Finite Group Databasebeta

← #27293624

Subgroups — #27293624

Order 13568.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#20278998C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272664C212yesmaximal
4#20272644C212yesmaximal
5C53×#1464C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic
7#16810676—21no—
8#16810561—21no—
9#16810554—21no—
10#16810507—21no—
11#16810507—21no—
12#16810495C411yescharacteristic
13#16810495C2211yescharacteristic
14#16810486C411yescharacteristic
15#16810484C411yescharacteristic
16#16810484C2211yescharacteristic
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C2×#11719919—21no—
26C2×#11719919—21no—
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—41no—
30C2×#11719919—41no—
31C2×C22⋊Dic53C2×C411yescharacteristic
32C2×C22⋊Dic53C2×C411yescharacteristic
33C4⋊#88322—41no—
34C22⋊#88322—22no—
35C4×#88322—41no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45#117×C53—41no—
46#117×C53—41no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53C2311yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55#97×C53—41no—
56C24×C53C22×C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—22no—
72C22×Dic53—42no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C2×C8×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C22⋊Dic53—22no—
86C22⋊Dic53—41no—
87#11719919—42no—
88#11719919—42no—
89#11719919—42no—
90#11719919—41no—
91#11719919—41no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#13112yes—
99C2×C4×C53—21no—
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—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#9911yescharacteristic
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—82no—
131C8×C53—82no—
132#88322—82no—
133#11544—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53#40811yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53#41311yescharacteristic
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151Dic53—42no—
152Dic53—81no—
153Dic53—81no—
154Dic53—81no—
155#2484—531no—
156#1464D5311yescharacteristic
157#1458—531no—
158#1326—532no—
159#1317—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53#146811yescharacteristic
164C2×C53#212011yescharacteristic
165C2×C53#212311yescharacteristic
166C2×#117C2×D5311yescharacteristic
167C22×C4⋊C4—531no—
168#432—21no—
169#431—1061no—
170#400—1061no—
171#398—1061no—
172#387—1061no—
173#387—1061no—
174#385—531no—
175#385—531no—
176#381—531no—
177#381—531no—
178#381—531no—
179#324—21no—
180#324—1061no—
181#323—532no—
182#323C2×D5312yes—
183C53#1154411yescharacteristic, complemented, Sylow
184C23×C4—531no—
185C23×C4—531no—
186C23×C4C4×D5311yescharacteristic
187#117—41no—
188#117—41no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—2121no—
194C2×C4⋊C4—2121no—
195C2×C4⋊C4D21211yescharacteristic
196C2×C4⋊C4C22⋊D5311yescharacteristic
197C2×C22⋊C4—531no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4C4×D5311yescharacteristic
200C2×C22⋊C4C22×D5311yescharacteristic
201C2×C42D21211yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—2121no—
205C22⋊C8—22no—
206C22⋊C8—1062no—
207#97—41no—
208C4×C8—2121no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C24C2×C4×D5311yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—1061no—
223C22×C4—1062no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4—2122no—
228C22×C4—2121no—
229C22×C4—2121no—
230C22×C4—2121no—
231C22×C4#1171992111yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4#1171992111yescharacteristic
234C22×C4#1171992111yescharacteristic
235C22×C4C2×D21211yescharacteristic
236C22×C4C2×C22⋊D5311yescharacteristic
237C2×C8—42no—
238C2×C8—2122no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—1062no—
249C22⋊C4—2121no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×#1171992111yescharacteristic
259C23#1313322512yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—1062no—
276C2×C4—4242no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2122no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4#1313313111yescharacteristic
291C2×C4#1313313111yescharacteristic
292C8—82no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1681059111yescharacteristic
302C22C2×#1313313111yescharacteristic
303C22#1681060511yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4241no—
312C4—4241no—
313C4—4241no—
314C4—2122no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2027306411yescharacteristic
319C2#2027635711yescharacteristic
320C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutator, prime radicalC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11544
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
183. C53#11544semidirect

Maximal subgroups

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

subgroupindex
#202789982
#202730292
#202726642
#202726442
C53×#14642
#1154453