Finite Group Databasebeta

← #20284677

Subgroups — #20284677

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811196C211yesmaximal, characteristic
2#16811104C211yesmaximal, characteristic
3#16810967C211yesmaximal, characteristic, complemented
4#16810965C211yesmaximal, characteristic, complemented
5#16810962C211yesmaximal, characteristic, complemented
6#16810961C211yesmaximal, characteristic, complemented
7#16810958C211yesmaximal, characteristic, complemented
8#16810957C211yesmaximal, characteristic, complemented
9#16810955C211yesmaximal, characteristic, complemented
10#16810662C211yesmaximal, characteristic, complemented
11#16810658C211yesmaximal, characteristic, complemented
12#16810630C211yesmaximal, characteristic, complemented
13#103×D53C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic
15C53×#483C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133216—21no—
21#13133203—21no—
22#13133202—21no—
23#13133199—21no—
24#13133197C2211yescharacteristic
25#13133196C2211yescharacteristic
26#13133193C2211yescharacteristic
27#13133182—21no—
28#13133179—21no—
29C4×C22⋊D53—21no—
30C4×C22⋊D53—21no—
31#13133176C2211yescharacteristic
32#13133175C2211yescharacteristic
33#13133175C2211yescharacteristic
34#13133175C2211yescharacteristic
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
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⋊D53D411yescharacteristic
63C2×C22⋊D53D411yescharacteristic
64Q8×D53—22no—
65Q8×D53—22no—
66#11719945—22no—
67#11719945—22no—
68C2×D212D411yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106D411yescharacteristic
76C2×Dic106C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107C8×D53—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53C2×D411yescharacteristic
110C22×D53C2×D411yescharacteristic
111Q8×C53—21no—
112Q8×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—22no—
123C22⋊D53—22no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130D212—22no—
131C4×D53—21no—
132C4×D53—21no—
133C4×D53—42no—
134C4×D53—81no—
135C4×D53—81no—
136C4×D53#4112yes—
137C4×D53#4112yes—
138Dic106—22no—
139Dic106—21no—
140Dic106—21no—
141Dic106—42no—
142C8×C53—21no—
143C8×C53—21no—
144#88322—21no—
145#88322—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53C22×D411yescharacteristic
150C2×D53—21no—
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53#12312yes—
156C4×C53—21no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#4111yescharacteristic
162C4×C53C2×#4111yescharacteristic
163Dic53—21no—
164Dic53—41no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53#12312yes—
169#3097—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53C2×#12311yescharacteristic
172C2×C53#57711yescharacteristic
173C2×C53#57811yescharacteristic
174D53—22no—
175D53—41no—
176#536—531no—
177#520—531no—
178#485—531no—
179#485—531no—
180#485—531no—
181#485—531no—
182#484—531no—
183#484—531no—
184#483—531no—
185#483D5311yescharacteristic, complemented
186#428—531no—
187#427—531no—
188#419—531no—
189C2×#103—531no—
190#408—531no—
191C53#309711yescharacteristic, complemented, Sylow
192C2×#41—531no—
193C22×Q8—531no—
194C2×M16—531no—
195C22×C8—531no—
196C4⋊Q8—1061no—
197#124—1061no—
198#123—1061no—
199#123—1061no—
200C22⋊Q8—531no—
201C22⋊Q8—531no—
202C22⋊Q8—531no—
203C22⋊Q8C2×D5311yescharacteristic
204C4×Q8—1061no—
205C4×D4—1061no—
206C4×D4—1061no—
207C4×D4—1061no—
208#117—531no—
209#117—531no—
210#117—531no—
211#117—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4C2×D5311yescharacteristic
215#107—531no—
216#107—531no—
217#107—531no—
218#107—531no—
219#106—531no—
220#106—531no—
221#106C2×D5311yescharacteristic
222#106C2×D5311yescharacteristic
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⋊C8—531no—
233C22⋊C8—531no—
234C22⋊C8C2×D5311yescharacteristic
235#41—1062no—
236#41—1062no—
237C2×Q8—531no—
238C2×Q8—531no—
239C2×Q8—1061no—
240C2×Q8—1062no—
241C2×Q8—1062no—
242C2×Q8C22×D5311yescharacteristic
243C2×D4—531no—
244C2×D4—531no—
245C2×D4—531no—
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4C22×D5311yescharacteristic
253M16—1062no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8—1062no—
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4C22×D5311yescharacteristic
270C4⋊C4C22×D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C42—1061no—
281C42—1061no—
282C42—1061no—
283C42—1061no—
284C23—531no—
285C23—531no—
286C23D4×D5311yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—1062no—
290Q8—1061no—
291Q8—1061no—
292Q8—2122no—
293D4—1062no—
294D4—1062no—
295D4—1062no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—531no—
304C2×C4—532no—
305C2×C4—532no—
306C2×C4—531no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4—2122no—
316C2×C4D4×D5311yescharacteristic
317C2×C4C23×D5311yescharacteristic
318C8—21no—
319C8—21no—
320C8—1061no—
321C8—1061no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—532no—
326C22—1061no—
327C22—1061no—
328C22—1061no—
329C22—1061no—
330C22—1061no—
331C22C2×D4×D5311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—532no—
338C4—1061no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C4—2121no—
343C4#41×D5311yescharacteristic
344C4#41×D5311yescharacteristic
345C2—41no—
346C2—1062no—
347C2—2121no—
348C2#123×D5311yescharacteristic
349C2#1681142911yescharacteristic
350C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#483C2
socleC22×C53C22×D4
hypercenter#483D53
hypocenterC53#3097
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
3. #16810967C2semidirect
4. #16810965C2semidirect
5. #16810962C2semidirect
6. #16810961C2semidirect
7. #16810958C2semidirect
8. #16810957C2semidirect
9. #16810955C2semidirect
10. #16810662C2semidirect
11. #16810658C2semidirect
12. #16810630C2semidirect
13. #103×D53C2semidirect
15. C53×#483C2semidirect
185. #483D53semidirect
191. C53#3097semidirect