Finite Group Databasebeta

← #27484053

Subgroups — #27484053

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285238C211yesmaximal, characteristic, complemented
2C2×#16810745C211yesmaximal, characteristic
3#20275889C212yesmaximal, complemented
4#20275885C212yesmaximal, complemented
5#20275614C211yesmaximal, characteristic
6#20274839C211yesmaximal, characteristic, complemented
7#20274838C211yesmaximal, characteristic, complemented
8#20274767C212yesmaximal, complemented
9#20274759C212yesmaximal, complemented
10#20274600C211yesmaximal, characteristic, complemented
11C53×#1976C211yesmaximal, characteristic
12C2×#13133224C2212yes—
13#16811176C2212yes—
14#16811031—22no—
15#16811030—22no—
16#16810745—22no—
17#16810745—22no—
18C2×#13133153C2211yescharacteristic
19C2×#13133152C2211yescharacteristic
20#16810661C2211yescharacteristic
21#16810659C2212yes—
22#16810659C2211yescharacteristic
23#16810640C2212yes—
24C2×#13133144C2212yes—
25#16810597C2212yes—
26#16810578C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28#16810394C2212yes—
29#16810393C2212yes—
30#16810384C2211yescharacteristic
31#16810382C2211yescharacteristic
32#16810379C2212yes—
33#16810379C2212yes—
34C2×#135×C53C2211yescharacteristic
35C53×#427C2211yescharacteristic
36C53×#421C2212yes—
37C53×#410C2212yes—
38C53×#405C2211yescharacteristic
39#13133224—22no—
40#13133224—22no—
41#13133224—22no—
42#13133224—22no—
43C2×#11719917—22no—
44C2×#11719917C2311yescharacteristic
45C2×#11719917C2312yes—
46C22×#88322—22no—
47C22×#88322C2311yescharacteristic
48#13133216—22no—
49#13133208—22no—
50#13133194—42no—
51C4×Dic106—42no—
52#13133176C2311yescharacteristic
53#13133176C2312yes—
54C2×C4×Dic53C2311yescharacteristic
55#13133153—21no—
56#13133153—21no—
57#13133152—22no—
58#13133151—21no—
59#13133150—22no—
60#13133150—21no—
61#13133149—21no—
62#13133148—22no—
63#13133148—21no—
64#13133144—24no—
65#13133144—42no—
66#13133132—42no—
67#13133127—21no—
68#13133127D412yes—
69#13133127D412yes—
70C8×Dic53—21no—
71C2×#41×C53C2312yes—
72#135×C53—21no—
73#135×C53—21no—
74#135×C53—21no—
75#135×C53—21no—
76C2×Q16×C53Q811yescharacteristic, complemented
77C2×SD16×C53Q812yescomplemented
78C2×D8×C53Q811yescharacteristic, complemented
79C2×M16×C53C2311yescharacteristic
80C22×C8×C53C2311yescharacteristic
81#117×C53C2311yescharacteristic
82#117×C53C2312yes—
83#107×C53—21no—
84#106×C53—21no—
85#104×C53—42no—
86#99×C53—42no—
87#97×C53—21no—
88C4×C8×C53—21no—
89#41×C53—22no—
90#41×C53—22no—
91#41×C53—22no—
92#41×C53—22no—
93C2×Q8×C53C2×Q812yes—
94C2×D4×C53—22no—
95C2×D4×C53C2×Q812yes—
96C22×C4×C53—22no—
97C22×C4×C53C2411yescharacteristic
98C22×Dic53—21no—
99C2×Dic106—42no—
100Q16×C53—21no—
101Q16×C53—21no—
102SD16×C53—22no—
103SD16×C53—22no—
104D8×C53—21no—
105D8×C53—21no—
106M16×C53—21no—
107M16×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×Q811yescharacteristic
112C2×C8×C53C2×Q811yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C2×C8×C53#4111yescharacteristic
115C4⋊C4×C53—22no—
116C4⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—22no—
119C42×C53—21no—
120C42×C53—22no—
121C22⋊Dic53—21no—
122C22⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—42no—
