Finite Group Databasebeta

← #27538732

Subgroups — #27538732

Order 13568.

546 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
1C4×#13133239C211yesmaximal, characteristic
2#20276876C212yesmaximal, complemented
3#20276869C212yesmaximal, complemented
4#20276864C211yesmaximal, characteristic, complemented
5#20276859C211yesmaximal, characteristic, complemented
6#20276854C211yesmaximal, characteristic, complemented
7#20276849C211yesmaximal, characteristic, complemented
8#20276497C212yesmaximal, complemented
9C2×C4⋊#11719908C211yesmaximal, characteristic, complemented
10#20276420C212yesmaximal
11C53×#3160C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C4×#11719945—21no—
14C22×#11719908C2212yes—
15C4×Dic212C2211yescharacteristic
16C4×Dic212C2211yescharacteristic
17C4×#11719913—21no—
18C4×#11719913—21no—
19C4×D424C2211yescharacteristic
20C4×D424C2211yescharacteristic
21#16810661C2211yescharacteristic
22#16810620—21no—
23#16810450C2212yes—
24#16810450C2212yes—
25#16810448C2212yes—
26#16810448C2212yes—
27C4⋊#11719908C2211yescharacteristic
28C4⋊#11719908C2211yescharacteristic
29C4⋊#11719908C2212yes—
30#16810426C2212yes—
31#16810426C2212yes—
32#16810423C2212yes—
33#16810423C2212yes—
34C53×#533—21no—
35C53×#500C2211yescharacteristic
36C53×#500C2211yescharacteristic
37C53×#499—21no—
38C53×#499C2212yes—
39C53×#498—21no—
40C53×#427—21no—
41C53×#427C2212yes—
42C2×C4×C8×C53C2211yescharacteristic
43C2×#11719945—21no—
44#13133239—22no—
45#13133239Q814yescomplemented
46C2×Dic212C2311yescharacteristic
47C2×#11719913—21no—
48C2×D424C2311yescharacteristic
49C4×Dic106—21no—
50C4×Dic106—21no—
51C4×D212—21no—
52C4×D212—21no—
53C4×C22⋊D53—41no—
54C4×C22⋊D53—41no—
55#13133176—21no—
56#13133174—41no—
57C42×D53—41no—
58C2×#11719908—24no—
59C2×#11719908C2312yes—
60C2×#11719908C2312yes—
61#13133151C2311yescharacteristic
62#13133151C2311yescharacteristic
63#13133149—21no—
64#13133142—21no—
65#13133142—21no—
66#13133139—21no—
67#13133139—21no—
68C22×C8×C53D411yescharacteristic
69C22×C8×C53C2311yescharacteristic
70C4⋊Q8×C53—21no—
71C4⋊Q8×C53—21no—
72#125×C53—22no—
73C22⋊Q8×C53—41no—
74C22⋊Q8×C53—41no—
75C4×Q8×C53—41no—
76C4×Q8×C53—41no—
77#117×C53—22no—
78C2×C42×C53D411yescharacteristic
79#107×C53—21no—
80#107×C53—21no—
81#107×C53C2312yes—
82#107×C53C2312yes—
83#106×C53—22no—
84#106×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—41no—
90C2×Q8×C53—41no—
91C22×C4×C53D811yescharacteristic
92C22×C4×C53D811yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—41no—
95#11719945—24no—
96#11719945—42no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×Dic106—21no—
102C2×C8×C53—22no—
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×Q812yes—
107C2×C8×C53#4112yes—
108C2×C8×C53C2411yescharacteristic
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—42no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—42no—
116C42×C53—42no—
117C42×C53D811yescharacteristic
118C42×C53D811yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C22⋊Dic53—41no—
122#11719921—41no—
123#11719921—41no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C4×Dic53—41no—
131Dic212—21no—
132Dic212C2×Q812yes—
133D424—21no—
134D424C2×Q812yes—
135#11719913—24no—
136#11719908—22no—
137#11719908—22no—
138C23×C53C2×D811yescharacteristic
139C22×D53—41no—
140Q8×C53—82no—
141Q8×C53—82no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53C22⋊Q812yes—
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53#13712yes—
153C2×C4×C53C22×D411yescharacteristic
154C22⋊D53—44no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158D212—22no—
159D212—41no—
160C4×D53—44no—
161C4×D53—82no—
162C4×D53—82no—
163Dic106—22no—
164Dic106—41no—
165C8×C53—21no—
166C8×C53—21no—
167C8×C53C22⋊Q812yes—
168C8×C53C22×Q811yescharacteristic
169C8×C53C2×#4111yescharacteristic
170#33626—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53#47412yes—
174C22×C53C22×D811yescharacteristic
175C2×D53—42no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—82no—
182C4×C53—82no—
183C4×C53#47412yes—
184C4×C53#50812yes—
185C4×C53C2×C22⋊Q811yescharacteristic
186C4×C53C2×#13711yescharacteristic
187Dic53—42no—
188Dic53—81no—
189Dic53—81no—
190#3160D5311yescharacteristic, complemented
191C4×#135—531no—
192#2229—532no—
193#2228—532no—
194#2227—531no—
195#2226—531no—
196#2225—531no—
197#2224—531no—
198#2158—532no—
199C2×C4⋊C16—531no—
