Finite Group Databasebeta

← #27288478

Subgroups — #27288478

Order 13568.

574 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285466C211yesmaximal, characteristic
2#20275551C212yesmaximal, complemented
3#20275143C212yesmaximal, complemented
4#20272583C211yesmaximal, characteristic, complemented
5C53×#1850C211yesmaximal, characteristic, complemented
6C2×#13133191C2212yes—
7#16810937—22no—
8#16810859—21no—
9#16810859—21no—
10#16810759—22no—
11C2×#13133139—22no—
12C2×#13133131—22no—
13C2×#13133131—22no—
14#16810483C2212yes—
15#16810386—22no—
16C23×C8×C53C2211yescharacteristic
17C2×C4⋊C8×C53—21no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C4×C22⋊C4×C53C2211yescharacteristic
20C53×#336—21no—
21C22×D212D412yes—
22C22×C22⋊D53—22no—
23C2×D424—21no—
24C2×D424—21no—
25#13133191—22no—
26#13133191—22no—
27#13133191—22no—
28#13133191—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53D412yes—
31C2×C22⋊Dic53—22no—
32C2×#11719921—22no—
33#13133151—21no—
34#13133151—21no—
35#13133139—22no—
36#13133139—22no—
37#13133139—44no—
38#13133131—44no—
39#13133131—44no—
40#13133124—22no—
41C23×C4×C53C2311yescharacteristic
42C22×C8×C53—21no—
43C22×C8×C53—22no—
44C22×C8×C53—42no—
45C22×C8×C53D411yescharacteristic
46C22×C8×C53D411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53—21no—
50C4⋊C8×C53—42no—
51C22⋊C8×C53—22no—
52#95×C53—21no—
53C24×C53C2×D411yescharacteristic
54C23×D53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53#4111yescharacteristic
62C22×C4×C53#4112yes—
63C2×C22⋊D53—42no—
64C2×C22⋊D53—42no—
65C2×C22⋊D53—42no—
66C2×C22⋊D53—42no—
67C2×C22⋊D53—42no—
68C2×C22⋊D53—42no—
69C22×Dic53—22no—
70C22×Dic53—42no—
71C2×D212—22no—
72C2×D212—22no—
73C2×D212—42no—
74C2×D212—42no—
75C2×C8×C53—22no—
76C2×C8×C53—24no—
77C2×C8×C53—21no—
78C2×C8×C53—21no—
79C2×C8×C53—42no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53D812yescomplemented
87C42×C53—41no—
88C42×C53—41no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91#11719921—42no—
92#11719921—42no—
93C4⋊Dic53—22no—
94C4⋊Dic53—22no—
95C4⋊Dic53—42no—
96C4⋊Dic53—42no—
97D424—81no—
98D424—81no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53#12111yescharacteristic
102C23×C53#12112yes—
103C23×C53C2×D811yescharacteristic
104C23×C53#13511yescharacteristic
105C23×C53AGL(1,ℤ/8)12yes—
106C22×D53—42no—
107C22×D53—42no—
108C22×D53—42no—
109C22×D53—82no—
110C22×D53—82no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12312yes—
125C2×C4×C53#12411yescharacteristic
126C22⋊D53—82no—
127C22⋊D53—82no—
128C22⋊D53—82no—
129C22⋊D53—82no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136D212—44no—
137D212—42no—
138D212—42no—
139C8×C53—41no—
140C8×C53—42no—
141C8×C53—41no—
142#10564—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—21no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—42no—
150C22×C53#39411yescharacteristic
151C22×C53#45912yes—
152C22×C53#46312yes—
153C22×C53#46611yescharacteristic
154C22×C53#46911yescharacteristic
155C2×D53—82no—
156C2×D53—82no—
157C2×D53—82no—
158C2×D53—162no—
159C2×D53—162no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—42no—
163C4×C53—81no—
164C4×C53B(4,𝔽2)11yescharacteristic
165C4×C53#45811yescharacteristic
166Dic53—82no—
167Dic53—162no—
168#3186—531no—
169#1964—532no—
170#1893—532no—
171#1850D5311yescharacteristic, complemented
172#1289—531no—
173C2×C53—22no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—22no—
177C2×C53#204411yescharacteristic
