Finite Group Databasebeta

← #27301106

Subgroups — #27301106

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284435C211yesmaximal, characteristic
2#20276450C211yesmaximal, characteristic, complemented
3C2×#16810432C211yesmaximal, characteristic, complemented
4#20276343C211yesmaximal, characteristic, complemented
5#20272831C211yesmaximal, characteristic, complemented
6#20272813C211yesmaximal, characteristic, complemented
7C53×#2912C211yesmaximal, characteristic, complemented
8#16811421—21no—
9C2×#13133239C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810898C412yescomplemented
13#16810893C412yescomplemented
14#16810886—22no—
15C2×#13133153C2211yescharacteristic
16#16810605—21no—
17#16810432—22no—
18#16810432—21no—
19#16810432—21no—
20#16810407—22no—
21C2×#131×C53C2211yescharacteristic
22C53×#520—21no—
23C53×#428—21no—
24C53×#428—22no—
25C53×#427—21no—
26C53×#427C2211yescharacteristic
27C2×#106×C53—21no—
28C2×#106×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133248—41no—
31#13133247—22no—
32#13133243—41no—
33#13133242—22no—
34#13133239—21no—
35#13133239—21no—
36#13133239—21no—
37#13133239—21no—
38#13133237—42no—
39C2×Dic212C2×C411yescharacteristic, complemented
40C2×#11719913—21no—
41C2×D424C2×C411yescharacteristic, complemented
42C2×#11719917—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157—21no—
46#13133157—21no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49#13133153—21no—
50#13133153C2×C412yescomplemented
51#13133135—22no—
52#13133133—22no—
53C2×#41×C53D411yescharacteristic
54#131×C53—21no—
55#131×C53—21no—
56#131×C53C2×C412yes—
57#131×C53D412yes—
58C2×M16×C53—21no—
59C2×M16×C53D411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62C4×Q8×C53—41no—
63C4×D4×C53—42no—
64C4×D4×C53—41no—
65#117×C53—22no—
66#117×C53—21no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—21no—
69#107×C53—22no—
70#107×C53—21no—
71#106×C53—22no—
72#106×C53—21no—
73#106×C53D411yescharacteristic
74#106×C53D412yes—
75#106×C53D411yescharacteristic
76#41×C53—21no—
77#41×C53—21no—
78#41×C53C22⋊C412yes—
79#41×C53SD1612yes—
80C2×Q8×C53D811yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53D811yescharacteristic
83C22×C4×C53—21no—
84C22×C4×C53—42no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C2×C22⋊D53—41no—
88#11719945—41no—
89#11719945—41no—
90#11719945—41no—
91#11719945—41no—
92C2×D212—21no—
93C2×C4×D53—41no—
94C2×Dic106—21no—
95M16×C53—21no—
96M16×C53—22no—
97M16×C53—22no—
98M16×C53C22⋊C412yes—
99C2×C8×C53—21no—
100C2×C8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53—41no—
115C42×C53—42no—
116C42×C53—41no—
117#11719925—42no—
118#11719924—42no—
119#11719923—42no—
120#11719922—42no—
121#11719917—22no—
122#11719917—41no—
123C2×#88322—41no—
124Dic212—21no—
125Dic212—21no—
126D424—21no—
127D424—21no—
128#11719913—41no—
129#11719913—41no—
130#11719908—41no—
131#11719908—41no—
132C23×C53—21no—
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C22×D53—41no—
135Q8×C53—21no—
136Q8×C53#10212yes—
137D4×C53—22no—
138D4×C53—22no—
139D4×C53—21no—
140D4×C53#10212yes—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—161no—
148C2×C4×C53—162no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C22⋊D53—81no—
156C22⋊D53—81no—
157C2×Dic53—41no—
158D212—41no—
159D212—41no—
160C4×D53—81no—
161C4×D53—81no—
162Dic106—41no—
163Dic106—41no—
164C8×C53—22no—
165C8×C53C4×D411yescharacteristic
166C8×C53C4×D411yescharacteristic
167C8×C53#12111yescharacteristic
168C8×C53#12111yescharacteristic
169#88322—42no—
170#13005—531nomaximal, Sylow
171C22×C53—22no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53#41711yescharacteristic
175C22×C53#44711yescharacteristic
176C22×C53#48111yescharacteristic
177C2×D53—81no—
178C2×D53—161no—
179C4×C53—22no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C53—82no—
183C4×C53#38611yescharacteristic
184C4×C53C2×#10211yescharacteristic
185C4×C53#44811yescharacteristic
186C4×C53#48011yescharacteristic
187Dic53—81no—
188#3067—531no—
189#2912D5311yescharacteristic, complemented
190#2145—531no—
191C2×#360—531no—
192#2117—531no—
193#1389—531no—
194#1383—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—42no—
199C2×C53#187411yescharacteristic
200D53—161no—
201#575—1061no—
202C2×#135—531no—
203C2×#131C2×D5311yescharacteristic
