Finite Group Databasebeta

← #27279329

Subgroups — #27279329

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283459C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic
4#20273118C211yesmaximal, characteristic, complemented
5#20272555C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#3167C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810701—21no—
10#16810695—22no—
11D4×#88322—21no—
12#16810676C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14#16810590—21no—
15C4×#11719917C2211yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17#16810382—21no—
18#16810382—21no—
19#16810357—22no—
20Q8⋊#88322—21no—
21Q8⋊#88322—21no—
22C53×#533C2211yescharacteristic
23C53×#532C2211yescharacteristic
24C53×#498C411yescharacteristic, complemented
25C53×#492C411yescharacteristic, complemented
26C53×#487C411yescharacteristic, complemented
27C53×#486C411yescharacteristic, complemented
28C53×#465—22no—
29C4×SD16×C53—22no—
30C2×C4×C8×C53C2211yescharacteristic
31#13133224—42no—
32#13133224—42no—
33C2×#11719917—22no—
34C2×#11719917D411yescharacteristic
35C2×#11719917D411yescharacteristic
36C22×#88322—22no—
37C22×#88322—21no—
38C2×C4⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44C22⋊#88322—22no—
45C22⋊#88322—22no—
46C22⋊#88322—21no—
47#13133128—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C2×SD16×C53—22no—
53C22×C8×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56C4⋊Q8×C53C2×C411yescharacteristic
57#127×C53C2×C411yescharacteristic
58#125×C53—21no—
59#124×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62#121×C53—21no—
63#121×C53D412yes—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53D412yes—
68#117×C53—21no—
69C2×C42×C53C2311yescharacteristic
70#106×C53—22no—
71#103×C53—22no—
72#102×C53—22no—
73C4×C8×C53—21no—
74C4×C8×C53C2×C411yescharacteristic
75C4×C8×C53C2×C411yescharacteristic
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C22⋊C412yes—
87C22×C4×C53—22no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×Dic53—41no—
92SD16×C53—44no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—42no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53C22⋊C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—41no—
107C42×C53—41no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C4⋊Dic53—22no—
113C4⋊Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116#11719917—21no—
117#11719917—21no—
118#11719917—21no—
119#11719917—44no—
120#11719917SD1612yescomplemented
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—44no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53#12111yescharacteristic
131Q8×C53—42no—
132Q8×C53—41no—
133Q8×C53—41no—
134Q8×C53—41no—
135Q8×C53—41no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×Dic53—42no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C8×C53—22no—
165C8×C53—41no—
166#88322—22no—
167#88322—41no—
168#88322—41no—
169#88322—41no—
170#88322—81no—
171#9055—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38611yescharacteristic
178C22×C53#46111yescharacteristic
179C22×C53#46311yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53C4×SD1611yescharacteristic
187C4×C53#44011yescharacteristic
188C4×C53#44311yescharacteristic
189C4×C53#44411yescharacteristic
190C4×C53#45011yescharacteristic
191C4×C53#45211yescharacteristic
192Dic53—82no—
193#3167D5311yescharacteristic
194#2935—531no—
195#1986—531no—
196#1502—531no—
197#1481—531no—
198#1279—531no—
199#1278—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—82no—
203C2×C53#187611yescharacteristic
204C2×C53#187711yescharacteristic
205C2×C53#188411yescharacteristic
206C2×#131—1061no—
207#533C2×D5311yescharacteristic
208#532C2×D5311yescharacteristic
209#498Dic5311yescharacteristic, complemented
210#492Dic5311yescharacteristic, complemented
211#487Dic5311yescharacteristic, complemented
