Finite Group Databasebeta

← #27301485

Subgroups — #27301485

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
2C2×#16810433C212yesmaximal, complemented
3#20274740C211yesmaximal, characteristic, complemented
4#20272820C212yesmaximal, complemented
5C53×#3050C211yesmaximal, characteristic
6C2×#13133157—24no—
7C2×#13133157—21no—
8C2×#13133157C2212yes—
9C22×#11719908—21no—
10C2×#13133149C2212yes—
11#16810483—21no—
12#16810433—24no—
13#16810386—21no—
14C2×#135×C53C2211yescharacteristic
15C23×C8×C53C2211yescharacteristic
16C53×#525—21no—
17C53×#467C411yescharacteristic, complemented
18C53×#467C411yescharacteristic, complemented
19C53×#466C411yescharacteristic, complemented
20C53×#466C411yescharacteristic, complemented
21C53×#465—21no—
22C53×#465—21no—
23C53×#427C2211yescharacteristic
24C53×#416—21no—
25C2×C4⋊Dic53—22no—
26C2×C22⋊Dic53—41no—
27#13133157—24no—
28#13133157—22no—
29#13133157D414yescomplemented
30C2×#11719908—24no—
31C2×#11719908—22no—
32#13133149—22no—
33#13133149—22no—
34#13133124—41no—
35C2×#41×C53—21no—
36C23×C4×C53D411yescharacteristic
37#135×C53—22no—
38#135×C53—22no—
39C2×Q16×C53C2×C411yescharacteristic
40C2×SD16×C53—21no—
41C2×D8×C53C2×C411yescharacteristic
42C22×C8×C53—22no—
43C22×C8×C53—22no—
44C22×C8×C53C2×C411yescharacteristic
45C22×C8×C53C2×C411yescharacteristic
46C22×C8×C53D411yescharacteristic
47C22×C8×C53D411yescharacteristic
48C22×C8×C53D411yescharacteristic
49C22×C8×C53C2311yescharacteristic
50#123×C53—41no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55#120×C53—41no—
56C4×D4×C53—41no—
57C4×D4×C53—41no—
58#117×C53—21no—
59#107×C53C2×C411yescharacteristic
60#107×C53C2×C411yescharacteristic
61#106×C53—21no—
62#103×C53—21no—
63#103×C53—21no—
64#102×C53—21no—
65#102×C53—21no—
66C24×C53D811yescharacteristic
67#41×C53—42no—
68#41×C53—42no—
69C2×Q8×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53D811yescharacteristic
80C22×C4×C53SD1611yescharacteristic
81C22×C4×C53SD1611yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—42no—
84Q16×C53—22no—
85SD16×C53—41no—
86SD16×C53—41no—
87D8×C53—22no—
88C2×C8×C53—22no—
89C2×C8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53C22⋊C411yescharacteristic
96C2×C8×C53C22⋊C411yescharacteristic
97C2×C8×C53C22×C411yescharacteristic
98C2×C8×C53C2×D412yes—
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4112yes—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—41no—
111C22⋊Dic53—82no—
112C4⋊Dic53—42no—
113C4⋊Dic53—42no—
114#11719908—24no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53#10211yescharacteristic
120C23×C53#10211yescharacteristic
121C23×C53C2×D811yescharacteristic
122Q8×C53—41no—
123Q8×C53—41no—
124D4×C53—41no—
125D4×C53—41no—
126D4×C53—82no—
127D4×C53—82no—
128D4×C53—82no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—82no—
140C2×C4×C53#10211yescharacteristic
141C2×C4×C53#10211yescharacteristic
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53#12112yes—
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53#13512yes—
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
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—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53#36012yescomplemented
167C22×C53C2×#10211yescharacteristic
168C22×C53#46612yes—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53#38611yescharacteristic
175C4×C53#41611yescharacteristic
176C4×C53#46512yes—
177Dic53—162no—
178C22×M32—531no—
179#3050D5311yescharacteristic
180C2×#360—532no—
181#1807—531no—
182#1386—532no—
183C2×C53—22no—
184C2×C53—22no—
185C2×C53—41no—
186C2×C53—161no—
187C2×C53#189311yescharacteristic
188C2×C53C2×#36011yescharacteristic
189C2×C53#214511yescharacteristic
190C2×#135C2×D5311yescharacteristic
191C23×C8C2×D5311yescharacteristic
192#525—21no—
193C2×M32—532no—
194C2×M32—1064no—
195C2×M32—1061no—
196C22×C16—1061no—
