Finite Group Databasebeta

← #28299540

Subgroups — #28299540

Order 13568.

406 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290643C211yesmaximal, characteristic
2#20290631C211yesmaximal, characteristic
3#20289536C212yesmaximal
4#20289533C212yesmaximal
5#20277186C211yesmaximal, characteristic
6C2×#13133284—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133182C411yescharacteristic
13#16810529C411yescharacteristic
14#16810528C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810510—21no—
17#16810504—21no—
18C4⋊C4×Dic53—21no—
19C4⋊C4×Dic53C411yescharacteristic
20#16810468—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22×Dic106C2×C411yescharacteristic
27#13133182—22no—
28#13133182C814yescomplemented
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×Dic106—21no—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C2×Dic106C2×C812yes—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—21no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53C2×C812yes—
76C42×C53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80#11719919—24no—
81C4×Dic53—24no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53—42no—
89C4×Dic53C2×C812yes—
90C4×Dic53M1612yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C22×C811yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53C22⋊C812yes—
123C2×Dic53C22⋊C812yes—
124C2×Dic53AGL(1,ℤ/8)11yescharacteristic
125C2×Dic53#13711yescharacteristic
126C2×Dic53#13712yes—
127Dic106—42no—
128Dic106—42no—
129#88324—82no—
130#88324—82no—
131#7596—531nomaximal, Sylow
132C22×C53C2×C22⋊C811yescharacteristic
133C22×C53#41011yescharacteristic
134C22×C53#41311yescharacteristic
135C22×C53#41711yescharacteristic
136C22×C53#41911yescharacteristic
137C22×C53#42111yescharacteristic
138C22×C53#42111yescharacteristic
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—41no—
144C4×C53—81no—
145Dic53—22no—
146Dic53—22no—
147Dic53—21no—
148Dic53—21no—
149Dic53—22no—
150Dic53—42no—
151Dic53—81no—
152#2295—531no—
153#1839—531no—
154#1831—531no—
155#1272—532no—
156#1271—532no—
157C2×C53#145811yescharacteristic
158C2×C53#147711yescharacteristic
159C2×C53#147911yescharacteristic
160C2×C53#151211yescharacteristic
161C2×C53#151511yescharacteristic
162C2×C53#153511yescharacteristic
163C2×C53#155911yescharacteristic
164C2×C4×Q8—531no—
165C2×C4⋊C8—1061no—
166#391—531no—
167#391—531no—
168#389—531no—
169#389#615711yescharacteristic
170#387—1061no—
171#387—1061no—
172C4×C4⋊C4—531no—
173C4×C4⋊C4—1061no—
174#376—1061no—
175C43—531no—
176#336—532no—
177#336—532no—
178#336—1061no—
179#336—1061no—
180#336—1061no—
181C53#759611yescharacteristic, complemented, Sylow
182C22×Q8—531no—
183C22×C8—1062no—
184C22×C8—1062no—
185C4×Q8—534no—
186C4×Q8—1062no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4C2×#615711yescharacteristic
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C42—531no—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—1061no—
200C2×C42—1061no—
201C2×C42C2×#615711yescharacteristic
202C4⋊C8—2122no—
203#95—21no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209#95—1061no—
210C2×Q8—532no—
211C2×Q8—1061no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—1064no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4D53⋊C812yes—
239C42—41no—
240C42—532no—
241C42—532no—
242C42—1061no—
243C42—1061no—
244C42—1062no—
245C42—1064no—
246C42—2121no—
247C42—2121no—
248C42—2121no—
249C42—2122no—
250C23C2×C22⋊#615711yescharacteristic
251Q8—2122no—
252Q8—2122no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—531no—
270C2×C4—531no—
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×C4C2×D53⋊C811yescharacteristic
287C2×C4D53⋊M1611yescharacteristic
288C8—4242no—
289C8—4242no—
290C22D53⋊(C22⋊C8)11yescharacteristic
291C22#1681160411yescharacteristic
292C22#1681161011yescharacteristic
293C22#1681161411yescharacteristic
294C22#1681161611yescharacteristic
295C22#1681162011yescharacteristic
296C22#1681162111yescharacteristic
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—81no—
303C4—1061no—
304C4—1061no—
305C4—1062no—
306C4—1062no—
307C4—1062no—
308C4—4241no—
309C4—2122no—
310C2#2028994911yescharacteristic
311C2#2029001111yescharacteristic
312C2#2029001811yescharacteristic
313C2#2029012911yescharacteristic
314C2#2029013811yescharacteristic
315C2#2029020011yescharacteristic
316C2#2029025111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7596
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
28. #13133182C8semidirect
181. C53#7596semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202906312
#202895362
#202895332
#202771862
#759653