Finite Group Databasebeta

← #27281833

Subgroups — #27281833

Order 13568.

610 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 93 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#20279187C211yesmaximal, characteristic
2#20277316C211yesmaximal, characteristic, complemented
3#20274793C211yesmaximal, characteristic, complemented
4#20273244C211yesmaximal, characteristic
5#20272569C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#1510C211yesmaximal, characteristic
8C2×#13133198—21no—
9C2×#13133198—21no—
10C22×C4⋊Dic53C2211yescharacteristic
11#16810587C2211yescharacteristic
12#16810559—22no—
13#16810526C412yescomplemented
14#16810508—22no—
15#16810506—22no—
16#16810492C412yescomplemented
17#16810486C2211yescharacteristic
18#16810481—21no—
19C22⋊C4×Dic53—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810391—22no—
22#16810357—22no—
23C2×#121×C53C2211yescharacteristic
24C53×#407C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C53×#327—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×#11719917—21no—
30#13133198—44no—
31#13133198—44no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C412yes—
37C2×C4⋊Dic53D412yes—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C4×Dic53—22no—
46C2×C4×Dic53C2×C412yes—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49#13133124—22no—
50#13133124—22no—
51#13133124—21no—
52#13133124—21no—
53#13133124D412yes—
54C22×D4×C53—21no—
55C22×D4×C53C2×C411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C2×M16×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53—21no—
68#95×C53D412yes—
69C24×C53—21no—
70C24×C53C2×D411yescharacteristic
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—42no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—42no—
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53#4111yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×Dic53—22no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—22no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53C22⋊C412yes—
96C22×Dic53C22⋊C412yes—
97M16×C53—42no—
98C2×C8×C53—41no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊Dic53—24no—
105C22⋊Dic53—44no—
106C22⋊Dic53—44no—
107C22⋊Dic53—42no—
108C4⋊Dic53—22no—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112#11719919—44no—
113#11719919—44no—
114#11719919—82no—
115C4×Dic53—42no—
116C4×Dic53—82no—
117#11719917—42no—
118C2×#88322—41no—
119C23×C53—22no—
120C23×C53—22no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53—41no—
126C23×C53—42no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12011yescharacteristic
129C23×C53#12311yescharacteristic
130D4×C53—81no—
131D4×C53—81no—
132D4×C53—82no—
133D4×C53—82no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×Dic53—24no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C8×C53—81no—
165#88322—81no—
166#9464—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—44no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53#38611yescharacteristic
181C22×C53#41012yes—
182C22×C53#45312yes—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—81no—
189Dic53—44no—
190Dic53—42no—
191Dic53—42no—
192Dic53—82no—
193#2501—531no—
194#2313—531no—
195#1821—531no—
196#1510—531no—
197#1510D5311yescharacteristic
198#1284—531no—
199#1284—531no—
200C2×C53—22no—
201C2×C53—22no—
202C2×C53—41no—
203C2×C53—161no—
204C2×C53#188711yescharacteristic
205C2×C53#189911yescharacteristic
206C2×C53#191211yescharacteristic
207C2×#123—1061no—
208C2×#123—1061no—
209C2×#121C2×D5311yescharacteristic
210C22×C4⋊C4—531no—
211#407—531no—
212#407C2×D5311yescharacteristic
213#399—1062no—
214#390—532no—
215#387—1062no—
216#387—1062no—
217#384—532no—
218#381—531no—
219#381C2×D5311yescharacteristic
220#380—1061no—
221C4×C22⋊C4—531no—
222C4×C22⋊C4—1061no—
223#343—1062no—
224#327—22no—
225#327—1062no—
226C53#946411yescharacteristic, complemented, Sylow
227C22×D4—21no—
228C22×D4C4×D5311yescharacteristic
229C23×C4—531no—
230C23×C4—531no—
231C23×C4C22×D5311yescharacteristic
232C2×M16—21no—
233C2×M16—1061no—
234#123—2124no—
235#123—2124no—
236#121—21no—
237#121—21no—
238#121—21no—
239#121—21no—
240C2×C4⋊C4—532no—
241C2×C4⋊C4—532no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1062no—
247C2×C4⋊C4—1062no—
248C2×C4⋊C4—1062no—
249C2×C4⋊C4C4×D5311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—1061no—
253C2×C22⋊C4—1062no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1061no—
256C2×C42—532no—
257C2×C42—1062no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—1061no—
261C22⋊C8—1061no—
262#95—21no—
263#95—532no—
264#95—1062no—
265#95—1062no—
266#95—1061no—
267#95—1061no—
268#95C22⋊D5312yes—
269C24—21no—
270C24D4×D5311yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1062no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1062no—
289C22×C4—2122no—
290C22×C4—2122no—
291C22×C4C2×C4×D5311yescharacteristic
292C22×C4#1171994511yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297M16—42no—
298M16—2122no—
299C2×C8—41no—
300C2×C8—2121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—4242no—
308C4⋊C4—2124no—
309C4⋊C4—2124no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—1064no—
314C22⋊C4—2124no—
315C22⋊C4—2124no—
316C22⋊C4—2122no—
317C42—4242no—
318C42—2122no—
319C23—22no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—41no—
326C23—42no—
327C23C22⋊C4×D5311yescharacteristic
328C23#1313318511yescharacteristic
329C23#1313319911yescharacteristic
330D4—81no—
331D4—81no—
332D4—82no—
333D4—82no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—1064no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313317711yescharacteristic
361C2×C4C4×C22⋊D5311yescharacteristic
362C2×C4#1313318811yescharacteristic
363C2×C4#1313319011yescharacteristic
364C8—81no—
365C8—4241no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—44no—
374C22—82no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#1681049911yescharacteristic
380C22#1681059512yes—
381C22#1681078912yes—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—81no—
388C4—4242no—
389C4—2124no—
390C4—2122no—
391C4—2122no—
392C2—22no—
393C2—22no—
394C2—41no—
395C2—161no—
396C2#2027511011yescharacteristic
397C2#2027517011yescharacteristic
398C2#2027524911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1510C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#1510D53
hypocenterC53#9464
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #20277316C2semidirect
3. #20274793C2semidirect
5. #20272569C2semidirect
6. #20272567C2semidirect
13. #16810526C4semidirect
16. #16810492C4semidirect
226. C53#9464semidirect

Maximal subgroups

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

subgroupindex
#202791872
#202773162
#202747932
#202732442
#202725692
#202725672
C53×#15102
#946453