Finite Group Databasebeta

← #27273263

Subgroups — #27273263

Order 13568.

404 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 83 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#20281676C211yesmaximal, characteristic
2#20276192C211yesmaximal, characteristic, complemented
3#20276119C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272535C211yesmaximal, characteristic
7C53×#2099C211yesmaximal, characteristic, complemented
8C2×#13133212—21no—
9C2×#13133148—21no—
10C2×#13133137—21no—
11#16810560—21no—
12#16810545—21no—
13#16810540—21no—
14#16810525C2211yescharacteristic
15#16810522—21no—
16#16810521C2211yescharacteristic
17#16810515—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C53×#389C2211yescharacteristic
23C53×#384C2211yescharacteristic
24C53×#340—21no—
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×D212D411yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30#13133212—42no—
31#13133212—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45#13133148—42no—
46#13133137—42no—
47#13133124—21no—
48#13133124—21no—
49C22×C8×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58C23×D53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—41no—
71C2×D212—41no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53SD1612yes—
85C42×C53—41no—
86#11719921—42no—
87#11719921—42no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91#11719921—41no—
92#11719921—81no—
93#11719921—81no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—81no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—42no—
103C2×#88322—22no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C23×C53#12011yescharacteristic
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—22no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C2×SD1611yescharacteristic
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147D212—42no—
148D212—81no—
149C8×C53—81no—
150#88322—42no—
151#88322—81no—
152#7808—531nomaximal, Sylow
153C22×C53#44911yescharacteristic
154C22×C53#44911yescharacteristic
155C22×C53#45011yescharacteristic
156C22×C53#45411yescharacteristic
157C22×C53#45511yescharacteristic
158C22×C53#45611yescharacteristic
159C22×C53#45811yescharacteristic
160C2×D53—81no—
161C2×D53—81no—
162C2×D53—81no—
163C2×D53—81no—
164C2×D53—81no—
165C2×D53—81no—
166C2×D53—81no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—81no—
175C4×C53—161no—
176Dic53—81no—
177Dic53—81no—
178Dic53—81no—
179#2727—531no—
180#2099D5311yescharacteristic, complemented
181#2084—531no—
182#2071—531no—
183#1271—531no—
184#1271—531no—
185#1270—531no—
186C2×C53#160911yescharacteristic
187C2×C53#161211yescharacteristic
188C2×C53#161611yescharacteristic
189C2×C53#162811yescharacteristic
190C2×C53#164211yescharacteristic
191C2×C53#165411yescharacteristic
192C2×C53#166411yescharacteristic
193D53—161no—
194C2×#126—1061no—
195C2×#106—1061no—
196C2×#102—1061no—
197#399—1061no—
198#396—1061no—
199#394—1061no—
200#390—531no—
201#389—531no—
202#389—1061no—
203#389C2×D5311yescharacteristic
204#388—1061no—
205#384C2×D5311yescharacteristic
206#340—21no—
207#340—21no—
208#340—1061no—
209#340—1061no—
210#336—531no—
211#336—531no—
212#336C2×D5311yescharacteristic
213C53#780811yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C22×C8—21no—
216C22×C8—1061no—
217C22×C8—1061no—
218#126—2122no—
219#126—2122no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C42—1061no—
239C2×C42C22×D5311yescharacteristic
240#106—2122no—
241#102—2122no—
242#95—21no—
243#95—1061no—
244#95—1061no—
245C24—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—2121no—
249C2×D4—2121no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4D4×D5311yescharacteristic
257C22×C4D4×D5311yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—1062no—
263C2×C8—2121no—
264C2×C8—2121no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—4241no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C4⋊C4—2122no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4#1171992412yes—
286C22⋊C4—4241no—
287C22⋊C4—4241no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2121no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C42—41no—
295C42—2122no—
296C23—2121no—
297C23—2121no—
298C23—2121no—
299C23—2121no—
300C23—2121no—
301C23—2121no—
302C23—2121no—
303C23#1313318511yescharacteristic
304D4—4241no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4C2×#1171992411yescharacteristic
331C2×C4#1313323611yescharacteristic
332C2×C4#1313323811yescharacteristic
333C2×C4#1313324111yescharacteristic
334C2×C4#1313324211yescharacteristic
335C2×C4#1313324411yescharacteristic
336C8—81no—
337C8—4241no—
338C8—2122no—
339C22—4241no—
340C22—4241no—
341C22—4241no—
342C22—4241no—
343C22—4241no—
344C22—4241no—
345C22—4241no—
346C22#1681076511yescharacteristic
347C22#1681076711yescharacteristic
348C22#1681077311yescharacteristic
349C22#1681079511yescharacteristic
350C22#1681079911yescharacteristic
351C22#1681080411yescharacteristic
352C22#1681081211yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—81no—
361C4—161no—
362C4—4241no—
363C4—4241no—
364C4—4241no—
365C2—8481no—
366C2#2027371111yescharacteristic
367C2#2027372611yescharacteristic
368C2#2027375111yescharacteristic
369C2#2027382211yescharacteristic
370C2#2027391411yescharacteristic
371C2#2027399811yescharacteristic
372C2#2027404611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2099C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#2099D53
hypocenterC53#7808
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
2. #20276192C2semidirect
4. #20272539C2semidirect
5. #20272538C2semidirect
7. C53×#2099C2semidirect
180. #2099D53semidirect
213. C53#7808semidirect

Maximal subgroups

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

subgroupindex
#202816762
#202761922
#202761192
#202725392
#202725382
#202725352
C53×#20992
#780853