Finite Group Databasebeta

← #22613471

Subgroups — #22613471

Order 9344.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556563C211yesmaximal, characteristic, complemented
2C2×#14556563C211yesmaximal, characteristic, complemented
3C2×#14556560C211yesmaximal, characteristic, complemented
4C2×D4×Dic73C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic73C211yesmaximal, characteristic, complemented
6#17988510C211yesmaximal, characteristic, complemented
7#17988503C211yesmaximal, characteristic, complemented
8#17988501C211yesmaximal, characteristic, complemented
9#17988495C211yesmaximal, characteristic, complemented
10#17988487C211yesmaximal, characteristic, complemented
11#17988481C211yesmaximal, characteristic, complemented
12#17988467C211yesmaximal, characteristic, complemented
13#17988464C211yesmaximal, characteristic, complemented
14#17988459C211yesmaximal, characteristic, complemented
15C2×#120×C73C211yesmaximal, characteristic, complemented
16C22×C22⋊D73C2211yescharacteristic
17C22×C22⋊D73C2211yescharacteristic
18C22×C22⋊D73C2211yescharacteristic
19C22×C22⋊D73C2211yescharacteristic, complemented
20C23×Dic73C2211yescharacteristic
21C23×Dic73C2211yescharacteristic
22C23×Dic73C2211yescharacteristic, complemented
23#14556563—24no—
24#14556563—24no—
25#14556560—24no—
26D4×Dic73—24no—
27C2×C4⋊Dic73C2211yescharacteristic, complemented
28C2×#11924007C2211yescharacteristic, complemented
29C2×#11924007C2211yescharacteristic, complemented
30C2×#11924007C2211yescharacteristic, complemented
31C2×C22⋊Dic73—24no—
32C2×C22⋊Dic73C2211yescharacteristic
33C2×C22⋊Dic73C2211yescharacteristic
34C2×C22⋊Dic73C2211yescharacteristic
35C2×C22⋊Dic73C2211yescharacteristic
36C2×C22⋊Dic73C2211yescharacteristic
37C2×C22⋊Dic73C2211yescharacteristic
38C2×C22⋊Dic73C2211yescharacteristic
39C2×C22⋊Dic73C2211yescharacteristic, complemented
40C2×C22⋊Dic73C2211yescharacteristic, complemented
41C2×#11924005C2211yescharacteristic, complemented
42C2×#11924005C2211yescharacteristic, complemented
43C2×#11924005C2211yescharacteristic, complemented
44C2×C4×Dic73C2211yescharacteristic, complemented
45#14556498C2211yescharacteristic, complemented
46#14556498C2211yescharacteristic, complemented
47#14556498C2211yescharacteristic, complemented
48#14556498C2211yescharacteristic, complemented
49C25×C73C2211yescharacteristic
50C22×D4×C73C2211yescharacteristic
51C22×D4×C73C2211yescharacteristic
52C22×D4×C73C2211yescharacteristic
53#120×C73—24no—
54C2×C22⋊C4×C73C2211yescharacteristic
55C2×C22⋊C4×C73C2211yescharacteristic
56C2×C22⋊C4×C73C2211yescharacteristic
57C23×D73C2311yescharacteristic
58C2×C22⋊D73—22no—
59C2×C22⋊D73—22no—
60C2×C22⋊D73—22no—
61C2×C22⋊D73—22no—
62C2×C22⋊D73—22no—
63C2×C22⋊D73—22no—
64C2×C22⋊D73—22no—
65C2×C22⋊D73—22no—
66C2×C22⋊D73—22no—
67C2×C22⋊D73—41no—
68C2×C22⋊D73—41no—
69C2×C22⋊D73—41no—
70C22×Dic73—22no—
71C22×Dic73—22no—
72C22×Dic73—22no—
73C22×Dic73—22no—
74C22×Dic73—22no—
75C22×Dic73—22no—
76C22×Dic73—41no—
77C22×Dic73—41no—
78C22×Dic73—41no—
79C22×Dic73C2311yescharacteristic
80C22×Dic73C2311yescharacteristic
81C22×Dic73C2311yescharacteristic
