Finite Group Databasebeta

← #28313916

Subgroups — #28313916

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291523C211yesmaximal, characteristic
2C4×#13133305C211yesmaximal, characteristic
3#20289939C212yesmaximal, complemented
4#20289933C212yesmaximal, complemented
5#20289927C211yesmaximal, characteristic, complemented
6#20289925C211yesmaximal, characteristic, complemented
7#20289921C211yesmaximal, characteristic, complemented
8#20289917C212yesmaximal, complemented
9#20289913C212yesmaximal, complemented
10C8×D53⋊C8C211yesmaximal, characteristic, complemented
11#20283680C211yesmaximal, characteristic
12C2×#13133305C412yes—
13#16811674C412yes—
14#16811657C412yes—
15#16811653C414yes—
16#16811642C2211yescharacteristic
17#16811641C2212yes—
18#16811588C414yes—
19#16811588C412yes—
20C4×#11719936C2212yes—
21C4×#11719936C2211yescharacteristic
22C4×#11719936C2212yes—
23C4×D53⋊C8C2211yescharacteristic
24C4×D53⋊C8C2211yescharacteristic
25C4×D53⋊C8C2212yes—
26#88324⋊C8C2212yes—
27#88324⋊C8C2212yes—
28#88324⋊C8C2212yes—
29#16811464C2212yes—
30#16811464C2212yes—
31C8×#88324C2211yescharacteristic
32C8×#88324C2211yescharacteristic
33C8×#88324C2211yescharacteristic
34C8×#88324C2211yescharacteristic
35C8×#88324C2212yes—
36C4×#11719949C2211yescharacteristic
37#16810754C411yescharacteristic
38#16810702C411yescharacteristic
39#16810700C411yescharacteristic
40#16810696C412yes—
41#16810693C412yes—
42#16810675C2212yes—
43#16810675C2211yescharacteristic
44#16810575C2212yes—
45C4×C8×D53C2211yescharacteristic
46C53×#446C411yescharacteristic
47#13133305—28no—
48C2×#11719936C2×C412yes—
49C2×#11719936C2×C414yes—
50C2×#11719936C2×C414yes—
51C2×D53⋊C8C2×C414yes—
52C2×D53⋊C8C2×C412yes—
53C2×D53⋊C8C2×C412yes—
54#13133284C2×C414yes—
55C4⋊#88324C2×C412yes—
56#13133274C2×C414yes—
57#13133274C2×C412yes—
58#13133272C2×C412yes—
59#13133272C2×C414yes—
60#13133272C2×C414yes—
61C4×#88324C2×C414yes—
62C4×#88324C2×C412yes—
63C4×#88324C2×C412yes—
64C4×#88324C2311yescharacteristic
65C4×#88324C2311yescharacteristic
66C4×#88324C2311yescharacteristic
67C4×#88324C2312yes—
68C4×#88324C2311yescharacteristic
69C4×#88324C2312yes—
70C2×#11719949C2×C411yescharacteristic
71C2×#11719912C2×C412yes—
72C2×#11719912C2×C412yes—
73C2×C8×D53C2×C411yescharacteristic
74C2×C8×D53C2×C411yescharacteristic
75Q8×Dic53C2×C411yescharacteristic
76C4×D212C2×C412yes—
77C4×D212C2×C411yescharacteristic
78#13133179C2×C412yes—
79#13133179C2×C411yescharacteristic
80#13133175C2×C412yes—
81#13133175C2×C411yescharacteristic
82C42×D53C2311yescharacteristic
83C42×D53C2312yes—
84C4⋊#88322C2×C412yes—
85C4⋊#88322C2×C411yescharacteristic
86#13133129C2×C412yes—
87#13133129C2×C412yes—
88#13133129C2×C412yes—
89#13133128C2×C412yes—
90C4×#88322C2×C411yescharacteristic
91C8×Dic53C2311yescharacteristic
92C8×Dic53C2311yescharacteristic
93C8×Dic53C2312yes—
94C4×Q8×C53C2×C411yescharacteristic
95C4⋊C8×C53C2×C412yes—
96C4⋊C8×C53C2×C411yescharacteristic
97#97×C53C2×C412yes—
98C4×C8×C53C2×C411yescharacteristic
99C2×Q8×C53C4211yescharacteristic
100#11719949—24no—
101C2×D212C4212yes—
102C2×D212C4211yescharacteristic
103C2×C4×D53C22×C411yescharacteristic
104C2×C4×D53C22×C412yes—
105C2×C4×D53C22×C412yes—
106C2×C4×D53C22×C411yescharacteristic
107C2×C4×D53C22×C412yes—
108C2×C4×D53C22×C411yescharacteristic
109C2×#88324C22×C414yes—
110C2×#88324C22×C412yes—
111C2×#88324C22×C414yes—
112C2×#88324C22×C412yes—
