Finite Group Databasebeta

← #27279614

Subgroups — #27279614

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283755C211yesmaximal, characteristic, complemented
2#20277437C211yesmaximal, characteristic
3#20273106C211yesmaximal, characteristic, complemented
4#20273035C211yesmaximal, characteristic, complemented
5#20272543C211yesmaximal, characteristic, complemented
6#20272543C211yesmaximal, characteristic, complemented
7C53×#1478C211yesmaximal, characteristic, complemented
8C2×#13133213C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810702—22no—
11#16810700—22no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810585—22no—
15#16810526C411yescharacteristic
16#16810524C411yescharacteristic
17#16810502—22no—
18#16810499—22no—
19#16810488C411yescharacteristic
20#16810469C2211yescharacteristic
21D4⋊#88322—22no—
22#16810346—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×D4⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C22⋊D53—22no—
29C22×C4×D53D412yes—
30C2×#11719912—24no—
31#13133213—22no—
32#13133213—22no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53—22no—
39C2×#11719921—22no—
40C2×#11719921D412yes—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47#13133129—24no—
48C4×#88322—21no—
49C4×#88322D412yescomplemented
50#13133124—21no—
51#13133124—21no—
52#13133124—22no—
53C22×D4×C53C2×C411yescharacteristic
54C23×C4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53D412yescomplemented
68C24×C53—21no—
69C23×D53C22⋊C412yes—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53M1612yescomplemented
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—82no—
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C2×C22⋊D53—28no—
84C2×C22⋊D53—82no—
85C2×C22⋊D53—82no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53C22⋊C412yes—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—21no—
94C2×D212—21no—
95C2×C4×D53—22no—
96C2×C4×D53—22no—
97C2×C4×D53—82no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53#4111yescharacteristic
107C42×C53#4111yescharacteristic
108C22⋊Dic53—82no—
109#11719921—22no—
110#11719921—22no—
111#11719921—82no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53—21no—
117C4×Dic53M1612yescomplemented
118C2×#88322—22no—
119C2×#88322—41no—
120#11719912—84no—
121C23×C53—22no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53C2×C22⋊C411yescharacteristic
127C22×D53—22no—
128C22×D53—22no—
129C22×D53—22no—
130C22×D53—42no—
131C22×D53—42no—
132D4×C53—22no—
133D4×C53—22no—
134D4×C53—41no—
135D4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53C2×M1611yescharacteristic
149C2×C4×C53C2×M1611yescharacteristic
150C2×C4×C53#13111yescharacteristic
151C2×C4×C53#13111yescharacteristic
152C22⋊D53—88no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161D212—41no—
162D212—41no—
163D212—41no—
164D212—41no—
165C4×D53—44no—
166C4×D53—82no—
167C8×C53—42no—
168C8×C53—81no—
169#88322—42no—
170#88322—81no—
171#9105—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—44no—
175C22×C53—42no—
176C22×C53—82no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53#38611yescharacteristic
180C22×C53#40711yescharacteristic
181C22×C53#40811yescharacteristic
182C2×D53—44no—
183C2×D53—42no—
184C2×D53—42no—
185C2×D53—42no—
186C2×D53—82no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53#43511yescharacteristic
193C4×C53#43511yescharacteristic
194C4×C53#43611yescharacteristic
195C4×C53#43611yescharacteristic
196Dic53—42no—
197Dic53—42no—
198Dic53—81no—
199#2975—531no—
200#2328—531no—
201#1478—531no—
202#1478D5311yescharacteristic, complemented
203#1460—531no—
204#1274—531no—
205#1274—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—82no—
209C2×C53#187011yescharacteristic
210C2×C53#191011yescharacteristic
211C2×C53#191111yescharacteristic
212D53—82no—
213C2×#127—531no—
214C2×C4×D4—531no—
215C2×C4×D4C2×D5311yescharacteristic
216#436—1062no—
217#436—1062no—
218#431—531no—
219#431—531no—
220#431C2×D5311yescharacteristic