82C22×Dic73C2311yescharacteristic
83C22×Dic73C2311yescharacteristic
84C22×Dic73C2311yescharacteristic
85C22×Dic73C2311yescharacteristic
86C4⋊Dic73—22no—
87#11924007—22no—
88#11924007—22no—
89#11924007—41no—
90C22⋊Dic73—22no—
91C22⋊Dic73—22no—
92C22⋊Dic73—22no—
93C22⋊Dic73—22no—
94C22⋊Dic73—22no—
95C22⋊Dic73—22no—
96C22⋊Dic73—22no—
97C22⋊Dic73D414yescomplemented
98C22⋊Dic73D414yescomplemented
99#11924005—22no—
100#11924005—22no—
101#11924005—41no—
102C4×Dic73—22no—
103C24×C73—22no—
104C24×C73—22no—
105C24×C73—22no—
106C24×C73—22no—
107C24×C73—22no—
108C24×C73—41no—
109C24×C73—41no—
110C24×C73C2311yescharacteristic
111C24×C73C2311yescharacteristic
112C24×C73C2311yescharacteristic
113C24×C73C2311yescharacteristic
114C2×D4×C73—22no—
115C2×D4×C73—22no—
116C2×D4×C73—22no—
117C2×D4×C73—22no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73—41no—
121C2×D4×C73—41no—
122C2×D4×C73D414yescomplemented
123C22×C4×C73C2311yescharacteristic
124C22×C4×C73C2311yescharacteristic
125C22×C4×C73C2311yescharacteristic
126C22⋊C4×C73—22no—
127C22⋊C4×C73—22no—
128C22⋊C4×C73—22no—
129C23×C73—21no—
130C23×C73—21no—
131C23×C73—21no—
132C23×C73—21no—
133C23×C73—28no—
134C23×C73—21no—
135C23×C73—21no—
136C23×C73—21no—
137C23×C73—21no—
138C23×C73—21no—
139C23×C73—21no—
140C23×C73—21no—
141C23×C73—21no—
142C23×C73—21no—
143C23×C73—21no—
144C23×C73—21no—
145C23×C73—21no—
146C23×C73—21no—
147C23×C73—21no—
148C23×C73—21no—
149C23×C73—21no—
150C23×C73—21no—
151C23×C73—21no—
152C23×C73—44no—
153C23×C73—44no—
154C23×C73—44no—
155C23×C73—44no—
156C23×C73—44no—
157C23×C73—44no—
158C23×C73C2×D412yes—
159C23×C73C2×D412yes—
160C23×C73C2×D412yes—
161C23×C73C2×D412yes—
162C23×C73#4112yes—
163C23×C73#4112yes—
164C23×C73C2411yescharacteristic
165C22×D73—21no—
166C22×D73—21no—
167C22×D73—21no—
168C22×D73—21no—
169C22×D73—21no—
170C22×D73—21no—
171C22×D73—21no—
172D4×C73—28no—
173D4×C73—44no—
174D4×C73—44no—
175C2×C4×C73—21no—
176C2×C4×C73—21no—
177C2×C4×C73—21no—
178C2×C4×C73—21no—
179C2×C4×C73—21no—
180C2×C4×C73—21no—
181C2×C4×C73—21no—
182C2×C4×C73—21no—
183C2×C4×C73C2×D412yes—
184C22⋊D73—44no—
185C22⋊D73—44no—
186C22⋊D73—44no—
187C22⋊D73—44no—
188C2×Dic73—21no—
189C2×Dic73—21no—
190C2×Dic73—21no—
191C2×Dic73—21no—
192C2×Dic73—21no—
193C2×Dic73—21no—
194C2×Dic73—21no—
195C2×Dic73—21no—
196C2×Dic73—21no—
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—21no—
204C2×Dic73—21no—
205C2×Dic73—44no—
206C2×Dic73—44no—
207C2×Dic73—44no—
208C2×Dic73C2×D412yes—
209C2×Dic73C2×D412yes—
210C2×Dic73C2×D412yes—
211C2×Dic73C2×D412yes—
212C22×C73—22no—
213C22×C73—22no—
214C22×C73—22no—
215C22×C73—22no—
216C22×C73—22no—
217C22×C73—22no—
218C22×C73—22no—
219C22×C73—22no—
220C22×C73—22no—
221C22×C73—22no—
222C22×C73—22no—
223C22×C73—22no—
224C22×C73—22no—
225C22×C73—22no—
226C22×C73—22no—
227C22×C73—416no—
228C22×C73—41no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—41no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—41no—
237C22×C73—41no—
238C22×C73—41no—
239C22×C73—41no—
240C22×C73#12114yescomplemented
241C22×C73C22×D411yescharacteristic
242C22×C73C22×D411yescharacteristic