113C2×#88324C22×C412yes—
114C2×#88324C22×C412yes—
115#11719936—24no—
116#11719936—28no—
117#11719936—28no—
118D53⋊C8—28no—
119D53⋊C8—24no—
120D53⋊C8M1618yescomplemented
121C2×C8×C53C22×C411yescharacteristic
122C2×C8×C53C22×C412yes—
123C2×C8×C53C22×C411yescharacteristic
124C4⋊C4×C53C4212yes—
125C4⋊C4×C53C4211yescharacteristic
126C42×C53C22×C411yescharacteristic
127C42×C53C22×C412yes—
128#11719921C4212yes—
129#11719921C4212yes—
130#11719921C4212yes—
131C4⋊Dic53C4212yes—
132C4⋊Dic53C4211yescharacteristic
133C4×Dic53C22×C411yescharacteristic
134C4×Dic53C22×C412yes—
135C4×Dic53C22×C411yescharacteristic
136C4×Dic53C22×C412yes—
137C4×Dic53C2411yescharacteristic
138C2×#88322C22×C411yescharacteristic
139C2×#88322C22×C412yes—
140C2×#88322C22×C411yescharacteristic
141#11719912—24no—
142#11719912—24no—
143C8×D53—22no—
144C8×D53—22no—
145C22×D53C2×C4212yes—
146C22×D53C2×C4211yescharacteristic
147Q8×C53—22no—
148C2×C4×C53C2×C4212yes—
149C2×C4×C53C2×C4211yescharacteristic
150C2×C4×C53C2×C4212yes—
151C2×C4×C53C2×C4211yescharacteristic
152C2×C4×C53C23×C411yescharacteristic
153C2×Dic53C2×C4211yescharacteristic
154C2×Dic53C2×C4212yes—
155C2×Dic53C23×C411yescharacteristic
156C2×Dic53C23×C411yescharacteristic
157D212—24no—
158D212—22no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—22no—
162C4×D53—22no—
163C4×D53—24no—
164C4×D53—24no—
165C4×D53—22no—
166C4×D53C2×M1614yes—
167#88324—24no—
168#88324—24no—
169#88324—22no—
170#88324C2×M1614yes—
171#88324C2×M1614yes—
172#88324#13114yescomplemented
173C8×C53—21no—
174C8×C53—22no—
175C8×C53#13112yescomplemented
176#88322—21no—
177#88322—22no—
178#88322#13112yescomplemented
179#17335—531nomaximal, Sylow
180C22×C53C22×C4211yescharacteristic
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—22no—
186C2×D53C4×M1612yes—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—21no—
190C4×C53C4×M1612yes—
191C4×C53C2×#13111yescharacteristic
192C4×C53C2×#13111yescharacteristic
193Dic53—21no—
194Dic53—22no—
195Dic53C22×M1612yes—
196Dic53C2×#13112yes—
197#2965—531no—
198#2965—531no—
199C4×#131—531no—
200#1455—532no—
201#1453—532no—
202#1451—531no—
203#1451—531no—
204#1450—531no—
205C8×M16—532no—
206C8×M16—532no—
207C2×C82—531no—
208C2×C53C2×C4×M1611yescharacteristic
209C2×C53C4×#13111yescharacteristic
210C2×C53C4×#13111yescharacteristic
211D53—22no—
212D53—42no—
213C2×#131—532no—
214C4×#41—531no—
215#446—532no—
216#446—531no—
217#446#615711yescharacteristic
218#436—532no—
219#436—531no—
220#436—531no—
221#435—534no—
222#435—532no—
223#435—532no—
224#431—532no—
225#431—531no—
226#431—532no—
227#431—531no—
228#405—532no—
229#405—534no—
230C4×M16—531no—
231C4×M16—532no—
232C4×M16—532no—
233C4×M16—532no—
234C2×C4×C8—531no—
235C2×C4×C8—532no—
236C2×C4×C8—531no—
237C2×C4×C8—531no—
238#322—532no—
239#322—532no—
240#322—532no—
241#322—532no—
242#322—532no—
243C82—531no—
244C82—531no—
245C82—531no—
246C82—531no—
247C82—532no—
248C53#1733511yescharacteristic, complemented, Sylow
249C2×#41—531no—
250#131—1068no—
251C2×M16—534no—
252C2×M16—532no—
253C2×M16—534no—
254C2×M16—532no—
255C2×M16—532no—
256C22×C8—532no—
257C22×C8—534no—
258C22×C8—532no—
259C22×C8—531no—
260C22×C8—531no—
261C4×Q8—531no—
262C4×Q8C2×#615711yescharacteristic
263C4×D4—532no—
264C4×D4—531no—
265C4×D4—532no—
266C4×D4—531no—
267#117—531no—
268#117—532no—
269C2×C42—531no—
270C2×C42—532no—
