Finite Group Databasebeta

← #20285818

Subgroups — #20285818

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#16811199C211yesmaximal, characteristic, complemented
2#16811018C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16810985C211yesmaximal, characteristic, complemented
5#16810975C211yesmaximal, characteristic, complemented
6#16810845C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810833C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810778C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic
13#16810761C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#542C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719924C2211yescharacteristic
25C2×#11719922C2211yescharacteristic
26#13133217C2211yescharacteristic
27#13133217C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133205C2211yescharacteristic
30#13133201—21no—
31#13133201—21no—
32#13133197—21no—
33#13133197—21no—
34C4⋊#88322C2211yescharacteristic
35C4⋊#88322C2211yescharacteristic
36#13133140C2211yescharacteristic
37#13133140C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48C4⋊Q8×C53C2211yescharacteristic
49#126×C53—21no—
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52#123×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58#117×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67#11719945—41no—
68#11719945—41no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53D411yescharacteristic
90C22⋊C4×C53D411yescharacteristic
91C42×C53—21no—
92C42×C53C2311yescharacteristic
93C42×C53C2311yescharacteristic
94#11719925—41no—
95#11719925—41no—
96#11719925—41no—
97#11719924—41no—
98#11719923—41no—
99#11719923—41no—
100#11719923—41no—
101#11719922—41no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—81no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—81no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140D212—21no—
141D212—21no—
142C4×D53—42no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—21no—
149Dic106—42no—
150#88322—21no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C22×D411yescharacteristic
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×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53H(5,𝔽2)11yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3225—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53—81no—
180C2×C53#53511yescharacteristic
181C2×C53#57711yescharacteristic
182C2×C53#57811yescharacteristic
183D53—81no—
184#542D5311yescharacteristic, complemented
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#433—531no—
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⋊Q8C2×D5311yescharacteristic
213#126—21no—
214#124—21no—
215#124—531no—
216#124—531no—
217#124C2×D5311yescharacteristic
218#123—21no—
219#123—1061no—
220#123—1061no—
221C22⋊Q8—21no—
222C22⋊Q8—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8C2×D5311yescharacteristic
225#121C2×D5311yescharacteristic
226C4×Q8C2×D5311yescharacteristic
227C4×D4C2×D5311yescharacteristic
228#117C2×D5311yescharacteristic
229C4⋊C8—531no—
230C4⋊C8—531no—
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⋊C8—531no—
240C22⋊C8—531no—
241#97—531no—
242C4×C8—531no—
243#41—2121no—
244#41—2121no—
245C2×Q8—21no—
246C2×Q8—531no—
247C2×Q8—531no—
248C2×Q8—531no—
249C2×Q8—1061no—
250C2×Q8—2121no—
251C2×Q8—2121no—
252C2×Q8C22×D5311yescharacteristic
253C2×D4—21no—
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4C22×D5311yescharacteristic
257C22×C4—21no—
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×C8—531no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8—531no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22⋊D5311yescharacteristic
281C4⋊C4C22⋊D5311yescharacteristic
282C4⋊C4C22×D5311yescharacteristic
283C4⋊C4C22×D5311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4C22⋊D5311yescharacteristic
294C22⋊C4C22⋊D5311yescharacteristic
295C42—21no—
296C42C22×D5311yescharacteristic
297C42C22×D5311yescharacteristic
298C23—21no—
299C23—1061no—
300C23C2×C22⋊D5311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—81no—
304Q8—1061no—
305Q8—1061no—
306Q8—1061no—
307Q8—1061no—
308Q8—1061no—
309Q8—1061no—
310Q8—2122no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—1061no—
315D4—1061no—
316D4—2122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4C2×C22⋊D5311yescharacteristic
329C2×C4C2×C22⋊D5311yescharacteristic
330C2×C4C2×C22⋊D5311yescharacteristic
331C2×C4C2×C22⋊D5311yescharacteristic
332C2×C4C2×C22⋊D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—1061no—
335C8—1061no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C22×C22⋊D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4#1313326311yescharacteristic
360C4#1313326311yescharacteristic
361C2—41no—
362C2—81no—
363C2—4241no—
364C2#1681118711yescharacteristic
365C2#1681143411yescharacteristic
366C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#542C2
socleC22×C53C22×D4
hypercenter#542D53
hypocenterC53#3225
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16811199C2semidirect
2. #16811018C2semidirect
3. #16811005C2semidirect
4. #16810985C2semidirect
5. #16810975C2semidirect
6. #16810845C2semidirect
7. #16810838C2semidirect
8. #16810833C2semidirect
9. #16810828C2semidirect
10. #16810783C2semidirect
11. #16810778C2semidirect
13. #16810761C2semidirect
14. #16810684C2semidirect
15. C53×#542C2semidirect
184. #542D53semidirect
199. C53#3225semidirect