243C22×C73C22×D411yescharacteristic
244C22×C73C2×#4111yescharacteristic
245C22×C73C2×#4111yescharacteristic
246C22×C73H(5,𝔽2)11yescharacteristic
247C22×C73H(5,𝔽2)11yescharacteristic
248C2×D73—41no—
249C2×D73—41no—
250C2×D73—41no—
251C2×D73—41no—
252C2×D73—41no—
253C2×D73—41no—
254C2×D73—41no—
255C4×C73—22no—
256C4×C73—41no—
257C4×C73—41no—
258Dic73—22no—
259Dic73—22no—
260Dic73—22no—
261Dic73—22no—
262Dic73—41no—
263Dic73—41no—
264Dic73—41no—
265C2×C73—24no—
266C2×C73—42no—
267C2×C73—42no—
268C2×C73—42no—
269C2×C73C2×#12111yescharacteristic
270C2×C73#52511yescharacteristic
271C2×C73#53811yescharacteristic
272C2×C73D4211yescharacteristic
273C2×C73D4211yescharacteristic
274C2×C73#54611yescharacteristic
275C2×C73#54611yescharacteristic
276D73—81no—
277#2424—731nomaximal, Sylow
278C73#242411yescharacteristic, complemented, Sylow
279C2×#121—731no—
280C2×#121—731no—
281C2×#120—731no—
282C2×#120D7311yescharacteristic, complemented
283C2×C4×D4—731no—
284C22×C22⋊C4—731no—
285#396—731no—
286#394—731no—
287#394—731no—
288#392—731no—
289#390—731no—
290#388—731no—
291#386—731no—
292#386—731no—
293#385—731no—
294C25C2×D7311yescharacteristic
295C22×D4—731no—
296C22×D4—731no—
297C22×D4—731no—
298C22×D4—731no—
299C22×D4C2×D7311yescharacteristic
300C22×D4C2×D7311yescharacteristic
301C22×D4C2×D7311yescharacteristic
302C23×C4—731no—
303C23×C4—731no—
304C23×C4—731no—
305#121—1464no—
306#121—1464no—
307#120—24no—
308#120—1464no—
309C4×D4—1464no—
310C2×C4⋊C4—731no—
311C2×C4⋊C4—731no—
312C2×C4⋊C4—731no—
313C2×C4⋊C4—731no—
314C2×C22⋊C4—731no—
315C2×C22⋊C4—731no—
316C2×C22⋊C4—731no—
317C2×C22⋊C4—731no—
318C2×C22⋊C4—731no—
319C2×C22⋊C4—731no—
320C2×C22⋊C4—731no—
321C2×C22⋊C4—731no—
322C2×C22⋊C4—731no—
323C2×C22⋊C4—731no—
324C2×C22⋊C4—731no—
325C2×C22⋊C4—731no—
326C2×C22⋊C4—1464no—
327C2×C22⋊C4C2×D7311yescharacteristic
328C2×C22⋊C4C2×D7311yescharacteristic
329C2×C22⋊C4C2×D7311yescharacteristic
330C2×C42—731no—
331#95—731no—
332#95—731no—
333#95—731no—
334#95—731no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—731no—
343C24C22×D7311yescharacteristic
344C24C22×D7311yescharacteristic
345C24C22×D7311yescharacteristic
346C24C22×D7311yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—2921no—
356C2×D4—2921no—
357C2×D4—2921no—
358C2×D4—1462no—
359C2×D4—1462no—
360C2×D4—1462no—
361C2×D4—1462no—
362C2×D4—1462no—
363C2×D4—1462no—
364C2×D4—1462no—
365C2×D4—1462no—
366C2×D4—1462no—
367C2×D4C22⋊D7314yescomplemented
368C22×C4—2921no—
369C22×C4—2921no—
370C22×C4—2921no—
371C22×C4—731no—
372C22×C4—731no—
373C22×C4—731no—
374C22×C4—731no—
375C22×C4—731no—
376C22×C4—731no—
377C22×C4—731no—
378C22×C4—1462no—
379C22×C4—1462no—
380C22×C4—1462no—
381C22×C4—1462no—
382C22×C4—1462no—
383C22×C4—1462no—
384C22×C4C22×D7311yescharacteristic
385C22×C4C22×D7311yescharacteristic
386C22×C4C22×D7311yescharacteristic
387C4⋊C4—2921no—
388C4⋊C4—1462no—
389C4⋊C4—1462no—
390C4⋊C4—1462no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—2921no—
395C22⋊C4—734no—
396C22⋊C4—734no—
397C22⋊C4—1462no—
398C22⋊C4—1462no—
399C22⋊C4—1462no—
400C22⋊C4—1462no—
401C22⋊C4—1462no—
402C22⋊C4—1462no—