221#431C2×D5311yescharacteristic
222#407—1062no—
223#390—531no—
224#390—531no—
225#386—1062no—
226#386—1062no—
227#382—531no—
228#376—531no—
229D4⋊C8—22no—
230D4⋊C8—1062no—
231#323—1062no—
232C53#910511yescharacteristic, complemented, Sylow
233C22×D4—531no—
234C22×D4—1062no—
235C22×D4C4×D5311yescharacteristic
236C23×C4—21no—
237C23×C4—532no—
238C2×M16—1064no—
239#127—1062no—
240#127—1062no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248C4×D4—1061no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4C4×D5311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—532no—
253C2×C22⋊C4—1062no—
254C2×C22⋊C4—1062no—
255C2×C42—531no—
256C2×C42—531no—
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—1061no—
261C4⋊C8—1061no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—1064no—
265C22⋊C8—1061no—
266C22⋊C8—1061no—
267C4×C8—21no—
268C4×C8—532no—
269C4×C8—1061no—
270C4×C8C22⋊D5312yescomplemented
271#95—1062no—
272#95—1061no—
273#95—1061no—
274C24—21no—
275C24—532no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—1068no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—4242no—
289C2×D4—4242no—
290C2×D4#1171991212yescomplemented
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—532no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—1061no—
300C22×C4—4242no—
301C22×C4C2×C4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304M16—4244no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—1062no—
308C2×C8—2121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—4242no—
318C22⋊C4—4242no—
319C42—532no—
320C42—1061no—
321C42—1061no—
322C42—1061no—
323C42#1171994511yescharacteristic
324C42D4×D5311yescharacteristic
325C42#1171994711yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C23—22no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—1062no—
333C23—1062no—
334C23—1062no—
335C23—2122no—
336C23—2122no—
337C23C22⋊C4×D5311yescharacteristic
338D4—22no—
339D4—22no—
340D4—41no—
341D4—41no—
342D4—4248no—
343D4—2121no—
344D4—2121no—
345D4—2121no—
346D4—2121no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—1062no—
354C2×C4—1062no—
355C2×C4—4242no—
356C2×C4—2124no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4#1313317711yescharacteristic
364C2×C4C4×C22⋊D5311yescharacteristic
365C2×C4#1313318511yescharacteristic
366C2×C4#1313318811yescharacteristic
367C2×C4#1313319011yescharacteristic
368C2×C4#1313319911yescharacteristic
369C2×C4M16×D5311yescharacteristic
370C2×C4C2×#1171991211yescharacteristic
371C2×C4#1313322311yescharacteristic
372C2×C4#1313322511yescharacteristic
373C8—42no—
374C8—81no—
375C8—4241no—
376C8—2122no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—42no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22—4242no—
385C22—2124no—
386C22—2122no—
387C22—2122no—
388C22—2122no—
389C22#1681049911yescharacteristic
390C22#1681058511yescharacteristic
391C22#1681058811yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—4241no—
398C4—2122no—
399C4—2122no—
400C4#1681069111yescharacteristic
401C4#1681069411yescharacteristic
402C4#1681069811yescharacteristic
403C4#1681069911yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—4242no—
408C2#2027500911yescharacteristic
409C2#2027523911yescharacteristic
410C2#2027524411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1478C2
socleC22×C53#386
prime radicalC2×#13133213C4
cosocleC2×C42×C53C23
hypercenter#1478D53
hypocenterC53#9105
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #20283755C2semidirect
3. #20273106C2semidirect
4. #20273035C2semidirect
5. #20272543C2semidirect
6. #20272543C2semidirect
7. C53×#1478C2semidirect
49. C4×#88322D4semidirect
67. C4×C8×C53D4semidirect
75. C2×D4×C53M16semidirect
117. C4×Dic53M16semidirect
202. #1478D53semidirect
232. C53#9105semidirect
270. C4×C8C22⋊D53semidirect
290. C2×D4#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837552
#202774372
#202731062
#202730352
#202725432
C53×#14782
#910553