126#11719919—42no—
127#11719919—42no—
128C4×Dic53—21no—
129C4×Dic53—21no—
130C4×Dic53—22no—
131C4×Dic53C2×D412yes—
132#11719917—22no—
133#11719917—21no—
134#11719917—22no—
135#11719917—21no—
136#11719917—42no—
137#11719917—42no—
138C2×#88322—21no—
139C2×#88322—22no—
140C2×#88322—42no—
141C2×#88322—42no—
142C2×#88322C2×D412yes—
143C2×#88322#4111yescharacteristic
144C2×#88322#4111yescharacteristic
145C2×#88322#4112yes—
146C23×C53—22no—
147C23×C53H(5,𝔽2)11yescharacteristic
148Q8×C53—22no—
149Q8×C53—22no—
150D4×C53—22no—
151D4×C53—22no—
152D4×C53—42no—
153D4×C53—42no—
154C2×C4×C53—22no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C22×Q811yescharacteristic
161C2×C4×C53C2×#4111yescharacteristic
162C2×C4×C53C2×#4111yescharacteristic
163C2×C4×C53C2×#4111yescharacteristic
164C2×C4×C53H(5,𝔽2)11yescharacteristic
165C2×Dic53—22no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—81no—
169Dic106—84no—
170C8×C53—41no—
171C8×C53C22⋊Q811yescharacteristic
172C8×C53C22⋊Q811yescharacteristic
173C8×C53C22⋊Q811yescharacteristic
174C8×C53C22⋊Q811yescharacteristic
175#88322—42no—
176#88322—41no—
177#88322#12412yes—
178#88322#12412yes—
179#29155—531nomaximal, Sylow
180C22×C53—41no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53#53511yescharacteristic
184C22×C53#55411yescharacteristic
185C22×C53#55511yescharacteristic
186C4×C53—42no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53C2×C22⋊Q811yescharacteristic
190C4×C53C2×#12411yescharacteristic
191C4×C53#55411yescharacteristic
192C4×C53#55511yescharacteristic
193Dic53—42no—
194Dic53—82no—
195Dic53—81no—
196#3162—531no—
197C2×#444—531no—
198#2031—532no—
199#2031—532no—
200#1976—531no—
201#1976D5311yescharacteristic
202#1833—531no—
203#1833—531no—
204#1814—532no—
205#1812—532no—
206#1773—531no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—82no—
211C2×C53#256311yescharacteristic
212C2×#135C2×D5311yescharacteristic
213C2×#131—532no—
214#533—532no—
215#501—1062no—
216#501—1062no—
217#444—1062no—
218#444—1062no—
219C2×#108—531no—
220C2×#108—531no—
221#427—531no—
222#427—531no—
223#427—532no—
224#427C2×D5311yescharacteristic
225#421—532no—
226#421C2×D5312yes—
227C2×#104—532no—
228#410—532no—
229#410C2×D5312yes—
230#405—531no—
231#405C2×D5311yescharacteristic
232C2×#97—531no—
233#344—532no—
234#344—532no—
235#339—531no—
236#339—531no—
237H(3,ℤ/4)—532no—
238H(3,ℤ/4)—532no—
239C53#2915511yescharacteristic, complemented, Sylow
240C2×#41C22×D5312yes—
241#135—21no—
242#135—21no—
243#135—21no—
244#135—21no—
245C2×Q16Dic10611yescharacteristic, complemented
246C2×SD16Dic10612yescomplemented
247C2×D8Dic10611yescharacteristic, complemented
248#131—1062no—
249#131—1062no—
250#131—1062no—
251#131—1062no—
252C2×M16—532no—
253C2×M16—531no—
254C2×M16—1062no—
255C2×M16C22×D5311yescharacteristic
256C22×C8—531no—
257C22×C8—1062no—
258C22×C8C22×D5311yescharacteristic
259C4⋊Q8—1062no—
260#125—1062no—
261C22⋊Q8—2122no—
262C4×Q8—2122no—
263#117—532no—
264#117—531no—
265#117C22×D5312yes—
266#117C22×D5311yescharacteristic
267C2×C42—531no—
268#108—1062no—
269#108—1061no—
270#108—1061no—
271#107—21no—
272#107—1061no—
273#107—1062no—
274#107—1061no—
275#106—21no—
