Finite Group Databasebeta

← #27538643

Subgroups — #27538643

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811063C211yesmaximal, characteristic, complemented
2C2×#16811040C211yesmaximal, characteristic, complemented
3#20284370C211yesmaximal, characteristic, complemented
4#20276826C212yesmaximal, complemented
5#20276821C212yesmaximal, complemented
6#20276809C212yesmaximal, complemented
7#20276804C212yesmaximal, complemented
8#20276501C211yesmaximal, characteristic, complemented
9#20276422C211yesmaximal, characteristic, complemented
10C2×#16810426C211yesmaximal, characteristic, complemented
11C53×#3061C211yesmaximal, characteristic
12C22×Dic212C2211yescharacteristic
13#16811194—21no—
14#16811063—24no—
15C2×#13133163C2212yes—
16C2×#13133161C2212yes—
17#16811040—24no—
18C2×#13133157C2212yes—
19C2×#13133157C2211yescharacteristic
20C22×#11719908C2212yes—
21C22×#11719908C2211yescharacteristic
22#16810881—22no—
23#16810864C2212yes—
24#16810864C2212yes—
25#16810661C2211yescharacteristic
26#16810633—21no—
27#16810450C2212yes—
28#16810448C2212yes—
29#16810427C2211yescharacteristic
30#16810427C2211yescharacteristic
31#16810426C2212yes—
32#16810426C2212yes—
33#16810424C2211yescharacteristic
34#16810424C2211yescharacteristic
35C2×#135×C53C2211yescharacteristic
36C2×#131×C53C2211yescharacteristic
37C53×#537—21no—
38C53×#468—22no—
39C53×#467C2212yes—
40C53×#466C2212yes—
41C2×#107×C53C2211yescharacteristic
42C53×#417—21no—
43C22×Dic106—21no—
44C2×Dic212—21no—
45C2×Dic212—22no—
46C2×Dic212—22no—
47C2×Dic212C2312yes—
48#13133215—41no—
49#13133204—41no—
50#13133198—42no—
51#13133196—22no—
52#13133196—22no—
53#13133176—21no—
54#13133163—24no—
55#13133161—24no—
56#13133157—24no—
57#13133157D414yescomplemented
58C2×#11719908—22no—
59C2×#11719908—24no—
60C2×#11719908C2311yescharacteristic
61C2×#11719908C2311yescharacteristic
62C2×#11719908C2312yes—
63#13133151C2311yescharacteristic
64#13133151C2311yescharacteristic
65#13133149—21no—
66#13133142—22no—
67#13133142—22no—
68C2×#41×C53—21no—
69C2×#41×C53D411yescharacteristic
70#135×C53—22no—
71#135×C53—22no—
72C2×Q16×C53C2311yescharacteristic
73C2×SD16×C53—21no—
74C2×D8×C53C2311yescharacteristic
75#131×C53—22no—
76#131×C53—22no—
77C2×M16×C53D412yes—
78C2×M16×C53D411yescharacteristic
79C22×C8×C53C2312yes—
80C22×C8×C53C2311yescharacteristic
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—41no—
83#121×C53—22no—
84#121×C53—42no—
85#121×C53—41no—
86C2×C4⋊C4×C53—21no—
87#107×C53—21no—
88#107×C53C2312yes—
89#103×C53—22no—
90#102×C53—22no—
91#41×C53—22no—
92#41×C53—22no—
93#41×C53—42no—
94#41×C53—42no—
95C2×Q8×C53—21no—
96C2×Q8×C53D811yescharacteristic
97C2×D4×C53—21no—
98C2×D4×C53—41no—
99C2×D4×C53—42no—
100C2×D4×C53D812yes—
101C2×D4×C53D811yescharacteristic
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×Dic53—41no—
107C2×Dic106—22no—
108C2×Dic106—42no—
109C2×Dic106—42no—
110C2×Dic106—41no—
111Q16×C53—21no—
112Q16×C53—21no—
113SD16×C53—42no—
114D8×C53—21no—
115D8×C53—21no—
116M16×C53—22no—
117M16×C53—24no—
118C2×C8×C53—22no—
119C2×C8×C53—22no—
120C2×C8×C53C2×D412yes—
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D412yes—
123C2×C8×C53#4112yes—
124C2×C8×C53C2411yescharacteristic
125C4⋊C4×C53—22no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—42no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—42no—
130C22⋊Dic53—41no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135#11719919—42no—
136#11719919—42no—
137C4×Dic53—41no—
138Dic212—22no—
139Dic212—22no—
140Dic212—42no—
141#11719908—21no—
142#11719908—21no—
143#11719908C2×D414yes—
144C23×C53—41no—
145C23×C53C2×D812yes—
146C23×C53C2×D811yescharacteristic
147Q8×C53—22no—
148Q8×C53—41no—
149Q8×C53—41no—
150D4×C53—24no—
151D4×C53—22no—
152D4×C53—41no—
153D4×C53—41no—
154D4×C53—82no—
155D4×C53—162no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53—161no—
162C2×C4×C53#12112yes—
163C2×C4×C53C2×D812yes—
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53#13512yes—
166C2×C4×C53C22×D411yescharacteristic
167C2×Dic53—42no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—82no—
171Dic106—42no—
172Dic106—42no—
173Dic106—82no—
174C8×C53—22no—
175C8×C53#12112yes—
176C8×C53C22×D411yescharacteristic
177C8×C53C2×#4111yescharacteristic
178#33613—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—22no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#46612yes—
188C22×C53C22×D811yescharacteristic
189C4×C53—22no—
190C4×C53—82no—
191C4×C53—81no—
192C4×C53#46612yes—
193C4×C53C2×#12111yescharacteristic
194C4×C53C2×#13511yescharacteristic
