Finite Group Databasebeta

← #28299773

Subgroups — #28299773

Order 13568.

426 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290631C211yesmaximal, characteristic
2C4×C4⋊#88324C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5#20283074C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133207—21no—
13#16810522C2211yescharacteristic
14#16810521C411yescharacteristic
15#16810520—21no—
16#16810519C2211yescharacteristic
17#16810518—21no—
18#16810492C411yescharacteristic
19#16810492C411yescharacteristic
20#16810487—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—24no—
27C4×#88324—42no—
28#13133207—42no—
29#13133207—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—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×C53—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
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×#88324—24no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C42×C53—22no—
74C4⋊Dic53—22no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78#11719919—42no—
79#11719919—42no—
80#11719919—42no—
81#11719919—42no—
82#11719919—42no—
83#11719919—41no—
84#11719919—41no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53D812yes—
90C4×Dic53SD1612yes—
91C4×Dic53SD1612yes—
92C4×Dic53Q1612yes—
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53#10212yes—
107C2×C4×C53#10312yes—
108C2×C4×C53#13112yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53#10112yes—
122C2×Dic53#10212yes—
123C2×Dic53#10312yes—
124C2×Dic53#10412yes—
125C2×Dic53C2×D811yescharacteristic
126C2×Dic53C2×SD1611yescharacteristic
127C2×Dic53C2×SD1611yescharacteristic
128C2×Dic53C2×Q1611yescharacteristic
129#88324—44no—
130#88324—82no—
131#7663—531nomaximal, Sylow
132C22×C53#40811yescharacteristic
133C22×C53#41011yescharacteristic
134C22×C53C2×#10111yescharacteristic
135C22×C53C2×#10211yescharacteristic
136C22×C53C2×#10311yescharacteristic
137C22×C53C2×#10411yescharacteristic
138C22×C53#42111yescharacteristic
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—81no—
143C4×C53—81no—
144Dic53—21no—
145Dic53—21no—
146Dic53—21no—
147Dic53—21no—
148Dic53—22no—
149Dic53—81no—
150Dic53—81no—
151Dic53#33214yes—
152#2859—531no—
153#1831—531no—
154C4×C4⋊C8—531no—
155#1271—532no—
156#1271—532no—
157C2×C53#147011yescharacteristic
158C2×C53#148011yescharacteristic
159C2×C53#149311yescharacteristic
160C2×C53#150211yescharacteristic
161C2×C53#151811yescharacteristic
162C2×C53#152811yescharacteristic
163C2×C53C2×#33211yescharacteristic
164C2×#125—1061no—
165C2×C4⋊C8—1061no—
166C2×C4⋊C8—1061no—
167C2×C4×C8—1061no—
168#389—531no—
169#389—531no—
170#389—531no—
171#389—1061no—
172#389—1061no—
173#389#615711yescharacteristic
174#384—531no—
175#384—531no—
176#382—1061no—
177C43—531no—
178#336—532no—
179#336—532no—
180#336—1061no—
181C53#766311yescharacteristic, complemented, Sylow
182C22×C8—1062no—
183C22×C8—1062no—
184#125—2122no—
185#125—2122no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4C2×#615711yescharacteristic
197C2×C4⋊C4C2×#615711yescharacteristic
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42C2×#615711yescharacteristic
205C4⋊C8—1064no—
206C4⋊C8—2122no—
207C4×C8—2122no—
208#95—21no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223C2×C8—1064no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—1062no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C4⋊C4—2121no—
240C4⋊C4—2122no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C42—22no—
244C42—532no—
245C42—532no—
246C42—532no—
247C42—532no—
248C42—1062no—
249C42—1062no—
250C42—2121no—
251C42—2121no—
252C23C2×C22⋊#615711yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—532no—
266C2×C4—532no—
267C2×C4—532no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—532no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4#1313327912yes—
286C2×C4#1313328012yes—
287C2×C4#1313330412yes—
288C8—4242no—
289C8—2124no—
290C22#1681159611yescharacteristic
291C22#1681160211yescharacteristic
292C22D53⋊#10111yescharacteristic
293C22C2×#1313327911yescharacteristic
294C22C2×#1313328011yescharacteristic
295C22D53⋊#10411yescharacteristic
296C22#1681162011yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—81no—
301C4—81no—
302C4—534no—
303C4—1061no—
304C4—1061no—
305C4—1061no—
306C4—1061no—
307C4—1062no—
308C4—4241no—
309C4—4241no—
310C2#2028998711yescharacteristic
311C2#2029002011yescharacteristic
312C2#2029006311yescharacteristic
313C2#2029009411yescharacteristic
314C2#2029014811yescharacteristic
315C2#2029017811yescharacteristic
316C2D53⋊#33211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7663
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
181. C53#7663semidirect

Maximal subgroups

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

subgroupindex
#202906312
C4×C4⋊#883242
#202895342
#202895332
#202830742
#766353