Finite Group Databasebeta

← #27272065

Subgroups — #27272065

Order 13568.

684 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 143 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#20277279C211yesmaximal, characteristic
2#20274563C212yesmaximal, complemented
3#20272536C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic, complemented
6C2×#13133213C411yescharacteristic
7C2×D4×Dic53C411yescharacteristic
8C2×#13133179C411yescharacteristic
9C2×#13133177—22no—
10C2×#13133146—22no—
11C2×C8×Dic53—22no—
12#16810525C2211yescharacteristic
13#16810523—22no—
14#16810492C411yescharacteristic
15C42×Dic53C2211yescharacteristic
16#16810378C2212yes—
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×D212C2×C411yescharacteristic
20C22×C22⋊D53—22no—
21C23×Dic53—21no—
22C22×C4×D53—21no—
23C22×#88322—22no—
24#13133213—22no—
25#13133213C814yescomplemented
26D4×Dic53—42no—
27#13133179—42no—
28#13133177—44no—
29#13133177—44no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—22no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39#13133146—28no—
40C8×Dic53—44no—
41C22×D4×C53C2×C411yescharacteristic
42C22×C8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C42×C53C2311yescharacteristic
46C24×C53—21no—
47C23×D53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53C2×C812yes—
52C22×C4×C53—21no—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D412yes—
55C2×C22⋊D53—28no—
56C2×C22⋊D53—44no—
57C2×C22⋊D53—44no—
58C22×Dic53—21no—
59C22×Dic53—42no—
60C22×Dic53—44no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C412yes—
63C22×Dic53C22⋊C411yescharacteristic
64C2×D212—21no—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212C2×C812yes—
68C2×C4×D53—42no—
69C2×C4×D53—44no—
70C2×C8×C53—24no—
71C2×C8×C53—42no—
72C2×C8×C53—42no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C42×C53—41no—
77C22⋊Dic53—42no—
78#11719921—42no—
79#11719921—42no—
80#11719921—42no—
81C4⋊Dic53—41no—
82#11719919—44no—
83C4×Dic53—22no—
84C4×Dic53—24no—
85C4×Dic53C2×C812yes—
86C4×Dic53M1612yes—
87C4×Dic53D814yes—
88C4×Dic53SD1614yes—
89C2×#88322—24no—
90C2×#88322—42no—
91C2×#88322—42no—
92C23×C53—22no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98C22×D53—22no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—42no—
102C22×D53—42no—
103D4×C53—42no—
104D4×C53—42no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53C22×C811yescharacteristic
112C2×C4×C53C2×M1611yescharacteristic
113C2×C4×C53C2×D812yes—
114C2×C4×C53C2×SD1612yes—
115C22⋊D53—416no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—42no—
120C2×Dic53—84no—
121C2×Dic53C22⋊C814yes—
122C2×Dic53#10012yes—
123C2×Dic53#10214yes—
124C2×Dic53#10214yes—
125C2×Dic53#10414yes—
126D212—42no—
127D212—42no—
128C4×D53—84no—
129C8×C53—44no—
130#88322—44no—
131#7569—531nomaximal, Sylow
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—81no—
135C22×C53—82no—
136C22×C53—81no—
137C22×C53C2×C22⋊C811yescharacteristic
138C22×C53#41011yescharacteristic
139C22×C53C2×#10011yescharacteristic
140C22×C53C2×#10212yes—
141C22×C53C2×#10412yes—
142C2×D53—42no—
143C2×D53—44no—
144C2×D53—81no—
145C2×D53—82no—
146C2×D53—81no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—81no—
152Dic53—41no—
153Dic53—81no—
154Dic53D4⋊C818yes—
155Dic53#33114yes—
156C4×#127—531no—
157C4×C4⋊C8—532no—
158#1270—531no—
159#1270—532no—
160#1270D5311yescharacteristic, complemented
