Finite Group Databasebeta

← #20286427

Subgroups — #20286427

Order 6784.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133239C211yesmaximal, characteristic, complemented
2C22×#11719913C211yesmaximal, characteristic, complemented
3#16811282C211yesmaximal, characteristic, complemented
4#16811269C211yesmaximal, characteristic, complemented
5#16811007C211yesmaximal, characteristic, complemented
6#16810852C211yesmaximal, characteristic, complemented
7#16810852C211yesmaximal, characteristic, complemented
8#16810846C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810781C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic, complemented
13#16810759C211yesmaximal, characteristic, complemented
14C4×#11719913C211yesmaximal, characteristic, complemented
15C8×D4×C53C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic, complemented
18C22×Dic106C2211yescharacteristic
19C22×D212C2211yescharacteristic
20#13133239—24no—
21C2×Dic212C2211yescharacteristic, complemented
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719913—24no—
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic
26C2×#11719913C2211yescharacteristic
27C2×#11719913C2211yescharacteristic
28C2×#11719913C2211yescharacteristic, complemented
29C2×D424C2211yescharacteristic
30C2×D424C2211yescharacteristic
31#13133205C2211yescharacteristic
32#13133205C2211yescharacteristic, complemented
33#13133203—21no—
34#13133202—21no—
35#13133196C2211yescharacteristic
36#13133196C2211yescharacteristic, complemented
37#13133195—21no—
38#13133191C2211yescharacteristic
39#13133191C2211yescharacteristic
40#13133190—21no—
41#13133189—21no—
42#13133185—21no—
43C4×Dic106C2211yescharacteristic, complemented
44C4×D212C2211yescharacteristic
45#13133177—21no—
46C22⋊C4×D53—21no—
47#13133149C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133142C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic
52#13133139C2211yescharacteristic
53#13133139C2211yescharacteristic, complemented
54C22×C8×C53C2211yescharacteristic
55C22×C8×C53C2211yescharacteristic, complemented
56C4×D4×C53C2211yescharacteristic
57C4⋊C8×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C22⋊C8×C53C2211yescharacteristic, complemented
60C4×C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×D4×C53D411yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70#11719945—22no—
71#11719945—22no—
72#11719945—41no—
73#11719945—41no—
74C2×D212—22no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×Dic106—22no—
84C2×Dic106—21no—
85C2×Dic106—41no—
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88C2×Dic106C2311yescharacteristic
89C2×C8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C2×C8×C53C2311yescharacteristic
94C2×C8×C53C2311yescharacteristic
95C4⋊C4×C53D411yescharacteristic
96C22⋊C4×C53D411yescharacteristic
97C22⋊C4×C53D411yescharacteristic
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108C4⋊Dic53C2311yescharacteristic
109C4⋊Dic53C2311yescharacteristic
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113C4×Dic53—21no—
114Dic212—22no—
115Dic212—41no—
116D424—22no—
117D424—41no—
118#11719913—22no—
119#11719913—22no—
120#11719913—22no—
121#11719913—22no—
122#11719913D414yescomplemented
123C23×C53C2×D411yescharacteristic
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—81no—
131D4×C53—22no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53—42no—
142C22⋊D53—42no—
143C22⋊D53—42no—
144C22⋊D53—81no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151D212—21no—
152D212—21no—
153D212—21no—
154D212—21no—
155D212—21no—
156D212—42no—
157D212C2×D412yes—
158C4×D53—42no—
159C4×D53—42no—
160C4×D53—81no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—21no—
165Dic106—21no—
166Dic106—42no—
167Dic106C2×D412yes—
168C8×C53—21no—
169C8×C53—21no—
170C8×C53—21no—
171C8×C53C2×D412yes—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53#13512yescomplemented
178C22×C53C22×D411yescharacteristic
179C2×D53—22no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—84no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53C22×D411yescharacteristic
194C4×C53H(5,𝔽2)11yescharacteristic
195Dic53—22no—
196Dic53—41no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200#3282—531nomaximal, Sylow
201C2×C53—22no—
202C2×C53—41no—
203C2×C53D4211yescharacteristic
204C2×C53C2×#13511yescharacteristic
205C2×C53#57711yescharacteristic
206D53—42no—
207D53—81no—
208D53—81no—
209C2×#135—531no—
210C22×SD16—531no—
211#548—531no—
212#546—531no—
213#495—531no—
214#465—531no—
215#465—531no—
216#464—531no—
217#463—531no—
