Finite Group Databasebeta

← #27277955

Subgroups — #27277955

Order 13568.

482 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#20283738C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic
3C4×#13133133C211yesmaximal, characteristic
4#20273468C211yesmaximal, characteristic, complemented
5#20273467C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#1278C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C8×D212—21no—
10C8×D212—21no—
11#16810688—21no—
12#16810688—21no—
13#16810676C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133133C411yescharacteristic
16C2×#13133133C411yescharacteristic
17C2×#13133129—22no—
18C22⋊C8×D53—22no—
19C4×#11719917C2211yescharacteristic
20#16810391C411yescharacteristic
21#16810391C411yescharacteristic
22#16810376—21no—
23#16810375—21no—
24#16810374—21no—
25#16810373—21no—
26C53×#431C2211yescharacteristic
27C4×M16×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53D412yes—
31C2×#11719917C2×C411yescharacteristic
32C2×#11719917C2×C411yescharacteristic
33C22×#88322—21no—
34C2×C8×D53—22no—
35C2×C8×D53—22no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719921Q812yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322D411yescharacteristic
45C4⋊#88322D411yescharacteristic
46#13133133—22no—
47#13133133C814yescomplemented
48C22⋊#88322—21no—
49#13133129—22no—
50#13133129—22no—
51#13133129—44no—
52#13133128—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55C2×M16×C53C2×C411yescharacteristic
56C2×M16×C53C2×C411yescharacteristic
57C22×C8×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53D411yescharacteristic
60C4⋊C8×C53D411yescharacteristic
61C22⋊C8×C53—21no—
62#97×C53—21no—
63C4×C8×C53—21no—
64C4×C8×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×D212C2×C812yes—
77C2×D212M1612yes—
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×C53C2×C812yescomplemented
86C2×C8×C53—21no—
87C2×C8×C53—22no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53#4111yescharacteristic
96C42×C53#4111yescharacteristic
97#11719921—22no—
98#11719921—22no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101#11719917—21no—
102#11719917—21no—
103#11719917—21no—
104#11719917C2×C812yescomplemented
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—42no—
110C2×#88322—41no—
111C8×D53—82no—
112C8×D53—82no—
113C23×C53C2×C4⋊C411yescharacteristic
114C22×D53—22no—
115C22×D53—22no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53C4⋊C814yes—
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×C53C22×C811yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×C4×C53#13111yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C2×Dic53—22no—
133C2×Dic53—42no—
134D212—22no—
135D212—22no—
136D212—22no—
137D212—22no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—82no—
141C8×C53—22no—
142C8×C53—42no—
143C8×C53—41no—
144C8×C53—41no—
145C8×C53—41no—
146#88322—22no—
147#88322—41no—
148#88322—41no—
149#88322—41no—
150#88322—81no—
151#8835—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53#38511yescharacteristic
155C22×C53C2×C4⋊C811yescharacteristic
156C22×C53#42411yescharacteristic
157C2×D53—24no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—44no—
161C2×D53—82no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53C8×D411yescharacteristic
166C4×C53C8×D411yescharacteristic
167C4×C53#43511yescharacteristic
168C4×C53#43511yescharacteristic
169C4×C53#45311yescharacteristic
170C4×C53#45511yescharacteristic
171Dic53—42no—
172#2972—531no—
173#1954—531no—
174C4×#100—531no—
175#1565—531no—
176#1565—531no—
177#1278—531no—
178#1278D5311yescharacteristic, complemented
179C2×C53—21no—
180C2×C53—21no—
181C2×C53#185011yescharacteristic
182C2×C53#185611yescharacteristic
183C2×C53#185811yescharacteristic
184D53—44no—
185C2×C4×D4—531no—
186C8×D4—1061no—
187C8×D4—1061no—
188C8×D4—1061no—
189C8×D4—1061no—
190#431—531no—
191#431C2×D5311yescharacteristic
192C2×C4⋊C8—531no—
193C2×#100—531no—
194C2×#100—531no—
195C2×C22⋊C8—1062no—
196C2×C22⋊C8—1062no—
197C4×M16—531no—
198C4×M16C2×D5311yescharacteristic
199C2×C4×C8C2×D5311yescharacteristic
200#343—531no—
201#343—531no—
202#335—1061no—
203#335—1061no—
204#334—1061no—
205#334—1061no—
206C53#883511yescharacteristic, complemented, Sylow
207C22×D4—531no—
208C23×C4—532no—
209C2×M16—531no—
210C2×M16—531no—
211C2×M16C4×D5311yescharacteristic
212C2×M16C4×D5311yescharacteristic
213C22×C8—21no—
214C22×C8—1062no—
215C22×C8—1062no—
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—531no—
225C4⋊C8—531no—
226C4⋊C8—1061no—
227C4⋊C8—1061no—
228C4⋊C8D21211yescharacteristic
229C4⋊C8D21211yescharacteristic
230#100—534no—
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—21no—
242C4×C8—21no—
243C4×C8—1061no—
244C4×C8—1061no—
245C24—532no—
246C2×D4—532no—
247C2×D4—532no—
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—
267M16C8×D5312yescomplemented
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1061no—
278C2×C8—1061no—
279C2×C8—4242no—
280C2×C8—4242no—
281C2×C8—2122no—
282C2×C8—2121no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C22⋊C4—1062no—
286C22⋊C4—1062no—
287C42C2×D21211yescharacteristic
288C42#1171994511yescharacteristic
289C42D4×D5311yescharacteristic
290C42#1171994711yescharacteristic
291C23—534no—
292C23—1062no—
293C23—1062no—
294C23—2122no—
295C23—2122no—
296C23C4⋊C4×D5311yescharacteristic
297D4—1062no—
298D4—1062no—
299D4—1062no—
300D4—1062no—
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×C4#1313317711yescharacteristic
310C2×C4C4×D21211yescharacteristic
311C2×C4#1313318711yescharacteristic
312C2×C4#1313319511yescharacteristic
313C2×C4#1313319711yescharacteristic
314C2×C4#1313319911yescharacteristic
315C2×C4C2×C8×D5311yescharacteristic
316C2×C4M16×D5311yescharacteristic
317C2×C4#1313322311yescharacteristic
318C2×C4#1313322511yescharacteristic
319C8—22no—
320C8—42no—
321C8—41no—
322C8—41no—
323C8—41no—
324C8—1062no—
325C8—4241no—
326C8—2121no—
327C8—2121no—
328C8—2121no—
329C22—21no—
330C22—21no—
331C22—1064no—
332C22—4242no—
333C22—2124no—
334C22—2122no—
335C22—2122no—
336C22#1681049711yescharacteristic
337C22C4⋊C8×D5311yescharacteristic
338C22#1681064811yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—2122no—
343C4#1681068611yescharacteristic
344C4C8×D21211yescharacteristic
345C4#1681069611yescharacteristic
346C4#1681069711yescharacteristic
347C4#1681079211yescharacteristic
348C4#1681080211yescharacteristic
349C2—21no—
350C2—21no—
351C2—2124no—
352C2#2027489411yescharacteristic
353C2#2027492611yescharacteristic
354C2#2027493611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1278C2
socleC22×C53#385
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1278D53
hypocenterC53#8835
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
4. #20273468C2semidirect
5. #20273467C2semidirect
6. #20272552C2semidirect
7. C53×#1278C2semidirect
47. #13133133C8semidirect
85. M16×C53C2×C8semidirect
104. #11719917C2×C8semidirect
178. #1278D53semidirect
206. C53#8835semidirect
267. M16C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202837382
#202755092
C4×#131331332
#202734682
#202734672
#202725522
C53×#12782
#883553