Finite Group Databasebeta

← #20279784

Subgroups — #20279784

Order 6784.

696 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133205C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic, complemented
3C2×#13133191C211yesmaximal, characteristic, complemented
4C2×#13133190C211yesmaximal, characteristic, complemented
5#16810544C211yesmaximal, characteristic, complemented
6#16810539C211yesmaximal, characteristic, complemented
7#16810533C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810525C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810514C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810501C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24#13133205—24no—
25#13133203—24no—
26#13133191—24no—
27#13133190—24no—
28C2×C4⋊Dic53C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C42×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—41no—
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—22no—
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—41no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—41no—
94C2×Dic106—22no—
95C2×Dic106—22no—
96C2×Dic106—41no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53D414yescomplemented
99C42×C53—22no—
100C42×C53—41no—
101C22⋊Dic53—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921D414yescomplemented
113C4⋊Dic53—22no—
114C4⋊Dic53—41no—
115#11719919—22no—
116C4×Dic53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53C2×D412yes—
124C23×C53C2411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53C2×D412yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—44no—
160C2×C4×C53C2×D412yes—
161C2×C4×C53C2×D412yes—
162C2×C4×C53C2×D412yes—
163C2×C4×C53#4112yes—
164C2×C4×C53#4112yes—
165C22⋊D53—44no—
166C22⋊D53—44no—
167C22⋊D53—44no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53C2×D412yes—
182C2×Dic53#4112yes—
183D212—44no—
184D212—44no—
185D212—44no—
186C4×D53—44no—
187Dic106—44no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53C22×D411yescharacteristic
196C22×C53C22×D411yescharacteristic
197C22×C53C2×#4111yescharacteristic
198C22×C53C2×#4111yescharacteristic
199C22×C53C2×#4111yescharacteristic
200C22×C53H(5,𝔽2)11yescharacteristic
201C22×C53H(5,𝔽2)11yescharacteristic
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—41no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C2×D53—41no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—22no—
226C4×C53—22no—
227C4×C53—22no—
228C4×C53—41no—
229Dic53—22no—
230Dic53—22no—
231Dic53—41no—
232Dic53—41no—
233Dic53—41no—
234#2555—531nomaximal, Sylow
235C2×C53—42no—
236C2×C53#53211yescharacteristic
237C2×C53#53211yescharacteristic
238C2×C53#53811yescharacteristic
239C2×C53D4211yescharacteristic
240C2×C53#54611yescharacteristic
241C2×C53#54611yescharacteristic
242C2×C53#55511yescharacteristic
243D53—42no—
244D53—81no—
245D53—81no—
246C2×#124—531no—
247C2×#124—531no—
248C2×#121—531no—
249C2×#121—531no—
250#395—531no—
251#394—531no—
252#392—531no—
253#392—531no—
254#390—531no—
255#388—531no—
256#388—531no—
257#386—531no—
258#386—531no—
259C4×C22⋊C4—531no—
260C4×C22⋊C4D5311yescharacteristic, complemented
261C53#255511yescharacteristic, complemented, Sylow
262C22×Q8—531no—
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4—531no—
267C22×D4—531no—
268C22×D4—531no—
269C23×C4—531no—
270C23×C4C2×D5311yescharacteristic
271#124—1064no—
272#124—1064no—
273#121—1064no—
274#121—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4C2×D5311yescharacteristic
291C2×C22⋊C4C2×D5311yescharacteristic
292C2×C42—531no—
293C2×C42C2×D5311yescharacteristic
294C2×C42C2×D5311yescharacteristic
295#95—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95C2×D5311yescharacteristic
300#95C2×D5311yescharacteristic
301C24—531no—
302C24—531no—
303C24—531no—
304C24C22×D5311yescharacteristic
305C2×Q8—1062no—
306C2×Q8—1062no—
307C2×Q8—2121no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—1062no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—2121no—
337C22×C4C22×D5311yescharacteristic
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—1062no—
344C4⋊C4—1062no—
345C4⋊C4—2121no—
346C22⋊C4—22no—
347C22⋊C4—534no—
348C22⋊C4—1062no—
349C22⋊C4—1062no—
350C22⋊C4—1062no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4D21214yescomplemented
360C42—22no—
361C42—41no—
362C42—1062no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—532no—
370C23—1061no—
371C23—1061no—
372C23—1061no—
373C23—1061no—
374C23—1061no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23C2×D21212yes—
391C23C23×D5311yescharacteristic
392Q8—2124no—
393D4—2124no—
394D4—2124no—
395D4—2124no—
396D4—2124no—
397D4—2124no—
398D4—2124no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—44no—
413C2×C4—532no—
414C2×C4—532no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—2124no—
429C2×C4C2×D21212yes—
430C2×C4C2×D21212yes—
431C2×C4#1171994512yes—
432C2×C4#1171994512yes—
433C2×C4D4×D5312yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—2121no—
445C22—2121no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22—2121no—
455C22—2121no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22—2121no—
460C22—2121no—
461C22—2121no—
462C22C2×D4×D5311yescharacteristic
463C22C22×D21211yescharacteristic
464C22#41×D5311yescharacteristic
465C22C2×#1171994511yescharacteristic
466C22C2×#1171994511yescharacteristic
467C22#1313326211yescharacteristic
468C22#1313326311yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—41no—
475C4—1062no—
476C4—1062no—
477C4—2121no—
478C4—2121no—
479C4—2121no—
480C2—42no—
481C2—4241no—
482C2—4241no—
483C2—2122no—
484C2#1681117011yescharacteristic
485C2#1681117311yescharacteristic
486C2#1681120611yescharacteristic
487C2D4×D21211yescharacteristic
488C2#1681126311yescharacteristic
489C2#1681126811yescharacteristic
490C2#1681133611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC4×C22⋊C4×C53C2
hypercenterC4×C22⋊C4D53
hypocenterC53#2555
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133205C2semidirect
2. C2×#13133203C2semidirect
3. C2×#13133191C2semidirect
4. C2×#13133190C2semidirect
5. #16810544C2semidirect
6. #16810539C2semidirect
7. #16810533C2semidirect
8. #16810532C2semidirect
9. #16810525C2semidirect
10. #16810515C2semidirect
11. #16810514C2semidirect
12. #16810502C2semidirect
13. #16810501C2semidirect
14. C4×#11719921C2semidirect
15. C4×C22⋊C4×C53C2semidirect
98. C22⋊C4×C53D4semidirect
112. #11719921D4semidirect
260. C4×C22⋊C4D53semidirect
261. C53#2555semidirect
359. C22⋊C4D212semidirect