Finite Group Databasebeta

← #27279545

Subgroups — #27279545

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810733C211yesmaximal, characteristic
2#20273107C211yesmaximal, characteristic, complemented
3#20273106C211yesmaximal, characteristic, complemented
4#20273035C211yesmaximal, characteristic, complemented
5#20272559C211yesmaximal, characteristic, complemented
6#20272545C211yesmaximal, characteristic, complemented
7C53×#2975C211yesmaximal, characteristic, complemented
8C22×#11719922C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810733—22no—
11#16810733—22no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14C2×#13133148C411yescharacteristic
15C2×#13133138C411yescharacteristic
16C2×#13133137C411yescharacteristic
17C2×#13133128C2211yescharacteristic
18#16810352—22no—
19D4⋊#88322—22no—
20#16810346—22no—
21C2×C4×D4×C53C2211yescharacteristic
22C53×#436—22no—
23C53×#436—22no—
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×#407—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719922—22no—
30C2×#11719922—22no—
31C2×#11719922—22no—
32C22×#88322C2×C411yescharacteristic
33C22×#88322C2×C411yescharacteristic
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719921—21no—
40#13133148—22no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43#13133138—22no—
44#13133137—22no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47#13133128—21no—
48#13133128—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53—21no—
53C2×M16×C53—24no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53D412yes—
57C4×D4×C53D412yes—
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C22⋊C8×C53—24no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53D412yes—
66C4×C8×C53—21no—
67C4×C8×C53D412yes—
68C24×C53—21no—
69C23×D53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53C22⋊C412yes—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—82no—
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212M1612yescomplemented
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—82no—
93M16×C53—84no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53C22⋊C412yes—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104#11719922—28no—
105#11719921—41no—
106#11719921—41no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322M1612yescomplemented
117C23×C53—22no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53C4×D411yescharacteristic
123C22×D53—22no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—81no—
127C22×D53—81no—
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—41no—
131D4×C53—41no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—44no—
136C2×C4×C53—82no—
137C2×C4×C53—81no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C2×M1611yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×C4×C53#13111yescharacteristic
147C2×C4×C53#13111yescharacteristic
148C2×Dic53—41no—
149C2×Dic53—81no—
150D212—22no—
151D212—22no—
152D212—41no—
153D212—41no—
154C4×D53—82no—
155C4×D53—82no—
156C8×C53—42no—
157C8×C53—42no—
158#88322—22no—
159#88322—41no—
160#88322—81no—
161#88322—81no—
162#9093—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—82no—
169C22×C53—82no—
170C22×C53#38611yescharacteristic
171C22×C53#43511yescharacteristic
172C22×C53#43611yescharacteristic
173C2×D53—42no—
174C2×D53—82no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—82no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53#40711yescharacteristic
184C4×C53#40811yescharacteristic
185C4×C53#43511yescharacteristic
186C4×C53#43611yescharacteristic
187Dic53—81no—
188#2975D5311yescharacteristic, complemented
189C2×#442—531no—
190#1478—531no—
191#1478—531no—
192#1460—531no—
193#1281—531no—
194#1275—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—82no—
198C2×C53#187011yescharacteristic
199C2×C53#190311yescharacteristic
200C2×C53#190411yescharacteristic
201D53—82no—
202C22×D8—531no—
203C2×C4×D4—531no—
204C2×C4×D4C2×D5311yescharacteristic
205#442—1062no—
206#442—1062no—
207#436—22no—
208#436—22no—
209#431—531no—
210#431—531no—
211#431C2×D5311yescharacteristic
212#431C2×D5311yescharacteristic
213C2×#106—531no—
214C2×#102—531no—
215C2×#102—531no—
216#407—22no—
217C2×#97—531no—
218D4⋊C8—1062no—
219D4⋊C8—1062no—
220#323—1062no—
221C53#909311yescharacteristic, complemented, Sylow
222C22×D4—531no—
223C22×D4C4×D5311yescharacteristic
224C23×C4—21no—
225C23×C4—1061no—
226C2×D8—1062no—
227C2×D8—1062no—
228C2×D8—1062no—
229C2×M16—24no—
230C22×C8—531no—
231C22×C8—531no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—1061no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4D21212yes—
239C4×D4C22⋊D5312yes—
240C2×C4⋊C4—531no—
241C2×C4⋊C4C4×D5311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1061no—
244C2×C42C22×D5311yescharacteristic
245#106—1062no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—1061no—
249C4⋊C8—1061no—
250#102—1062no—
251#102—1062no—
252C22⋊C8—24no—
253C22⋊C8—21no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C22⋊C8C22⋊D5312yes—
257#97—1061no—
258#97—1061no—
259C4×C8—21no—
260C4×C8—1061no—
261C4×C8—1061no—
262C4×C8C22⋊D5312yes—
263C24—21no—
264C24—1061no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—532no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4#1171992112yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1061no—
282C22×C4—4242no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4C2×C4×D5311yescharacteristic
286C22×C4#1171994511yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288D8—1068no—
289M16—84no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—532no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—2121no—
299C2×C8—2121no—
300C4⋊C4—21no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4#1171992112yes—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42C2×D21211yescharacteristic
309C42#1171994511yescharacteristic
310C42C2×C22⋊D5311yescharacteristic
311C42C2×C22⋊D5311yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23—81no—
316C23—81no—
317C23—1062no—
318C23—4241no—
319C23—4241no—
320C23—2121no—
321C23—2121no—
322C23C4×C22⋊D5311yescharacteristic
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327D4—1062no—
328D4—1062no—
329D4—2121no—
330D4—2121no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—4242no—
338C2×C4—4242no—
339C2×C4—4241no—
340C2×C4—2121no—
341C2×C4C2×#1171992111yescharacteristic
342C2×C4C4×C22⋊D5311yescharacteristic
343C2×C4#1313318611yescharacteristic
344C2×C4#1313319111yescharacteristic
345C2×C4#1313319111yescharacteristic
346C2×C4#1313320011yescharacteristic
347C2×C4M16×D5311yescharacteristic
348C2×C4M16×D5311yescharacteristic
349C2×C4#1313322511yescharacteristic
350C2×C4#1313322511yescharacteristic
351C8—42no—
352C8—42no—
353C8—1062no—
354C8—4241no—
355C8—4241no—
356C8—2121no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—4242no—
365C22—4242no—
366C22—4241no—
367C22—4241no—
368C22—2122no—
369C22#1681050211yescharacteristic
370C22#1681069211yescharacteristic
371C22#1681069911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—4241no—
378C4#1681058611yescharacteristic
379C4#1681059111yescharacteristic
380C4#1681069211yescharacteristic
381C4#1681069911yescharacteristic
382C2—21no—
383C2—21no—
384C2—82no—
385C2—4242no—
386C2#2027501211yescharacteristic
387C2#2027519511yescharacteristic
388C2#2027520211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2975C2
socleC22×C53#386
prime radicalC22×#11719922C4
cosocleC2×C42×C53C23
hypercenter#2975D53
hypocenterC53#9093
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #20273107C2semidirect
3. #20273106C2semidirect
4. #20273035C2semidirect
5. #20272559C2semidirect
6. #20272545C2semidirect
7. C53×#2975C2semidirect
89. C2×D212M16semidirect
116. C2×#88322M16semidirect
188. #2975D53semidirect
221. C53#9093semidirect

Maximal subgroups

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

subgroupindex
C2×#168107332
#202731072
#202731062
#202730352
#202725592
#202725452
C53×#29752
#909353