Finite Group Databasebeta

← #27277409

Subgroups — #27277409

Order 13568.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283691C211yesmaximal, characteristic
2C4×#13133150C211yesmaximal, characteristic, complemented
3#20273476C211yesmaximal, characteristic, complemented
4#20273240C211yesmaximal, characteristic
5#20272571C211yesmaximal, characteristic, complemented
6#20272570C211yesmaximal, characteristic, complemented
7C53×#1509C211yesmaximal, characteristic
8#16810754C411yescharacteristic, complemented
9Q8×#88322C411yescharacteristic, complemented
10#16810701C411yescharacteristic, complemented
11#16810695—21no—
12D4×#88322—21no—
13D4×#88322C411yescharacteristic, complemented
14#16810684—21no—
15#16810680C2211yescharacteristic
16#16810680C2211yescharacteristic
17C2×#13133150—21no—
18#16810579—21no—
19C2×C4×#88322C2211yescharacteristic
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810375—21no—
23#16810373—21no—
24#16810360—21no—
25#16810357—21no—
26C53×#529C2211yescharacteristic
27C53×#432C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C53×#328—21no—
30C53×#327—21no—
31C2×#11719917—21no—
32C22×#88322—21no—
33C22×#88322D411yescharacteristic
34C22×#88322D411yescharacteristic
35C2×C4⋊Dic53—21no—
36C2×C4×Dic53—21no—
37#13133150—24no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322C2×C411yescharacteristic
42C4⋊#88322C2×C411yescharacteristic
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47#13133128—21no—
48#13133128—21no—
49#13133128—21no—
50C4×#88322—21no—
51C4×#88322C2×C411yescharacteristic
52C4×#88322C2×C411yescharacteristic
53C4×#88322D411yescharacteristic
54C4×#88322D411yescharacteristic
55#13133124—21no—
56#126×C53—21no—
57#125×C53Q811yescharacteristic
58#124×C53Q811yescharacteristic
59#123×C53—21no—
60C22⋊Q8×C53D411yescharacteristic
61#121×C53D411yescharacteristic
62C4×Q8×C53C2×C411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53C2×C411yescharacteristic
65#117×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71#97×C53—21no—
72#95×C53—21no—
73C2×Q8×C53C4⋊C411yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53C4⋊C411yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—41no—
82C2×C8×C53—41no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53C4⋊C411yescharacteristic
90C4⋊C4×C53C4⋊C411yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C42×C53—21no—
97C42×C53—81no—
98C42×C53C22×C411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×Q811yescharacteristic
101C42×C53#4111yescharacteristic
102C4⋊Dic53—22no—
103C4⋊Dic53—41no—
104C4×Dic53—81no—
105#11719917—42no—
106C2×#88322—22no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—41no—
110C2×#88322—42no—
111C2×#88322—41no—
112C2×#88322C22⋊C411yescharacteristic
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322C22⋊C411yescharacteristic
115C2×#88322C22⋊C411yescharacteristic
116C23×C53—21no—
117C23×C53#12111yescharacteristic
118Q8×C53—41no—
119D4×C53—41no—
120D4×C53—41no—
121D4×C53—41no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C2×C4⋊C411yescharacteristic
137C2×C4×C53#11711yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×Dic53—42no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C8×C53—81no—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148#88322—41no—
149#88322—41no—
150#88322—81no—
151#88322#10412yes—
152#8697—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53#38011yescharacteristic
159C22×C53#47211yescharacteristic
160C22×C53#47311yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—82no—
170C4×C53C2×#10411yescharacteristic
171C4×C53#42111yescharacteristic
172Dic53—82no—
173#2966—531no—
174C4×#107—531no—
175#1568—531no—
176#1509—531no—
177#1509D5311yescharacteristic
178#1285—531no—
