Finite Group Databasebeta

← #27301482

Subgroups — #27301482

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133157C211yesmaximal, characteristic, complemented
2#20284296C211yesmaximal, characteristic
3C2×#16810432C212yesmaximal, complemented
4#20272822C212yesmaximal, complemented
5C53×#1807C211yesmaximal, characteristic, complemented
6C2×#13133239C2211yescharacteristic
7#16811134—21no—
8C2×#13133157—24no—
9C2×#13133157—21no—
10C2×#13133157C2212yes—
11C22×#11719908—21no—
12#16810864C411yescharacteristic, complemented
13#16810864C411yescharacteristic, complemented
14#16810859C411yescharacteristic, complemented
15#16810859C411yescharacteristic, complemented
16#16810852—21no—
17#16810852—21no—
18#16810661C2211yescharacteristic
19#16810620—21no—
20#16810432—24no—
21C23×C8×C53C2211yescharacteristic
22C2×#106×C53C2212yes—
23C53×#380—21no—
24C53×#340—21no—
25C2×#11719945—21no—
26#13133239—22no—
27#13133239—22no—
28C2×Dic212C2×C411yescharacteristic
29C2×#11719913—21no—
30C2×D424C2×C411yescharacteristic
31#13133200—41no—
32#13133196—21no—
33#13133196—21no—
34#13133191—21no—
35#13133191—21no—
36#13133186—41no—
37C4×C22⋊D53—41no—
38C4×C22⋊D53—41no—
39#13133176—21no—
40#13133157—24no—
41#13133157—22no—
42#13133157D414yescomplemented
43C2×#11719908—24no—
44C2×#11719908—22no—
45#13133151C2×C411yescharacteristic
46#13133151C2×C411yescharacteristic
47#13133149—21no—
48#13133142—21no—
49#13133142—21no—
50#13133139—21no—
51#13133139—21no—
52C23×C4×C53D411yescharacteristic
53C22×C8×C53—22no—
54C22×C8×C53—22no—
55C22×C8×C53C2×C411yescharacteristic
56C22×C8×C53C2×C411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59C22×C8×C53D411yescharacteristic
60C22×C8×C53C2311yescharacteristic
61C2×C4⋊C4×C53—22no—
62C2×C22⋊C4×C53—41no—
63#106×C53—22no—
64#106×C53—22no—
65#95×C53—41no—
66C24×C53D811yescharacteristic
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53D811yescharacteristic
73C22×C4×C53SD1611yescharacteristic
74C22×C4×C53SD1611yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79#11719945—42no—
80#11719945—42no—
81C2×D212—21no—
82C2×C4×D53—41no—
83C2×Dic106—21no—
84C2×C8×C53—22no—
85C2×C8×C53—22no—
86C2×C8×C53—22no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53C22⋊C411yescharacteristic
92C2×C8×C53C22⋊C411yescharacteristic
93C2×C8×C53C22×C411yescharacteristic
94C2×C8×C53C2×D412yes—
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53#4112yes—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—42no—
99C22⋊C4×C53—82no—
100C22⋊Dic53—41no—
101C22⋊Dic53—41no—
102C22⋊Dic53—41no—
103#11719921—41no—
104#11719921—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107#11719919—41no—
108#11719919—41no—
109C4×Dic53—41no—
110Dic212—22no—
111D424—22no—
112#11719913—41no—
113#11719913—41no—
114#11719908—24no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53#10211yescharacteristic
119C23×C53#10211yescharacteristic
120C23×C53C2×D811yescharacteristic
121C22×D53—41no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×C4×C53—82no—
132C2×C4×C53#10211yescharacteristic
133C2×C4×C53#10211yescharacteristic
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53#12112yes—
136C2×C4×C53C2×SD1611yescharacteristic
137C2×C4×C53#13512yes—
138C22⋊D53—82no—
139C22⋊D53—82no—
140C22⋊D53—82no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144D212—41no—
145D212—41no—
146C4×D53—82no—
147Dic106—41no—
148Dic106—41no—
149C8×C53—21no—
150C8×C53—21no—
151C8×C53C4×D412yes—
152C8×C53#12011yescharacteristic
153C8×C53#12311yescharacteristic
154#13065—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—44no—
163C22×C53#36012yescomplemented
164C22×C53C2×#10211yescharacteristic
165C22×C53#46612yes—
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—162no—
172C4×C53#38611yescharacteristic
173C4×C53#41611yescharacteristic
174C4×C53#46512yes—
175Dic53—81no—
176Dic53—81no—
177Dic53—81no—
178C22×M32—531no—
179#3050—531no—
180C2×#360—532no—
181#1807D5311yescharacteristic, complemented
182#1386—532no—
183C2×C53—22no—
184C2×C53—22no—
185C2×C53—41no—
186C2×C53#189311yescharacteristic
187C2×C53C2×#36011yescharacteristic
