Finite Group Databasebeta

← #27301275

Subgroups — #27301275

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20276451C211yesmaximal, characteristic, complemented
3C2×#16810432C211yesmaximal, characteristic, complemented
4#20275613C211yesmaximal, characteristic, complemented
5#20272822C211yesmaximal, characteristic, complemented
6#20272801C211yesmaximal, characteristic, complemented
7C53×#2952C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811064—21no—
10#16811063C412yescomplemented
11#16811059—21no—
12#16811058C412yescomplemented
13#16811053—22no—
14C2×#13133157C2211yescharacteristic
15C2×#13133157C2211yescharacteristic
16#16810661C2211yescharacteristic
17#16810639—21no—
18#16810596—21no—
19#16810433—22no—
20#16810432—22no—
21C2×#135×C53C2211yescharacteristic
22C53×#520—21no—
23C53×#442—21no—
24C53×#441—21no—
25C4×SD16×C53—22no—
26C2×#106×C53C2211yescharacteristic
27C53×#417—21no—
28C53×#405C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic, complemented
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic, complemented
37#13133176—21no—
38#13133163—22no—
39#13133161—22no—
40#13133160—22no—
41#13133158—22no—
42#13133157—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157C2×C412yescomplemented
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133151—21no—
49#13133149—21no—
50#13133143—42no—
51#13133131—42no—
52C2×#41×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53C2×C412yes—
56#135×C53D412yes—
57C2×Q16×C53D411yescharacteristic
58C2×SD16×C53—21no—
59C2×D8×C53D411yescharacteristic
60C2×M16×C53D411yescharacteristic
61C22×C8×C53C2311yescharacteristic
62C4×Q8×C53—22no—
63C4×D4×C53—22no—
64C4×D4×C53—42no—
65#117×C53—41no—
66#117×C53D411yescharacteristic
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—41no—
69#106×C53—21no—
70#106×C53D412yes—
71#103×C53—22no—
72#102×C53—22no—
73#97×C53—21no—
74C4×C8×C53—21no—
75#41×C53—22no—
76#41×C53—22no—
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C22×C4×C53—42no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53—41no—
87#11719945—41no—
88#11719945—41no—
89#11719945—41no—
90#11719945—41no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—21no—
94Q16×C53—21no—
95Q16×C53C22⋊C412yes—
96SD16×C53—22no—
97SD16×C53—22no—
98D8×C53—21no—
99D8×C53C22⋊C412yes—
100M16×C53—21no—
101M16×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53C22×C411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53D812yes—
115C42×C53—21no—
116C42×C53—42no—
117C22⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4×Dic53—41no—
120Dic212—21no—
121Dic212—21no—
122D424—21no—
123D424—21no—
124#11719913—41no—
125#11719913—41no—
126#11719908—22no—
127#11719908—41no—
128C23×C53—41no—
129C23×C53C2×D811yescharacteristic
130C22×D53—41no—
131Q8×C53—22no—
132Q8×C53—41no—
133D4×C53—22no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—162no—
143C2×C4×C53—161no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C22⋊D53—81no—
151C22⋊D53—81no—
152C2×Dic53—41no—
153C2×Dic53—81no—
154D212—41no—
155D212—41no—
156C4×D53—81no—
157C4×D53—81no—
158Dic106—41no—
159Dic106—41no—
160C8×C53—41no—
161C8×C53C2×C22⋊C411yescharacteristic
162C8×C53C4×D411yescharacteristic
163C8×C53#12111yescharacteristic
164C8×C53#12311yescharacteristic
165#13030—531nomaximal, Sylow
166C22×C53—41no—
167C22×C53—42no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53C4×D811yescharacteristic
171C22×C53#44711yescharacteristic
172C22×C53#45911yescharacteristic
173C2×D53—81no—
174C2×D53—161no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—82no—
178C4×C53#38611yescharacteristic
179C4×C53#44211yescharacteristic
180C4×C53#44911yescharacteristic
181C4×C53#46311yescharacteristic
182Dic53—81no—
183Dic53—161no—
184#3414—531no—
185#2952D5311yescharacteristic, complemented
186#2145—531no—
187C2×#360—531no—
188#1976—531no—
189#1386—531no—
190#1378—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—82no—
195C2×C53#187911yescharacteristic
196D53—161no—
197C2×#135—531no—
198C2×#135C2×D5311yescharacteristic
199#520—21no—
200#510—532no—
201#510—1061no—
202#509—532no—
