Finite Group Databasebeta

← #20286798

Subgroups — #20286798

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811330C211yesmaximal, characteristic, complemented
2#16811269C211yesmaximal, characteristic
3#16810983C211yesmaximal, characteristic, complemented
4#16810959C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810904C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810864C211yesmaximal, characteristic, complemented
9#16810846C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810776C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×Dic212C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21#13133217C2211yescharacteristic, complemented
22#13133208—21no—
23#13133205C2211yescharacteristic
24#13133202—21no—
25#13133196C2211yescharacteristic
26#13133196C2211yescharacteristic
27#13133196C2211yescharacteristic, complemented
28#13133194—21no—
29#13133194—21no—
30#13133191C2211yescharacteristic
31#13133190—21no—
32#13133185—21no—
33C4×Dic106C2211yescharacteristic, complemented
34C4×D212C2211yescharacteristic
35D4×Dic53—21no—
36C2×C4⋊Dic53C2211yescharacteristic
37C2×C4⋊Dic53C2211yescharacteristic, complemented
38C22⋊C4×D53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133151C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133142C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53D411yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67#11719945—22no—
68#11719945—41no—
69C2×D212—22no—
70C2×D212—21no—
71C2×D212—41no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79M16×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53D411yescharacteristic
86C22⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108Dic212—41no—
109#11719913—22no—
110#11719913—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118D4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—42no—
142D212#4112yes—
143C4×D53—42no—
144C4×D53—81no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—21no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#13712yes—
160C22×C53C22×D411yescharacteristic
161C2×D53—22no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3314—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—41no—
183C2×C53#54611yescharacteristic
184C2×C53C2×#13711yescharacteristic
185C2×C53#57611yescharacteristic
186D53—42no—
187D53—81no—
188#554—531no—
189#546—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#474—531no—
194#468—531no—
195#467—531no—
196#464—531no—
197#452—531no—
198#450—531no—
199#440—531no—
200#435D5311yescharacteristic, complemented
201#417—531no—
202C2×#102—531no—
203C53#331411yescharacteristic, complemented, Sylow
204C2×#41—531no—
205C22×D4—531no—
206C2×Q16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×M16C2×D5311yescharacteristic, complemented
210C22×C8C2×D5311yescharacteristic
211C4⋊Q8—531no—
212#125—1061no—
213#124—531no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—531no—
218C22⋊Q8—1061no—
219C22⋊Q8—1061no—
220#121—531no—
221#121—1061no—
222#120—1061no—
223C4×Q8—531no—
224C4×D4—531no—
225C4×D4—1061no—
226C4×D4C2×D5311yescharacteristic
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C22⋊C4—1061no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8C2×D5311yescharacteristic, complemented
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8C2×D5311yescharacteristic
245C22⋊C8C2×D5311yescharacteristic, complemented
246#97C2×D5311yescharacteristic, complemented
247C24—1061no—
248#41—1062no—
249#41—2121no—
250C2×Q8—531no—
251C2×Q8—531no—
252C2×Q8—1061no—
253C2×D4—531no—
254C2×D4—531no—
255C2×D4—1062no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4—2121no—
260C2×D4D21211yescharacteristic
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267Q16—2121no—
268SD16—1062no—
269SD16—2121no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D5311yescharacteristic
273C2×C8C22×D5311yescharacteristic
274C2×C8C22×D5311yescharacteristic
275C2×C8C22×D5311yescharacteristic
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4D21211yescharacteristic
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4D21211yescharacteristic
296C22⋊C4D21211yescharacteristic
297C42—1061no—
298C42C22×D5311yescharacteristic
299C23—1061no—
300C23—1061no—
301C23—4241no—
302C23—2122no—
303C23—2122no—
304C23C2×D21211yescharacteristic
305C23C2×D21211yescharacteristic
306Q8—1061no—
307Q8—1061no—
308Q8—1061no—
309Q8—1061no—
310Q8—4241no—
311D4—22no—
312D4—532no—
313D4—1061no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—2122no—
333C2×C4C2×D21211yescharacteristic
334C2×C4C2×D21211yescharacteristic
335C2×C4C2×D21211yescharacteristic
336C2×C4C2×D21211yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—1062no—
348C22—4244no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22#1313325012yes—
355C22C22×D21211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4#41×D5311yescharacteristic
367C4#1313326211yescharacteristic
368C2—22no—
369C2—41no—
370C2—4241no—
371C2—2122no—
372C2#1681126811yescharacteristic
373C2C2×#1313325011yescharacteristic
374C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3314
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811330C2semidirect
3. #16810983C2semidirect
4. #16810959C2semidirect
5. #16810937C2semidirect
6. #16810904C2semidirect
7. #16810869C2semidirect
8. #16810864C2semidirect
9. #16810846C2semidirect
10. #16810787C2semidirect
11. #16810776C2semidirect
12. #16810725C2semidirect
13. #16810625C2semidirect
14. C2×#13133139C2semidirect
15. C53×#435C2semidirect
18. C2×Dic212C22semidirect
21. #13133217C22semidirect
27. #13133196C22semidirect
33. C4×Dic106C22semidirect
37. C2×C4⋊Dic53C22semidirect
39. #13133151C22semidirect
40. #13133151C22semidirect
41. #13133151C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133142C22semidirect
52. C2×M16×C53C22semidirect
55. C4⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. #97×C53C22semidirect
200. #435D53semidirect
203. C53#3314semidirect
209. C2×M16C2×D53semidirect
233. C4⋊C8C2×D53semidirect
245. C22⋊C8C2×D53semidirect
246. #97C2×D53semidirect