Finite Group Databasebeta

← #28299686

Subgroups — #28299686

Order 13568.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290432C211yesmaximal, characteristic
2#20290432C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5C4×#13133216C211yesmaximal, characteristic
6C2×#13133272—21no—
7C2×#13133272—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×C4×Dic106—21no—
14#16810527—21no—
15#16810522C2211yescharacteristic
16#16810520—21no—
17#16810519C2211yescharacteristic
18#16810492C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133272—24no—
26#13133272—24no—
27C22×Dic106—21no—
28#13133216—22no—
29#13133216—22no—
30C4×Dic106—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52#95×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×Dic106—24no—
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×#88324—24no—
68C2×#88324—24no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53M1612yes—
76C42×C53—22no—
77C42×C53—42no—
78C4⋊Dic53—21no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53M1612yes—
82#11719919—24no—
83#11719919—42no—
84#11719919—42no—
85#11719919—42no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—41no—
93C4×Dic53—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53#10112yes—
109C2×C4×C53C2×M1611yescharacteristic
110C2×C4×C53C2×M1611yescharacteristic
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53#10412yes—
126C2×Dic53#10412yes—
127C2×Dic53AGL(1,ℤ/8)12yes—
128C2×Dic53#13712yes—
129Dic106—48no—
130#88324—44no—
131#88324—44no—
132#7638—531nomaximal, Sylow
133C22×C53#40711yescharacteristic
134C22×C53#41011yescharacteristic
135C22×C53C2×#10111yescharacteristic
136C22×C53#41811yescharacteristic
137C22×C53#41911yescharacteristic
138C22×C53C2×#10411yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—22no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—42no—
144C4×C53—41no—
145C4×C53—81no—
146Dic53—22no—
147Dic53—22no—
148Dic53—22no—
149Dic53—22no—
150Dic53—42no—
151Dic53—81no—
152C4×C4⋊Q8—531no—
153#1745—531no—
154#1745—531no—
155#1271—532no—
156#1271—532no—
157C2×C53#146711yescharacteristic
158C2×C53#149011yescharacteristic
159C2×C53#149111yescharacteristic
160C2×C53#150511yescharacteristic
161C2×C53#150611yescharacteristic
162C2×C53#152611yescharacteristic
163C2×C53#155611yescharacteristic
164C2×C4⋊Q8—531no—
165C2×C4×Q8—1061no—
166C2×#97—1061no—
167C2×#97—1061no—
168#391—1061no—
169#389—531no—
170#389—531no—
171#389—1061no—
172#384—531no—
173C4×C4⋊C4—531no—
174C4×C4⋊C4—1061no—
175C4×C4⋊C4#615711yescharacteristic
176C43—531no—
177#336—532no—
178#336—532no—
179#336—1061no—
180#336—1061no—
181C53#763811yescharacteristic, complemented, Sylow
182C22×Q8—1061no—
183C22×C8—1062no—
184C22×C8—1062no—
185C4⋊Q8—1062no—
186C4⋊Q8—1062no—
187C4×Q8—2124no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4C2×#615711yescharacteristic
196C2×C4⋊C4C2×#615711yescharacteristic
197C2×C42—21no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—1061no—
205C2×C42C2×#615711yescharacteristic
206#97—1064no—
207#97—1064no—
208#95—21no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212C2×Q8—1064no—
213C2×Q8—2122no—
214C2×Q8—2122no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226C2×C8—1064no—
227C2×C8—1064no—
228C2×C8—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C2×C8—2122no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—532no—
235C4⋊C4—1064no—
236C4⋊C4—1061no—
237C4⋊C4—2121no—
238C4⋊C4—2122no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C4⋊C4—2122no—
242C4⋊C4#1171993612yes—
243C42—22no—
244C42—42no—
245C42—1062no—
246C42—1062no—
247C42—1062no—
248C42—1062no—
249C42—1061no—
250C42—1061no—
251C42—2121no—
252C42—2122no—
253C23C2×C22⋊#615711yescharacteristic
254Q8—2128no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—532no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4#1313327812yes—
287C2×C4D53⋊M1611yescharacteristic
288C2×C4C2×#1171993611yescharacteristic
289C8—2124no—
290C8—2124no—
291C22#1681159411yescharacteristic
292C22#1681160211yescharacteristic
293C22C2×#1313327811yescharacteristic
294C22#1681161511yescharacteristic
295C22#1681161611yescharacteristic
296C22D53⋊#10411yescharacteristic
297C22D53⋊#10411yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—42no—
302C4—41no—
303C4—81no—
304C4—1062no—
305C4—1062no—
306C4—1062no—
307C4—1062no—
308C4—4241no—
309C4—2122no—
310C2#2028997711yescharacteristic
311C2#2029005511yescharacteristic
312C2#2029005811yescharacteristic
313C2#2029010511yescharacteristic
314C2#2029010811yescharacteristic
315C2#2029017211yescharacteristic
316C2#2029024411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7638
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
181. C53#7638semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202895342
#202895332
C4×#131332162
#763853