Finite Group Databasebeta

← #20287775

Subgroups — #20287775

Order 6784.

368 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#16811346C211yesmaximal, characteristic, complemented
2#16811018C211yesmaximal, characteristic, complemented
3#16811010C211yesmaximal, characteristic
4#16811005C211yesmaximal, characteristic, complemented
5#16810845C211yesmaximal, characteristic, complemented
6#16810842C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810820C211yesmaximal, characteristic, complemented
11#16810754C211yesmaximal, characteristic, complemented
12#16810733C211yesmaximal, characteristic, complemented
13#16810728C211yesmaximal, characteristic, complemented
14C4×#11719923C211yesmaximal, characteristic, complemented
15C53×#559C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25#13133217C2211yescharacteristic
26#13133212—21no—
27#13133212—21no—
28#13133205C2211yescharacteristic
29#13133205C2211yescharacteristic
30#13133193—21no—
31C4×Dic106C2211yescharacteristic
32C4×Dic106C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133175—21no—
35#13133148C2211yescharacteristic
36C4⋊#88322C2211yescharacteristic
37C4⋊#88322C2211yescharacteristic
38C4⋊#88322C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133128C2211yescharacteristic
46#13133128C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48#127×C53C2211yescharacteristic
49#124×C53—21no—
50#124×C53C2211yescharacteristic
51#124×C53C2211yescharacteristic
52#123×C53—21no—
53#123×C53—21no—
54C4×Q8×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4×D4×C53—21no—
57C4×D4×C53C2211yescharacteristic
58C4×D4×C53C2211yescharacteristic
59C2×Q8×C53D411yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×Dic106C2311yescharacteristic
70C2×Dic106C2311yescharacteristic
71C4⋊C4×C53—21no—
72C4⋊C4×C53D411yescharacteristic
73C4⋊C4×C53D411yescharacteristic
74C4⋊C4×C53D411yescharacteristic
75C4⋊C4×C53C2311yescharacteristic
76C22⋊C4×C53—21no—
77C22⋊C4×C53—21no—
78C22⋊C4×C53—21no—
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C42×C53—21no—
83C42×C53C2311yescharacteristic
84C42×C53C2311yescharacteristic
85C42×C53C2311yescharacteristic
86#11719925—22no—
87#11719925—41no—
88#11719924—41no—
89#11719923—22no—
90#11719923—41no—
91#11719923—41no—
92#11719923—41no—
93#11719922—22no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C22×D53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—41no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—42no—
122D4×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137D212—21no—
138D212—21no—
139C4×D53—42no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144#88322—21no—
145#88322—21no—
146#88322—21no—
147#88322#4112yes—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53H(5,𝔽2)11yescharacteristic
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#3401—531nomaximal, Sylow
172C2×C53—81no—
173C2×C53—81no—
174C2×C53#55011yescharacteristic
175C2×C53#57511yescharacteristic
176C2×C53#57711yescharacteristic
177D53—81no—
178#559D5311yescharacteristic, complemented
179#556—531no—
180#497—531no—
181#496—531no—
182#495—531no—
183#464—531no—
184#464—531no—
185#463—531no—
186#463—531no—
187#461—531no—
188#460—531no—
189#446—531no—
190#442—531no—
191#441—531no—
192C4×SD16—531no—
193C53#340111yescharacteristic, complemented, Sylow
194C2×Q16—531no—
195C2×Q16—531no—
196C2×SD16—531no—
197C2×SD16—531no—
198C2×SD16—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×D8—531no—
202C2×D8—531no—
203C4⋊Q8—531no—
204#127C2×D5311yescharacteristic
205#126—1061no—
206#126—1061no—
207#124—21no—
208#124—531no—
209#124—531no—
210#124C2×D5311yescharacteristic
211#124C2×D5311yescharacteristic
212#123—21no—
213#123—21no—
214C22⋊Q8—1061no—
215C4×Q8—531no—
216C4×Q8—531no—
217C4×Q8C2×D5311yescharacteristic
218C4×Q8C2×D5311yescharacteristic
219C4×D4—21no—
220C4×D4—531no—
221C4×D4C2×D5311yescharacteristic
222C4×D4C2×D5311yescharacteristic
223#117—1061no—
224#106—531no—
225C4⋊C8—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#97—531no—
235#97—531no—
236C4×C8—531no—
237C2×Q8—531no—
238C2×Q8—531no—
239C2×Q8C22⋊D5311yescharacteristic
240C2×Q8C22×D5311yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—531no—
244C2×D4C22×D5311yescharacteristic
245C2×D4C22×D5311yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—1061no—
249Q16—1062no—
250Q16—2121no—
251SD16—1062no—
252SD16—2121no—
253SD16—2121no—
254SD16—2121no—
255SD16—2121no—
256D8—1062no—
257D8—2121no—
258C2×C8—531no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—531no—
262C4⋊C4—21no—
263C4⋊C4—531no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4C22⋊D5311yescharacteristic
269C4⋊C4C22⋊D5311yescharacteristic
270C4⋊C4C22⋊D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—21no—
284C42—1061no—
285C42—1061no—
286C42C22×D5311yescharacteristic
287C42C22×D5311yescharacteristic
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—21no—
291C23—1061no—
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—1061no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—81no—
305D4—1061no—
306D4—1061no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2122no—
315C2×C4C2×C22⋊D5311yescharacteristic
316C2×C4C2×C22⋊D5311yescharacteristic
317C2×C4C2×C22⋊D5311yescharacteristic
318C2×C4C2×C22⋊D5311yescharacteristic
319C2×C4C2×C22⋊D5311yescharacteristic
320C2×C4C2×C22⋊D5311yescharacteristic
321C2×C4C23×D5311yescharacteristic
322C8—532no—
323C8—1061no—
324C8—1061no—
325C8—1061no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22C22×C22⋊D5311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4#41×D5311yescharacteristic
348C4#1313326311yescharacteristic
349C2—81no—
350C2—81no—
351C2—4241no—
352C2#1681129211yescharacteristic
353C2#1681142111yescharacteristic
354C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#559C2
socleC22×C53C22×D4
hypercenter#559D53
hypocenterC53#3401
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #16811346C2semidirect
2. #16811018C2semidirect
4. #16811005C2semidirect
5. #16810845C2semidirect
6. #16810842C2semidirect
7. #16810838C2semidirect
8. #16810835C2semidirect
9. #16810828C2semidirect
10. #16810820C2semidirect
11. #16810754C2semidirect
12. #16810733C2semidirect
13. #16810728C2semidirect
14. C4×#11719923C2semidirect
15. C53×#559C2semidirect
178. #559D53semidirect
193. C53#3401semidirect