179#1285—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—81no—
183C2×C53#180611yescharacteristic
184C2×C53#181211yescharacteristic
185C2×C53#181611yescharacteristic
186#529C2×D5311yescharacteristic
187#446—531no—
188C8×Q8—531no—
189#436—531no—
190#435—1061no—
191C8×D4—531no—
192C8×D4—1061no—
193#433—1061no—
194#432—531no—
195#432—531no—
196#432C2×D5311yescharacteristic
197C2×#107—1061no—
198#405—1061no—
199C2×C4×C8—531no—
200C4×C4⋊C4—531no—
201C4×C4⋊C4C2×D5311yescharacteristic
202#340—1061no—
203#335—1061no—
204#334—1061no—
205#328—21no—
206#328—1061no—
207#327—21no—
208#327—1061no—
209C53#869711yescharacteristic, complemented, Sylow
210C2×M16—1061no—
211C22×C8—531no—
212C22×C8—531no—
213C22×C8—1061no—
214#126—21no—
215#125Dic10611yescharacteristic
216#124Dic10611yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D5311yescharacteristic
219#121C22⋊D5311yescharacteristic
220C4×Q8C4×D5311yescharacteristic
221C4×D4—21no—
222C4×D4C4×D5311yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1061no—
226C2×C42—21no—
227C2×C42—1061no—
228C2×C42C22×D5311yescharacteristic
229#107—1064no—
230C4⋊C8—21no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233C4⋊C8—1061no—
234C4⋊C8—1061no—
235C4⋊C8—1061no—
236C22⋊C8—21no—
237C22⋊C8—1061no—
238C22⋊C8—1061no—
239C22⋊C8—1061no—
240C22⋊C8—1061no—
241#97—21no—
242#97—1061no—
243#97—1061no—
244#97—1061no—
245C4×C8—531no—
246C4×C8—531no—
247C4×C8—531no—
248C4×C8—531no—
249C4×C8—1061no—
250#95—21no—
251#95—1061no—
252C2×Q8#1171991911yescharacteristic
253C2×D4—21no—
254C2×D4#1171991911yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—2121no—
258C22×C4D4×D5311yescharacteristic
259C22×C4#1171994711yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261M16—2122no—
262C2×C8—41no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8—531no—
267C2×C8—1062no—
268C2×C8—1061no—
269C2×C8—1061no—
270C2×C8—2121no—
271C2×C8—2122no—
272C2×C8—2121no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—1062no—
280C4⋊C4—2121no—
281C4⋊C4#1171991911yescharacteristic
282C4⋊C4#1171991911yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—4241no—
291C42C2×Dic10611yescharacteristic
292C42C2×C4×D5311yescharacteristic
293C42D4×D5311yescharacteristic
294C42#1171994711yescharacteristic
295C23—21no—
296C23#1313318911yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—4241no—
315C2×C4—4241no—
316C2×C4—2122no—
317C2×C4C22⋊C4×D5311yescharacteristic
318C2×C4C2×#1171991911yescharacteristic
319C2×C4#1313317311yescharacteristic
320C2×C4#1313318911yescharacteristic
321C2×C4#1313319411yescharacteristic
322C2×C4#1313319411yescharacteristic
323C8—81no—
324C8—532no—
325C8—1061no—
326C8—1061no—
327C8—1061no—
328C8—4241no—
329C8—2121no—
330C8—2121no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1681048111yescharacteristic
337C22#1681089111yescharacteristic
338C22#1681089611yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—4242no—
349C4#104×D5311yescharacteristic
350C4#1681063811yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2027473311yescharacteristic
355C2#2027475911yescharacteristic
356C2#2027477711yescharacteristic
subgroupG/H
centerC22#16810481
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1509C2
socleC22×C53#380
prime radical#121×C53D4
cosocleC2×C42×C53C23
hypercenter#1509D53
hypocenterC53#8697
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#13133150C2semidirect
3. #20273476C2semidirect
5. #20272571C2semidirect
6. #20272570C2semidirect
8. #16810754C4semidirect
9. Q8×#88322C4semidirect
10. #16810701C4semidirect
13. D4×#88322C4semidirect
209. C53#8697semidirect

Maximal subgroups

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

subgroupindex
#202836912
C4×#131331502
#202734762
#202732402
#202725712
#202725702
C53×#15092
#869753