Finite Group Databasebeta

← #20284914

Subgroups — #20284914

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811228C211yesmaximal, characteristic, complemented
2#16810941C211yesmaximal, characteristic, complemented
3#16810936C211yesmaximal, characteristic, complemented
4#16810930C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810823C211yesmaximal, characteristic, complemented
7#16810818C211yesmaximal, characteristic, complemented
8#16810768C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810733C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12C4×#11719924C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#520C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133214C2211yescharacteristic
21#13133209C2211yescharacteristic
22#13133202—21no—
23#13133202—21no—
24#13133191C2211yescharacteristic
25#13133191C2211yescharacteristic
26#13133190—21no—
27#13133190—21no—
28C4×D212C2211yescharacteristic
29C4×D212C2211yescharacteristic
30#13133150C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133148C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35C4⋊#88322C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C4×Q8×C53C2211yescharacteristic
50C4×Q8×C53C2211yescharacteristic
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#117×C53—21no—
56#117×C53C2211yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—22no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C42×C53—21no—
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91C42×C53C2311yescharacteristic
92#11719924—22no—
93#11719924—22no—
94#11719922—22no—
95#11719922—22no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111C22×D53—21no—
112Q8×C53—21no—
113Q8×C53#4112yes—
114D4×C53—24no—
115D4×C53—21no—
116D4×C53#4112yes—
117C2×C4×C53—24no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—81no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—81no—
141D212—81no—
142C4×D53—81no—
143C4×D53—81no—
144#88322—21no—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—22no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53C2×#4111yescharacteristic
171Dic53—41no—
172Dic53—41no—
173#3119—531nomaximal, Sylow
174C2×C53—41no—
175C2×C53—42no—
176C2×C53#52511yescharacteristic
177C2×C53#57611yescharacteristic
178C2×C53#57711yescharacteristic
179D53—81no—
180D53—81no—
181#540—531no—
182#520D5311yescharacteristic, complemented
183#481—531no—
184#480—531no—
185#479—531no—
186#478—531no—
187#460—531no—
188#459—531no—
189#449—531no—
190#449—531no—
191#442—531no—
192#440—531no—
193C4×SD16—531no—
194C4×D8—531no—
195#433—531no—
196C53#311911yescharacteristic, complemented, Sylow
197C2×#41C2×D5311yescharacteristic
198C2×SD16—531no—
199C2×SD16—531no—
200C2×D8—531no—
201C2×D8—531no—
202#127—531no—
203#125—531no—
204#123—1061no—
205#123—1061no—
206#121—531no—
207#121—531no—
208#121—1061no—
209#121—1061no—
210C4×Q8C2×D5311yescharacteristic
211C4×Q8C2×D5311yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4C2×D5311yescharacteristic
217C4×D4C2×D5311yescharacteristic
218#117—21no—
219#117C2×D5311yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D5311yescharacteristic
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236C22⋊C8—531no—
237C22⋊C8—531no—
238#97—531no—
239C4×C8—531no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D5311yescharacteristic
243C2×D4—21no—
244C2×D4—531no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4C22×D5311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258SD16—1062no—
259SD16—1062no—
260D8—1062no—
261D8—1062no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—531no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4C22⋊D5311yescharacteristic
274C4⋊C4C22⋊D5311yescharacteristic
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
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—
287C42C22×D5311yescharacteristic
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—1061no—
291C23—1061no—
292C23C2×C22⋊D5311yescharacteristic
293Q8—21no—
294Q8#1171994512yes—
295D4—24no—
296D4—21no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—1061no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305D4#1171994512yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4C2×C22⋊D5311yescharacteristic
320C2×C4C2×C22⋊D5311yescharacteristic
321C2×C4C2×C22⋊D5311yescharacteristic
322C2×C4C2×C22⋊D5311yescharacteristic
323C2×C4C2×C22⋊D5311yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—1061no—
326C8—1061no—
327C8—1061no—
328C8—1061no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—2121no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22C22×C22⋊D5311yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—2121no—
351C4—2121no—
352C4C2×#1171994511yescharacteristic
353C4C2×#1171994511yescharacteristic
354C2—41no—
355C2—42no—
356C2—4241no—
357C2—4241no—
358C2#1681113411yescharacteristic
359C2#1681142611yescharacteristic
360C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#520C2
socleC22×C53C22×D4
hypercenter#520D53
hypocenterC53#3119
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16811228C2semidirect
2. #16810941C2semidirect
3. #16810936C2semidirect
4. #16810930C2semidirect
5. #16810923C2semidirect
6. #16810823C2semidirect
7. #16810818C2semidirect
8. #16810768C2semidirect
9. #16810768C2semidirect
10. #16810733C2semidirect
11. #16810721C2semidirect
12. C4×#11719924C2semidirect
13. C4×#11719922C2semidirect
14. #16810684C2semidirect
15. C53×#520C2semidirect
182. #520D53semidirect
196. C53#3119semidirect