Finite Group Databasebeta

← #27294326

Subgroups — #27294326

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284296C211yesmaximal, characteristic
2C2×#16810386C211yesmaximal, characteristic, complemented
3#20272728C212yesmaximal, complemented
4#20272725C212yesmaximal, complemented
5C53×#2112C211yesmaximal, characteristic, complemented
6C2×#13133239C411yescharacteristic, complemented
7#16811134—21no—
8C22×C4⋊Dic53—21no—
9#16810864C411yescharacteristic, complemented
10#16810864C2211yescharacteristic
11#16810859C411yescharacteristic, complemented
12#16810859C2211yescharacteristic
13#16810852—21no—
14#16810852—21no—
15#16810661C411yescharacteristic, complemented
16#16810620—21no—
17#16810386—21no—
18#16810386C2212yes—
19C23×C8×C53C2211yescharacteristic
20C2×M32×C53—21no—
21C53×C22⋊C16—21no—
22C53×C22⋊C16C2212yes—
23C2×#11719945—21no—
24C23×Dic53—41no—
25#13133239—42no—
26C2×Dic212C2×C411yescharacteristic, complemented
27C2×#11719913—21no—
28C2×D424C2×C411yescharacteristic, complemented
29#13133200—41no—
30#13133196—21no—
31#13133196—21no—
32#13133191—21no—
33#13133191—21no—
34#13133186—41no—
35C4×C22⋊D53—41no—
36C4×C22⋊D53—41no—
37#13133176—21no—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—42no—
41C2×C4⋊Dic53—41no—
42#13133151C2×C411yescharacteristic, complemented
43#13133151C2×C411yescharacteristic, complemented
44#13133149—21no—
45#13133142—21no—
46#13133142—21no—
47#13133139—21no—
48#13133139—21no—
49C23×C4×C53D411yescharacteristic
50C22×C8×C53—42no—
51C22×C8×C53C2×C411yescharacteristic
52C22×C8×C53C2×C411yescharacteristic
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C22×C8×C53D411yescharacteristic
56C22×C8×C53C2311yescharacteristic
57M32×C53—42no—
58C2×C16×C53—22no—
59C24×C53D811yescharacteristic
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53D811yescharacteristic
65C22×C4×C53SD1611yescharacteristic
66C22×C4×C53SD1611yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C2×C22⋊D53—41no—
69C2×C22⋊D53—41no—
70C2×C22⋊D53—41no—
71C22×Dic53—44no—
72C22×Dic53—42no—
73C22×Dic53—82no—
74C22×Dic53—82no—
75#11719945—42no—
76#11719945—42no—
77C2×D212—21no—
78C2×C4×D53—41no—
79C2×Dic106—21no—
80C2×C8×C53—22no—
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53C22⋊C411yescharacteristic
87C2×C8×C53C22⋊C411yescharacteristic
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—41no—
93#11719921—41no—
94#11719921—41no—
95C4⋊Dic53—24no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—44no—
99C4⋊Dic53—42no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—41no—
103Dic212—41no—
104D424—41no—
105#11719913—42no—
106C16×C53—42no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#10211yescharacteristic
112C23×C53#10211yescharacteristic
113C23×C53C2×D811yescharacteristic
114C22×D53—41no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53#9912yes—
123C2×C4×C53#10211yescharacteristic
124C2×C4×C53#10211yescharacteristic
125C2×C4×C53#10412yes—
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53C2×SD1611yescharacteristic
128C22⋊D53—82no—
129C22⋊D53—82no—
130C22⋊D53—81no—
131C22⋊D53—81no—
132C2×Dic53—44no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—88no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139D212—41no—
140D212—41no—
141C4×D53—82no—
142Dic106—41no—
143Dic106—41no—
144C8×C53—21no—
145C8×C53—21no—
146C8×C53—21no—
147C8×C53—21no—
148#11704—531nomaximal, Sylow
149C22×C53—24no—
150C22×C53—24no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53#32712yes—
158C22×C53#36012yes—
159C22×C53C2×#10211yescharacteristic
160C22×C53#50911yescharacteristic
161C22×C53#51011yescharacteristic
162C2×D53—81no—
163C2×D53—81no—
164C2×D53—81no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53#32712yes—
168C4×C53C2×#9911yescharacteristic
169C4×C53C2×#10411yescharacteristic
170Dic53—84no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174#3050—531no—
175#2112D5311yescharacteristic, complemented
176C2×#340—531no—
177#1348—532no—
178#1347—532no—
179C2×C53—21no—
180C2×C53—22no—
181C2×C53—22no—
182C2×C53—21no—
183C2×C53C2×#32711yescharacteristic
184C2×C53#213911yescharacteristic
