Finite Group Databasebeta

← #28303477

Subgroups — #28303477

Order 13568.

422 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290581C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20289613C212yesmaximal
4#20289568C212yesmaximal
5#20277632C211yesmaximal, characteristic
6#16811640C2212yes—
7#16811626C2212yes—
8C2×#13133278—21no—
9#16811503—21no—
10#16811469—21no—
11#16811468—21no—
12C2×C4×Dic106C2211yescharacteristic
13C2×#13133182—21no—
14#16810520—21no—
15#16810520—21no—
16#16810519—21no—
17#16810487C411yescharacteristic
18#16810487C2211yescharacteristic
19C4×C4⋊Dic53C411yescharacteristic
20#16810474C411yescharacteristic
21#16810474C2211yescharacteristic
22C53×#382C411yescharacteristic
23C2×#11719940—22no—
24#13133284—22no—
25#13133284—22no—
26C4⋊#88324—22no—
27#13133278—42no—
28C22⋊#88324—22no—
29C4×#88324—22no—
30C22×Dic106D411yescharacteristic
31C4×Dic106—22no—
32C4×Dic106—22no—
33#13133182—44no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919D411yescharacteristic
41C2×#11719919D411yescharacteristic
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124C2×C411yescharacteristic
50#13133124C2×C411yescharacteristic
51C2×C4⋊C4×C53—21no—
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×Dic106—22no—
68C2×Dic106—22no—
69C2×Dic106SD1612yes—
70C2×Dic106Q1612yes—
71#11719940—42no—
72#11719940—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C4⋊C4×C53—42no—
76C42×C53—21no—
77C42×C53—21no—
78C4⋊Dic53—22no—
79C4⋊Dic53—42no—
80#11719919—22no—
81#11719919—22no—
82#11719919—42no—
83#11719919—42no—
84#11719919—42no—
85#11719919—42no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—42no—
89C4×Dic53—42no—
90C4×Dic53—42no—
91C4×Dic53#4112yes—
92C4×Dic53#4112yes—
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53#10312yes—
106C2×C4×C53#11711yescharacteristic
107C2×C4×C53#11711yescharacteristic
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53#10312yes—
120C2×Dic53#12312yes—
121C2×Dic53#12312yes—
122C2×Dic53C2×SD1611yescharacteristic
123C2×Dic53C2×Q1611yescharacteristic
124C2×Dic53AGL(1,ℤ/8)12yes—
125Dic106—24no—
126Dic106—24no—
127#88324—82no—
128#88324—82no—
129#8767—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53#38111yescharacteristic
133C22×C53C2×#10311yescharacteristic
134C22×C53#41811yescharacteristic
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—81no—
140C4×C53—81no—
141Dic53—22no—
142Dic53—22no—
143Dic53—42no—
144Dic53—41no—
145Dic53—81no—
146Dic53#48112yes—
147Dic53#48412yes—
148#2351—531no—
149#1801—531no—
150#1461—531no—
151#1303—532no—
152#1286—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53#182411yescharacteristic
156C2×C53#182711yescharacteristic
157C2×C53#183011yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×Q8—1061no—
160#431—532no—
161#423—532no—
162C2×#101—1061no—
163#389—1061no—
164#389—1061no—
165#389—1061no—
166#382—531no—
167#382—531no—
168#382#615711yescharacteristic
169C4×C4⋊C4—531no—
170C4×C4⋊C4—531no—
171C4×C4⋊C4—531no—
172#343—1061no—
173#324—1061no—
174#324—1061no—
175C53#876711yescharacteristic, complemented, Sylow
176C22×Q8—531no—
177C2×M16—1062no—
178C4×Q8—1062no—
179C4×Q8—1062no—
180C4×Q8—2124no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1061no—
193C2×C42—1061no—
194C2×C42—1061no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198C4⋊C8—1062no—
199#101—2122no—
200C22⋊C8—1062no—
201C4×C8—1062no—
202#95—21no—
203#95—531no—
204#95—531no—
205#95—1061no—
206#95—1061no—
207#95C2×#615711yescharacteristic
208#95C2×#615711yescharacteristic
209C2×Q8—532no—
210C2×Q8—532no—
211C2×Q8—1062no—
212C2×Q8—1062no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22×#615711yescharacteristic
224M16—2122no—
225M16—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—42no—
229C4⋊C4—1062no—
230C4⋊C4—1062no—
231C4⋊C4—1062no—
232C4⋊C4—2122no—
233C4⋊C4—2122no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C42—21no—
238C42—21no—
239C42—532no—
240C42—532no—
241C42—1061no—
242C42—1061no—
243C42—2122no—
244C42—2122no—
245C42—2122no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—1064no—
248Q8—1064no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—532no—
261C2×C4—531no—
262C2×C4—531no—
263C2×C4—532no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4#1313328012yes—
278C2×C4#1313329611yescharacteristic
279C2×C4#1313329611yescharacteristic
280C8—4242no—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22#1681156611yescharacteristic
285C22C2×#1313328011yescharacteristic
286C22#1681161511yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—532no—
294C4—532no—
295C4—1062no—
296C4—1062no—
297C4—4241no—
298C4—2122no—
299C4—2121no—
300C2—21no—
301C2—21no—
302C2#2029061911yescharacteristic
303C2#2029062511yescharacteristic
304C2#2029062911yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC22×C53#381
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#8767
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
175. C53#8767semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202896132
#202895682
#202776322
#876753