197#467Dic5311yescharacteristic, complemented
198#467Dic5311yescharacteristic, complemented
199#466Dic5311yescharacteristic, complemented
200#466Dic5311yescharacteristic, complemented
201#465—21no—
202#465—21no—
203#427C2×D5311yescharacteristic
204C2×#106—532no—
205#416—21no—
206#380—1061no—
207#360—1064no—
208#340—1061no—
209C53#1306511yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C23×C4C22⋊D5311yescharacteristic
212#135—22no—
213#135—22no—
214C2×Q16C2×Dic5311yescharacteristic
215C2×SD16—21no—
216C2×D8C2×Dic5311yescharacteristic
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C2×Dic5311yescharacteristic
220C22×C8C2×Dic5311yescharacteristic
221C22×C8C22⋊D5311yescharacteristic
222C22×C8C22⋊D5311yescharacteristic
223C22×C8C22⋊D5311yescharacteristic
224C22×C8C22×D5311yescharacteristic
225#123—41no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#121—21no—
229#121—21no—
230#120—41no—
231C4×D4—41no—
232C4×D4—41no—
233#117—21no—
234C2×C4⋊C4—1062no—
235C2×C22⋊C4—2121no—
236M32—534no—
237M32—1064no—
238M32—1062no—
239C2×C16—1064no—
240C2×C16—1062no—
241#107C2×Dic5311yescharacteristic
242#107C2×Dic5311yescharacteristic
243#106—21no—
244#106—1062no—
245#106—1062no—
246#103—21no—
247#103—21no—
248#102—21no—
249#102—21no—
250#95—2121no—
251C24#1171992211yescharacteristic
252#41—42no—
253#41—42no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—2122no—
263C22×C4#1171992211yescharacteristic
264C22×C4#1171992311yescharacteristic
265C22×C4#1171992311yescharacteristic
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269Q16—22no—
270SD16—41no—
271SD16—41no—
272D8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22⋊Dic5311yescharacteristic
281C2×C8C22⋊Dic5311yescharacteristic
282C2×C8D4×D5312yes—
283C2×C8#1171994712yes—
284C2×C8C22×Dic5311yescharacteristic
285C2×C8C2×C22⋊D5311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—4242no—
298C42—41no—
299C16—1064no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23#1313313811yescharacteristic
305C23#1313313811yescharacteristic
306C23C2×#1171992211yescharacteristic
307Q8—41no—
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—82no—
312D4—82no—
313D4—82no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—4242no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4#1313313811yescharacteristic
329C2×C4#1313313811yescharacteristic
330C2×C4C2×C22⋊Dic5311yescharacteristic
331C2×C4#1313318912yes—
332C2×C4C2×#1171992311yescharacteristic
333C2×C4#1313323712yes—
334C8—21no—
335C8—21no—
336C8D4×Dic5312yes—
337C8#1313318511yescharacteristic
338C8#1313319811yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#1681043312yescomplemented
351C22C2×#1313313811yescharacteristic
352C22#1681085712yes—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—8482no—
359C4#1681050311yescharacteristic
360C4#1681061911yescharacteristic
361C4#1681085012yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—161no—
366C2#2027514411yescharacteristic
367C2C2×#1681043311yescharacteristic
368C2#2027645111yescharacteristic
subgroupG/H
centerC22C2×#13133138
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3050C2
socleC22×C53C2×#102
cosocleC22×C8×C53C23
hypercenter#3050D53
hypocenterC53#13065
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#13133157C2semidirect
2. C2×#16810433C2semidirect
3. #20274740C2semidirect
4. #20272820C2semidirect
17. C53×#467C4semidirect
18. C53×#467C4semidirect
19. C53×#466C4semidirect
20. C53×#466C4semidirect
29. #13133157D4semidirect
166. C22×C53#360semidirect
197. #467Dic53semidirect
198. #467Dic53semidirect
199. #466Dic53semidirect
200. #466Dic53semidirect
209. C53#13065semidirect
350. C22#16810433semidirect

Maximal subgroups

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

subgroupindex
C22×#131331572
C2×#168104332
#202747402
#202728202
C53×#30502
#1306553