271C4⋊C8—534no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—531no—
275C4⋊C8C2×#615712yes—
276C4⋊C8C2×#615711yescharacteristic
277C22⋊C8—532no—
278C22⋊C8—534no—
279C22⋊C8—532no—
280C22⋊C8—532no—
281C22⋊C8—532no—
282#97—534no—
283#97—532no—
284#97—534no—
285#97—532no—
286#97C2×#615712yes—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—531no—
290C4×C8—531no—
291C4×C8—532no—
292C4×C8—532no—
293C4×C8—532no—
294C4×C8—534no—
295C4×C8—532no—
296C4×C8—531no—
297C4×C8—531no—
298C4×C8—532no—
299C4×C8—531no—
300C4×C8C2×#615711yescharacteristic
301#41—1064no—
302C2×Q8C4×#615711yescharacteristic
303C2×D4—532no—
304C2×D4—531no—
305C22×C4—531no—
306C22×C4—531no—
307C22×C4—532no—
308C22×C4—532no—
309C22×C4—531no—
310C22×C4—532no—
311M16—1064no—
312M16—1068no—
313M16—1068no—
314M16—1064no—
315M16—1064no—
316C2×C8—532no—
317C2×C8—532no—
318C2×C8—532no—
319C2×C8—538no—
320C2×C8—531no—
321C2×C8—534no—
322C2×C8—532no—
323C2×C8—534no—
324C2×C8—531no—
325C2×C8—532no—
326C2×C8—1062no—
327C2×C8—1068no—
328C2×C8—1064no—
329C2×C8—1062no—
330C2×C8C22×#615711yescharacteristic
331C2×C8C22×#615712yes—
332C2×C8C22×#615711yescharacteristic
333C4⋊C4—532no—
334C4⋊C4—531no—
335C4⋊C4C4×#615712yes—
336C4⋊C4C4×#615711yescharacteristic
337C22⋊C4—532no—
338C22⋊C4—532no—
339C22⋊C4—532no—
340C42—531no—
341C42—532no—
342C42—531no—
343C42—532no—
344C42—531no—
345C42C22×#615711yescharacteristic
346C42C22×#615712yes—
347C23—531no—
348C23—532no—
349Q8—22no—
350D4—1064no—
351D4—1062no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—534no—
355C2×C4—531no—
356C2×C4—532no—
357C2×C4—1064no—
358C2×C4—1062no—
359C2×C4—1064no—
360C2×C4—1061no—
361C2×C4—1062no—
362C2×C4—1061no—
363C2×C4—1062no—
364C2×C4C2×C4×#615711yescharacteristic
365C2×C4C2×C4×#615712yes—
366C2×C4C2×C4×#615712yes—
367C2×C4C2×C4×#615711yescharacteristic
368C2×C4C23×#615711yescharacteristic
369C8—21no—
370C8—22no—
371C8—534no—
372C8—534no—
373C8—534no—
374C8—532no—
375C8—1062no—
376C8—1064no—
377C8—1064no—
378C8—1061no—
379C8—1062no—
380C8#1313330512yescomplemented
381C22—532no—
382C22—1061no—
383C22—1061no—
384C22—1062no—
385C22—1062no—
386C22—1062no—
387C22C22×C4×#615711yescharacteristic
388C4—22no—
389C4—21no—
390C4—22no—
391C4—532no—
392C4—532no—
393C4—1061no—
394C4—1062no—
395C4M16×#615712yes—
396C4D53⋊#13111yescharacteristic
397C4C2×#1313330511yescharacteristic
398C2—1062no—
399C2—2122no—
400C2C2×M16×#615711yescharacteristic
401C2#131×#615711yescharacteristic
402C2C4×#1313330511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#6157
commutator, socleC22×C53C22×C42
FittingC53×#446C4
prime radicalC2×#11719949C2×C4
cosocleC4×Dic53C24
hypercenter#446#6157
hypocenterC53#17335
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #446

Complements

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

NH ≅ G/Ntype
3. #20289939C2semidirect
4. #20289933C2semidirect
5. #20289927C2semidirect
6. #20289925C2semidirect
7. #20289921C2semidirect
8. #20289917C2semidirect
9. #20289913C2semidirect
10. C8×D53⋊C8C2semidirect
120. D53⋊C8M16semidirect
172. #88324#131semidirect
175. C8×C53#131semidirect
178. #88322#131semidirect
248. C53#17335semidirect
380. C8#13133305semidirect

Maximal subgroups

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

subgroupindex
#202915232
C4×#131333052
#202899392
#202899332
#202899272
#202899252
#202899212
#202899172
#202899132
C8×D53⋊C82
#202836802
#1733553