Finite Group Databasebeta

← #20286891

Subgroups — #20286891

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#16811323C211yesmaximal, characteristic
2Q8×D212C211yesmaximal, characteristic, complemented
3#16810992C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810960C211yesmaximal, characteristic, complemented
6#16810930C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810851C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810760C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133217—21no—
23#13133217C2211yescharacteristic
24#13133210—21no—
25#13133209C2211yescharacteristic
26#13133200—21no—
27#13133197—21no—
28#13133197—21no—
29#13133197C2211yescharacteristic
30#13133197C2211yescharacteristic
31#13133190—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36C4×C22⋊D53—21no—
37#13133175C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133148C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133140C2211yescharacteristic
49#13133140C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64Q8×D53—22no—
65Q8×D53—22no—
66Q8×D53—41no—
67Q8×D53—41no—
68C2×D212D411yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78SD16×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719923—41no—
90#11719923—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C23×C53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53#4112yes—
116D4×C53—21no—
117D4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—81no—
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×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—22no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—81no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—42no—
146Dic106—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53#13712yes—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
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—41no—
174#3320—531nomaximal, Sylow
175C2×C53—81no—
176C2×C53#54611yescharacteristic
177C2×C53C2×#13711yescharacteristic
178C2×C53#57711yescharacteristic
179D53—22no—
180D53—41no—
181#553—531no—
182D4×Q8—531no—
183#491—531no—
184#485—531no—
185#484—531no—
186#479—531no—
187#468—531no—
188#465—531no—
189#461—531no—
190#448—531no—
191#448—531no—
192#440D5311yescharacteristic, complemented
193#435—531no—
194#419—531no—
195C2×#103—531no—
196C53#332011yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C22×Q8—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16C2×D5311yescharacteristic
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205C4⋊Q8—1061no—
206#126—1061no—
207#125—531no—
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213#121—531no—
214#121—531no—
215#121—1061no—
216C4×Q8C2×D5311yescharacteristic
217C4×D4—531no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222C2×C4⋊C4—531no—
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—
239#97C2×D5311yescharacteristic
240C2×Q8—531no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8—1061no—
244C2×Q8—1062no—
245C2×Q8—1062no—
246C2×Q8—2121no—
247C2×Q8—2121no—
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4C22×D5311yescharacteristic
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259SD16—22no—
260SD16—2121no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—2121no—
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⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—531no—
283C22⋊C4—531no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—21no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—21no—
293C23—531no—
294C23—531no—
295Q8—21no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299Q8—1061no—
300Q8—2122no—
301Q8—2122no—
302Q8#1171994512yes—
303D4—21no—
304D4—21no—
305D4—1062no—
306D4—4241no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
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×C4D4×D5311yescharacteristic
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—532no—
336C22—1061no—
337C22—1061no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22C2×D4×D5311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994511yescharacteristic
354C4#1313326311yescharacteristic
355C2—81no—
356C2—1062no—
357C2—2121no—
358C2#1681126911yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3320
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. Q8×D212C2semidirect
3. #16810992C2semidirect
4. #16810966C2semidirect
5. #16810960C2semidirect
6. #16810930C2semidirect
7. #16810868C2semidirect
8. #16810851C2semidirect
9. #16810828C2semidirect
10. #16810760C2semidirect
11. #16810760C2semidirect
12. #16810696C2semidirect
13. #16810630C2semidirect
14. #103×D53C2semidirect
15. C53×#440C2semidirect
192. #440D53semidirect
196. C53#3320semidirect