403C22⋊C4—1462no—
404C22⋊C4—1462no—
405C22⋊C4—1462no—
406C42—1462no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—1461no—
437C23—1461no—
438C23—1461no—
439C23—1461no—
440C23—1461no—
441C23—1461no—
442C23—1461no—
443C23D4×D7312yes—
444C23D4×D7312yes—
445C23C2×C22⋊D7312yes—
446C23C2×C22⋊D7312yes—
447C23#1192402712yes—
448C23#1192402712yes—
449C23C23×D7311yescharacteristic
450D4—28no—
451D4—44no—
452D4—44no—
453D4—2924no—
454D4—2924no—
455D4—2924no—
456D4—2924no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—2924no—
466C2×C4—2924no—
467C2×C4—2924no—
468C2×C4—732no—
469C2×C4—732no—
470C2×C4—732no—
471C2×C4—732no—
472C2×C4—1461no—
473C2×C4—1461no—
474C2×C4—1461no—
475C2×C4—1461no—
476C2×C4—1461no—
477C2×C4—1461no—
478C2×C4—1461no—
479C2×C4—1461no—
480C2×C4—1461no—
481C2×C4—1461no—
482C2×C4—1461no—
483C2×C4—1461no—
484C2×C4—1461no—
485C2×C4—1461no—
486C2×C4—1461no—
487C2×C4—1461no—
488C2×C4—1461no—
489C2×C4C2×C22⋊D7312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—416no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—2921no—
519C22—2921no—
520C22—2921no—
521C22—2921no—
522C22—2921no—
523C22—2921no—
524C22—2921no—
525C22#1455656314yescomplemented
526C22C2×D4×D7311yescharacteristic
527C22C2×D4×D7311yescharacteristic
528C22C22×C22⋊D7311yescharacteristic
529C22C2×#1192402711yescharacteristic
530C22C2×#1192402711yescharacteristic
531C22#1455663611yescharacteristic
532C22#1455663611yescharacteristic
533C4—22no—
534C4—41no—
535C4—41no—
536C4—2921no—
537C4—2921no—
538C4—2921no—
539C4—1462no—
540C4—1462no—
541C4—1462no—
542C4—1462no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—42no—
547C2—5841no—
548C2C2×#1455656311yescharacteristic
549C2#1798909311yescharacteristic
550C2#1798917211yescharacteristic
551C2#1798921911yescharacteristic
552C2D4×C22⋊D7311yescharacteristic
553C2#1798922411yescharacteristic
554C2#1798923111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×#120×C73C2
hypercenterC2×#120D73
hypocenterC73#2424
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14556563C2semidirect
2. C2×#14556563C2semidirect
3. C2×#14556560C2semidirect
4. C2×D4×Dic73C2semidirect
5. C22×C22⋊Dic73C2semidirect
6. #17988510C2semidirect
7. #17988503C2semidirect
8. #17988501C2semidirect
9. #17988495C2semidirect
10. #17988487C2semidirect
11. #17988481C2semidirect
12. #17988467C2semidirect
13. #17988464C2semidirect
14. #17988459C2semidirect
15. C2×#120×C73C2semidirect
19. C22×C22⋊D73C22semidirect
22. C23×Dic73C22semidirect
27. C2×C4⋊Dic73C22semidirect
28. C2×#11924007C22semidirect
29. C2×#11924007C22semidirect
30. C2×#11924007C22semidirect
39. C2×C22⋊Dic73C22semidirect
40. C2×C22⋊Dic73C22semidirect
41. C2×#11924005C22semidirect
42. C2×#11924005C22semidirect
43. C2×#11924005C22semidirect
44. C2×C4×Dic73C22semidirect
45. #14556498C22semidirect
46. #14556498C22semidirect
47. #14556498C22semidirect
48. #14556498C22semidirect
97. C22⋊Dic73D4semidirect
98. C22⋊Dic73D4semidirect
122. C2×D4×C73D4semidirect
240. C22×C73#121semidirect
278. C73#2424semidirect
282. C2×#120D73semidirect
367. C2×D4C22⋊D73semidirect
525. C22#14556563semidirect