200#2138—532no—
201C2×C53—22no—
202C2×C53C2×#47411yescharacteristic
203C2×C53C2×#50811yescharacteristic
204C2×C53#341411yescharacteristic
205D53—82no—
206C2×#135—531no—
207#533—21no—
208C4×#41—1061no—
209C22×C16—532no—
210#500C2×D5311yescharacteristic
211#500C2×D5311yescharacteristic
212#499—21no—
213#499C2×D5312yes—
214#498—21no—
215C4×Q16—531no—
216C4×Q16—531no—
217C4×SD16—1061no—
218C4×SD16—1061no—
219C4×D8—531no—
220C4×D8—531no—
221#427—21no—
222#427—531no—
223#427C2×D5312yes—
224#416—1061no—
225C2×C4×C8C2×D5311yescharacteristic
226#367—532no—
227#367—532no—
228#366—532no—
229#366—532no—
230C4⋊C16—531no—
231C4⋊C16—531no—
232C4⋊C16—532no—
233#358—532no—
234#358—532no—
235#357—532no—
236#357—532no—
237C53#3362611yescharacteristic, complemented, Sylow
238C2×#41—1061no—
239#135—534no—
240#135—1062no—
241C2×Q16—531no—
242C2×SD16—1061no—
243C2×D8—531no—
244C22×C8C22⋊D5311yescharacteristic
245C22×C8C22×D5311yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248#125—22no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—1061no—
254C4×Q8—1061no—
255C4×D4—1061no—
256C4×D4—1061no—
257C4×D4—2121no—
258C4×D4—2121no—
259#117—22no—
260#117—1061no—
261#117—2121no—
262C2×C42—2121no—
263C2×C42C22⋊D5311yescharacteristic
264C2×C16—532no—
265C2×C16—532no—
266C2×C16—1064no—
267#107—21no—
268#107—21no—
269#107—531no—
270#107—531no—
271#107C22×D5312yes—
272#107C22×D5312yes—
273#106—22no—
274#106—22no—
275#106—1061no—
276#103—1061no—
277#103—1061no—
278#102—1061no—
279#102—1061no—
280C4×C8C22⋊D5311yescharacteristic
281C4×C8C22⋊D5311yescharacteristic
282C4×C8C22×D5311yescharacteristic
283C4×C8C22×D5311yescharacteristic
284#41—1064no—
285#41—2122no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—1061no—
289C2×D4—1061no—
290C2×D4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4#1171992211yescharacteristic
295C22×C4#1171992211yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—532no—
298Q16—1061no—
299SD16—1064no—
300D8—532no—
301D8—1061no—
302C2×C8—22no—
303C2×C8Q8×D5312yes—
304C2×C8#1171994912yes—
305C2×C8C2×C22⋊D5311yescharacteristic
306C2×C8C2×C22⋊D5311yescharacteristic
307C2×C8C2×C22⋊D5311yescharacteristic
308C2×C8C23×D5311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C22⋊C4—42no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—42no—
324C42—2121no—
325C42—2121no—
326C42—2121no—
327C42#1171992211yescharacteristic
328C42#1171992211yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C42C2×C22⋊D5311yescharacteristic
331C16—1062no—
332C16—1062no—
333C23—2121no—
334C23C2×#1171992211yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—1062no—
338Q8—2121no—
339D4—1062no—
340D4—2124no—
341D4—2121no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—4242no—
347C2×C4—4242no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2124no—
352C2×C4#1313319312yes—
353C2×C4C2×#1171992211yescharacteristic
354C2×C4C2×#1171992211yescharacteristic
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992211yescharacteristic
357C2×C4C2×#1171992211yescharacteristic
358C2×C4#1313324812yes—
359C2×C4C22×C22⋊D5311yescharacteristic
360C8—21no—
361C8—21no—
362C8#1313319312yes—
363C8C2×Q8×D5311yescharacteristic
364C8C2×#1171994911yescharacteristic
365C22—21no—
366C22—21no—
367C22—4241no—
368C22—4241no—
369C22—2122no—
370C22#1681090012yes—
371C22C22×#1171992211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—4241no—
378C4—4241no—
379C4—2122no—
380C4#1681090012yes—
381C4#1681105312yes—
382C4C2×#1313319311yescharacteristic
383C4C2×#1313324811yescharacteristic
384C2—22no—
385C2—4242no—
386C2C2×#1681090011yescharacteristic
387C2C2×#1681105311yescharacteristic
388C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3160C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#3160D53
hypocenterC53#33626
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. #20276876C2semidirect
3. #20276869C2semidirect
4. #20276864C2semidirect
5. #20276859C2semidirect
6. #20276854C2semidirect
7. #20276849C2semidirect
8. #20276497C2semidirect
9. C2×C4⋊#11719908C2semidirect
11. C53×#3160C2semidirect
45. #13133239Q8semidirect
190. #3160D53semidirect
237. C53#33626semidirect

Maximal subgroups

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

subgroupindex
C4×#131332392
#202768762
#202768692
#202768642
#202768592
#202768542
#202768492
#202764972
C2×C4⋊#117199082
#202764202
C53×#31602
#3362653