Finite Group Databasebeta

← #28306242

Subgroups — #28306242

Order 13568.

518 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290630C211yesmaximal, characteristic
2C4×#13133272C211yesmaximal, characteristic
3#20289551C212yesmaximal
4#20289551C212yesmaximal
5#20277286C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×#13133272—21no—
8C2×#13133272—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12C2×#13133216C411yescharacteristic
13C2×#13133182—21no—
14#16810527—21no—
15#16810521C411yescharacteristic
16#16810518—21no—
17#16810492C411yescharacteristic
18C4⋊C4×Dic53C2211yescharacteristic
19C4⋊C4×Dic53C2211yescharacteristic
20#16810474—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26#13133272—24no—
27#13133272—24no—
28C4×#88324—22no—
29C4×#88324Q814yescomplemented
30C22×Dic106—21no—
31#13133216—22no—
32#13133216—22no—
33#13133182—44no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53C2311yescharacteristic
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—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×Dic106—24no—
68C2×Dic106—42no—
69C2×Dic106—42no—
70C2×#88324—24no—
71C2×#88324—24no—
72C2×#88324—24no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53M1612yes—
80C42×C53—22no—
81C4⋊Dic53—21no—
82C4⋊Dic53—22no—
83C4⋊Dic53M1612yes—
84#11719919—24no—
85#11719919—42no—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—22no—
94C4×Dic53—42no—
95C4×Dic53—42no—
96C4×Dic53C2×Q812yes—
97C4×Dic53#4112yes—
98C23×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53C4×Q812yes—
112C2×C4×C53C2×M1611yescharacteristic
113C2×C4×C53C2×M1611yescharacteristic
114C2×C4×C53#13112yes—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—22no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53#10412yes—
128C2×Dic53#10412yes—
129C2×Dic53C22⋊Q812yes—
130C2×Dic53#12411yescharacteristic
131C2×Dic53C4⋊Q811yescharacteristic
132C2×Dic53AGL(1,ℤ/8)12yes—
133C2×Dic53#13712yes—
134Dic106—48no—
135#88324—28no—
136#88324—82no—
137#9877—531nomaximal, Sylow
138C22×C53#39111yescharacteristic
139C22×C53#40711yescharacteristic
140C22×C53#40811yescharacteristic
141C22×C53#41811yescharacteristic
142C22×C53#41911yescharacteristic
143C22×C53C2×#10411yescharacteristic
144C22×C53C2×#10411yescharacteristic
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—42no—
148C4×C53—81no—
149C4×C53#44612yes—
150C4×C53#44612yes—
151Dic53—22no—
152Dic53—22no—
153Dic53—22no—
154Dic53—42no—
155Dic53—81no—
156Dic53#48912yes—
157Dic53#50112yes—
158C4×C4⋊Q8—531no—
159#1831—531no—
160C4×#97—531no—
161#1279—532no—
162#1279—532no—
163C2×C53#148411yescharacteristic
164C2×C53#148511yescharacteristic
165C2×C53#149011yescharacteristic
166C2×C53#149111yescharacteristic
167C2×C53#198511yescharacteristic
168C2×C53#199311yescharacteristic
169C2×C53#199411yescharacteristic
170C2×C4⋊Q8—531no—
171C2×C4×Q8—1061no—
172C2×C4⋊C8—532no—
173C2×#97—1061no—
174C2×#97—1061no—
175C2×C4×C8—532no—
176#391—1061no—
177#389—531no—
178#389—1061no—
179#389#615711yescharacteristic
180#384—531no—
181C4×C4⋊C4—531no—
182C4×C4⋊C4—531no—
183C4×C4⋊C4—1061no—
184C43—531no—
185#336—1061no—
186#336—1061no—
187C53#987711yescharacteristic, complemented, Sylow
188C22×Q8—1061no—
189C22×C8—1062no—
190C22×C8—1062no—
191C4⋊Q8—1062no—
192C4⋊Q8—1062no—
193C4×Q8—2124no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4C2×#615711yescharacteristic
202C2×C4⋊C4C2×#615711yescharacteristic
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—1061no—
210C2×C42—1061no—
211C2×C42C2×#615711yescharacteristic
212C4⋊C8—1064no—
213#97—1064no—
214#97—1064no—
215C4×C8—534no—
216C4×C8—1062no—
217#95—21no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221C2×Q8—1064no—
222C2×Q8—2122no—
223C2×Q8—2122no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—1061no—
231C22×C4—1061no—
232C22×C4C22⋊#615711yescharacteristic
233C22×C4C22⋊#615711yescharacteristic
234C22×C4C22×#615711yescharacteristic
235C2×C8—1064no—
236C2×C8—1064no—
237C2×C8—1064no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—532no—
245C4⋊C4—1064no—
246C4⋊C4—1061no—
247C4⋊C4—1062no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C4⋊C4—2122no—
251C4⋊C4#1171993612yes—
252C42—22no—
253C42—532no—
254C42—532no—
255C42—1062no—
256C42—1062no—
257C42—1062no—
258C42—1062no—
259C42—1061no—
260C42—1061no—
261C42—2122no—
262C42—2122no—
263C23C2×C22⋊#615711yescharacteristic
264Q8—2128no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—531no—
278C2×C4—531no—
279C2×C4—532no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—532no—
283C2×C4—532no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1062no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4Q8×#615712yes—
297C2×C4D53⋊M1611yescharacteristic
298C2×C4C2×#1171993611yescharacteristic
299C2×C4#1313330412yes—
300C8—1068no—
301C8—4242no—
302C22(C2×#6157)⋊Q811yescharacteristic
303C22#1681159411yescharacteristic
304C22#1681159611yescharacteristic
305C22#1681161511yescharacteristic
306C22#1681161611yescharacteristic
307C22D53⋊#10411yescharacteristic
308C22D53⋊#10411yescharacteristic
309C4—42no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—532no—
314C4—532no—
315C4—1062no—
316C4—1062no—
317C4—1062no—
318C4—4241no—
319C4—2122no—
320C4#1681167312yes—
321C4#1681167512yes—
322C2#2029003411yescharacteristic
323C2#2029003711yescharacteristic
324C2#2029005511yescharacteristic
325C2#2029005811yescharacteristic
326C2#2029088811yescharacteristic
327C2#2029089911yescharacteristic
328C2#2029090111yescharacteristic
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#9877
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
29. C4×#88324Q8semidirect
187. C53#9877semidirect

Maximal subgroups

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

subgroupindex
#202906302
C4×#131332722
#202895512
#202772862
#987753