Finite Group Databasebeta

← #20280958

Subgroups — #20280958

Order 6784.

682 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×#13133202C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3C2×#13133192C211yesmaximal, characteristic, complemented
4C2×#13133192C211yesmaximal, characteristic, complemented
5#16810558C211yesmaximal, characteristic, complemented
6#16810541C211yesmaximal, characteristic, complemented
7#16810541C211yesmaximal, characteristic, complemented
8#16810540C211yesmaximal, characteristic, complemented
9#16810533C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810494C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14#16810482C211yesmaximal, characteristic, complemented
15C53×#398C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×D212C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25#13133202—24no—
26#13133192—24no—
27#13133192—24no—
28#13133192—24no—
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44C2×#11719921C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C23×D53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×D212—22no—
71C2×D212—22no—
72C2×D212—22no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212D414yescomplemented
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101#11719921—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—41no—
110#11719921—41no—
111#11719921—41no—
112#11719921—41no—
113#11719921D414yescomplemented
114#11719919—22no—
115#11719919—41no—
116#11719919—41no—
117C23×C53C2411yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
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×D53C2×D412yes—
144C22×D53C2×D412yes—
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—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53C2×D412yes—
165C2×C4×C53C2×D412yes—
166C2×C4×C53#4112yes—
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×Dic53C2×D412yes—
179D212—44no—
180D212—44no—
181D212—44no—
182D212—44no—
183D212—44no—
184D212—44no—
185C4×D53—44no—
186C4×D53—44no—
187C4×D53—44no—
188C22×C53C22×D411yescharacteristic
189C22×C53C22×D411yescharacteristic
190C22×C53C2×#4111yescharacteristic
191C22×C53H(5,𝔽2)11yescharacteristic
192C22×C53H(5,𝔽2)11yescharacteristic
193C22×C53H(5,𝔽2)11yescharacteristic
194C22×C532−1+411yescharacteristic
195C2×D53—22no—
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—22no—
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—41no—
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—41no—
227C4×C53—41no—
228C4×C53—41no—
229C4×C53—41no—
230Dic53—22no—
231Dic53—41no—
232Dic53—41no—
233Dic53—41no—
234#2671—531nomaximal, Sylow
235C2×C53#53411yescharacteristic
236C2×C53#53711yescharacteristic
237C2×C53#54011yescharacteristic
238C2×C53D4211yescharacteristic
239C2×C53#54611yescharacteristic
240C2×C53#54711yescharacteristic
241C2×C53#56211yescharacteristic
242D53—42no—
243D53—42no—
244D53—42no—
245D53—81no—
246C2×#123—531no—
247C2×#121—531no—
248C2×#121—531no—
249C2×#121—531no—
250#399—531no—
251#398D5311yescharacteristic, complemented
252#394—531no—
253#394—531no—
254#394—531no—
255#392—531no—
256#386—531no—
257#386—531no—
258#385—531no—
259#385—531no—
260#380—531no—
261C53#267111yescharacteristic, 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—
273#121—1064no—
274#121—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4C2×D5311yescharacteristic
279C2×C4⋊C4C2×D5311yescharacteristic
280C2×C4⋊C4C2×D5311yescharacteristic
281C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4—531no—
293C2×C22⋊C4—531no—
294C2×C22⋊C4—531no—
295#95—531no—
296#95—531no—
297#95—531no—
298#95C2×D5311yescharacteristic
299#95C2×D5311yescharacteristic
300#95C2×D5311yescharacteristic
301C24—531no—
302C24—531no—
303C24—531no—
304C24—531no—
305C2×D4—534no—
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—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
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—1062no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4C22×D5311yescharacteristic
337C22×C4C22×D5311yescharacteristic
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—1062no—
348C4⋊C4—2121no—
349C4⋊C4—2121no—
350C22⋊C4—534no—
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⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C23—532no—
364C23—532no—
365C23—532no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
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—
390C23—1061no—
391C23C23×D5311yescharacteristic
392D4—2124no—
393D4—2124no—
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—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—532no—
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×C4D4×D5312yes—
432C2×C4D4×D5312yes—
433C2×C4#1171994912yes—
434C22—1062no—
435C22—1062no—
436C22—1062no—
437C22—1062no—
438C22—1062no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22—2121no—
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
463C22C2×D4×D5311yescharacteristic
464C22C2×#1171994911yescharacteristic
465C22#1313326311yescharacteristic
466C22#1313326311yescharacteristic
467C22#1313326311yescharacteristic
468C22#1313326511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—1062no—
477C4—2121no—
478C4—2121no—
479C4—2121no—
480C2—4241no—
481C2—2122no—
482C2—2122no—
483C2—2122no—
484C2#1681118411yescharacteristic
485C2#1681120311yescharacteristic
486C2#1681122911yescharacteristic
487C2#1681125711yescharacteristic
488C2#1681126411yescharacteristic
489C2#1681127611yescharacteristic
490C2#1681137111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2671
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133192C2semidirect
3. C2×#13133192C2semidirect
4. C2×#13133192C2semidirect
5. #16810558C2semidirect
6. #16810541C2semidirect
7. #16810541C2semidirect
8. #16810540C2semidirect
9. #16810533C2semidirect
10. #16810501C2semidirect
11. #16810501C2semidirect
12. #16810494C2semidirect
13. #16810494C2semidirect
14. #16810482C2semidirect
15. C53×#398C2semidirect
87. C2×D212D4semidirect
113. #11719921D4semidirect
251. #398D53semidirect
261. C53#2671semidirect