218#452—531no—
219#451—531no—
220#449—531no—
221#447—531no—
222C4×SD16—531no—
223C8×D4D5311yescharacteristic, complemented
224C53#328211yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41—531no—
227C22×Q8—531no—
228C22×D4—531no—
229#135—1064no—
230C2×Q16—531no—
231C2×Q16—531no—
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×SD16—1064no—
238C2×D8—531no—
239C2×D8—531no—
240C22×C8C2×D5311yescharacteristic
241C22×C8C2×D5311yescharacteristic, complemented
242#124—531no—
243#124—531no—
244#124—1061no—
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—531no—
248C22⋊Q8—1061no—
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8—531no—
255C4×D4—531no—
256C4×D4—1061no—
257C4×D4C2×D5311yescharacteristic
258C2×C22⋊C4—1061no—
259#106—531no—
260C4⋊C8C2×D5311yescharacteristic, complemented
261#103—531no—
262#103—531no—
263#103—531no—
264#102—531no—
265#102—531no—
266#102—531no—
267C22⋊C8C2×D5311yescharacteristic
268C22⋊C8C2×D5311yescharacteristic, complemented
269C4×C8C2×D5311yescharacteristic, complemented
270C24—1061no—
271#41—1062no—
272#41—1062no—
273#41—2121no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—531no—
277C2×Q8—531no—
278C2×Q8—1062no—
279C2×Q8—1061no—
280C2×Q8—2121no—
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1062no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—2121no—
291C2×D4D21211yescharacteristic
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4C22×D5311yescharacteristic
297C22×C4C22×D5311yescharacteristic
298Q16—1062no—
299Q16—2121no—
300SD16—534no—
301SD16—1062no—
302SD16—1062no—
303SD16—1062no—
304SD16—1062no—
305D8—1062no—
306D8—2121no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8C22×D5311yescharacteristic
310C2×C8C22×D5311yescharacteristic
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—531no—
314C4⋊C4—531no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4D21211yescharacteristic
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4D21211yescharacteristic
329C22⋊C4D21211yescharacteristic
330C42—1061no—
331C42C22×D5311yescharacteristic
332C23—1061no—
333C23—1061no—
334C23—1061no—
335C23—4241no—
336C23—2122no—
337C23—2122no—
338C23C2×D21211yescharacteristic
339C23C2×D21211yescharacteristic
340Q8—532no—
341Q8—1061no—
342Q8—1061no—
343Q8—1061no—
344Q8—1061no—
345Q8—1061no—
346Q8—2122no—
347D4—22no—
348D4—532no—
349D4—1061no—
350D4—1061no—
351D4—1061no—
352D4—1061no—
353D4—1061no—
354D4—4241no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358D4—2122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—4241no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4C2×D21211yescharacteristic
373C2×C4C2×D21211yescharacteristic
374C2×C4C2×D21211yescharacteristic
375C2×C4C2×D21211yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D5312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—1062no—
387C22—4244no—
388C22—2121no—
389C22—2121no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22#1313323912yescomplemented
397C22C22×D21211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—1062no—
403C4—2121no—
404C4—2121no—
405C4—2121no—
406C4—2121no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326311yescharacteristic
409C2—22no—
410C2—41no—
411C2—4241no—
412C2—4241no—
413C2—2122no—
414C2D4×D21211yescharacteristic
415C2C2×#1313323911yescharacteristic
416C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC8×D4×C53C2
socleC22×C53C22×D4
hypercenterC8×D4D53
hypocenterC53#3282
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133239C2semidirect
2. C22×#11719913C2semidirect
3. #16811282C2semidirect
4. #16811269C2semidirect
5. #16811007C2semidirect
6. #16810852C2semidirect
7. #16810852C2semidirect
8. #16810846C2semidirect
9. #16810839C2semidirect
10. #16810787C2semidirect
11. #16810781C2semidirect
12. #16810770C2semidirect
13. #16810759C2semidirect
14. C4×#11719913C2semidirect
15. C8×D4×C53C2semidirect
17. C2×#11719945C22semidirect
21. C2×Dic212C22semidirect
22. C2×Dic212C22semidirect
28. C2×#11719913C22semidirect
32. #13133205C22semidirect
36. #13133196C22semidirect
43. C4×Dic106C22semidirect
47. #13133149C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
50. #13133142C22semidirect
53. #13133139C22semidirect
55. C22×C8×C53C22semidirect
57. C4⋊C8×C53C22semidirect
59. C22⋊C8×C53C22semidirect
60. C4×C8×C53C22semidirect
122. #11719913D4semidirect
177. C22×C53#135semidirect
223. C8×D4D53semidirect
224. C53#3282semidirect
241. C22×C8C2×D53semidirect
260. C4⋊C8C2×D53semidirect
268. C22⋊C8C2×D53semidirect
269. C4×C8C2×D53semidirect
396. C22#13133239semidirect