178C2×C53#205311yescharacteristic
179C2×C53#205411yescharacteristic
180D53—162no—
181C23×C8C2×D5311yescharacteristic
182C2×#121—532no—
183#480—1062no—
184#466—1061no—
185#466—1061no—
186#447—1062no—
187C2×C4⋊C8—21no—
188C2×#102—1062no—
189C2×#99—1062no—
190C2×#99—1062no—
191C2×C22⋊C8C2×D5311yescharacteristic
192#380—532no—
193C4×C22⋊C4C2×D5311yescharacteristic
194#340—1062no—
195#336—21no—
196C53#1056411yescharacteristic, complemented, Sylow
197C22×D4—532no—
198C22×D4—1062no—
199C23×C4C22×D5311yescharacteristic
200C2×D8—1061no—
201C2×D8—1061no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8C22⋊D5311yescharacteristic
206C22×C8C22⋊D5311yescharacteristic
207#121—1062no—
208#121—1062no—
209#121—1062no—
210#121—1062no—
211C2×C4⋊C4—532no—
212C2×C4⋊C4—1062no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4D21211yescharacteristic
216C2×C22⋊C4D21211yescharacteristic
217C2×C42—21no—
218#107—1061no—
219#107—1061no—
220C4⋊C8—42no—
221#102—1062no—
222#102—1062no—
223#102—2124no—
224#99—2124no—
225#99—2124no—
226C22⋊C8—22no—
227#95—21no—
228#95—1062no—
229C24—1062no—
230C24C2×D21211yescharacteristic
231C2×D4—1062no—
232C2×D4—1062no—
233C2×D4—2122no—
234C2×D4—2122no—
235C2×D4—2122no—
236C2×D4—2122no—
237C2×D4—2122no—
238C2×D4—2122no—
239C2×D4—2122no—
240C2×D4—2122no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—1062no—
246C22×C4—2122no—
247C22×C4#1171994511yescharacteristic
248C22×C4D4×D5312yes—
249C22×C4#1171994912yes—
250C22×C4C2×C22⋊D5311yescharacteristic
251D8—4241no—
252D8—4241no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—1062no—
263C4⋊C4—1062no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C22⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C22⋊C4D42412yescomplemented
273C42—41no—
274C42—41no—
275C23—41no—
276C23—41no—
277C23—4242no—
278C23—4242no—
279C23—2122no—
280C23—2122no—
281C23—2122no—
282C23#1313319111yescharacteristic
283C23#1313319212yes—
284C23C2×D42411yescharacteristic
285C23#1313323911yescharacteristic
286C23#1313324512yes—
287D4—4242no—
288D4—4242no—
289D4—4242no—
290D4—4242no—
291D4—2124no—
292D4—2122no—
293D4—2122no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4#1313318511yescharacteristic
313C2×C4#1313320212yes—
314C2×C4#1313320611yescharacteristic
315C8—42no—
316C8—41no—
317C8—41no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—8482no—
326C22—8482no—
327C22—4242no—
328C22—4242no—
329C22—4242no—
330C22#1681054011yescharacteristic
331C22#1681081912yes—
332C22#1681083912yes—
333C22#1681085911yescharacteristic
334C22#1681087611yescharacteristic
335C4—22no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4—8482no—
340C4—4242no—
341C4#1681080911yescharacteristic
342C4#1681081211yescharacteristic
343C2—21no—
344C2—21no—
345C2—22no—
346C2—22no—
347C2—8482no—
348C2#2027596611yescharacteristic
349C2#2027602111yescharacteristic
350C2#2027602611yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1850C2
socleC22×C53#394
prime radical#20285466C2
hypercenter#1850D53
hypocenterC53#10564
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
2. #20275551C2semidirect
3. #20275143C2semidirect
4. #20272583C2semidirect
5. C53×#1850C2semidirect
86. C22⋊C4×C53D8semidirect
171. #1850D53semidirect
196. C53#10564semidirect
272. C22⋊C4D424semidirect

Maximal subgroups

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

subgroupindex
#202854662
#202755512
#202751432
#202725832
C53×#18502
#1056453