Finite Group Databasebeta

← #20278604

Subgroups — #20278604

Order 6784.

718 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×#13133201C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic, complemented
3C2×C4×D212C211yesmaximal, characteristic, complemented
4C2×#13133177C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810524C211yesmaximal, characteristic, complemented
10#16810514C211yesmaximal, characteristic, complemented
11#16810508C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810500C211yesmaximal, characteristic, complemented
14#16810497C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C23×Dic53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25#13133201—24no—
26#13133190—24no—
27C4×D212—24no—
28#13133177—24no—
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
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
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—22no—
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53—41no—
73C2×C22⋊D53—41no—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—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—22no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—22no—
94C2×C4×D53—22no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C4⋊C4×C53—41no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53D414yescomplemented
101C42×C53—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—41no—
111#11719921—41no—
112C4⋊Dic53—22no—
113C4⋊Dic53D414yescomplemented
114#11719919—22no—
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#4112yes—
145C22×D53#4112yes—
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—21no—
160C2×C4×C53—21no—
161C2×C4×C53C2×D412yes—
162C2×C4×C53C2×D412yes—
163C2×C4×C53C2×D412yes—
164C22⋊D53—28no—
165C22⋊D53—44no—
166C22⋊D53—44no—
167C2×Dic53—21no—
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—44no—
180C2×Dic53C2×D412yes—
181C2×Dic53C2×D412yes—
182C2×Dic53#4112yes—
183D212—28no—
184D212—44no—
185D212—44no—
186C4×D53—44no—
187C4×D53—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—22no—
206C2×D53—22no—
207C2×D53—22no—
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—41no—
227C4×C53—41no—
228C4×C53—41no—
229Dic53—22no—
230Dic53—22no—
231Dic53—22no—
232Dic53—41no—
233Dic53—41no—
234#2456—531nomaximal, Sylow
235C2×C53—42no—
236C2×C53#52511yescharacteristic
237C2×C53#52511yescharacteristic
238C2×C53#53811yescharacteristic
239C2×C53D4211yescharacteristic
240C2×C53#54611yescharacteristic
241C2×C53#55011yescharacteristic
242C2×C53#55311yescharacteristic
243D53—42no—
244D53—42no—
245D53—81no—
246C2×#123—531no—
247C2×#121—531no—
248C2×C4×D4—531no—
249C2×C4×D4—531no—
250#396—531no—
251#392—531no—
252#392—531no—
253#392—531no—
254#390—531no—
255#388—531no—
256#388D5311yescharacteristic, complemented
257#387—531no—
258#386—531no—
259#386—531no—
260#385—531no—
261C53#245611yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4—531no—
267C22×D4—531no—
268C23×C4—531no—
269C23×C4—531no—
270C23×C4—531no—
271#123—1064no—
272#121—1064no—
273C4×D4—1064no—
274C4×D4—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4C2×D5311yescharacteristic
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×C22⋊C4C2×D5311yescharacteristic
293C2×C42—531no—
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×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
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—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4—2121no—
337C22×C4C22×D5311yescharacteristic
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—41no—
344C4⋊C4—534no—
345C4⋊C4—1062no—
346C4⋊C4—1062no—
347C4⋊C4—1062no—
348C22⋊C4—22no—
349C22⋊C4—41no—
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—2121no—
359C22⋊C4—2121no—
360C22⋊C4D21214yescomplemented
361C42—22no—
362C42—1062no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—532no—
370C23—532no—
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
392D4—1068no—
393D4—1068no—
394D4—2124no—
395D4—2124no—
396D4—2124no—
397D4—2124no—
398C2×C4—21no—
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—21no—
413C2×C4—532no—
414C2×C4—532no—
415C2×C4—532no—
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×C4—2124no—
430C2×C4—2124no—
431C2×C4C2×D21212yes—
432C2×C4C2×D21212yes—
433C2×C4D4×D5312yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22—1062no—
446C22—1062no—
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
465C22#41×D5311yescharacteristic
466C22#41×D5311yescharacteristic
467C22#1313326211yescharacteristic
468C22#1313326311yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—1062no—
476C4—1062no—
477C4—1062no—
478C4—2121no—
479C4—2121no—
480C2—42no—
481C2—4241no—
482C2—2122no—
483C2—2122no—
484C2#1681113111yescharacteristic
485C2#1681113511yescharacteristic
486C2#1681120711yescharacteristic
487C2D4×D21211yescharacteristic
488C2#1681126811yescharacteristic
489C2#1681129111yescharacteristic
490C2#1681132411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2456
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
2. C2×#13133190C2semidirect
3. C2×C4×D212C2semidirect
4. C2×#13133177C2semidirect
5. #16810545C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810532C2semidirect
9. #16810524C2semidirect
10. #16810514C2semidirect
11. #16810508C2semidirect
12. #16810501C2semidirect
13. #16810500C2semidirect
14. #16810497C2semidirect
15. C53×#388C2semidirect
100. C22⋊C4×C53D4semidirect
113. C4⋊Dic53D4semidirect
256. #388D53semidirect
261. C53#2456semidirect
360. C22⋊C4D212semidirect