185C2×C53C2×#36011yescharacteristic
186D53—161no—
187C2×#135—531no—
188C23×C8C2×D5311yescharacteristic
189#525—1061no—
190C22×C4⋊C4—1061no—
191C2×M32—21no—
192#467—531no—
193#467—531no—
194#466—531no—
195#466—531no—
196#465—1061no—
197#465—1061no—
198#427—531no—
199#416—1061no—
200C22⋊C16—21no—
201C22⋊C16C2×D5312yes—
202#340—532no—
203#340—1061no—
204C53#1170411yescharacteristic, complemented, Sylow
205C2×#41—1061no—
206C23×C4—2121no—
207C23×C4D21211yescharacteristic
208#135—2122no—
209C2×Q16—531no—
210C2×SD16—1061no—
211C2×D8—531no—
212C22×C8—42no—
213C22×C8C4×D5311yescharacteristic
214C22×C8C4×D5311yescharacteristic
215C22×C8D21211yescharacteristic
216C22×C8C22⋊D5311yescharacteristic
217C22×C8C22⋊D5311yescharacteristic
218C22×C8C22×D5311yescharacteristic
219#123—2121no—
220C22⋊Q8—1061no—
221C22⋊Q8—1061no—
222#121—1061no—
223#121—1061no—
224#120—2121no—
225C4×D4—2121no—
226C4×D4—2121no—
227#117—1061no—
228C2×C4⋊C4—1064no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—2122no—
231C2×C4⋊C4—2121no—
232M32—42no—
233C2×C16—22no—
234#107—531no—
235#107—531no—
236#106—1061no—
237#103—1061no—
238#103—1061no—
239#102—1061no—
240#102—1061no—
241C24D42411yescharacteristic
242#41—2122no—
243#41—2122no—
244C2×Q8—1061no—
245C2×D4—1061no—
246C2×D4—2121no—
247C2×D4—2121no—
248C2×D4—2121no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—4242no—
252C22×C4—4242no—
253C22×C4—2124no—
254C22×C4—2122no—
255C22×C4—2121no—
256C22×C4#1171991311yescharacteristic
257C22×C4#1171991311yescharacteristic
258C22×C4D42411yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4C2×D21211yescharacteristic
262Q16—2121no—
263SD16—2122no—
264D8—2121no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8#1171992111yescharacteristic
272C2×C8#1171992111yescharacteristic
273C2×C8C2×C4×D5311yescharacteristic
274C2×C8C2×C22⋊D5311yescharacteristic
275C4⋊C4—1064no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—2122no—
279C4⋊C4—2124no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C22⋊C4—2121no—
283C22⋊C4—2121no—
284C22⋊C4—2121no—
285C22⋊C4—2121no—
286C22⋊C4—2121no—
287C42—2121no—
288C16—42no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—2121no—
294C23#1313313911yescharacteristic
295C23#1313313911yescharacteristic
296C23C2×D42411yescharacteristic
297Q8—2121no—
298Q8—2121no—
299D4—4241no—
300D4—4241no—
301D4—4242no—
302D4—4242no—
303D4—2121no—
304D4—2121no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—4248no—
313C2×C4—4242no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—2124no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4#1313313112yes—
321C2×C4#1313313911yescharacteristic
322C2×C4#1313313911yescharacteristic
323C2×C4#1313314312yes—
324C2×C4C2×#1171992111yescharacteristic
325C2×C4C2×#1171991311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—4241no—
339C22—4241no—
340C22—4241no—
341C22#1681035812yes—
342C22#1681043412yes—
343C22C2×#1313313911yescharacteristic
344C22#1681106111yescharacteristic
345C22#1681106611yescharacteristic
346C4—21no—
347C4—21no—
348C4—4244no—
349C4—4241no—
350C4—4241no—
351C4—4241no—
352C4#1681035812yes—
353C4C2×#1313313111yescharacteristic
354C4C2×#1313314311yescharacteristic
355C2—21no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—8481no—
360C2C2×#1681035811yescharacteristic
361C2#2027642811yescharacteristic
362C2C2×#1681043411yescharacteristic
subgroupG/H
centerC22C2×#13133139
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2112C2
socleC22×C53C2×#102
prime radical#20284296C2
cosocleC22×C8×C53C23
hypercenter#2112D53
hypocenterC53#11704
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×#16810386C2semidirect
3. #20272728C2semidirect
4. #20272725C2semidirect
5. C53×#2112C2semidirect
6. C2×#13133239C4semidirect
9. #16810864C4semidirect
11. #16810859C4semidirect
15. #16810661C4semidirect
26. C2×Dic212C2×C4semidirect
28. C2×D424C2×C4semidirect
42. #13133151C2×C4semidirect
43. #13133151C2×C4semidirect
175. #2112D53semidirect
204. C53#11704semidirect

Maximal subgroups

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

subgroupindex
#202842962
C2×#168103862
#202727282
#202727252
C53×#21122
#1170453