276#106—1061no—
277#106—1062no—
278#106—1061no—
279#104—42no—
280#104—1064no—
281#104—2122no—
282#99—42no—
283#99—2122no—
284#97—21no—
285#97—532no—
286#97—532no—
287#97—1061no—
288C4×C8—21no—
289C4×C8—1061no—
290#41—22no—
291#41—22no—
292#41—22no—
293#41—22no—
294C2×Q8—2122no—
295C2×Q8C2×Dic10612yes—
296C2×D4—22no—
297C2×D4C2×Dic10612yes—
298C22×C4—22no—
299C22×C4—1061no—
300C22×C4C23×D5311yescharacteristic
301Q16—21no—
302Q16—21no—
303SD16—22no—
304SD16—22no—
305D8—21no—
306D8—21no—
307M16—21no—
308M16—21no—
309M16—1062no—
310M16—1061no—
311M16—1062no—
312M16—1061no—
313M16—2122no—
314M16—2122no—
315C2×C8—21no—
316C2×C8—532no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—532no—
320C2×C8—1062no—
321C2×C8—1061no—
322C2×C8—2122no—
323C2×C8—2122no—
324C2×C8C2×Dic10611yescharacteristic
325C2×C8C2×Dic10611yescharacteristic
326C2×C8D4×D5311yescharacteristic
327C2×C8D4×D5311yescharacteristic
328C2×C8#1171994711yescharacteristic
329C2×C8#1171994711yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—1062no—
333C4⋊C4—1061no—
334C4⋊C4—2122no—
335C4⋊C4—2122no—
336C4⋊C4—2122no—
337C22⋊C4—21no—
338C22⋊C4—22no—
339C22⋊C4—1061no—
340C22⋊C4—1062no—
341C42—21no—
342C42—22no—
343C42—532no—
344C42—1061no—
345C42—1061no—
346C42—1062no—
347C23—22no—
348C23#1313326211yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—4244no—
352D4—22no—
353D4—22no—
354D4—42no—
355D4—42no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—1062no—
362C2×C4—4241no—
363C2×C4—2121no—
364C2×C4—2122no—
365C2×C4C2×D4×D5311yescharacteristic
366C2×C4C22×Dic10611yescharacteristic
367C2×C4#41×D5311yescharacteristic
368C2×C4#41×D5311yescharacteristic
369C2×C4C2×#1171994711yescharacteristic
370C2×C4#1313326211yescharacteristic
371C8—41no—
372C8—532no—
373C8—532no—
374C8—2121no—
375C8—2122no—
376C8#1313319411yescharacteristic
377C8#1313319411yescharacteristic
378C8#1313319411yescharacteristic
379C8#1313319411yescharacteristic
380C22—41no—
381C22—42no—
382C22—82no—
383C22#1681118611yescharacteristic
384C22#1681133211yescharacteristic
385C22#1681133711yescharacteristic
386C4—42no—
387C4—81no—
388C4—82no—
389C4—4241no—
390C4—4242no—
391C4—2122no—
392C4C2×#1313319411yescharacteristic
393C4#124×D5311yescharacteristic
394C4#1681133211yescharacteristic
395C4#1681133711yescharacteristic
396C2—21no—
397C2—21no—
398C2—21no—
399C2—82no—
400C2#2027988611yescharacteristic
subgroupG/H
centerC4#16811337
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1976C2
socleC2×C53#2563
prime radicalC2×#135×C53C22
hypercenter#1976D53
hypocenterC53#29155
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
1. #20285238C2semidirect
3. #20275889C2semidirect
4. #20275885C2semidirect
6. #20274839C2semidirect
7. #20274838C2semidirect
8. #20274767C2semidirect
9. #20274759C2semidirect
10. #20274600C2semidirect
76. C2×Q16×C53Q8semidirect
77. C2×SD16×C53Q8semidirect
78. C2×D8×C53Q8semidirect
239. C53#29155semidirect
245. C2×Q16Dic106semidirect
246. C2×SD16Dic106semidirect
247. C2×D8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852382
C2×#168107452
#202758892
#202758852
#202756142
#202748392
#202748382
#202747672
#202747592
#202746002
C53×#19762
#2915553