204#520—21no—
205C2×M32—531no—
206C2×M32—531no—
207#473—532no—
208#472—532no—
209#471—1062no—
210C2×#108—531no—
211#428—21no—
212#428—22no—
213#427—21no—
214#427C2×D5311yescharacteristic
215C2×#106—21no—
216C2×#106C2×D5311yescharacteristic
217#413—1061no—
218#360—1062no—
219#360—1061no—
220#360—1061no—
221#350—1062no—
222C53#1300511yescharacteristic, complemented, Sylow
223C2×#41—1061no—
224C2×#41C22⋊D5311yescharacteristic
225#137—1062no—
226#137—2121no—
227AGL(1,ℤ/8)—1062no—
228AGL(1,ℤ/8)—2121no—
229#135—1061no—
230#135—1061no—
231#135—1061no—
232#135—1061no—
233#135—2122no—
234C2×Q16—531no—
235C2×SD16—1061no—
236C2×D8—531no—
237#131—21no—
238#131—21no—
239#131C4×D5312yes—
240#131D21212yes—
241C2×M16—21no—
242C2×M16—1061no—
243C2×M16C22⋊D5311yescharacteristic
244C22×C8—21no—
245C22×C8C22×D5311yescharacteristic
246C4×Q8—41no—
247C4×D4—42no—
248C4×D4—41no—
249#117—22no—
250#117—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253M32—1061no—
254M32—1061no—
255M32—1061no—
256M32—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259#108—532no—
260#108—1061no—
261#107—22no—
262#107—21no—
263#106—22no—
264#106—21no—
265#106C22⋊D5311yescharacteristic
266#106C22⋊D5312yes—
267#106C22⋊D5311yescharacteristic
268#101—1062no—
269#100—1062no—
270#41—21no—
271#41—21no—
272#41—2121no—
273#41—2121no—
274#41—2121no—
275#41—2121no—
276#41#1171992112yes—
277#41#1171992412yes—
278C2×Q8—1061no—
279C2×Q8#1171992211yescharacteristic
280C2×D4—21no—
281C2×D4—1061no—
282C2×D4—2121no—
283C2×D4#1171992211yescharacteristic
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—2121no—
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—1061no—
290Q16—1061no—
291Q16—2122no—
292SD16—2122no—
293SD16—2122no—
294SD16—2121no—
295SD16—2121no—
296D8—1061no—
297D8—1061no—
298D8—2122no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—1062no—
303M16—2121no—
304M16#1171992112yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—2121no—
309C2×C8C2×C4×D5311yescharacteristic
310C2×C8C2×D21211yescharacteristic
311C2×C8#1171994511yescharacteristic
312C2×C8#1171994511yescharacteristic
313C2×C8C2×C22⋊D5311yescharacteristic
314C2×C8C2×C22⋊D5311yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C42—42no—
323C42—41no—
324C16—2121no—
325C16—2121no—
326C23—21no—
327C23—2121no—
328C23#1313324311yescharacteristic
329Q8—21no—
330Q8—2121no—
331Q8—2121no—
332Q8#1313313712yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—4241no—
337D4—4241no—
338D4—2121no—
339D4—2121no—
340D4#1313313712yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—161no—
348C2×C4—162no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2121no—
352C2×C4C2×#1171992111yescharacteristic
353C2×C4#1313318611yescharacteristic
354C2×C4#1313320011yescharacteristic
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992411yescharacteristic
357C2×C4#1313324811yescharacteristic
358C8—22no—
359C8—2122no—
360C8C4×C22⋊D5311yescharacteristic
361C8C4×C22⋊D5311yescharacteristic
362C8#1313319111yescharacteristic
363C8#1313319111yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—8481no—
368C22—4241no—
369C22#1681062211yescharacteristic
370C22#1681075711yescharacteristic
371C22#1681094111yescharacteristic
372C4—22no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—4241no—
377C4#1681050211yescharacteristic
378C4C2×#1313313711yescharacteristic
379C4#1681076211yescharacteristic
380C4#1681093611yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—8481no—
386C2#2027503211yescharacteristic
subgroupG/H
centerC2#20275032
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2912C2
socleC2×C53#1874
prime radical#20284435C2
cosocleC22×C8×C53C23
hypercenter#2912D53
hypocenterC53#13005
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #20276450C2semidirect
3. C2×#16810432C2semidirect
4. #20276343C2semidirect
5. #20272831C2semidirect
6. #20272813C2semidirect
7. C53×#2912C2semidirect
12. #16810898C4semidirect
13. #16810893C4semidirect
39. C2×Dic212C2×C4semidirect
41. C2×D424C2×C4semidirect
50. #13133153C2×C4semidirect
189. #2912D53semidirect
222. C53#13005semidirect

Maximal subgroups

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

subgroupindex
#202844352
#202764502
C2×#168104322
#202763432
#202728312
#202728132
C53×#29122
#1300553