Finite Group Databasebeta

← #27281838

Subgroups — #27281838

Order 13568.

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

normal subgroups only

This group has 554 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#20279208C211yesmaximal, characteristic
2#20277316C211yesmaximal, characteristic, complemented
3#20274793C211yesmaximal, characteristic, complemented
4#20273244C211yesmaximal, characteristic
5#20272569C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#1510C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810548—22no—
11#16810526C412yescomplemented
12#16810508—22no—
13#16810506—22no—
14#16810503—21no—
15#16810503—21no—
16#16810488C412yescomplemented
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—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C412yes—
35C2×C4⋊Dic53D412yes—
36C2×#11719919—22no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—22no—
41C2×C4×Dic53—22no—
42C2×C4×Dic53C2×C412yes—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133124—22no—
46#13133124—22no—
47#13133124—21no—
48#13133124—21no—
49#13133124—22no—
50#13133124—22no—
51#13133124D412yes—
52C22×D4×C53—21no—
53C22×D4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—21no—
56#121×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67C24×C53—21no—
68C24×C53C2×D411yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—42no—
72C2×D4×C53—44no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—42no—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—81no—
91C22×Dic53—81no—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94M16×C53—42no—
95C2×C8×C53—41no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊Dic53—42no—
102C22⋊Dic53—42no—
103C22⋊Dic53—82no—
104C22⋊Dic53—82no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107C4⋊Dic53—22no—
108C4⋊Dic53—22no—
109#11719919—82no—
110C4×Dic53—42no—
111C4×Dic53—82no—
112#11719917—42no—
113C2×#88322—41no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12011yescharacteristic
123C23×C53#12311yescharacteristic
124D4×C53—81no—
125D4×C53—81no—
126D4×C53—82no—
127D4×C53—82no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157C8×C53—81no—
158#88322—81no—
159#9465—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—44no—
168C22×C53—84no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C22×C53#41012yes—
174C22×C53#45412yes—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—82no—
180C4×C53—81no—
181Dic53—42no—
182Dic53—42no—
183Dic53—82no—
184Dic53—82no—
185#2504—531no—
186#2313—531no—
187#1821—531no—
188#1510—531no—
189#1510D5311yescharacteristic
190#1284—531no—
191#1284—531no—
192C2×C53—22no—
193C2×C53—22no—
194C2×C53—41no—
195C2×C53—161no—
196C2×C53#188811yescharacteristic
197C2×C53#189911yescharacteristic
198C2×C53#191111yescharacteristic
199C2×#121C2×D5311yescharacteristic
200C22×C4⋊C4—531no—
201#407—531no—
202#407C2×D5311yescharacteristic
203#397—1062no—
204#390—532no—
205#387—1062no—
206#387—1062no—
207#386—1061no—
208#386—1061no—
209#382—532no—
210#381—531no—
211#381C2×D5311yescharacteristic
212#380—1061no—
213C4×C22⋊C4—531no—
214C4×C22⋊C4—1061no—
215#343—1062no—
216#327—22no—
217#327—1062no—
218C53#946511yescharacteristic, complemented, Sylow
219C22×D4—21no—
220C22×D4C4×D5311yescharacteristic
221C23×C4—531no—
222C23×C4—531no—
223C23×C4C22×D5311yescharacteristic
224C2×M16—21no—
225C2×M16—1061no—
226#121—21no—
227#121—21no—
228#121—21no—
229#121—21no—
230C2×C4⋊C4—532no—
231C2×C4⋊C4—532no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42—532no—
245C2×C42—1062no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—1061no—
249C22⋊C8—1061no—
250#95—21no—
251#95—532no—
252#95—1062no—
253#95—1062no—
254#95—1062no—
255#95—1062no—
256#95—1061no—
257#95—1061no—
258#95C22⋊D5312yes—
259C24—21no—
260C24D4×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—4241no—
279C22×C4—4241no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4#1171994511yescharacteristic
282C22×C4D4×D5311yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4#1171994711yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—42no—
287M16—2122no—
288C2×C8—41no—
289C2×C8—2121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—4242no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—4242no—
301C22⋊C4—4242no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C42—4242no—
305C42—2122no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23C22⋊C4×D5311yescharacteristic
314C23#1313318511yescharacteristic
315C23#1313319911yescharacteristic
316D4—81no—
317D4—81no—
318D4—82no—
319D4—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—1062no—
332C2×C4—1062no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4#1313317711yescharacteristic
346C2×C4C4×C22⋊D5311yescharacteristic
347C2×C4#1313318811yescharacteristic
348C2×C4#1313319011yescharacteristic
349C8—81no—
350C8—4241no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—44no—
359C22—84no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#1681049911yescharacteristic
364C22#1681059512yes—
365C22#1681079412yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—81no—
372C4—4242no—
373C4—4242no—
374C4—2122no—
375C4—2122no—
376C2—22no—
377C2—22no—
378C2—41no—
379C2—161no—
380C2#2027511511yescharacteristic
381C2#2027517011yescharacteristic
382C2#2027524411yescharacteristic
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#9465
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
11. #16810526C4semidirect
16. #16810488C4semidirect
218. C53#9465semidirect

Maximal subgroups

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

subgroupindex
#202792082
#202773162
#202747932
#202732442
#202725692
#202725672
C53×#15102
#946553