Finite Group Databasebeta

← #28309570

Subgroups — #28309570

Order 13568.

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

normal subgroups only

This group has 372 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#20290581C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20289716C212yesmaximal
4#20289714C212yesmaximal
5#20280081C211yesmaximal, characteristic
6#16811640—21no—
7#16811626—21no—
8C2×#13133278—21no—
9#16811503C2212yes—
10#16811469C2212yes—
11#16811468—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810561C411yescharacteristic
14#16810561C2211yescharacteristic
15#16810542C411yescharacteristic
16#16810542C2211yescharacteristic
17#16810529—21no—
18#16810521—21no—
19#16810504—21no—
20#16810504—21no—
21#16810489C411yescharacteristic
22C2×C4×Q8×C53C411yescharacteristic
23C2×#11719940—22no—
24#13133284—41no—
25#13133284—41no—
26C4⋊#88324—41no—
27#13133278—42no—
28C22⋊#88324—22no—
29C4×#88324—41no—
30C22×Dic106D411yescharacteristic
31C4×Dic106—41no—
32C4×Dic106—41no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919C2×C411yescharacteristic
38C2×#11719919C2311yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124C2×C411yescharacteristic
49#13133124C2×C411yescharacteristic
50C22×Q8×C53C2×C411yescharacteristic
51C4×Q8×C53—41no—
52C4×Q8×C53—41no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57C2×Q8×C53—42no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106SD1612yes—
73C2×Dic106Q1612yes—
74#11719940—42no—
75#11719940—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—42no—
84C4⋊Dic53—41no—
85C4⋊Dic53—42no—
86#11719919—41no—
87#11719919—41no—
88#11719919—42no—
89#11719919—42no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#10312yes—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53#10312yes—
125C2×Dic53C2×SD1611yescharacteristic
126C2×Dic53C2×Q1611yescharacteristic
127Dic106—42no—
128Dic106—42no—
129#88324—82no—
130#88324—82no—
131#11905—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10311yescharacteristic
136C22×C53#42111yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—42no—
141C4×C53—41no—
142C4×C53—81no—
143Dic53—21no—
144Dic53—22no—
145Dic53—21no—
146Dic53—41no—
147Dic53—41no—
148Dic53—81no—
149Dic53—81no—
150#2581—531no—
151#1801—531no—
152#1461—531no—
153#1354—532no—
154#1353—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53#152011yescharacteristic
158C2×C53#212011yescharacteristic
159C2×C53#213311yescharacteristic
160C2×C4×Q8—531no—
161C2×C4×Q8#615711yescharacteristic
162#431—1061no—
163#423—1061no—
164C2×#101—1061no—
165#400—531no—
166#400—531no—
167#395—531no—
168#395—531no—
169#391—1061no—
170#389—1061no—
171#387—1061no—
172#387—1061no—
173#382—531no—
174#343—532no—
175#324—532no—
176#324—1061no—
177C53#1190511yescharacteristic, complemented, Sylow
178C22×Q8—531no—
179C22×Q8C2×#615711yescharacteristic
180C2×M16—1062no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—2121no—
184C4×Q8—2121no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C42—21no—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42C2×#615711yescharacteristic
197C4⋊C8—2121no—
198C4⋊C8—2121no—
199C4⋊C8—2121no—
200#101—2122no—
201C22⋊C8—1062no—
202C4×C8—2121no—
203#95—531no—
204#95—531no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—532no—
215C2×Q8—532no—
216C2×Q8—2121no—
217C2×Q8—2121no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229M16—2122no—
230M16—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C4⋊C4—2122no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—1061no—
246C42—1061no—
247C42—1061no—
248C42—1061no—
249C23C2×C22⋊#615711yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—2122no—
253Q8—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—532no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4#1313327511yescharacteristic
282C2×C4#1313327511yescharacteristic
283C2×C4#1313328012yes—
284C8—4242no—
285C8—4242no—
286C22—21no—
287C22—21no—
288C22C2×#1313327511yescharacteristic
289C22C2×#1313328011yescharacteristic
290C22#1681162211yescharacteristic
291C4—22no—
292C4—21no—
293C4—21no—
294C4—42no—
295C4—41no—
296C4—81no—
297C4—1061no—
298C4—1062no—
299C4—1061no—
300C4—4241no—
301C4—4241no—
302C4—2121no—
303C4—2121no—
304C2—21no—
305C2—21no—
306C2#2029015511yescharacteristic
307C2#2029096611yescharacteristic
308C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C4×Q8#6157
hypocenterC53#11905
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
177. C53#11905semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202897162
#202897142
#202800812
#1190553