Finite Group Databasebeta

← #28302249

Subgroups — #28302249

Order 13568.

836 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 146 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#20290570C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3C2×#16811473C214yesmaximal, complemented
4C2×#16811178C211yesmaximal, characteristic
5C2×C4⋊#88324C2212yes—
6C2×#13133274—21no—
7C2×#13133274—21no—
8C2×C4×#88324C2212yes—
9#16811489—21no—
10#16811489—21no—
11#16811473—28no—
12#16811178—28no—
13C2×#13133217C411yescharacteristic
14C2×#13133208—21no—
15C2×#13133197C412yes—
16C2×#13133193—21no—
17C2×Q8×Dic53—21no—
18C2×#13133182C2212yes—
19C2×#13133175—21no—
20C2×C42×D53C2211yescharacteristic
21C2×C4⋊Q8×C53C411yescharacteristic
22C22×#88324—22no—
23C22×#88324—22no—
24C4⋊#88324—24no—
25#13133274—24no—
26#13133274—42no—
27C4×#88324—24no—
28C22×Dic106C2×C412yes—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133217—24no—
33#13133208—28no—
34#13133197—24no—
35#13133197—24no—
36#13133193—28no—
37Q8×Dic53—28no—
38#13133182—24no—
39#13133182D418yes—
40#13133175—28no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—22no—
44C2×#11719919—22no—
45C2×#11719921—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50C42×D53—24no—
51C22×Q8×C53—21no—
52C4⋊Q8×C53—24no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C412yes—
55C2×C42×C53C2×C411yescharacteristic
56C23×D53C22⋊C411yescharacteristic
57C2×Q8×C53—24no—
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53SD1612yes—
77C2×C4×D53Q1612yes—
78C2×Dic106—24no—
79C2×Dic106—24no—
80C2×Dic106—22no—
81C2×Dic106C22⋊C414yes—
82C2×#88324—24no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53C22⋊C414yes—
91C42×C53—22no—
92#11719921—28no—
93C4⋊Dic53—24no—
94C4⋊Dic53—24no—
95C4⋊Dic53—28no—
96#11719919—28no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—28no—
100C4×Dic53C2×D414yes—
101C23×C53C2×C22⋊C411yescharacteristic
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53#10312yes—
109Q8×C53—48no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—24no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53#10312yes—
118C2×C4×C53#10412yes—
119C2×C4×C53C2×C22⋊C412yes—
120C2×C4×C53#13112yes—
121C2×C4×C53#13112yes—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—24no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53#12012yes—
131C2×Dic53#12012yes—
132C2×Dic53C2×SD1611yescharacteristic
133C2×Dic53C2×Q1611yescharacteristic
134C2×Dic53#13712yes—
135C4×D53—24no—
136C4×D53—24no—
137C4×D53—44no—
138C4×D53—44no—
139Dic106—28no—
140Dic106—44no—
141#88324—44no—
142#88324—82no—
143#8314—531nomaximal, Sylow
144C22×C53#37911yescharacteristic
145C22×C53#40811yescharacteristic
146C22×C53#40811yescharacteristic
147C22×C53C2×#10311yescharacteristic
148C22×C53#41911yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—44no—
161C4×C53#40814yes—
162Dic53—22no—
163Dic53—44no—
164Dic53—41no—
165Dic53#44812yes—
166Dic53#45112yes—
167C2×#533—531no—
168#1797—531no—
169C4×C22⋊C8—531no—
170C2×Q8⋊C8—534no—
171C2×C53#148111yescharacteristic
172C2×C53#148511yescharacteristic
173C2×C53#149311yescharacteristic
174C2×C53#149711yescharacteristic
175C2×C53#178011yescharacteristic
176C2×C53#178711yescharacteristic
177C2×C53#178911yescharacteristic
178D53—42no—
179#533—1068no—
180C2×C4⋊Q8—531no—
181C2×C4⋊Q8#615711yescharacteristic
182C2×#125—1061no—
183C2×C22⋊Q8—532no—
184C2×C22⋊Q8—1061no—
185C2×C4×Q8—532no—
186C2×C4×Q8—1061no—
187C2×#117—1061no—
188C22×C42—531no—
189C2×C4⋊C8—532no—
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192C2×C4×C8—532no—
193#336—1061no—
194#336—1061no—
195Q8⋊C8—1068no—
196C53#831411yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C22×Q8—532no—
199C23×C4—531no—
200C23×C4—531no—
201C23×C4—531no—
202C22×C8—1062no—
203C22×C8—1062no—
204C4⋊Q8—24no—
205C4⋊Q8—1064no—
206#125—1068no—
207C22⋊Q8—1064no—
208C22⋊Q8—1068no—
209C22⋊Q8—1064no—
210C4×Q8—538no—
211C4×Q8—1064no—
212C4×Q8—1068no—
213#117—1068no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C2×#615712yes—
220C2×C22⋊C4—1062no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—1064no—
225C2×C42—1062no—
226C2×C42C2×#615711yescharacteristic
227C4⋊C8—1064no—
228C22⋊C8—1064no—
229C22⋊C8—2122no—
230C4×C8—1064no—
231C24—531no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—534no—
236C2×Q8—1062no—
237C2×Q8—1064no—
238C2×Q8—1064no—
239C22×C4—22no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—532no—
243C22×C4—532no—
244C22×C4—531no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—1062no—
253C22×C4—2121no—
254C22×C4C22⋊#615711yescharacteristic
255C22×C4C22⋊#615711yescharacteristic
256C22×C4C22×#615711yescharacteristic
257C2×C8—1064no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—1068no—
266C4⋊C4—1064no—
267C4⋊C4—1068no—
268C4⋊C4—1064no—
269C4⋊C4C22⋊#615714yes—
270C22⋊C4—1068no—
271C42—22no—
272C42—534no—
273C42—1062no—
274C42—1062no—
275C42—1068no—
276C23—532no—
277C23—1061no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23—1061no—
282C23—1061no—
283C23C2×C22⋊#615711yescharacteristic
284Q8—48no—
285Q8—1068no—
286Q8—2124no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—532no—
295C2×C4—531no—
296C2×C4—531no—
297C2×C4—532no—
298C2×C4—532no—
299C2×C4—1064no—
300C2×C4—1064no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1064no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—2124no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2124no—
311C2×C4#1313328012yes—
312C2×C4#1313328112yes—
313C2×C4C2×C22⋊#615712yes—
314C2×C4#1313330512yes—
315C2×C4#1313330512yes—
316C8—4242no—
317C8—2124no—
318C22—1062no—
319C22—1062no—
320C22—1062no—
321C22—2121no—
322C22—2121no—
323C22—2121no—
324C22—2121no—
325C22#1681156411yescharacteristic
326C22#1681159811yescharacteristic
327C22#1681159811yescharacteristic
328C22C2×#1313328011yescharacteristic
329C22#1681161611yescharacteristic
330C22C2×#1313328111yescharacteristic
331C22#1681162111yescharacteristic
332C4—22no—
333C4—22no—
334C4—44no—
335C4—532no—
336C4—532no—
337C4—1062no—
338C4—2121no—
339C4—2124no—
340C4#1681159814yes—
341C2—2122no—
342C2#2029002511yescharacteristic
343C2#2029003911yescharacteristic
344C2#2029006511yescharacteristic
345C2#2029007911yescharacteristic
346C2#2029054011yescharacteristic
347C2#2029055411yescharacteristic
348C2#2029055611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#8314
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16811473C2semidirect
196. C53#8314semidirect

Maximal subgroups

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

subgroupindex
#202905702
C4×#131332742
C2×#168114732
C2×#168111782
#831453