Finite Group Databasebeta

← #20286775

Subgroups — #20286775

Order 6784.

380 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#16811282C211yesmaximal, characteristic
2#16810985C211yesmaximal, characteristic, complemented
3#16810964C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810915C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810778C211yesmaximal, characteristic, complemented
11#16810728C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810631C211yesmaximal, characteristic, complemented
14#16810617C211yesmaximal, characteristic, complemented
15C53×#556C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133203—21no—
25#13133196C2211yescharacteristic
26#13133195—21no—
27#13133191C2211yescharacteristic
28#13133189—21no—
29C4×Dic106C2211yescharacteristic
30#13133177—21no—
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133140C2211yescharacteristic
37#13133140C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C4⋊Q8×C53C2211yescharacteristic
49#126×C53—21no—
50#126×C53—21no—
51#124×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic
53C22⋊Q8×C53C2211yescharacteristic
54C4×Q8×C53—21no—
55C4×Q8×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic
57#117×C53C2211yescharacteristic
58#117×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53D411yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67#11719945—22no—
68#11719945—41no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×Dic106—22no—
72C2×Dic106—21no—
73C2×Dic106—41no—
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53D411yescharacteristic
89C22⋊C4×C53D411yescharacteristic
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95#11719925—22no—
96#11719925—41no—
97#11719924—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—42no—
117D4×C53—41no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—42no—
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137D212—21no—
138D212—21no—
139C4×D53—42no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—42no—
144Dic106#4112yes—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53C22×D411yescharacteristic
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53H(5,𝔽2)11yescharacteristic
169Dic53—22no—
170Dic53—41no—
171Dic53—41no—
172#3313—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53—41no—
175C2×C53#54611yescharacteristic
176C2×C53#57511yescharacteristic
177C2×C53#57811yescharacteristic
178D53—81no—
179#556D5311yescharacteristic, complemented
180#548—531no—
181#489—531no—
182#485—531no—
183#482—531no—
184#477—531no—
185#468—531no—
186#467—531no—
187#464—531no—
188#452—531no—
189#451—531no—
190#441—531no—
191#435—531no—
192#419—531no—
193#416—531no—
194C53#331311yescharacteristic, complemented, Sylow
195C2×#41—531no—
196C22×Q8—531no—
197C2×Q16—531no—
198C2×Q16—531no—
199C2×SD16—531no—
200C2×M16—531no—
201C22×C8—531no—
202C4⋊Q8C2×D5311yescharacteristic
203#126—21no—
204#126—21no—
205#124—21no—
206#124—531no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—1061no—
210C22⋊Q8C2×D5311yescharacteristic
211C22⋊Q8C2×D5311yescharacteristic
212#121—531no—
213#121—1061no—
214C4×Q8—21no—
215C4×Q8—531no—
216C4×Q8C2×D5311yescharacteristic
217C4×D4—1061no—
218C4×D4C2×D5311yescharacteristic
219#117C2×D5311yescharacteristic
220#117C2×D5311yescharacteristic
221#107—531no—
222#107—531no—
223#106—531no—
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235C22⋊C8—531no—
236C22⋊C8—531no—
237#97—531no—
238#41—1062no—
239#41—2121no—
240C2×Q8—21no—
241C2×Q8—531no—
242C2×Q8—531no—
243C2×Q8—1062no—
244C2×Q8—1061no—
245C2×Q8—2121no—
246C2×Q8C22×D5311yescharacteristic
247C2×D4—531no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4C22⋊D5311yescharacteristic
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255Q16—1062no—
256Q16—2121no—
257SD16—2121no—
258M16—2121no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8—2121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—531no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4C22⋊D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4C22⋊D5311yescharacteristic
284C22⋊C4C22⋊D5311yescharacteristic
285C42—21no—
286C42—21no—
287C42—21no—
288C42—1061no—
289C42C22×D5311yescharacteristic
290C23—1061no—
291C23C2×C22⋊D5311yescharacteristic
292C23C2×C22⋊D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—532no—
297Q8—1061no—
298Q8—1061no—
299Q8—1061no—
300Q8—2122no—
301D4—41no—
302D4—1061no—
303D4—1061no—
304D4—2122no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4C2×C22⋊D5311yescharacteristic
320C2×C4C2×C22⋊D5311yescharacteristic
321C2×C4C2×C22⋊D5311yescharacteristic
322C2×C4C2×C22⋊D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—1061no—
325C8—1061no—
326C8—1061no—
327C8—1061no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C22×C22⋊D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—1062no—
347C4—2121no—
348C4—2121no—
349C4#41×D5311yescharacteristic
350C4#1313326311yescharacteristic
351C2—41no—
352C2—41no—
353C2—4241no—
354C2#1681126111yescharacteristic
355C2#1681142111yescharacteristic
356C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#556C2
socleC22×C53C22×D4
hypercenter#556D53
hypocenterC53#3313
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #16810985C2semidirect
3. #16810964C2semidirect
4. #16810947C2semidirect
5. #16810915C2semidirect
6. #16810871C2semidirect
7. #16810862C2semidirect
8. #16810845C2semidirect
9. #16810783C2semidirect
10. #16810778C2semidirect
11. #16810728C2semidirect
12. #16810695C2semidirect
13. #16810631C2semidirect
14. #16810617C2semidirect
15. C53×#556C2semidirect
179. #556D53semidirect
194. C53#3313semidirect