Finite Group Databasebeta

← #20287029

Subgroups — #20287029

Order 6784.

388 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#16811328C211yesmaximal, characteristic, complemented
2#16811275C211yesmaximal, characteristic
3#16810972C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810912C211yesmaximal, characteristic, complemented
6#16810882C211yesmaximal, characteristic, complemented
7#16810879C211yesmaximal, characteristic, complemented
8#16810832C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810710C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810622C211yesmaximal, characteristic, complemented
15C53×#483C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C2×#11719912C2211yescharacteristic
23#13133216C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133207—21no—
26#13133202—21no—
27#13133199—21no—
28#13133197C2211yescharacteristic
29#13133197C2211yescharacteristic
30#13133196—21no—
31#13133196C2211yescharacteristic
32#13133195—21no—
33#13133190—21no—
34#13133182C2211yescharacteristic
35#13133179C2211yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38C4⋊C4×D53—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#106×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53D411yescharacteristic
64#11719949—22no—
65#11719949—41no—
66#11719945—22no—
67#11719945—41no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719925—41no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108Dic212—41no—
109#11719913—22no—
110#11719912—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—81no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—22no—
126C22⋊D53—42no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—42no—
135D212#4112yes—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—81no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—22no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53H(5,𝔽2)11yescharacteristic
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53#13712yes—
174#3331—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53#54611yescharacteristic
177C2×C53C2×#13711yescharacteristic
178C2×C53#57711yescharacteristic
179D53—41no—
180D53—42no—
181#554—531no—
182#547—531no—
183#487—531no—
184#483—531no—
185#483D5311yescharacteristic, complemented
186#476—531no—
187#470—531no—
188#470—531no—
189#462—531no—
190#452—531no—
191#452—531no—
192C4×SD16—531no—
193#436—531no—
194#417—531no—
195#417—531no—
196C53#333111yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×#41—531no—
199C2×Q16—531no—
200C2×Q16—531no—
201C2×SD16—531no—
202C2×M16—531no—
203C2×M16—531no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206#125—1061no—
207#123—1061no—
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—531no—
212C22⋊Q8—1061no—
213C22⋊Q8—1061no—
214C22⋊Q8C2×D5311yescharacteristic
215#121—1061no—
216C4×Q8—531no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4C2×D5311yescharacteristic
223#106—531no—
224#106C2×D5311yescharacteristic
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103C2×D5311yescharacteristic
233#103C2×D5311yescharacteristic
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239C4×C8—531no—
240#41—1062no—
241#41—1062no—
242#41—2121no—
243#41—2121no—
244C2×Q8—531no—
245C2×Q8—531no—
246C2×Q8—1061no—
247C2×Q8C22×D5311yescharacteristic
248C2×D4—531no—
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C22×D5311yescharacteristic
259Q16—2121no—
260Q16—2121no—
261SD16—1062no—
262M16—2121no—
263M16—2121no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C4⋊C4C22×D5311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—531no—
285C22⋊C4—531no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C42—531no—
291C42—1061no—
292C23—531no—
293C23—1061no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—1061no—
298Q8—1061no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302D4—532no—
303D4—1062no—
304D4—1061no—
305D4—2122no—
306D4—2122no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—4241no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—1061no—
336C22—1062no—
337C22—1061no—
338C22—1061no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—532no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4#41×D5311yescharacteristic
354C4#1313326311yescharacteristic
355C2—41no—
356C2—2121no—
357C2—2122no—
358C2#1681126511yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#483C2
socleC22×C53C22×D4
hypercenter#483D53
hypocenterC53#3331
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #16811328C2semidirect
3. #16810972C2semidirect
4. #16810955C2semidirect
5. #16810912C2semidirect
6. #16810882C2semidirect
7. #16810879C2semidirect
8. #16810832C2semidirect
9. #16810788C2semidirect
10. #16810787C2semidirect
11. #16810710C2semidirect
12. #16810698C2semidirect
13. #16810623C2semidirect
14. #16810622C2semidirect
15. C53×#483C2semidirect
185. #483D53semidirect
196. C53#3331semidirect