203#509—1061no—
204#508—1062no—
205C2×M32—531no—
206C2×M32—531no—
207#442—21no—
208#441—21no—
209C4×SD16—22no—
210#427—531no—
211C2×#106C2×D5311yescharacteristic
212#421—1061no—
213#417—21no—
214#410—1061no—
215#405C2×D5311yescharacteristic
216#360—1062no—
217#360—1062no—
218C53#1303011yescharacteristic, complemented, Sylow
219C2×#41—21no—
220C2×#41—1061no—
221#135—21no—
222#135—21no—
223#135—1061no—
224#135—1061no—
225#135—1061no—
226#135—1061no—
227#135C4×D5312yes—
228#135D21212yes—
229C2×Q16—531no—
230C2×Q16C22⋊D5311yescharacteristic
231C2×SD16—21no—
232C2×SD16—1061no—
233C2×D8—531no—
234C2×D8C22⋊D5311yescharacteristic
235C2×M16C22⋊D5311yescharacteristic
236C22×C8C22×D5311yescharacteristic
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—42no—
240#117—41no—
241#117—1061no—
242#117C22⋊D5311yescharacteristic
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245Q32—1062no—
246SD32—1062no—
247SD32—1062no—
248D16—1062no—
249M32—532no—
250M32—1061no—
251M32—1061no—
252M32—1061no—
253C2×C16—1061no—
254C2×C16—1061no—
255#107—1061no—
256#106—21no—
257#106—1061no—
258#106C22⋊D5312yes—
259#104—2122no—
260#103—22no—
261#102—22no—
262#99—2122no—
263#97—21no—
264C4×C8—21no—
265#41—22no—
266#41—22no—
267#41—41no—
268#41—41no—
269#41—2121no—
270#41—2121no—
271#41—2121no—
272#41—2121no—
273C2×Q8—21no—
274C2×Q8—1061no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—1061no—
278C2×D4—2121no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—2121no—
283C22×C4C2×C22⋊D5311yescharacteristic
284Q16—21no—
285Q16—1061no—
286Q16—1061no—
287Q16#1171992112yes—
288SD16—22no—
289SD16—22no—
290SD16—2121no—
291SD16—2121no—
292D8—21no—
293D8—1061no—
294D8—1061no—
295D8#1171992112yes—
296M16—21no—
297M16—21no—
298C2×C8—21no—
299C2×C8C2×C4×D5311yescharacteristic
300C2×C8C2×D21211yescharacteristic
301C2×C8#1171994511yescharacteristic
302C2×C8#1171994511yescharacteristic
303C2×C8C2×C22⋊D5311yescharacteristic
304C2×C8C2×C22⋊D5311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—2121no—
310C22⋊C4—42no—
311C22⋊C4—2121no—
312C22⋊C4#1171992212yes—
313C42—21no—
314C42—42no—
315C42—2121no—
316C16—1062no—
317C16—2121no—
318C23—41no—
319C23—2121no—
320C23C2×#1171992211yescharacteristic
321Q8—22no—
322Q8—41no—
323Q8—2121no—
324Q8—2121no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—4241no—
330D4—4241no—
331D4—2121no—
332D4—2121no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—162no—
339C2×C4—161no—
340C2×C4—4241no—
341C2×C4—4241no—
342C2×C4—4241no—
343C2×C4—2121no—
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4#1313318611yescharacteristic
346C2×C4#1313319111yescharacteristic
347C2×C4#1313323711yescharacteristic
348C2×C4#1313324211yescharacteristic
349C2×C4#1313324311yescharacteristic
350C8—41no—
351C8C2×#1171992111yescharacteristic
352C8C4×C22⋊D5311yescharacteristic
353C8#1313319111yescharacteristic
354C8#1313320011yescharacteristic
355C22—41no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—8481no—
360C22—4241no—
361C22C4×#1171992211yescharacteristic
362C22#1681075711yescharacteristic
363C22#1681081811yescharacteristic
364C4—42no—
365C4—42no—
366C4—82no—
367C4—8481no—
368C4—4241no—
369C4#1681050211yescharacteristic
370C4#1681073311yescharacteristic
371C4#1681076811yescharacteristic
372C4#1681083811yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—8481no—
378C2#2027506311yescharacteristic
subgroupG/H
centerC2#20275063
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2952C2
socleC2×C53#1879
prime radical#20287849C2
cosocleC22×C8×C53C23
hypercenter#2952D53
hypocenterC53#13030
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. #20276451C2semidirect
3. C2×#16810432C2semidirect
4. #20275613C2semidirect
5. #20272822C2semidirect
6. #20272801C2semidirect
7. C53×#2952C2semidirect
10. #16811063C4semidirect
12. #16811058C4semidirect
34. C2×Dic212C2×C4semidirect
36. C2×D424C2×C4semidirect
45. #13133157C2×C4semidirect
185. #2952D53semidirect
218. C53#13030semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202764512
C2×#168104322
#202756132
#202728222
#202728012
C53×#29522
#1303053