Finite Group Databasebeta

← #28305711

Subgroups — #28305711

Order 13568.

558 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290632C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3C2×Q8⋊#88324C212yesmaximal, complemented
4C2×D4⋊#88324C212yesmaximal
5#124×Dic53C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C4×#88324C2212yes—
8#16811489—22no—
9#16811489—22no—
10Q8⋊#88324—24no—
11D4⋊#88324—24no—
12C2×Q8×Dic53C2211yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810526C411yescharacteristic
16#16810516—22no—
17#16810489C411yescharacteristic
18C22⋊C4×Dic53—22no—
19C42×Dic53C2211yescharacteristic
20C2×#124×C53C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4×#88324—22no—
26C4×#88324D414yes—
27C23×Dic53—21no—
28Q8×Dic53—22no—
29Q8×Dic53—22no—
30D4×Dic53—22no—
31D4×Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D412yes—
40C2×C4×Dic53D412yes—
41C2×C4×Dic53C2311yescharacteristic
42#13133124—22no—
43#13133124—22no—
44C22×Q8×C53C2×C411yescharacteristic
45C22×D4×C53C2×C411yescharacteristic
46#124×C53—22no—
47#124×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C42×C53C2×C411yescharacteristic
50C24×C53—21no—
51C2×Q8×C53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53M1612yes—
55C2×D4×C53—22no—
56C2×D4×C53—21no—
57C2×D4×C53—22no—
58C2×D4×C53M1612yes—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—44no—
67C22×Dic53C22⋊C412yes—
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×#88324—24no—
71C2×#88324—44no—
72C2×#88324—42no—
73C2×#88324—44no—
74C22⋊C4×C53—28no—
75C42×C53—21no—
76C42×C53—21no—
77C22⋊Dic53—44no—
78C22⋊Dic53—42no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C4⋊Dic53—42no—
82C4×Dic53—22no—
83C4×Dic53—24no—
84C4×Dic53—42no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53C2×D412yes—
88C4×Dic53#4112yes—
89C23×C53—22no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—24no—
96Q8×C53—42no—
97D4×C53—24no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53C4×D412yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53#13112yes—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—82no—
118C2×Dic53—82no—
119C2×Dic53#12112yes—
120C2×Dic53#12411yescharacteristic
121C2×Dic53#12711yescharacteristic
122C2×Dic53#13514yes—
123#88324—44no—
124#88324—82no—
125#9645—531nomaximal, Sylow
126C22×C53—42no—
127C22×C53—44no—
128C22×C53—81no—
129C22×C53—81no—
130C22×C53—82no—
131C22×C53#39011yescharacteristic
132C22×C53#40711yescharacteristic
133C22×C53#40811yescharacteristic
134C22×C53#41612yes—
135C22×C53#42112yes—
136C4×C53—42no—
137C4×C53—42no—
138C4×C53#43512yes—
139C4×C53#43512yes—
140Dic53—22no—
141Dic53—42no—
142Dic53—81no—
143Dic53—81no—
144Dic53#48812yes—
145Dic53#49512yes—
146C4×#124—531no—
147#1831—531no—
148#1744—531no—
149C2×Q8⋊C8—532no—
150C2×D4⋊C8—532no—
151C2×C53—82no—
152C2×C53#148212yes—
153C2×C53#148812yes—
154C2×C53#195511yescharacteristic
155C2×C53#195811yescharacteristic
156C2×C53#196311yescharacteristic
157C2×#124#615711yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×D4—531no—
160C2×C4⋊C8—532no—
161C2×C4×C8—532no—
162#391—531no—
163#390—531no—
164#388—1062no—
165#382—531no—
166C4×C22⋊C4—1062no—
167C43—531no—
168#336—1062no—
169#336—1062no—
170Q8⋊C8—1064no—
171D4⋊C8—1064no—
172C53#964511yescharacteristic, complemented, Sylow
173C22×Q8C2×#615711yescharacteristic
174C22×D4C2×#615711yescharacteristic
175C23×C4—1061no—
176C22×C8—1062no—
177C22×C8—1062no—
178#124—22no—
179#124—22no—
180C4×Q8—1062no—
181C4×Q8—1062no—
182C4×D4—1062no—
183C4×D4—1062no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1061no—
189C2×C22⋊C4—1062no—
190C2×C42—531no—
191C2×C42—532no—
192C2×C42—531no—
193C2×C42—532no—
194C2×C42—1061no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198C4×C8—534no—
199C4×C8—1062no—
200#95—1062no—
201#95—1062no—
202C24—21no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8#1171994012yes—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4#1171994012yes—
211C22×C4—21no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—532no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4—2124no—
220C22×C4C22⋊#615712yes—
221C22×C4C22×#615711yescharacteristic
222C2×C8—1064no—
223C2×C8—2122no—
224C2×C8—2124no—
225C2×C8—2124no—
226C4⋊C4—1062no—
227C4⋊C4—1062no—
228C4⋊C4—2122no—
229C22⋊C4—28no—
230C22⋊C4—2122no—
231C22⋊C4—2124no—
232C42—21no—
233C42—21no—
234C42—532no—
235C42—532no—
236C42—1064no—
237C42—1062no—
238C42—2122no—
239C42—2121no—
240C42—2121no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—24no—
248Q8—42no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—531no—
258C2×C4—531no—
259C2×C4—534no—
260C2×C4—532no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—4242no—
266C2×C4—4242no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4D4×#615712yes—
272C2×C4C2×#1171994011yescharacteristic
273C2×C4C2×#1171994011yescharacteristic
274C2×C4#1313330412yes—
275C8—4242no—
276C8—2124no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—81no—
281C22—82no—
282C22#1681157511yescharacteristic
283C22#1681159311yescharacteristic
284C22#1681159711yescharacteristic
285C22#1681161312yes—
286C22#1681162012yes—
287C4—42no—
288C4—42no—
289C4—532no—
290C4—532no—
291C4—1062no—
292C4—4241no—
293C4—4241no—
294C4—2122no—
295C4#1681165412yes—
296C4#1681165412yes—
297C2—82no—
298C2#2029002712yes—
299C2#2029004612yes—
300C2#2029083911yescharacteristic
301C2#2029084411yescharacteristic
302C2#2029085211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#9645
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20290632C2semidirect
2. #20290430C2semidirect
3. C2×Q8⋊#88324C2semidirect
172. C53#9645semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202904302
C2×Q8⋊#883242
C2×D4⋊#883242
#124×Dic532
#964553