Finite Group Databasebeta

← #27277962

Subgroups — #27277962

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283755C211yesmaximal, characteristic
2#20275508C211yesmaximal, characteristic
3C4×#13133133C211yesmaximal, characteristic
4#20273465C211yesmaximal, characteristic, complemented
5#20272553C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1565C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810702—21no—
10#16810702—21no—
11#16810700—21no—
12#16810700—21no—
13#16810676C2211yescharacteristic
14C2×#13133133C411yescharacteristic
15C2×#13133133C411yescharacteristic
16#16810585—22no—
17C2×#13133129—22no—
18C4×#11719917C2211yescharacteristic
19C2×C4×#88322C2211yescharacteristic
20#16810391C411yescharacteristic
21#16810391C411yescharacteristic
22#16810375—21no—
23#16810373—21no—
24C53×#431C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C4×M16×C53C2211yescharacteristic
27C53×#335—21no—
28C53×#334—21no—
29C22×D212C2×C411yescharacteristic
30C22×C4×D53D412yes—
31C2×#11719917C2×C411yescharacteristic
32C2×#11719917C2×C411yescharacteristic
33C2×#11719912—22no—
34C2×#11719912—22no—
35C22×#88322—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719921Q812yes—
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44#13133133—22no—
45#13133133—22no—
46C22⋊#88322—21no—
47#13133129—22no—
48#13133129—22no—
49#13133129—44no—
50#13133128—21no—
51C4×#88322—21no—
52C4×#88322—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55C2×M16×C53C2×C411yescharacteristic
56C2×M16×C53C2×C411yescharacteristic
57C22×C8×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D411yescharacteristic
62C4⋊C8×C53D411yescharacteristic
63C22⋊C8×C53—21no—
64#97×C53—21no—
65C4×C8×C53—21no—
66C4×C8×C53—21no—
67C23×D53C4⋊C412yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×Q811yescharacteristic
71C22×Dic53C4⋊C412yes—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—42no—
81C2×C4×D53—42no—
82M16×C53—21no—
83M16×C53—21no—
84M16×C53—21no—
85M16×C53M1612yescomplemented
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—42no—
91C2×C8×C53—41no—
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53#4111yescharacteristic
95C42×C53#4111yescharacteristic
96#11719921—22no—
97#11719921—22no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100#11719917—21no—
101#11719917—21no—
102#11719917—21no—
103#11719917M1612yescomplemented
104C2×#88322—21no—
105C2×#88322—22no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—41no—
110C2×#88322—41no—
111#11719912—82no—
112#11719912—82no—
113C23×C53C2×C4⋊C411yescharacteristic
114C22×D53—22no—
115C22×D53—22no—
116C22×D53—22no—
117C22×D53—42no—
118C22×D53—42no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53C4×D411yescharacteristic
123C2×C4×C53C4×D411yescharacteristic
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22⋊Q811yescharacteristic
126C2×C4×C53C22⋊Q811yescharacteristic
127C2×C4×C53#12311yescharacteristic
128C2×C4×C53C2×M1611yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×C4×C53#13111yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C2×Dic53—22no—
133C2×Dic53—42no—
134D212—41no—
135D212—41no—
136D212—41no—
137D212—41no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—82no—
141C8×C53—22no—
142C8×C53—41no—
143C8×C53—41no—
144C8×C53—41no—
145C8×C53—81no—
146#88322—22no—
147#88322—42no—
148#88322—41no—
149#88322—41no—
150#88322—41no—
151#8837—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53#38511yescharacteristic
155C22×C53#42311yescharacteristic
156C22×C53#42411yescharacteristic
157C2×D53—44no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—82no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53#43511yescharacteristic
166C4×C53#43511yescharacteristic
167C4×C53#43611yescharacteristic
168C4×C53#43611yescharacteristic
169C4×C53#45311yescharacteristic
170C4×C53#45511yescharacteristic
171Dic53—42no—
172#2975—531no—
173#1954—531no—
174C4×#100—531no—
175#1565—531no—
176#1565D5311yescharacteristic, complemented
177#1278—531no—
178#1278—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53#185111yescharacteristic
