Finite Group Databasebeta

← #28299477

Subgroups — #28299477

Order 13568.

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290607C211yesmaximal, characteristic
2#20290376C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20277196C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133175C411yescharacteristic
13C2×#13133175C411yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810497—21no—
19#16810494—21no—
20C4⋊C4×Dic53—21no—
21#16810474—21no—
22C53×#389C411yescharacteristic
23C2×D53⋊C8—24no—
24C2×D53⋊C8—42no—
25C2×D53⋊C8—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28#13133274—42no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133175—22no—
33#13133175—42no—
34#13133175C814yescomplemented
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47C42×D53—42no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55#95×C53—21no—
56C23×D53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53C2×C812yes—
77C2×C4×D53M1612yes—
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84D53⋊C8—48no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53C2×C812yescomplemented
89C42×C53—41no—
90#11719921—24no—
91#11719921—42no—
92C4⋊Dic53—21no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53C2×C812yescomplemented
96#11719919—42no—
97#11719919—41no—
98#11719919—41no—
99C4×Dic53—24no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C4×Dic53—42no—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53C22⋊C812yes—
110C22×D53#9912yes—
111C22×D53#10112yes—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10412yes—
126C2×C4×C53C22×C811yescharacteristic
127C2×C4×C53C2×M1611yescharacteristic
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53C22⋊C812yes—
141C2×Dic53#13512yes—
142C2×Dic53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53#13711yescharacteristic
144C4×D53—24no—
145C4×D53—24no—
146C4×D53—24no—
147C4×D53—24no—
148C4×D53—44no—
149#88324—44no—
150#88324—82no—
151#7579—531nomaximal, Sylow
152C22×C53C2×C22⋊C811yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53C2×#10111yescharacteristic
155C22×C53#41611yescharacteristic
156C22×C53#41711yescharacteristic
157C22×C53C2×#10411yescharacteristic
158C22×C53#42111yescharacteristic
159C2×D53—22no—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53#32414yes—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—81no—
172Dic53—22no—
173Dic53—21no—
174Dic53—21no—
175Dic53—22no—
176Dic53—42no—
177Dic53—81no—
178#2296—531no—
179#1817—531no—
180C2×#336—531no—
181#1271—532no—
182#1271—532no—
183C2×C53C2×#32411yescharacteristic
184C2×C53#147611yescharacteristic
185C2×C53#147711yescharacteristic
186C2×C53#150411yescharacteristic
187C2×C53#151611yescharacteristic
188C2×C53#153111yescharacteristic
189C2×C53#155211yescharacteristic
190D53—24no—
191C23×C8—1061no—
192C2×#117—531no—
193C2×#117—531no—
194C22×C42—531no—
195C2×C22⋊C8—1061no—
196#389—531no—
197#389—531no—
198#389—531no—
199#389#615711yescharacteristic
200#385—1061no—
201#385—1061no—
202C4×C4⋊C4—1061no—
203C4×C4⋊C4—1061no—
204#336—532no—
205#336—532no—
206#336—1061no—
207#336—1061no—
208C53#757911yescharacteristic, complemented, Sylow
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—531no—
212C22×C8—1062no—
213C22×C8—1064no—
214C22×C8—1062no—
215C22×C8—2122no—
216C22×C8—2122no—
217#117—534no—
218#117—1062no—
219#117—2122no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C2×#615711yescharacteristic
227C2×C4⋊C4C2×#615711yescharacteristic
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42—2122no—
236C2×C42C2×#615711yescharacteristic
237C22⋊C8—2122no—
238#95—21no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24—531no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—532no—
249C22×C4—532no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—2121no—
261C22×C4C22⋊#615711yescharacteristic
262C22×C4C22⋊#615711yescharacteristic
263C22×C4C22×#615711yescharacteristic
264C2×C8—1064no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2128no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—532no—
275C4⋊C4—1061no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2122no—
281C4⋊C4D53⋊C812yescomplemented
282C22⋊C4—1064no—
283C22⋊C4—2122no—
284C42—41no—
285C42—1064no—
286C42—2121no—
287C42—2121no—
288C42—2121no—
289C42—2122no—
290C23—532no—
291C23—532no—
292C23—532no—
293C23—1061no—
294C23—1061no—
295C23—1061no—
296C23—1061no—
297C23C2×C22⋊#615711yescharacteristic
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1064no—
316C2×C4—1064no—
317C2×C4—1064no—
318C2×C4—1064no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—2124no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4#1313328112yes—
333C2×C4C2×D53⋊C811yescharacteristic
334C2×C4D53⋊M1611yescharacteristic
335C8—4242no—
336C8—2124no—
337C22—534no—
338C22—1062no—
339C22—1062no—
340C22—1062no—
341C22—1062no—
342C22—1062no—
343C22—1062no—
344C22D53⋊(C22⋊C8)11yescharacteristic
345C22D53⋊#9911yescharacteristic
346C22D53⋊#10111yescharacteristic
347C22#1681161311yescharacteristic
348C22#1681161411yescharacteristic
349C22C2×#1313328111yescharacteristic
350C22#1681162211yescharacteristic
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—42no—
356C4—81no—
357C4—1061no—
358C4—1061no—
359C4—1062no—
360C4—1062no—
361C4—4241no—
362C4—2122no—
363C2—1064no—
364C2D53⋊#32411yescharacteristic
365C2#2029000711yescharacteristic
366C2#2029001311yescharacteristic
367C2#2029010011yescharacteristic
368C2#2029014211yescharacteristic
369C2#2029018811yescharacteristic
370C2#2029023711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7579
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
34. #13133175C8semidirect
88. C4⋊C4×C53C2×C8semidirect
95. C4⋊Dic53C2×C8semidirect
208. C53#7579semidirect
281. C4⋊C4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202903762
#202895342
#202895332
#202771962
#757953