Finite Group Databasebeta

← #27272357

Subgroups — #27272357

Order 13568.

636 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 103 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#20274462C212yesmaximal, complemented
3#20272536C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic, complemented
6C2×#13133213C411yescharacteristic
7C2×D4×Dic53C411yescharacteristic
8C2×#13133179C411yescharacteristic
9C2×#13133177—22no—
10C2×#13133127—22no—
11#16810525C2211yescharacteristic
12#16810523—22no—
13#16810492C411yescharacteristic
14C42×Dic53C2211yescharacteristic
15#16810378C2212yes—
16#16810377—22no—
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#13133213—22no—
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#13133127—28no—
40C22×D4×C53C2×C411yescharacteristic
41C22×C8×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C42×C53C2311yescharacteristic
45C24×C53—21no—
46C23×D53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53M1612yes—
51C22×C4×C53—21no—
52C22×C4×C53C22×C411yescharacteristic
53C22×C4×C53C2×D412yes—
54C2×C22⋊D53—28no—
55C2×C22⋊D53—44no—
56C2×C22⋊D53—44no—
57C22×Dic53—21no—
58C22×Dic53—44no—
59C22×Dic53—42no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C412yes—
63C2×D212—21no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212M1612yes—
67C2×C4×D53—44no—
68C2×C4×D53—42no—
69C2×C8×C53—24no—
70C2×C8×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C42×C53—41no—
76C22⋊Dic53—42no—
77#11719921—42no—
78#11719921—42no—
79#11719921—42no—
80C4⋊Dic53—41no—
81#11719919—44no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—24no—
88C2×#88322—24no—
89C2×#88322—42no—
90C2×#88322—42no—
91C23×C53—22no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97C22×D53—22no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—42no—
101C22×D53—42no—
102D4×C53—42no—
103D4×C53—42no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53C2×M1611yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×C4×C53AGL(1,ℤ/8)12yes—
113C2×C4×C53AGL(1,ℤ/8)12yes—
114C22⋊D53—416no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—42no—
122C2×Dic53—84no—
123C2×Dic53#10012yes—
124C2×Dic53#10414yes—
125D212—42no—
126D212—42no—
127C4×D53—84no—
128C8×C53—44no—
129#88322—44no—
130#7637—531nomaximal, Sylow
131C22×C53—42no—
132C22×C53—44no—
133C22×C53—81no—
134C22×C53—82no—
135C22×C53—81no—
136C22×C53#40711yescharacteristic
137C22×C53#41011yescharacteristic
138C22×C53C2×#10011yescharacteristic
139C22×C53#41812yes—
140C22×C53C2×#10412yes—
141C2×D53—42no—
142C2×D53—44no—
143C2×D53—81no—
144C2×D53—82no—
145C2×D53—81no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—81no—
151Dic53—22no—
152Dic53—24no—
153Dic53—41no—
154Dic53—81no—
155C4×#127—531no—
156#1745—532no—
157#1270—531no—
158#1270—532no—
159#1270D5311yescharacteristic, complemented
160C2×C53—82no—
161C2×C53#146611yescharacteristic
162C2×C53#149012yes—
163C2×C53#150512yes—
164C2×C53#152511yescharacteristic
165C2×C53#155511yescharacteristic
166D53—82no—
167C2×#127—531no—
168C2×C4×D4—531no—
169C2×C4×D4—531no—
170C2×C4×D4—1062no—
171C2×#97—1062no—
172#390—531no—
173#390—1062no—
174#390C2×D5311yescharacteristic
175#384—531no—
176C43—531no—
177#336—532no—
178#336—1062no—
179#336C2×D5312yes—
180C53#763711yescharacteristic, complemented, Sylow
181C22×D4—531no—
182C22×D4—1062no—
183C22×D4C4×D5311yescharacteristic
184C23×C4—1061no—
185C23×C4—1061no—
186C22×C8—22no—
187C22×C8—1062no—
188#127—1062no—
189#127—1062no—
190C4×D4—2124no—
191C4×D4—2124no—
192C4×D4—2122no—
193C4×D4—2122no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4C4×D5311yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1062no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—532no—
204C2×C42—532no—
205C2×C42C22×D5311yescharacteristic
206#97—1068no—
207C24—21no—
208C24—1061no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—532no—
213C2×D4—1068no—
214C2×D4—1061no—
215C2×D4—2121no—
216C2×D4—2121no—
217C2×D4—2124no—
218C2×D4—2124no—
219C2×D4#1171991212yes—
220C22×C4—21no—
221C22×C4—532no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—1061no—
225C22×C4—2122no—
226C22×C4—2124no—
227C22×C4—2122no—
228C22×C4—2124no—
229C22×C4C2×C4×D5311yescharacteristic
230C22×C4D4×D5312yes—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—1064no—
235C2×C8—2122no—
236C2×C8—2122no—
237C4⋊C4—41no—
238C4⋊C4—2121no—
239C4⋊C4—2124no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—41no—
247C42—1064no—
248C42—1062no—
249C42—1061no—
250C42—1061no—
251C42—1062no—
252C42—1062no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—1062no—
259C23—2121no—
260C23—2121no—
261C23—2122no—
262C23—2122no—
263C23C22⋊C4×D5311yescharacteristic
264D4—42no—
265D4—42no—
266D4—2122no—
267D4—2122no—
268D4—21216no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—532no—
276C2×C4—534no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—1062no—
280C2×C4—1061no—
281C2×C4—1062no—
282C2×C4—4244no—
283C2×C4—4244no—
284C2×C4—2121no—
285C2×C4—2122no—
286C2×C4M16×D5311yescharacteristic
287C2×C4C2×#1171991211yescharacteristic
288C2×C4#1313324112yes—
289C2×C4#1313324412yes—
290C8—44no—
291C8—2124no—
292C22—44no—
293C22—42no—
294C22—81no—
295C22—82no—
296C22—81no—
297C22—4241no—
298C22—4242no—
299C22—4241no—
300C22—2124no—
301C22—2122no—
302C22#1681058511yescharacteristic
303C22#1681059511yescharacteristic
304C22#100×D5311yescharacteristic
305C22#1681062612yes—
306C22#104×D5312yes—
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—1064no—
313C4—1062no—
314C4—4241no—
315C4—2121no—
316C2—82no—
317C2—4242no—
318C2#2027305411yescharacteristic
319C2#2027315612yes—
320C2#2027322112yes—
321C2#2027330511yescharacteristic
322C2#2027342711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C42×C53C23
hypercenter#1270D53
hypocenterC53#7637
LCSG ▷ C22×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. #20274462C2semidirect
3. #20272536C2semidirect
4. #20272535C2semidirect
5. C53×#1270C2semidirect
159. #1270D53semidirect
180. C53#7637semidirect

Maximal subgroups

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

subgroupindex
#202772792
#202744622
#202725362
#202725352
C53×#12702
#763753