188C2×C53#214511yescharacteristic
189D53—161no—
190C2×#135—531no—
191C23×C8C2×D5311yescharacteristic
192#525—1061no—
193C2×M32—532no—
194C2×M32—1064no—
195C2×M32—1061no—
196C22×C16—1061no—
197#467—531no—
198#467—531no—
199#466—531no—
200#466—531no—
201#465—1061no—
202#465—1061no—
203#427—531no—
204C2×#106C2×D5312yes—
205#416—1061no—
206#380—21no—
207#360—1064no—
208#340—21no—
209C53#1306511yescharacteristic, complemented, Sylow
210C2×#41—1061no—
211C23×C4C22⋊D5311yescharacteristic
212#135—1062no—
213#135—1062no—
214C2×Q16—531no—
215C2×SD16—1061no—
216C2×D8—531no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D5311yescharacteristic
220C22×C8C4×D5311yescharacteristic
221C22×C8D21211yescharacteristic
222C22×C8D21211yescharacteristic
223C22×C8C22⋊D5311yescharacteristic
224C22×C8C22×D5311yescharacteristic
225#123—2121no—
226C22⋊Q8—1061no—
227C22⋊Q8—1061no—
228#121—1061no—
229#121—1061no—
230#120—2121no—
231C4×D4—2121no—
232C4×D4—2121no—
233#117—1061no—
234C2×C4⋊C4—22no—
235C2×C22⋊C4—41no—
236M32—534no—
237M32—1064no—
238M32—1062no—
239C2×C16—1064no—
240C2×C16—1062no—
241#107—531no—
242#107—531no—
243#106—22no—
244#106—22no—
245#106—1061no—
246#103—1061no—
247#103—1061no—
248#102—1061no—
249#102—1061no—
250#95—41no—
251C24#1171992211yescharacteristic
252#41—2122no—
253#41—2122no—
254C2×Q8—1061no—
255C2×D4—1061no—
256C2×D4—2121no—
257C2×D4—2121no—
258C2×D4—2121no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—2121no—
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4#1171992211yescharacteristic
266C22×C4#1171992411yescharacteristic
267C22×C4#1171992411yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269Q16—1062no—
270SD16—2121no—
271SD16—2121no—
272D8—1062no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8#1171992111yescharacteristic
281C2×C8#1171992111yescharacteristic
282C2×C8C2×C4×D5311yescharacteristic
283C2×C8C2×D21211yescharacteristic
284C2×C8D4×D5312yes—
285C2×C8#1171994712yes—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C22⋊C4—82no—
293C22⋊C4—2121no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C42—2121no—
299C16—1064no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2121no—
304C23#1313313711yescharacteristic
305C23#1313313711yescharacteristic
306C23C2×#1171992211yescharacteristic
307Q8—2121no—
308Q8—2121no—
309D4—4242no—
310D4—4242no—
311D4—4242no—
312D4—2121no—
313D4—2121no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—4242no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4#1313313711yescharacteristic
329C2×C4#1313313711yescharacteristic
330C2×C4C2×#1171992111yescharacteristic
331C2×C4#1313318912yes—
332C2×C4C2×#1171992411yescharacteristic
333C2×C4#1313323712yes—
334C8—21no—
335C8—21no—
336C8#1313317712yes—
337C8#1313318711yescharacteristic
338C8#1313320111yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—4241no—
348C22—4241no—
349C22—4241no—
350C22#1681043212yescomplemented
351C22C2×#1313313711yescharacteristic
352C22#1681085712yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—4241no—
357C4—4241no—
358C4—4241no—
359C4#1681050011yescharacteristic
360C4#1681061711yescharacteristic
361C4#1681085412yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—8481no—
366C2#2027514111yescharacteristic
367C2C2×#1681043211yescharacteristic
368C2#2027645011yescharacteristic
subgroupG/H
centerC22C2×#13133137
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1807C2
socleC22×C53C2×#102
prime radical#20284296C2
cosocleC22×C8×C53C23
hypercenter#1807D53
hypocenterC53#13065
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#13133157C2semidirect
3. C2×#16810432C2semidirect
4. #20272822C2semidirect
5. C53×#1807C2semidirect
12. #16810864C4semidirect
13. #16810864C4semidirect
14. #16810859C4semidirect
15. #16810859C4semidirect
42. #13133157D4semidirect
163. C22×C53#360semidirect
181. #1807D53semidirect
209. C53#13065semidirect
350. C22#16810432semidirect

Maximal subgroups

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

subgroupindex
C22×#131331572
#202842962
C2×#168104322
#202728222
C53×#18072
#1306553