Finite Group Databasebeta

← #20285826

Subgroups — #20285826

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811244C211yesmaximal, characteristic, complemented
2#16811199C211yesmaximal, characteristic, complemented
3#16811017C211yesmaximal, characteristic, complemented
4#16811007C211yesmaximal, characteristic, complemented
5#16810983C211yesmaximal, characteristic, complemented
6#16810974C211yesmaximal, characteristic, complemented
7#16810846C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810834C211yesmaximal, characteristic, complemented
10#16810829C211yesmaximal, characteristic, complemented
11#16810787C211yesmaximal, characteristic, complemented
12#16810781C211yesmaximal, characteristic, complemented
13#16810770C211yesmaximal, characteristic
14#16810764C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×Dic212C2211yescharacteristic
20C2×Dic212C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719913C2211yescharacteristic
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×D424C2211yescharacteristic
26#13133217C2211yescharacteristic
27#13133217C2211yescharacteristic
28#13133217C2211yescharacteristic
29#13133212—21no—
30#13133205C2211yescharacteristic
31#13133205C2211yescharacteristic
32#13133205C2211yescharacteristic
33#13133203—21no—
34#13133201—21no—
35#13133201—21no—
36#13133199—21no—
37#13133197—21no—
38#13133197—21no—
39#13133196C2211yescharacteristic
40#13133195—21no—
41#13133191C2211yescharacteristic
42C4×Dic106C2211yescharacteristic
43C4×D212C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52#117×C53C2211yescharacteristic
53C4⋊C8×C53C2211yescharacteristic
54C4⋊C8×C53C2211yescharacteristic
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C2×C22⋊D53—21no—
61C2×C22⋊D53—21no—
62C22×Dic53—21no—
63#11719945—41no—
64#11719945—41no—
65C2×D212C2311yescharacteristic
66C2×D212C2311yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×Dic106—21no—
70C2×Dic106—21no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73C2×Dic106C2311yescharacteristic
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53D411yescharacteristic
83C22⋊C4×C53D411yescharacteristic
84C22⋊C4×C53D411yescharacteristic
85C42×C53C2311yescharacteristic
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106Dic212—41no—
107Dic212—41no—
108Dic212—41no—
109D424—41no—
110#11719913—41no—
111#11719913—41no—
112#11719913—41no—
113#11719913—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134D212—21no—
135D212—21no—
136D212—21no—
137D212—21no—
138C4×D53—42no—
139C4×D53—81no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—42no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53H(5,𝔽2)11yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3225—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53#53511yescharacteristic
180C2×C53#57711yescharacteristic
181C2×C53#57811yescharacteristic
182D53—81no—
183D53—81no—
184#542—531no—
185#536—531no—
186#497—531no—
187#495—531no—
188#489—531no—
189#487—531no—
190#464—531no—
191#463—531no—
192#462—531no—
193#461—531no—
194#452—531no—
195#451—531no—
196#449—531no—
197#448—531no—
198#433D5311yescharacteristic, complemented
199C53#322511yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C22×Q8—531no—
202C2×Q16—531no—
203C2×Q16—531no—
204C2×Q16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C4⋊Q8—531no—
211C4⋊Q8—531no—
212C4⋊Q8—531no—
213#126—1061no—
214#124—531no—
215#124—531no—
216#124—531no—
217#124—1061no—
218#123—1061no—
219#123—1061no—
220#123—1061no—
221C22⋊Q8—531no—
222C22⋊Q8—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8—1061no—
225#121—531no—
226C4×Q8—531no—
227C4×D4—531no—
228#117C2×D5311yescharacteristic
229C4⋊C8C2×D5311yescharacteristic
230C4⋊C8C2×D5311yescharacteristic
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#102—531no—
238#102—531no—
239C22⋊C8C2×D5311yescharacteristic
240C22⋊C8C2×D5311yescharacteristic
241#97C2×D5311yescharacteristic
242C4×C8C2×D5311yescharacteristic
243#41—2121no—
244#41—2121no—
245C2×Q8—531no—
246C2×Q8—531no—
247C2×Q8—531no—
248C2×Q8—531no—
249C2×Q8—1061no—
250C2×Q8—1061no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C2×D4—531no—
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4C22×D5311yescharacteristic
261Q16—2121no—
262Q16—2121no—
263Q16—2121no—
264SD16—2121no—
265SD16—2121no—
266SD16—2121no—
267SD16—2121no—
268D8—2121no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4D21211yescharacteristic
283C4⋊C4D21211yescharacteristic
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4D21211yescharacteristic
294C22⋊C4D21211yescharacteristic
295C42—1061no—
296C42C22×D5311yescharacteristic
297C42C22×D5311yescharacteristic
298C23—1061no—
299C23—1061no—
300C23C2×D21211yescharacteristic
301Q8—1061no—
302Q8—1061no—
303Q8—1061no—
304Q8—1061no—
305Q8—1061no—
306Q8—1061no—
307Q8—1061no—
308Q8—1061no—
309Q8—4241no—
310Q8—2122no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—4241no—
316D4—2122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4C2×D21211yescharacteristic
329C2×C4C2×D21211yescharacteristic
330C2×C4C2×D21211yescharacteristic
331C2×C4C2×D21211yescharacteristic
332C2×C4C2×D21211yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C22×D21211yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—2121no—
354C4—2121no—
355C4—2121no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4#1313326311yescharacteristic
360C4#1313326311yescharacteristic
361C2—41no—
362C2—4241no—
363C2—4241no—
364C2#1681119211yescharacteristic
365C2#1681143511yescharacteristic
366C2#1681144011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3225
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16811199C2semidirect
3. #16811017C2semidirect
4. #16811007C2semidirect
5. #16810983C2semidirect
6. #16810974C2semidirect
7. #16810846C2semidirect
8. #16810839C2semidirect
9. #16810834C2semidirect
10. #16810829C2semidirect
11. #16810787C2semidirect
12. #16810781C2semidirect
14. #16810764C2semidirect
15. C53×#433C2semidirect
198. #433D53semidirect
199. C53#3225semidirect