212#486Dic5311yescharacteristic, complemented
213#465—22no—
214C4×SD16—22no—
215#436—1061no—
216#435—1062no—
217C8×D4—1061no—
218#431—531no—
219C2×C4⋊C8—531no—
220#408—1061no—
221C4×M16—531no—
222C2×C4×C8C2×D5311yescharacteristic
223C4×C4⋊C4—531no—
224#339—1061no—
225#339—1061no—
226#327—1062no—
227Q8⋊C8—1061no—
228Q8⋊C8—1061no—
229C53#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—22no—
232#131—2122no—
233#131—2122no—
234C2×M16—531no—
235C2×M16—531no—
236C2×M16—1062no—
237C22×C8—21no—
238C22×C8—1062no—
239C22×C8—1061no—
240C4⋊Q8C2×Dic5311yescharacteristic
241C4⋊Q8C2×Dic5311yescharacteristic
242C4⋊Q8C2×Dic5311yescharacteristic
243#127C2×Dic5311yescharacteristic
244#125—21no—
245#124—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121C22⋊D5312yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4C22⋊D5312yes—
254#117—21no—
255C2×C4⋊C4—1061no—
256C2×C42—1061no—
257C2×C42C22×D5311yescharacteristic
258#106—22no—
259C4⋊C8—531no—
260C4⋊C8—531no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263#103—22no—
264#102—22no—
265C22⋊C8—1062no—
266C22⋊C8—1062no—
267C22⋊C8—1061no—
268#97—1061no—
269C4×C8—21no—
270C4×C8—1061no—
271C4×C8—1061no—
272C4×C8C2×Dic5311yescharacteristic
273C4×C8C2×Dic5311yescharacteristic
274#95—1061no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊Dic5312yes—
286C22×C4—22no—
287C22×C4—2121no—
288C22×C4D4×D5311yescharacteristic
289C22×C4#1171994711yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291SD16—44no—
292M16—532no—
293M16—1061no—
294M16—1061no—
295M16—1061no—
296M16—2124no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—1061no—
304C2×C8—2124no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1062no—
316C4⋊C4—2121no—
317C4⋊C4C22⋊Dic5312yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C42—41no—
322C42—2121no—
323C42—2121no—
324C42D4×D5311yescharacteristic
325C42#1171994711yescharacteristic
326C42C22×Dic5311yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C23—22no—
329C23#1313318911yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—4241no—
351C2×C4—4241no—
352C2×C4—2122no—
353C2×C4C2×C22⋊Dic5311yescharacteristic
354C2×C4D4×Dic5311yescharacteristic
355C2×C4D4×Dic5311yescharacteristic
356C2×C4#1313318511yescharacteristic
357C2×C4#1313318911yescharacteristic
358C2×C4#1313319811yescharacteristic
359C2×C4SD16×D5311yescharacteristic
360C2×C4#1313323811yescharacteristic
361C2×C4#1313324411yescharacteristic
362C2×C4#1313324611yescharacteristic
363C8—22no—
364C8—41no—
365C8—1062no—
366C8—4241no—
367C8—2121no—
368C8—2121no—
369C8—2121no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1681050311yescharacteristic
376C22#1681082611yescharacteristic
377C22#1681084011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4—4242no—
385C4SD16×Dic5311yescharacteristic
386C4#1681072411yescharacteristic
387C4#1681074011yescharacteristic
388C4#1681074511yescharacteristic
389C4#1681077211yescharacteristic
390C4#1681078211yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2027504411yescharacteristic
395C2#2027505111yescharacteristic
396C2#2027509911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3167C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3167D53
hypocenterC53#9055
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #20275660C2semidirect
4. #20273118C2semidirect
5. #20272555C2semidirect
6. #20272552C2semidirect
24. C53×#498C4semidirect
25. C53×#492C4semidirect
26. C53×#487C4semidirect
27. C53×#486C4semidirect
120. #11719917SD16semidirect
209. #498Dic53semidirect
210. #492Dic53semidirect
211. #487Dic53semidirect
212. #486Dic53semidirect
229. C53#9055semidirect

Maximal subgroups

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

subgroupindex
#202834592
#202756602
#202732102
#202731182
#202725552
#202725522
C53×#31672
#905553