Finite Group Databasebeta

← #20286690

Subgroups — #20286690

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811326C211yesmaximal, characteristic, complemented
2#16811276C211yesmaximal, characteristic, complemented
3#16810962C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810872C211yesmaximal, characteristic, complemented
7#16810855C211yesmaximal, characteristic, complemented
8#16810825C211yesmaximal, characteristic, complemented
9#16810771C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810736C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#491C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133212—21no—
24#13133202—21no—
25#13133202—21no—
26#13133202—21no—
27#13133193C2211yescharacteristic
28#13133193C2211yescharacteristic
29#13133192—21no—
30#13133192—21no—
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33C4×D212C2211yescharacteristic
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic
36#13133179—21no—
37#13133175C2211yescharacteristic
38C4⋊C4×D53—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43C4⋊#88322C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C4⋊Q8×C53C2211yescharacteristic
53#125×C53C2211yescharacteristic
54#103×C53C2211yescharacteristic
55#103×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61#11719949—22no—
62#11719949—22no—
63#11719949—41no—
64#11719949—41no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212D411yescharacteristic
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C42×C53C2311yescharacteristic
88#11719924—41no—
89#11719924—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921D411yescharacteristic
98#11719921D411yescharacteristic
99C4⋊Dic53D411yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—22no—
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130D212—21no—
131D212—21no—
132D212—21no—
133D212—41no—
134D212—42no—
135D212—42no—
136D212—42no—
137D212—81no—
138D212#4112yes—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—81no—
147C4×D53—81no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—22no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177#3307—531nomaximal, Sylow
178C2×C53#54611yescharacteristic
179C2×C53#57511yescharacteristic
180C2×C53#57711yescharacteristic
181D53—41no—
182D53—41no—
183D53—42no—
184D53—81no—
185#553—531no—
186#547—531no—
187#491D5311yescharacteristic, complemented
188#485—531no—
189#483—531no—
190#478—531no—
191#468—531no—
192#465—531no—
193#461—531no—
194#449—531no—
195#449—531no—
196#442—531no—
197#435—531no—
198#417—531no—
199#416—531no—
200C53#330711yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C2×#41—531no—
203C2×SD16—531no—
204C2×SD16—531no—
205C2×D8—531no—
206C2×M16—531no—
207C22×C8—531no—
208C4⋊Q8C2×D5311yescharacteristic
209#126—1061no—
210#125C2×D5311yescharacteristic
211#123—1061no—
212#123—1061no—
213#123—1061no—
214C22⋊Q8—531no—
215C22⋊Q8—531no—
216#121—531no—
217#121—531no—
218#121—1061no—
219#121—1061no—
220C4×D4—531no—
221C4×D4—531no—
222C4×D4—531no—
223C4×D4—1061no—
224#117—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—1061no—
227#107—531no—
228#106—531no—
229#106—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103C2×D5311yescharacteristic
234#103C2×D5311yescharacteristic
235#103C2×D5311yescharacteristic
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243#97C2×D5311yescharacteristic
244#41—1062no—
245#41—1062no—
246#41—2121no—
247#41—2121no—
248C2×Q8C22×D5311yescharacteristic
249C2×Q8C22×D5311yescharacteristic
250C2×D4—531no—
251C2×D4—531no—
252C2×D4—531no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—2121no—
265SD16—2121no—
266D8—1062no—
267M16—2121no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—2121no—
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4C22×D5311yescharacteristic
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—531no—
286C22⋊C4—531no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—1061no—
295C42C22×D5311yescharacteristic
296C23—531no—
297C23—531no—
298C23—1061no—
299C23—1061no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—532no—
305D4—1061no—
306D4—1061no—
307D4—1061no—
308D4—4241no—
309D4—2121no—
310D4—2122no—
311D4—2122no—
312D4—2122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4D4×D5311yescharacteristic
331C2×C4D4×D5311yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—21no—
334C8—21no—
335C8—1061no—
336C8—1061no—
337C22—1061no—
338C22—1061no—
339C22—1062no—
340C22—1061no—
341C22—1061no—
342C22—1061no—
343C22—1061no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—1061no—
357C4—1061no—
358C4—2121no—
359C4—2121no—
360C4#41×D5311yescharacteristic
361C4#1313326311yescharacteristic
362C2—4241no—
363C2—2121no—
364C2—2121no—
365C2—2122no—
366C2#1681126611yescharacteristic
367C2#1681142211yescharacteristic
368C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#491C2
socleC22×C53C22×D4
hypercenter#491D53
hypocenterC53#3307
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #16811326C2semidirect
2. #16811276C2semidirect
3. #16810962C2semidirect
4. #16810952C2semidirect
5. #16810920C2semidirect
6. #16810872C2semidirect
7. #16810855C2semidirect
8. #16810825C2semidirect
9. #16810771C2semidirect
10. #16810771C2semidirect
11. #16810736C2semidirect
12. #16810696C2semidirect
13. #16810623C2semidirect
14. #16810618C2semidirect
15. C53×#491C2semidirect
187. #491D53semidirect
200. C53#3307semidirect