161C2×C53—82no—
162C2×C53#146011yescharacteristic
163C2×C53C2×D4⋊C812yes—
164C2×C53#150212yes—
165C2×C53#152511yescharacteristic
166C2×C53C2×#33111yescharacteristic
167D53—82no—
168C2×#127—531no—
169C2×C4×D4—531no—
170C2×C4×D4—531no—
171C2×C4×D4—1062no—
172C2×C4⋊C8—1062no—
173C2×C4×C8—1062no—
174#390—531no—
175#390—1062no—
176#390C2×D5311yescharacteristic
177#384—531no—
178C43—531no—
179#336—532no—
180#336C2×D5312yes—
181C53#756911yescharacteristic, complemented, Sylow
182C22×D4—531no—
183C22×D4—1062no—
184C22×D4C4×D5311yescharacteristic
185C23×C4—1061no—
186C23×C4—1061no—
187C22×C8—22no—
188C22×C8—1062no—
189#127—534no—
190#127—1062no—
191C4×D4—2124no—
192C4×D4—2124no—
193C4×D4—2122no—
194C4×D4—2122no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4C4×D5311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1061no—
201C2×C22⋊C4—1062no—
202C2×C42—531no—
203C2×C42—532no—
204C2×C42—531no—
205C2×C42—532no—
206C2×C42C22×D5311yescharacteristic
207C4⋊C8—1068no—
208C4×C8—2124no—
209C24—21no—
210C24—1061no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—532no—
215C2×D4—1068no—
216C2×D4—1061no—
217C2×D4—2121no—
218C2×D4—2121no—
219C2×D4—2124no—
220C2×D4—2124no—
221C2×D4C8×D5312yes—
222C22×C4—21no—
223C22×C4—532no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—1061no—
227C22×C4—2124no—
228C22×C4—2122no—
229C22×C4—2124no—
230C22×C4—2122no—
231C22×C4C2×C4×D5311yescharacteristic
232C22×C4D4×D5312yes—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—1064no—
237C2×C8—2122no—
238C2×C8—2122no—
239C4⋊C4—41no—
240C4⋊C4—2121no—
241C4⋊C4—2124no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C42—41no—
249C42—532no—
250C42—532no—
251C42—534no—
252C42—534no—
253C42—1064no—
254C42—1062no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—1062no—
261C23—2121no—
262C23—2121no—
263C23—2122no—
264C23—2122no—
265C23C22⋊C4×D5311yescharacteristic
266D4—42no—
267D4—42no—
268D4—2122no—
269D4—2122no—
270D4—21216no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—534no—
278C2×C4—532no—
279C2×C4—534no—
280C2×C4—534no—
281C2×C4—534no—
282C2×C4—1061no—
283C2×C4—1062no—
284C2×C4—4244no—
285C2×C4—4244no—
286C2×C4—2121no—
287C2×C4—2122no—
288C2×C4C2×C8×D5311yescharacteristic
289C2×C4M16×D5311yescharacteristic
290C2×C4D8×D5312yes—
291C2×C4SD16×D5312yes—
292C8—44no—
293C8—2124no—
294C22—42no—
295C22—44no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22—4242no—
300C22—4241no—
301C22—4241no—
302C22—2122no—
303C22—2124no—
304C22C22⋊C8×D5311yescharacteristic
305C22#1681059511yescharacteristic
306C22#100×D5311yescharacteristic
307C22#102×D5312yes—
308C22#104×D5312yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—538no—
315C4—534no—
316C4—4241no—
317C4—2121no—
318C2—82no—
319C2—4242no—
320C2#2027303411yescharacteristic
321C2D4⋊C8×D5312yes—
322C2#2027320812yes—
323C2#2027330511yescharacteristic
324C2#331×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C42×C53C23
hypercenter#1270D53
hypocenterC53#7569
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20274563C2semidirect
3. #20272536C2semidirect
4. #20272535C2semidirect
5. C53×#1270C2semidirect
25. #13133213C8semidirect
160. #1270D53semidirect
181. C53#7569semidirect

Maximal subgroups

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

subgroupindex
#202772792
#202745632
#202725362
#202725352
C53×#12702
#756953