Finite Group Databasebeta

← #27294757

Subgroups — #27294757

Order 13568.

486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274792C211yesmaximal, characteristic, complemented
2#20273049C211yesmaximal, characteristic, complemented
3#20272715C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2392C211yesmaximal, characteristic
6#16810680—21no—
7#16810486—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×D4×C53—21no—
13C2×C4×D4×C53C411yescharacteristic, complemented
14C2×#117×C53C2211yescharacteristic
15C53×#398C411yescharacteristic, complemented
16C53×#397C2211yescharacteristic
17C53×#396—21no—
18C53×#394—21no—
19C53×#394C411yescharacteristic, complemented
20C53×#392C2211yescharacteristic
21C4×C4⋊C4×C53C411yescharacteristic, complemented
22C4×C22⋊C4×C53—21no—
23C2×C22⋊Dic53—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26#13133128—41no—
27#13133124—41no—
28#13133124—41no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53D411yescharacteristic
33C4×D4×C53—44no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36#117×C53—41no—
37#117×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53C2311yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—42no—
60C2×D4×C53—41no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—42no—
101C2×#88322—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—22no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53AGL(1,ℤ/8)12yes—
115D4×C53—42no—
116D4×C53—48no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—82no—
143C2×C4×C53—82no—
144C2×C4×C53#9911yescharacteristic
145C2×C4×C53#9911yescharacteristic
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149#88322—82no—
150#11798—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53C2×#9911yescharacteristic
166C22×C53#41811yescharacteristic
167C22×C53#42111yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—81no—
177C4×C53—81no—
178Dic53—162no—
179#2392D5311yescharacteristic
180#1820—531no—
181#1464—531no—
182#1344—532no—
183#1343—532no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53—82no—
188C2×C53#151011yescharacteristic
189C2×C53#212011yescharacteristic
190C2×C53#212611yescharacteristic
191C2×C4×D4—21no—
192C2×C4×D4Dic5311yescharacteristic, complemented
193C2×#117C2×D5311yescharacteristic
194#432—1061no—
195#398Dic5311yescharacteristic, complemented
196#397C2×D5311yescharacteristic
197#396—21no—
198#394—21no—
199#394Dic5311yescharacteristic, complemented
200#392C2×D5311yescharacteristic
201#381—1061no—
202C4×C4⋊C4Dic5311yescharacteristic, complemented
203C4×C22⋊C4—21no—
204#342—532no—
205#342—1061no—
206#324—1061no—
207#323—532no—
208C53#1179811yescharacteristic, complemented, Sylow
209C22×D4—21no—
210C22×D4C2×Dic5311yescharacteristic
211C23×C4—21no—
212C23×C4C22⋊D5311yescharacteristic
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—44no—
216#117—41no—
217#117—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C4⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4C22⋊D5311yescharacteristic
230C2×C22⋊C4C22×D5311yescharacteristic
231C2×C42—21no—
232C2×C42C2×Dic5311yescharacteristic
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—2121no—
235C22⋊C8—1062no—
236#97—2121no—
237#95—21no—
238#95—21no—
239#95—2121no—
240#95—2121no—
241#95C2×Dic5311yescharacteristic
242#95C2×Dic5311yescharacteristic
243C24—21no—
244C24C2×C22⋊D5311yescharacteristic
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—2122no—
262C22×C4C22⋊Dic5311yescharacteristic
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22×Dic5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268C2×C8—2122no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic5311yescharacteristic
300C23#1313324312yes—
301D4—42no—
302D4—48no—
303D4—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—4242no—
331C2×C4—4242no—
332C2×C4—4242no—
333C2×C4#1313313211yescharacteristic
334C2×C4#1313313211yescharacteristic
335C8—4242no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×#1313313211yescharacteristic
351C22#1681062811yescharacteristic
352C22#1681064011yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—8482no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—82no—
368C2#2027324411yescharacteristic
369C2#2027635611yescharacteristic
370C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2392C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2392D53
hypocenterC53#11798
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #20274792C2semidirect
2. #20273049C2semidirect
3. #20272715C2semidirect
13. C2×C4×D4×C53C4semidirect
15. C53×#398C4semidirect
19. C53×#394C4semidirect
21. C4×C4⋊C4×C53C4semidirect
192. C2×C4×D4Dic53semidirect
195. #398Dic53semidirect
199. #394Dic53semidirect
202. C4×C4⋊C4Dic53semidirect
208. C53#11798semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730492
#202727152
#202727122
C53×#23922
#1179853