Finite Group Databasebeta

← #20286769

Subgroups — #20286769

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811312C211yesmaximal, characteristic
2#16811286C211yesmaximal, characteristic
3#16810990C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810929C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810865C211yesmaximal, characteristic, complemented
8#16810832C211yesmaximal, characteristic, complemented
9#16810781C211yesmaximal, characteristic, complemented
10#16810780C211yesmaximal, characteristic, complemented
11#16810729C211yesmaximal, characteristic, complemented
12#16810697C211yesmaximal, characteristic
13#16810631C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×Dic212C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133216—21no—
24#13133215C2211yescharacteristic
25#13133210—21no—
26#13133209—21no—
27#13133208C2211yescharacteristic
28#13133199—21no—
29#13133197C2211yescharacteristic
30#13133197C2211yescharacteristic
31#13133196C2211yescharacteristic
32#13133195—21no—
33#13133194—21no—
34#13133182C2211yescharacteristic
35#13133182C2211yescharacteristic
36C4×C22⋊D53C2211yescharacteristic
37#13133177—21no—
38#13133174—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
49#13133140C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53D411yescharacteristic
62C22×Dic53—21no—
63Q8×D53—22no—
64Q8×D53—41no—
65C2×C4×D53—21no—
66C2×C4×D53—21no—
67C2×C4×D53C2311yescharacteristic
68C2×Dic106—22no—
69C2×Dic106—21no—
70C2×Dic106—21no—
71C2×Dic106—41no—
72C2×Dic106C2311yescharacteristic
73C2×Dic106C2311yescharacteristic
74C2×Dic106C2311yescharacteristic
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53D411yescharacteristic
86#11719925—41no—
87#11719921—21no—
88#11719921—21no—
89#11719921D411yescharacteristic
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94#11719919—21no—
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108Dic212—22no—
109Dic212—41no—
110C8×D53—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53C2×D411yescharacteristic
113Q8×C53—21no—
114Q8×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—22no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—42no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—42no—
135C4×D53—42no—
136Dic106—21no—
137Dic106—21no—
138Dic106—21no—
139Dic106—21no—
140Dic106—21no—
141Dic106—42no—
142Dic106—42no—
143Dic106#4112yes—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53#13712yes—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53C2×#4111yescharacteristic
161C4×C53H(5,𝔽2)11yescharacteristic
162Dic53—22no—
163Dic53—21no—
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3312—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53#54611yescharacteristic
172C2×C53C2×#13711yescharacteristic
173C2×C53#57811yescharacteristic
174D53—22no—
175#552—531no—
176D4×Q8—531no—
177#491—531no—
178#485D5311yescharacteristic, complemented
179#483—531no—
180#479—531no—
181#470—531no—
182#467—531no—
183#462—531no—
184#451—531no—
185#451—531no—
186#441—531no—
187#435—531no—
188#419—531no—
189C2×#103—531no—
190C53#331211yescharacteristic, complemented, Sylow
191C22×Q8—531no—
192C22×Q8—531no—
193C2×Q16—531no—
194C2×Q16—531no—
195C2×Q16—531no—
196C2×M16—531no—
197C22×C8—531no—
198C4⋊Q8—531no—
199C4⋊Q8—1061no—
200#126—1061no—
201#125—531no—
202#125—1061no—
203#123—1061no—
204C22⋊Q8—531no—
205C22⋊Q8—531no—
206C22⋊Q8—531no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209C22⋊Q8C2×D5311yescharacteristic
210C4×Q8—531no—
211C4×Q8—531no—
212C4×D4—531no—
213C4×D4—1061no—
214#117—1061no—
215#117C2×D5311yescharacteristic
216C2×C4⋊C4—531no—
217#107C2×D5311yescharacteristic
218#106—531no—
219#106C2×D5311yescharacteristic
220C4⋊C8—531no—
221#103—531no—
222#103—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103C2×D5311yescharacteristic
230#103C2×D5311yescharacteristic
231C22⋊C8—531no—
232C22⋊C8C2×D5311yescharacteristic
233#97—531no—
234C2×Q8—531no—
235C2×Q8—531no—
236C2×Q8—531no—
237C2×Q8—1061no—
238C2×Q8—1062no—
239C2×Q8—1061no—
240C2×Q8—1062no—
241C2×Q8—2121no—
242C2×Q8—2121no—
243C2×Q8C22×D5311yescharacteristic
244C2×D4—531no—
245C22×C4—531no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4C22×D5311yescharacteristic
250Q16—1062no—
251Q16—2121no—
252Q16—2121no—
253M16—2121no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8—2121no—
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—531no—
278C22⋊C4—531no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C42—21no—
283C42—531no—
284C42—1061no—
285C42—1061no—
286C23—531no—
287C23D4×D5311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—532no—
291Q8—1061no—
292Q8—1061no—
293Q8—1061no—
294Q8—1061no—
295Q8—1061no—
296Q8—2122no—
297Q8—2122no—
298D4—1062no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—531no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2122no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4D4×D5311yescharacteristic
318C2×C4C23×D5311yescharacteristic
319C8—21no—
320C8—21no—
321C8—1061no—
322C8—1061no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—532no—
327C22—1061no—
328C22—1061no—
329C22C2×D4×D5311yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—1061no—
336C4—1062no—
337C4—1061no—
338C4—1061no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C4#41×D5311yescharacteristic
343C4#1313326311yescharacteristic
344C2—41no—
345C2—1062no—
346C2#1681126511yescharacteristic
347C2#137×D5311yescharacteristic
348C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3312
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #16810990C2semidirect
4. #16810955C2semidirect
5. #16810929C2semidirect
6. #16810879C2semidirect
7. #16810865C2semidirect
8. #16810832C2semidirect
9. #16810781C2semidirect
10. #16810780C2semidirect
11. #16810729C2semidirect
13. #16810631C2semidirect
14. #103×D53C2semidirect
15. C53×#485C2semidirect
178. #485D53semidirect
190. C53#3312semidirect