Finite Group Databasebeta

← #20286778

Subgroups — #20286778

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#16811347C211yesmaximal, characteristic, complemented
2#16811285C211yesmaximal, characteristic
3#16810985C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810950C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810865C211yesmaximal, characteristic, complemented
8#16810846C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810780C211yesmaximal, characteristic, complemented
11#16810728C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#477C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133212—21no—
24#13133212—21no—
25#13133206—21no—
26#13133205—21no—
27#13133205C2211yescharacteristic
28#13133197—21no—
29#13133197C2211yescharacteristic
30#13133197C2211yescharacteristic
31#13133197C2211yescharacteristic
32#13133192—21no—
33#13133192C2211yescharacteristic
34Q8×Dic53—21no—
35C4×Dic106C2211yescharacteristic
36C4×D212—21no—
37C4×D212C2211yescharacteristic
38#13133175C2211yescharacteristic
39#13133175C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C4⋊Q8×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61#11719949—22no—
62#11719949—41no—
63Q8×D53—22no—
64Q8×D53—41no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212D411yescharacteristic
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53C2311yescharacteristic
72C2×C4×D53C2311yescharacteristic
73C2×Dic106—21no—
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C42×C53—21no—
84C42×C53C2311yescharacteristic
85#11719925—22no—
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53D411yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107Dic212—41no—
108#11719913—41no—
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—42no—
116Q8×C53#4112yes—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53C2×D411yescharacteristic
128C2×Dic53C2×D411yescharacteristic
129D212—21no—
130D212—21no—
131D212—41no—
132D212—42no—
133D212—42no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—42no—
139C4×D53—42no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—42no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53C22×D411yescharacteristic
150C2×D53—21no—
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3313—531nomaximal, Sylow
173C2×C53#54611yescharacteristic
174C2×C53#57511yescharacteristic
175C2×C53#57811yescharacteristic
176D53—41no—
177D53—41no—
178D53—81no—
179#556—531no—
180#548—531no—
181#489—531no—
182#485—531no—
183#482—531no—
184#477D5311yescharacteristic, complemented
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—1061no—
204#126—1061no—
205#124—531no—
206#124—1061no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—1061no—
214C4×Q8—531no—
215C4×Q8—1061no—
216C4×Q8C2×D5311yescharacteristic
217C4×D4—531no—
218C4×D4—1061no—
219#117—531no—
220#117—531no—
221#107C2×D5311yescharacteristic
222#107C2×D5311yescharacteristic
223#106—531no—
224C4⋊C8C2×D5311yescharacteristic
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
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—1061no—
244C2×Q8—1062no—
245C2×Q8—2121no—
246C2×Q8C22×D5311yescharacteristic
247C2×D4—531no—
248C2×D4—531no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—1061no—
254C22×C4—1061no—
255Q16—1062no—
256Q16—2121no—
257SD16—2121no—
258M16—2121no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—531no—
276C22⋊C4—531no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—21no—
286C42—1061no—
287C42—1061no—
288C42—1061no—
289C42C22×D5311yescharacteristic
290C23—531no—
291C23—531no—
292C23—1061no—
293Q8—21no—
294Q8—42no—
295Q8—1061no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299Q8—2122no—
300Q8#1171994912yes—
301D4—1061no—
302D4—1061no—
303D4—2121no—
304D4—2122no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4D4×D5311yescharacteristic
322C2×C4D4×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—21no—
325C8—21no—
326C8—1061no—
327C8—1061no—
328C22—1061no—
329C22—1061no—
330C22—1061no—
331C22—1061no—
332C22—1061no—
333C22—1061no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1061no—
345C4—1061no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4C2×#1171994911yescharacteristic
350C4#1313326311yescharacteristic
351C2—4241no—
352C2—2121no—
353C2—2121no—
354C2#1681126411yescharacteristic
355C2#1681141911yescharacteristic
356C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#477C2
socleC22×C53C22×D4
hypercenter#477D53
hypocenterC53#3313
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #16811347C2semidirect
3. #16810985C2semidirect
4. #16810966C2semidirect
5. #16810950C2semidirect
6. #16810870C2semidirect
7. #16810865C2semidirect
8. #16810846C2semidirect
9. #16810788C2semidirect
10. #16810780C2semidirect
11. #16810728C2semidirect
12. #16810693C2semidirect
13. #16810630C2semidirect
14. #16810618C2semidirect
15. C53×#477C2semidirect
184. #477D53semidirect
194. C53#3313semidirect