Finite Group Databasebeta

← #27277653

Subgroups — #27277653

Order 13568.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274703C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272594C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#2353C211yesmaximal, characteristic
6C2×#13133132—21no—
7C2×C22⋊#88322C2212yes—
8#16810483C2212yes—
9#16810390—21no—
10#16810349—21no—
11#16810347—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C53×#388—21no—
14C53×#387—21no—
15C53×#386—21no—
16C53×#383C411yescharacteristic, complemented
17C53×#381C411yescharacteristic, complemented
18C53×#380C2211yescharacteristic
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53C411yescharacteristic, complemented
21C53×#376C411yescharacteristic, complemented
22C2×#95×C53C2211yescharacteristic
23C22×#88322—22no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26#13133132—42no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C4×D4×C53—22no—
35C4×D4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46#95×C53—21no—
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53C2×C411yescharacteristic
52#95×C53C2×C411yescharacteristic
53#95×C53C2×C411yescharacteristic
54#95×C53C2×C411yescharacteristic
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4112yes—
78C22×C4×C53#4112yes—
79C22×Dic53—42no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—81no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—81no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—42no—
91C42×C53—81no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—82no—
95C2×#88322—42no—
96C2×#88322—82no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12312yes—
106C23×C53#13512yes—
107C23×C53AGL(1,ℤ/8)11yescharacteristic
108C23×C53#13711yescharacteristic
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—84no—
137C2×C4×C53#11711yescharacteristic
138C2×C4×C53#11711yescharacteristic
139C2×C4×C53#12312yes—
140C2×Dic53—82no—
141C2×Dic53—82no—
142C2×Dic53—82no—
143#88322—82no—
144#8763—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—21no—
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53#38111yescharacteristic
158C22×C53#41611yescharacteristic
159C22×C53#41711yescharacteristic
160C22×C53#48212yes—
161C22×C53#48512yes—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—42no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—81no—
172Dic53—162no—
173#2353D5311yescharacteristic
174#1799—531no—
175#1459—531no—
176#1294—532no—
177#1289—532no—
178C2×C53—22no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—22no—
182C2×C53—81no—
183C2×C53#182511yescharacteristic
184C2×C53#182711yescharacteristic
185C2×C53#182811yescharacteristic
186C2×C4×D4C2×D5311yescharacteristic
187C2×#99—1061no—
188C2×C22⋊C8—532no—
189#388—21no—
190#387—21no—
191#386—21no—
192#383Dic5311yescharacteristic, complemented
193#381Dic5311yescharacteristic, complemented
194#380—532no—
195#380C2×D5311yescharacteristic
196C4×C22⋊C4—21no—
197C4×C22⋊C4Dic5311yescharacteristic, complemented
198#376Dic5311yescharacteristic, complemented
199C2×#95C2×D5311yescharacteristic
200#342—1061no—
201#324—1061no—
202#323—1061no—
203C53#876311yescharacteristic, complemented, Sylow
204C22×D4C22⋊D5311yescharacteristic
205C23×C4—21no—
206C23×C4C2×Dic5311yescharacteristic
207C23×C4C22×D5311yescharacteristic
208C22×C8—1062no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4C22⋊D5311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4C22⋊D5311yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×Dic5311yescharacteristic
223#99—2122no—
224C22⋊C8—1062no—
225C22⋊C8—1062no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—1062no—
232#95C2×Dic5311yescharacteristic
233#95C2×Dic5311yescharacteristic
234#95C2×Dic5311yescharacteristic
235#95C2×Dic5311yescharacteristic
236C24C22⋊Dic5311yescharacteristic
237C24C2×C22⋊D5311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—2122no—
254C22×C4C22⋊Dic5311yescharacteristic
255C22×C4C22⋊Dic5311yescharacteristic
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4#1171994512yes—
258C22×C4#1171994512yes—
259C22×C4C22×Dic5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C2×C8—4242no—
262C2×C8—2122no—
263C4⋊C4—22no—
264C4⋊C4—81no—
265C4⋊C4—4242no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊Dic5311yescharacteristic
286C23#1313320012yes—
287C23#1313323712yes—
288C23#1313324211yescharacteristic
289C23#1313324711yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
318C2×C4—4242no—
319C2×C4—4242no—
320C2×C4—4242no—
321C2×C4#1313317611yescharacteristic
322C2×C4#1313317611yescharacteristic
323C2×C4#1313320012yes—
324C8—4242no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1681048611yescharacteristic
338C22#1681061911yescharacteristic
339C22#1681062411yescharacteristic
340C22#1681094712yes—
341C22#1681096412yes—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—8482no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2—81no—
358C2#2027481011yescharacteristic
359C2#2027481911yescharacteristic
360C2#2027482311yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2353C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2353D53
hypocenterC53#8763
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #20272594C2semidirect
4. #20272582C2semidirect
16. C53×#383C4semidirect
17. C53×#381C4semidirect
20. C4×C22⋊C4×C53C4semidirect
21. C53×#376C4semidirect
192. #383Dic53semidirect
193. #381Dic53semidirect
197. C4×C22⋊C4Dic53semidirect
198. #376Dic53semidirect
203. C53#8763semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730332
#202725942
#202725822
C53×#23532
#876353