182C2×C53#185611yescharacteristic
183C2×C53#185811yescharacteristic
184D53—82no—
185C2×C4×D4—531no—
186#436—1061no—
187#436—1061no—
188#436—1061no—
189#436—1061no—
190#431—531no—
191#431C2×D5311yescharacteristic
192C2×C4⋊C8C2×D5311yescharacteristic
193C2×#100—531no—
194C2×#100—531no—
195#407—1062no—
196C2×C22⋊C8—1062no—
197C4×M16—531no—
198C4×M16C2×D5311yescharacteristic
199C2×C4×C8—531no—
200#343—531no—
201#343—531no—
202#335—21no—
203#335—1061no—
204#334—21no—
205#334—1061no—
206C53#883711yescharacteristic, complemented, Sylow
207C22×D4—531no—
208C23×C4—532no—
209C2×M16—531no—
210C2×M16—531no—
211C2×M16—1062no—
212C2×M16—1062no—
213C2×M16C4×D5311yescharacteristic
214C2×M16C4×D5311yescharacteristic
215C22×C8—21no—
216C22×C8—1061no—
217C4×D4—1061no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4—1061no—
221C2×C4⋊C4—531no—
222C2×C22⋊C4—532no—
223C2×C42C22×D5311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228C4⋊C8C22⋊D5311yescharacteristic
229C4⋊C8C22⋊D5311yescharacteristic
230#100—1062no—
231#100—1062no—
232C22⋊C8—21no—
233C22⋊C8—1062no—
234C22⋊C8—1062no—
235C22⋊C8—1061no—
236C22⋊C8—2124no—
237#97—21no—
238#97—1061no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—1061no—
242C4×C8—1061no—
243C4×C8—1061no—
244C4×C8—1061no—
245C24—532no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C22×C4—532no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4C2×C4×D5311yescharacteristic
258C22×C4D4×D5311yescharacteristic
259C22×C4Q8×D5311yescharacteristic
260M16—21no—
261M16—21no—
262M16—21no—
263M16—532no—
264M16—1061no—
265M16—1061no—
266M16—1061no—
267M16—4242no—
268M16—4242no—
269M16#1171991212yescomplemented
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—1061no—
277C2×C8—1062no—
278C2×C8—1061no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—2121no—
282C2×C8—2121no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C22⋊C4—1062no—
286C22⋊C4—1062no—
287C42#1171994511yescharacteristic
288C42D4×D5311yescharacteristic
289C42#1171994911yescharacteristic
290C42C2×C22⋊D5311yescharacteristic
291C23—1062no—
292C23—1062no—
293C23—1062no—
294C23—2122no—
295C23—2122no—
296C23C4⋊C4×D5311yescharacteristic
297D4—2121no—
298D4—2121no—
299D4—2121no—
300D4—2121no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—1062no—
305C2×C4—4242no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4C4×C22⋊D5311yescharacteristic
310C2×C4#1313317911yescharacteristic
311C2×C4#1313318511yescharacteristic
312C2×C4#1313319311yescharacteristic
313C2×C4#1313319511yescharacteristic
314C2×C4#1313320211yescharacteristic
315C2×C4M16×D5311yescharacteristic
316C2×C4C2×#1171991211yescharacteristic
317C2×C4#1313322311yescharacteristic
318C2×C4#1313322511yescharacteristic
319C8—22no—
320C8—41no—
321C8—41no—
322C8—41no—
323C8—81no—
324C8—1062no—
325C8—2122no—
326C8—2121no—
327C8—2121no—
328C8—2121no—
329C22—21no—
330C22—21no—
331C22—4242no—
332C22—2124no—
333C22—2122no—
334C22—2122no—
335C22—2122no—
336C22#1681049411yescharacteristic
337C22#1681064511yescharacteristic
338C22#1681064811yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—2122no—
343C4#1681069311yescharacteristic
344C4#1681069411yescharacteristic
345C4#1681069911yescharacteristic
346C4#1681070011yescharacteristic
347C4#1681078911yescharacteristic
348C4#1681079911yescharacteristic
349C2—21no—
350C2—21no—
351C2—4242no—
352C2#2027489611yescharacteristic
353C2#2027492311yescharacteristic
354C2#2027493311yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1565C2
socleC22×C53#385
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1565D53
hypocenterC53#8837
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
4. #20273465C2semidirect
5. #20272553C2semidirect
6. #20272551C2semidirect
7. C53×#1565C2semidirect
85. M16×C53M16semidirect
103. #11719917M16semidirect
176. #1565D53semidirect
206. C53#8837semidirect
269. M16#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837552
#202755082
C4×#131331332
#202734652
#202725532
#202725512
C53×#15652
#883753