Finite Group Databasebeta

← #28305716

Subgroups — #28305716

Order 13568.

610 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 130 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#20290647C211yesmaximal, characteristic, complemented
2C4×#13133272C211yesmaximal, characteristic, complemented
3C2×#16811473C212yesmaximal, complemented
4C2×D4⋊#88324C212yesmaximal
5#20277262C211yesmaximal, characteristic
6C2×C4⋊#88324—22no—
7C2×C4⋊#88324C2212yes—
8C2×#13133272—22no—
9C2×C4×#88324C2212yes—
10#16811473—24no—
11D4⋊#88324—24no—
12C2×#13133204C411yescharacteristic
13C2×#13133182C2211yescharacteristic
14C2×D4×Dic53C2211yescharacteristic
15#16810529C411yescharacteristic
16#16810511—22no—
17#16810488C411yescharacteristic
18C22⋊C4×Dic53—22no—
19C42×Dic53C2211yescharacteristic
20C53×#390C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4⋊#88324—44no—
26#13133272—28no—
27C4×#88324—22no—
28C4×#88324D414yes—
29C22×Dic106C2×C411yescharacteristic
30C23×Dic53—21no—
31#13133204—22no—
32#13133204—22no—
33#13133182—22no—
34#13133182—22no—
35D4×Dic53—22no—
36D4×Dic53—22no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D412yes—
44C2×C4×Dic53D412yes—
45C2×C4×Dic53C2311yescharacteristic
46#13133124—22no—
47#13133124—22no—
48C22×D4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52C24×C53—21no—
53C2×D4×C53—22no—
54C2×D4×C53—21no—
55C2×D4×C53—22no—
56C2×D4×C53M1612yes—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53C22⋊C412yes—
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×Dic106—22no—
70C2×Dic106—22no—
71C2×Dic106—21no—
72C2×Dic106M1612yes—
73C2×#88324—28no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—44no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—44no—
79C42×C53—22no—
80C22⋊Dic53—28no—
81C22⋊Dic53—42no—
82C4⋊Dic53—22no—
83#11719919—42no—
84C4×Dic53—24no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53C2×D412yes—
92C4×Dic53#4112yes—
93C23×C53—22no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53C2×C22⋊C411yescharacteristic
99D4×C53—24no—
100D4×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53#10414yes—
106C2×C4×C53C4×D412yes—
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×C4×C53#13112yes—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—82no—
120C2×Dic53—81no—
121C2×Dic53—81no—
122C2×Dic53#12112yes—
123C2×Dic53#12411yescharacteristic
124C2×Dic53#12711yescharacteristic
125C2×Dic53AGL(1,ℤ/8)12yes—
126C2×Dic53#13712yes—
127Dic106—24no—
128Dic106—42no—
129#88324—28no—
130#88324—82no—
131#9646—531nomaximal, Sylow
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53—82no—
137C22×C53#39011yescharacteristic
138C22×C53#40711yescharacteristic
139C22×C53#40811yescharacteristic
140C22×C53#41712yes—
141C22×C53C2×#10412yes—
142C4×C53—24no—
143C4×C53—81no—
144C4×C53#43512yes—
145C4×C53#43512yes—
146Dic53—22no—
147Dic53—42no—
148Dic53—42no—
149Dic53—81no—
150Dic53#48912yes—
151Dic53#49611yescharacteristic
152Dic53#49711yescharacteristic
153C4×#124—531no—
154#1842—531no—
155C4×#97—531no—
156C2×Q8⋊C8—532no—
157C2×D4⋊C8—532no—
158C2×C53—82no—
159C2×C53#148312yes—
160C2×C53#148912yes—
161C2×C53#195511yescharacteristic
162C2×C53#195911yescharacteristic
163C2×C53#196211yescharacteristic
164C2×#124—531no—
165C2×C4×Q8—531no—
166C2×C4×D4—531no—
167C2×C4⋊C8—532no—
168C2×C4⋊C8—1062no—
169C2×#97—1062no—
170C2×C4×C8—532no—
171#391—531no—
172#390#615711yescharacteristic
173#388—1062no—
174#382—531no—
175C4×C22⋊C4—1062no—
176C43—531no—
177Q8⋊C8—1064no—
178D4⋊C8—1064no—
179C53#964611yescharacteristic, complemented, Sylow
180C22×Q8—531no—
181C22×D4C2×#615711yescharacteristic
182C23×C4—1061no—
183C22×C8—1062no—
184C22×C8—1062no—
185#124—1062no—
186#124—1062no—
187C4×Q8—1062no—
188C4×Q8—1062no—
189C4×D4—1062no—
190C4×D4—1062no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C2×#615711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1061no—
197C2×C42—531no—
198C2×C42—532no—
199C2×C42—532no—
200C2×C42—531no—
201C2×C42—1061no—
202C2×C42C2×#615711yescharacteristic
203C4⋊C8—1062no—
204C4⋊C8—1062no—
205C4⋊C8—2124no—
206#97—1068no—
207C4×C8—534no—
208C4×C8—1062no—
209#95—1062no—
210#95—1062no—
211C24—21no—
212C2×Q8—532no—
213C2×Q8—1061no—
214C2×Q8—1062no—
215C2×Q8—1062no—
216C2×D4—22no—
217C2×D4—21no—
218C2×D4—22no—
219C2×D4#1171993612yes—
220C22×C4—21no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—532no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4C22⋊#615712yes—
231C22×C4C22×#615711yescharacteristic
232C2×C8—1064no—
233C2×C8—1068no—
234C2×C8—2122no—
235C2×C8—2124no—
236C4⋊C4—22no—
237C4⋊C4—1062no—
238C4⋊C4—2122no—
239C22⋊C4—44no—
240C22⋊C4—1068no—
241C22⋊C4—2122no—
242C42—22no—
243C42—532no—
244C42—532no—
245C42—1062no—
246C42—1062no—
247C42—1064no—
248C42—1061no—
249C42—1061no—
250C42—2121no—
251C42—2121no—
252C23—22no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23C2×C22⋊#615711yescharacteristic
258Q8—1064no—
259Q8—2122no—
260D4—24no—
261D4—42no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—531no—
267C2×C4—531no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—4241no—
277C2×C4—4242no—
278C2×C4—4241no—
279C2×C4—2121no—
280C2×C4—2122no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4#1313328114yes—
284C2×C4D4×#615712yes—
285C2×C4D53⋊M1611yescharacteristic
286C2×C4C2×#1171993611yescharacteristic
287C2×C4#1313330412yes—
288C8—1068no—
289C8—4242no—
290C22—44no—
291C22—42no—
292C22—81no—
293C22—81no—
294C22—82no—
295C22#1681157511yescharacteristic
296C22#1681159411yescharacteristic
297C22#1681159611yescharacteristic
298C22#1681161412yes—
299C22C2×#1313328112yes—
300C4—24no—
301C4—81no—
302C4—531no—
303C4—532no—
304C4—531no—
305C4—1062no—
306C4—4241no—
307C4—2122no—
308C4—2122no—
309C4#1681165212yes—
310C4#1681165512yes—
311C2—82no—
312C2#2029003012yes—
313C2#2029005112yes—
314C2#2029084011yescharacteristic
315C2#2029084511yescharacteristic
316C2#2029084911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#9646
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290647C2semidirect
2. C4×#13133272C2semidirect
3. C2×#16811473C2semidirect
179. C53#9646semidirect

Maximal subgroups

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

subgroupindex
#202906472
C4×#131332722
C2×#168114732
C2×D4⋊#883242
#202772622
#964653