195Dic53—81no—
196Dic53—81no—
197Dic53—82no—
198C2×#510—531no—
199C2×#504—531no—
200#3061D5311yescharacteristic
201#3059—531no—
202#2220—532no—
203#2219—532no—
204#2217—532no—
205#2216—532no—
206#2159—531no—
207#2138—531no—
208C2×#358—531no—
209C2×C53—22no—
210C2×C53—42no—
211C2×C53—161no—
212C2×C53C2×#46611yescharacteristic
213C2×C53#341611yescharacteristic
214C2×C53#341711yescharacteristic
215C2×#135C2×D5311yescharacteristic
216C22×Q16—531no—
217C2×#131C2×D5311yescharacteristic
218#537—21no—
219#536—1061no—
220#510—1064no—
221C2×Q32—532no—
222C2×SD32—532no—
223#504—1064no—
224C2×M32—531no—
225C2×M32—532no—
226C22×C16—531no—
227C22×C16—532no—
228#470—1062no—
229#468—22no—
230#467—532no—
231#467—532no—
232#467C2×D5312yes—
233#466C2×D5312yes—
234#427—531no—
235C2×#107C2×D5311yescharacteristic
236#419—1061no—
237#417—21no—
238#367—532no—
239#366—532no—
240#358—531no—
241#358—531no—
242#358—532no—
243#358—532no—
244#357—531no—
245#357—531no—
246C53#3361311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41C22⋊D5311yescharacteristic
249C22×Q8—1061no—
250#135—22no—
251#135—22no—
252C2×Q16—532no—
253C2×Q16—1062no—
254C2×Q16—1062no—
255C2×Q16—1061no—
256C2×Q16C22×D5311yescharacteristic
257C2×SD16—21no—
258C2×D8C22×D5311yescharacteristic
259#131—22no—
260#131—22no—
261C2×M16C22⋊D5311yescharacteristic
262C2×M16C22⋊D5312yes—
263C22×C8C22×D5312yes—
264C22×C8C22×D5311yescharacteristic
265C4⋊Q8—2121no—
266#124—2121no—
267#123—2122no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270C22⋊Q8—1062no—
271C22⋊Q8—1062no—
272#121—22no—
273#121—41no—
274#121—42no—
275#117—1061no—
276C2×C4⋊C4—21no—
277Q32—1064no—
278SD32—1064no—
279M32—534no—
280M32—1064no—
281C2×C16—532no—
282C2×C16—531no—
283C2×C16—531no—
284C2×C16—1064no—
285C2×C16—1062no—
286#107—21no—
287#107—531no—
288#107—531no—
289#107C22×D5312yes—
290#106—1061no—
291#103—22no—
292#103—1062no—
293#103—1062no—
294#102—22no—
295#41—22no—
296#41—22no—
297#41—42no—
298#41—42no—
299C2×Q8—21no—
300C2×Q8—1062no—
301C2×Q8—2122no—
302C2×Q8—2122no—
303C2×Q8—2121no—
304C2×Q8#1171992211yescharacteristic
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4#1171992212yes—
309C2×D4#1171992211yescharacteristic
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—2121no—
313C22×C4C2×C22⋊D5312yes—
314C22×C4C2×C22⋊D5311yescharacteristic
315Q16—21no—
316Q16—21no—
317Q16—1062no—
318Q16—1062no—
319Q16—2122no—
320SD16—42no—
321D8—21no—
322D8—21no—
323M16—22no—
324M16—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8D4×D5312yes—
328C2×C8#1171994712yes—
329C2×C8C2×C22⋊D5312yes—
330C2×C8C2×C22⋊D5311yescharacteristic
331C2×C8C23×D5311yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—2122no—
338C4⋊C4—2122no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—2121no—
342C22⋊C4—2122no—
343C22⋊C4—2122no—
344C42—2121no—
345C16—534no—
346C16—1061no—
347C16—1061no—
348C23—41no—
349C23C2×#1171992212yes—
350C23C2×#1171992211yescharacteristic
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—4242no—
355Q8—2122no—
356Q8—2122no—
357D4—24no—
358D4—22no—
359D4—41no—
360D4—41no—
361D4—82no—
362D4—162no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—82no—
368C2×C4—161no—
369C2×C4—4242no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2122no—
373C2×C4#1313318912yes—
374C2×C4C2×#1171992212yes—
375C2×C4C2×#1171992211yescharacteristic
376C2×C4#1313323712yes—
377C2×C4C22×C22⋊D5311yescharacteristic
378C8—22no—
379C8#1313318912yes—
380C8C2×D4×D5311yescharacteristic
381C8C2×#1171994711yescharacteristic
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1681085712yes—
391C22C22×#1171992211yescharacteristic
392C4—22no—
393C4—82no—
394C4—81no—
395C4—4241no—
396C4—4242no—
397C4—4241no—
398C4#1681085712yes—
399C4C2×#1313318911yescharacteristic
400C4C2×#1313323711yescharacteristic
401C2—22no—
402C2—42no—
403C2—161no—
404C2C2×#1681085711yescharacteristic
405C2#2028786211yescharacteristic
406C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3061C2
socleC22×C53C22×D8
hypercenter#3061D53
hypocenterC53#33613
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
2. C2×#16811040C2semidirect
3. #20284370C2semidirect
4. #20276826C2semidirect
5. #20276821C2semidirect
6. #20276809C2semidirect
7. #20276804C2semidirect
8. #20276501C2semidirect
9. #20276422C2semidirect
10. C2×#16810426C2semidirect
57. #13133157D4semidirect
246. C53#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#168110632
C2×#168110402
#202843702
#202768262
#202768212
#202768092
#202768042
#202765012
#202764222
C2×#168104262
C53×#30612
#3361353