Finite Group Databasebeta

← #27273266

Subgroups — #27273266

Order 13568.

404 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 81 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#20281674C211yesmaximal, characteristic
2#20276274C211yesmaximal, characteristic, complemented
3#20276193C211yesmaximal, characteristic, complemented
4#20276118C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×#13133210—21no—
9C2×#13133149—21no—
10C2×#13133138—21no—
11#16810559—21no—
12#16810546—21no—
13#16810537—21no—
14#16810521C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810518—21no—
17#16810516—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810378C2211yescharacteristic
24C53×#390C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28#13133210—42no—
29#13133210—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133149—42no—
47#13133138—42no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53D411yescharacteristic
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C24×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C2×C8×C53—22no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C42×C53—41no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—81no—
87C22⋊Dic53—81no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4⋊Dic53SD1612yes—
99#11719919—42no—
100#11719919—41no—
101#11719919—41no—
102#11719919—41no—
103#11719919—41no—
104#11719919—81no—
105C4×Dic53—42no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53#12011yescharacteristic
117D4×C53—42no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53C2×SD1611yescharacteristic
129C2×C4×C53#13511yescharacteristic
130C2×C4×C53#13511yescharacteristic
131C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
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—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C8×C53—42no—
150C8×C53—81no—
151#88322—81no—
152#7808—531nomaximal, Sylow
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#44911yescharacteristic
161C22×C53#44911yescharacteristic
162C22×C53#45011yescharacteristic
163C22×C53#45411yescharacteristic
164C22×C53#45511yescharacteristic
165C22×C53#45611yescharacteristic
166C22×C53#45811yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—81no—
174Dic53—42no—
175Dic53—81no—
176Dic53—81no—
177Dic53—81no—
178Dic53—161no—
179#2727—531no—
180#2099—531no—
181#2084—531no—
182#2071—531no—
183#1271—531no—
184#1271—531no—
185#1270D5311yescharacteristic
186C2×C53—161no—
187C2×C53#160911yescharacteristic
188C2×C53#161211yescharacteristic
189C2×C53#161611yescharacteristic
190C2×C53#162811yescharacteristic
191C2×C53#164211yescharacteristic
192C2×C53#165411yescharacteristic
193C2×C53#166411yescharacteristic
194C2×#126—1061no—
195C2×#106—1061no—
196C2×#102—1061no—
197#399—1061no—
198#396—1061no—
199#394—1061no—
200#390C2×D5311yescharacteristic
201#389—531no—
202#389—531no—
203#389—1061no—
204#388—1061no—
205#384—531no—
206#340—1061no—
207#340—1061no—
208#340—1061no—
209#340—1061no—
210#336—531no—
211#336C2×D5311yescharacteristic
212#336C2×D5311yescharacteristic
213C53#780811yescharacteristic, complemented, Sylow
214C22×D4D21211yescharacteristic
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—1061no—
218#126—2122no—
219#126—2122no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4D21211yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
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—1061no—
243#95—1061no—
244#95—1061no—
245C24—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—2121no—
256C22×C4C2×D21211yescharacteristic
257C22×C4D4×D5311yescharacteristic
258C22×C4D4×D5311yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—41no—
269C4⋊C4—532no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—4241no—
274C4⋊C4—2122no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—4241no—
291C22⋊C4—4241no—
292C22⋊C4—2122no—
293C22⋊C4—2122no—
294C42—41no—
295C42—2122no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#1313318711yescharacteristic
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—1062no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—2121no—
320C2×C4—2121no—
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×C4SD16×D5311yescharacteristic
331C2×C4#1313323811yescharacteristic
332C2×C4#1313323911yescharacteristic
333C2×C4#1313324111yescharacteristic
334C2×C4#1313324111yescharacteristic
335C2×C4#1313324511yescharacteristic
336C8—42no—
337C8—81no—
338C8—4241no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1681076911yescharacteristic
347C22#1681077011yescharacteristic
348C22#1681077511yescharacteristic
349C22#1681079711yescharacteristic
350C22#1681080311yescharacteristic
351C22#1681080711yescharacteristic
352C22#1681081411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—8481no—
361C4—4241no—
362C4—4241no—
363C4—4241no—
364C4—2122no—
365C2—161no—
366C2#2027371311yescharacteristic
367C2#2027373111yescharacteristic
368C2#2027375311yescharacteristic
369C2#2027382711yescharacteristic
370C2#2027391611yescharacteristic
371C2#2027400111yescharacteristic
372C2#2027405211yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7808
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20276274C2semidirect
3. #20276193C2semidirect
5. #20272539C2semidirect
6. #20272537C2semidirect
213. C53#7808semidirect

Maximal subgroups

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

subgroupindex
#202816742
#202762742
#202761932
#202761182
#202725392
#202725372
C53×#12702
#780853