Finite Group Databasebeta

← #20287004

Subgroups — #20287004

Order 6784.

386 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#16811337C211yesmaximal, characteristic
2#16811286C211yesmaximal, characteristic, complemented
3#16810972C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810961C211yesmaximal, characteristic, complemented
6#16810917C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810826C211yesmaximal, characteristic, complemented
9#16810761C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810714C211yesmaximal, characteristic, complemented
12#16810699C211yesmaximal, characteristic, complemented
13#16810633C211yesmaximal, characteristic, complemented
14#16810630C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21#13133216—21no—
22#13133216C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133204—21no—
25#13133203—21no—
26#13133197C2211yescharacteristic
27#13133195—21no—
28#13133194—21no—
29#13133189—21no—
30#13133188C2211yescharacteristic
31#13133182C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133177—21no—
35#13133176C2211yescharacteristic
36#13133175C2211yescharacteristic
37#13133173—21no—
38#13133150C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67Q8×D53—22no—
68Q8×D53—41no—
69C2×C4×D53—21no—
70C2×C4×D53C2311yescharacteristic
71C2×Dic106—22no—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—41no—
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78SD16×C53—41no—
79M16×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719923—22no—
92#11719923—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—22no—
127C22⋊D53—81no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—42no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—42no—
144Dic106—42no—
145Dic106—81no—
146Dic106#4112yes—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—22no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53#13712yes—
174#3329—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—81no—
177C2×C53#54611yescharacteristic
178C2×C53C2×#13711yescharacteristic
179C2×C53#57711yescharacteristic
180D53—41no—
181#555—531no—
182D4×Q8—531no—
183#487—531no—
184#485—531no—
185#485—531no—
186#477—531no—
187#468—531no—
188#468D5311yescharacteristic, complemented
189#461—531no—
190#448—531no—
191#448—531no—
192C4×SD16—531no—
193#436—531no—
194#419—531no—
195#419—531no—
196C53#332911yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C22×Q8—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16C2×D5311yescharacteristic
202C2×M16—531no—
203C2×M16C2×D5311yescharacteristic
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206C4⋊Q8—1061no—
207#124—1061no—
208#124—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—1061no—
211C22⋊Q8—1061no—
212C22⋊Q8C2×D5311yescharacteristic
213#121—531no—
214#121—1061no—
215#121C2×D5311yescharacteristic
216C4×Q8—531no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220#117—531no—
221#117—531no—
222#117—1061no—
223#107—531no—
224#107C2×D5311yescharacteristic
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103C2×D5311yescharacteristic
236#102C2×D5311yescharacteristic
237C22⋊C8—531no—
238C22⋊C8—531no—
239C4×C8—531no—
240C2×Q8—531no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8—1061no—
244C2×Q8—1062no—
245C2×Q8—1061no—
246C2×Q8—1062no—
247C2×Q8—2121no—
248C2×Q8—2121no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—21no—
251C2×D4—531no—
252C2×D4—1061no—
253C2×D4C22×D5311yescharacteristic
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C22×D5311yescharacteristic
259SD16—41no—
260SD16—1062no—
261SD16—2121no—
262M16—41no—
263M16—2121no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—531no—
282C22⋊C4—531no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C42—531no—
289C42—1061no—
290C42—1061no—
291C42—1061no—
292C23—21no—
293C23—531no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—532no—
298Q8—1061no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302Q8—2122no—
303Q8—2122no—
304D4—21no—
305D4—21no—
306D4—81no—
307D4—1062no—
308D4—4241no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
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—1061no—
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—41no—
336C22—41no—
337C22—41no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—532no—
347C4—1061no—
348C4—1061no—
349C4—1062no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4#41×D5311yescharacteristic
354C4#1313326211yescharacteristic
355C2—41no—
356C2—81no—
357C2—2121no—
358C2#1681126211yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3329
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #16811286C2semidirect
3. #16810972C2semidirect
4. #16810966C2semidirect
5. #16810961C2semidirect
6. #16810917C2semidirect
7. #16810868C2semidirect
8. #16810826C2semidirect
9. #16810761C2semidirect
10. #16810760C2semidirect
11. #16810714C2semidirect
12. #16810699C2semidirect
13. #16810633C2semidirect
14. #16810630C2semidirect
15. C53×#468C2semidirect
188. #468D53semidirect
196. C53#3329semidirect