22no—
34#13133124—22no—
35C2×#41×C53—22no—
36C2×#41×C53—22no—
37C2×#41×C53—42no—
38C2×#41×C53—42no—
39C22×Q8×C53C2×C411yescharacteristic
40C22×D4×C53—21no—
41C22×D4×C53C2×C411yescharacteristic
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C24×C53—21no—
45C24×C53C22⋊C411yescharacteristic
46#41×C53—48no—
47#41×C53—48no—
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53M1612yescomplemented
52C2×D4×C53—24no—
53C2×D4×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53M1612yescomplemented
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—24no—
64C22×C4×C53—21no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×Dic53—24no—
70C22×Dic53—24no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53C22⋊C412yes—
76C22×Dic53C2×D412yes—
77#11719940—416no—
78C2×#88324—28no—
79C2×#88324—44no—
80C2×#88324—44no—
81C22⋊Dic53—42no—
82C22⋊Dic53—44no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53—42no—
89C4×Dic53—41no—
90C4×Dic53—42no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53#9912yes—
101C23×C53C2×C22⋊C411yescharacteristic
102Q8×C53—42no—
103Q8×C53—42no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53—48no—
107C2×C4×C53—24no—
108C2×C4×C53—24no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—48no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53#10012yes—
118C2×C4×C53C2×M1611yescharacteristic
119C2×C4×C53C2×M1611yescharacteristic
120C2×Dic53—28no—
121C2×Dic53—28no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53#10414yes—
131C2×Dic53AGL(1,ℤ/8)12yes—
132C2×Dic53#13712yes—
133#88324—48no—
134#7631—531nomaximal, Sylow
135C22×C53—22no—
136C22×C53—24no—
137C22×C53—41no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—42no—
141C22×C53—44no—
142C22×C53—81no—
143C22×C53—82no—
144C22×C53—81no—
145C22×C53#40711yescharacteristic
146C22×C53C2×#9911yescharacteristic
147C22×C53C2×#10011yescharacteristic
148C22×C53#41712yes—
149C22×C53C2×#10412yes—
150C4×C53—22no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—42no—
154Dic53—22no—
155Dic53—24no—
156Dic53—22no—
157Dic53—81no—
158Dic53—81no—
159#2297—531no—
160#1728—532no—
161#1272—532no—
162#1270—532no—
163C2×C53—42no—
164C2×C53—82no—
165C2×C53#146211yescharacteristic
166C2×C53#148912yes—
167C2×C53#150412yes—
168C2×C53#152511yescharacteristic
169C2×C53#156011yescharacteristic
170C22×#41#615711yescharacteristic
171C22×M16—1062no—
172C2×#117—531no—
173C22×C42—531no—
174#391—531no—
175#391—531no—
176#390—531no—
177#390—531no—
178#386—1062no—
179C4×C22⋊C4—1062no—
180#336—534no—
181#336—1062no—
182C53#763111yescharacteristic, complemented, Sylow
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—42no—
186C2×#41—42no—
187C22×Q8C2×#615711yescharacteristic
188C22×D4—21no—
189C22×D4C2×#615711yescharacteristic
190C23×C4—21no—
191C23×C4—532no—
192C23×C4C2×#615711yescharacteristic
193C2×M16—1068no—
194C2×M16—2124no—
195C2×M16—2124no—
196C22×C8—1064no—
197#117—2122no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1061no—
203C2×C42—532no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—1061no—
207C2×C42—2122no—
208#95—1062no—
209#95—1062no—
210C24—21no—
211C24C22⋊#615711yescharacteristic
212#41—48no—
213#41—48no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8#1171994012yescomplemented
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4#1171994012yescomplemented
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—532no—
234C22×C4—532no—
235C22×C4—1064no—
236C22×C4—1062no—
237C22×C4—1062no—
238C22×C4—1064no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4C22⋊#615711yescharacteristic
242C22×C4C22×#615711yescharacteristic
243M16—21216no—
244C2×C8—1068no—
245C2×C8—2124no—
246C2×C8—2124no—
247C4⋊C4—2121no—
248C4⋊C4—2121no—
249C22⋊C4—2122no—
250C22⋊C4—2122no—
251C22⋊C4—2124no—
252C22⋊C4—2122no—
253C42—2122no—
254C42—2121no—
255C42—2121no—
256C42—2122no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23#1313327512yes—
267C23C2×C22⋊#615711yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—42no—
271D4—42no—
272D4—48no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—534no—
286C2×C4—1068no—
287C2×C4—1068no—
288C2×C4—1062no—
289C2×C4—1062no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2122no—
295C2×C4—2122no—
296C2×C4#1313327712yes—
297C2×C4C2×#1171994011yescharacteristic
298C2×C4C2×#1171994011yescharacteristic
299C8—2128no—
300C22—24no—
301C22—22no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—44no—
306C22—42no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22#1681159311yescharacteristic
311C22C2×#1313327511yescharacteristic
312C22C2×#1313327711yescharacteristic
313C22#1681161412yes—
314C22D53⋊#10412yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—1062no—
320C4—1062no—
321C4—1064no—
322C4—4241no—
323C4—4241no—
324C2—42no—
325C2—82no—
326C2#2028996011yescharacteristic
327C2#2029005012yes—
328C2#2029010112yes—
329C2#2029016911yescharacteristic
330C2#2029025211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC22×#41×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×#41#6157
hypocenterC53#7631
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #20290378C2semidirect
2. #20289535C2semidirect
3. #20289531C2semidirect
51. C2×Q8×C53M16semidirect
58. C2×D4×C53M16semidirect
182. C53#7631semidirect
217. C2×Q8#11719940semidirect
224. C2×D4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202903782
#202895352
#202895312
#202772062
#763153