Finite Group Databasebeta

← #20286132

Subgroups — #20286132

Order 6784.

358 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#16811245C211yesmaximal, characteristic, complemented
2#16811229C211yesmaximal, characteristic, complemented
3#16810920C211yesmaximal, characteristic, complemented
4#16810912C211yesmaximal, characteristic, complemented
5#16810902C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810872C211yesmaximal, characteristic, complemented
8#16810860C211yesmaximal, characteristic, complemented
9#16810855C211yesmaximal, characteristic, complemented
10#16810841C211yesmaximal, characteristic, complemented
11#16810841C211yesmaximal, characteristic, complemented
12#16810823C211yesmaximal, characteristic, complemented
13#16810819C211yesmaximal, characteristic, complemented
14#16810648C211yesmaximal, characteristic, complemented
15C53×#479C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133214C2211yescharacteristic
24#13133211—21no—
25#13133206C2211yescharacteristic
26#13133205—21no—
27#13133202—21no—
28#13133202—21no—
29#13133202—21no—
30#13133195—21no—
31#13133193C2211yescharacteristic
32#13133192—21no—
33#13133192—21no—
34#13133192C2211yescharacteristic
35#13133192C2211yescharacteristic
36#13133192C2211yescharacteristic
37C4×D212C2211yescharacteristic
38#13133179C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133174C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133148C2211yescharacteristic
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133137C2211yescharacteristic
51#13133137C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D212—21no—
61C2×D212—21no—
62C2×D212—21no—
63C2×D212—21no—
64C2×D212—21no—
65C2×D212C2311yescharacteristic
66C2×D212C2311yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53C2311yescharacteristic
72C2×Dic106—21no—
73C2×C8×C53C2311yescharacteristic
74C2×C8×C53C2311yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C42×C53—21no—
82C42×C53C2311yescharacteristic
83#11719924—41no—
84#11719924—41no—
85#11719921—21no—
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⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919D411yescharacteristic
100#11719919D411yescharacteristic
101C4×Dic53C2311yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104D424—41no—
105D424—41no—
106#11719912—41no—
107C8×D53—41no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112Q8×C53—21no—
113Q8×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2411yescharacteristic
121C2×Dic53—21no—
122C2×Dic53C2×D411yescharacteristic
123C2×Dic53C2×D411yescharacteristic
124C2×Dic53C2×D411yescharacteristic
125D212—21no—
126D212—21no—
127D212—21no—
128D212—21no—
129D212—21no—
130D212—21no—
131D212—81no—
132D212—81no—
133D212—81no—
134D212—81no—
135D212—81no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—81no—
139C4×D53—81no—
140C4×D53—81no—
141Dic106—81no—
142C8×C53—21no—
143C8×C53—21no—
144#88322—21no—
145#88322—21no—
146C22×C53C22×D411yescharacteristic
147C2×D53—21no—
148C2×D53—21no—
149C2×D53—21no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53H(5,𝔽2)11yescharacteristic
166C4×C532−1+411yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171#3257—531nomaximal, Sylow
172C2×C53#53711yescharacteristic
173C2×C53#57611yescharacteristic
174C2×C53#57711yescharacteristic
175D53—41no—
176D53—81no—
177D53—81no—
178D53—81no—
179#542—531no—
180#540—531no—
181#479D5311yescharacteristic, complemented
182#478—531no—
183#476—531no—
184#474—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#463—531no—
190#463—531no—
191#460—531no—
192#459—531no—
193#424—531no—
194C53#325711yescharacteristic, complemented, Sylow
195C2×SD16—531no—
196C2×SD16—531no—
197C2×D8—531no—
198C2×D8—531no—
199C2×M16—531no—
200C22×C8—531no—
201C4⋊Q8—531no—
202#127—531no—
203#126—1061no—
204#125C2×D5311yescharacteristic
205#124—531no—
206#124—1061no—
207#123—1061no—
208#123—1061no—
209#123—1061no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—531no—
214#121—531no—
215#121—1061no—
216#121—1061no—
217C4×Q8C2×D5311yescharacteristic
218C4×D4—531no—
219C4×D4—531no—
220C4×D4—531no—
221#117—531no—
222#107—531no—
223#107C2×D5311yescharacteristic
224#106—531no—
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229C4⋊C8C2×D5311yescharacteristic
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238C2×Q8—1061no—
239C2×Q8C22×D5311yescharacteristic
240C2×D4—531no—
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252SD16—2121no—
253SD16—2121no—
254D8—2121no—
255D8—2121no—
256M16—2121no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—2121no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4C22×D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C22⋊C4—531no—
274C22⋊C4—531no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C42—21no—
285C42—531no—
286C42C22×D5311yescharacteristic
287C23—531no—
288C23—1061no—
289C23—1061no—
290C23—1061no—
291Q8—21no—
292Q8—21no—
293Q8—4241no—
294D4—1061no—
295D4—1061no—
296D4—1061no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4D4×D5311yescharacteristic
319C2×C4D4×D5311yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—21no—
322C8—21no—
323C8—1061no—
324C8—1061no—
325C22—1061no—
326C22—1061no—
327C22—1061no—
328C22—2121no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1061no—
345C4—1061no—
346C4—1061no—
347C4—2121no—
348C4#1313326311yescharacteristic
349C4#1313326511yescharacteristic
350C2—4241no—
351C2—4241no—
352C2—4241no—
353C2—2121no—
354C2#1681120311yescharacteristic
355C2#1681142511yescharacteristic
356C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#479C2
socleC22×C53C22×D4
hypercenter#479D53
hypocenterC53#3257
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #16811245C2semidirect
2. #16811229C2semidirect
3. #16810920C2semidirect
4. #16810912C2semidirect
5. #16810902C2semidirect
6. #16810877C2semidirect
7. #16810872C2semidirect
8. #16810860C2semidirect
9. #16810855C2semidirect
10. #16810841C2semidirect
11. #16810841C2semidirect
12. #16810823C2semidirect
13. #16810819C2semidirect
14. #16810648C2semidirect
15. C53×#479C2semidirect
181. #479D53semidirect
194. C53#3257semidirect