Finite Group Databasebeta

← #28308260

Subgroups — #28308260

Order 13568.

444 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289978C212yesmaximal, complemented
2#20289655C212yesmaximal, complemented
3#20289643C212yesmaximal
4#124×Dic53C211yesmaximal, characteristic
5#16811643—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×Q8×Dic53C2211yescharacteristic
9C2×D4×Dic53C411yescharacteristic
10#16810530C411yescharacteristic
11#16810526C2211yescharacteristic
12#16810516—22no—
13#16810489C2211yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic
16C2×#124×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C23×Dic53—21no—
21Q8×Dic53—41no—
22Q8×Dic53—41no—
23D4×Dic53—41no—
24D4×Dic53—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53C2311yescharacteristic
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35#13133124—22no—
36#13133124—22no—
37C22×Q8×C53C2×C411yescharacteristic
38C22×D4×C53C2×C411yescharacteristic
39#124×C53—22no—
40#124×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C24×C53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53M1612yes—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53—44no—
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D412yes—
63C2×#88324—44no—
64C22⋊C4×C53—28no—
65C42×C53—21no—
66C42×C53M1612yes—
67C22⋊Dic53—44no—
68C22⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85Q8×C53—42no—
86Q8×C53—42no—
87D4×C53—42no—
88D4×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×M1611yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—41no—
109C2×Dic53—82no—
110C2×Dic53—82no—
111C2×Dic53#9911yescharacteristic
112C2×Dic53#9911yescharacteristic
113#88324—84no—
114#11530—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53—82no—
120C22×C53—81no—
121C22×C53—81no—
122C22×C53#40711yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41311yescharacteristic
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53—42no—
129Dic53—21no—
130Dic53—42no—
131Dic53—42no—
132Dic53—41no—
133Dic53—81no—
134Dic53#35512yes—
135C4×#124—531no—
136#1467—532no—
137#1324—532no—
138#1318—532no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—82no—
142C2×C53#146511yescharacteristic
143C2×C53#212611yescharacteristic
144C2×C53C2×#35511yescharacteristic
145C2×#124#615711yescharacteristic
146C2×C4×Q8—531no—
147C2×C4×D4—531no—
148#432—1062no—
149#391—531no—
150#390—531no—
151#388—1062no—
152#382—531no—
153C4×C22⋊C4—1062no—
154C43—531no—
155#324—534no—
156#324—1062no—
157C53#1153011yescharacteristic, complemented, Sylow
158C22×Q8C2×#615711yescharacteristic
159C22×D4C2×#615711yescharacteristic
160C23×C4—1061no—
161#124—22no—
162#124—22no—
163C4×Q8—2121no—
164C4×Q8—2121no—
165C4×D4—2121no—
166C4×D4—2121no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—531no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—1061no—
172C2×C22⋊C4—1062no—
173C2×C42—532no—
174C2×C42—531no—
175C2×C42—531no—
176C2×C42—1061no—
177C2×C42—1061no—
178C2×C42C2×#615711yescharacteristic
179C4⋊C8—2122no—
180C22⋊C8—1064no—
181#97—2122no—
182#95—1062no—
183#95—1062no—
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4#1171994012yes—
193C22×C4—21no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—532no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—2121no—
200C22×C4—2121no—
201C22×C4—2124no—
202C22×C4C22⋊#615712yes—
203C22×C4C22×#615711yescharacteristic
204C2×C8—2124no—
205C4⋊C4—2121no—
206C4⋊C4—2122no—
207C4⋊C4—2121no—
208C22⋊C4—28no—
209C22⋊C4—2122no—
210C22⋊C4—2124no—
211C42—21no—
212C42—1062no—
213C42—1062no—
214C42—1061no—
215C42—1061no—
216C42—2121no—
217C42—2122no—
218C42—2122no—
219C42#1171994012yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊#615711yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—531no—
239C2×C4—531no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—1062no—
244C2×C4—4242no—
245C2×C4—4242no—
246C2×C4—2122no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2122no—
252C2×C4C2×#1171994011yescharacteristic
253C2×C4C2×#1171994011yescharacteristic
254C8—4244no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22#1681159311yescharacteristic
263C22D53⋊#9911yescharacteristic
264C22#1681161011yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—42no—
269C4—532no—
270C4—1061no—
271C4—4241no—
272C4—2122no—
273C4—2122no—
274C4—2121no—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#2028997111yescharacteristic
279C2#2029098711yescharacteristic
280C2D53⋊#35511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#124#6157
hypocenterC53#11530
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20289978C2semidirect
2. #20289655C2semidirect
157. C53#11530semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202896552
#202896432
#124×Dic532
#1153053