Finite Group Databasebeta

← #34034817

Subgroups — #34034817

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989708C211yesmaximal, characteristic, complemented
2C2×#17989706C211yesmaximal, characteristic, complemented
3#22627146C211yesmaximal, characteristic, complemented
4#22627145C211yesmaximal, characteristic, complemented
5#22627144C211yesmaximal, characteristic, complemented
6#22627142C211yesmaximal, characteristic, complemented
7#22627142C211yesmaximal, characteristic, complemented
8#22627141C211yesmaximal, characteristic, complemented
9#22627140C211yesmaximal, characteristic, complemented
10#22627139C211yesmaximal, characteristic, complemented
11#22627122C211yesmaximal, characteristic, complemented
12#22627101C211yesmaximal, characteristic, complemented
13C2×#17989684C211yesmaximal, characteristic, complemented
14#22627092C211yesmaximal, characteristic, complemented
15#22626648C211yesmaximal, characteristic
16#17989708—24no—
17C2×#14556688C2211yescharacteristic
18C2×#14556688C2211yescharacteristic
19C2×#14556688C2211yescharacteristic, complemented
20#17989706—24no—
21C2×#14556687C2211yescharacteristic
22C2×#14556687C2211yescharacteristic
23C2×#14556687C2211yescharacteristic, complemented
24C2×#14556686C2211yescharacteristic
25C2×#14556686C2211yescharacteristic
26C2×#14556686C2211yescharacteristic, complemented
27C22×#11924040C2211yescharacteristic
28C22×#11924040C2211yescharacteristic
29C22×#11924040C2211yescharacteristic, complemented
30#17989690C2211yescharacteristic, complemented
31#17989690C2211yescharacteristic, complemented
32C4⋊#11924040C2211yescharacteristic, complemented
33C4⋊#11924040C2211yescharacteristic, complemented
34#17989684C2211yescharacteristic
35#17989684C2211yescharacteristic
36#17989684C2211yescharacteristic
37#17989684C2211yescharacteristic
38#17989684C2211yescharacteristic, complemented
39#17989684C2211yescharacteristic, complemented
40C22⋊#11924040C2211yescharacteristic, complemented
41C22⋊#11924040C2211yescharacteristic, complemented
42#17989681C2211yescharacteristic, complemented
43#17989681C2211yescharacteristic, complemented
44C4×#11924040C2211yescharacteristic, complemented
45C4×#11924040C2211yescharacteristic, complemented
46C2×#14556678C2211yescharacteristic
47C22×#11924036C2211yescharacteristic
48#17989619C411yescharacteristic
49#17989619C411yescharacteristic
50#17989617C411yescharacteristic
51#17989617C411yescharacteristic
52#17989615C2211yescharacteristic
53#17989615C2211yescharacteristic
54#17989614C2211yescharacteristic
55#17989614C2211yescharacteristic
56#17989593C411yescharacteristic
57#17989560C411yescharacteristic
58#17989559C411yescharacteristic
59#17989551C2211yescharacteristic
60#17989095C411yescharacteristic
61#14556688—22no—
62#14556688—22no—
63#14556688—22no—
64#14556687—22no—
65#14556687—22no—
66#14556687D414yescomplemented
67#14556686—22no—
68#14556686—22no—
69#14556686—22no—
70C2×#11924040—22no—
71C2×#11924040—22no—
72C2×#11924040—22no—
73C2×#11924040C2311yescharacteristic
74C2×#11924040C2311yescharacteristic
75C2×#11924040C2311yescharacteristic
76C2×#11924040C2311yescharacteristic
77C2×#11924040C2311yescharacteristic
78C2×#11924040C2311yescharacteristic
79C2×#11924040C2311yescharacteristic
80C2×#11924040C2311yescharacteristic
81#14556678—22no—
82#14556678—22no—
83C2×#11924038C2×C411yescharacteristic
84C2×#11924037C2×C411yescharacteristic
85C2×#11924037C2×C411yescharacteristic
86C2×#11924037C2×C411yescharacteristic
87C2×#11924036—22no—
88C2×#11924036—22no—
89C2×#11924036C2×C411yescharacteristic
90C2×#11924036C2×C411yescharacteristic
91C2×#11924036C2311yescharacteristic
92C2×#11924036C2311yescharacteristic
93C2×#11924036C2311yescharacteristic
94C22×#10596833C2×C411yescharacteristic
95C22×#10596833C2311yescharacteristic
96C22×#10596833C2311yescharacteristic
97#14556658C2×C411yescharacteristic
98#14556658C2×C411yescharacteristic
99C4⋊#10596833C2×C411yescharacteristic
100C4⋊#10596833C2×C411yescharacteristic
101#14556648C2×C411yescharacteristic
102#14556648C2×C411yescharacteristic
103#14556648C2×C411yescharacteristic
104#14556648C2×C411yescharacteristic
105#14556648C2×C411yescharacteristic
106#14556648C2×C411yescharacteristic
107C22⋊#10596833C2×C411yescharacteristic
108C22⋊#10596833C2×C411yescharacteristic
109#14556646C2×C411yescharacteristic
110#14556646C2×C411yescharacteristic
111C4×#10596833C2311yescharacteristic
112C4×#10596833C2311yescharacteristic
113C2×#11924027C2×C411yescharacteristic
114C22×C4×D73C2×C411yescharacteristic
115#14556574C811yescharacteristic
116#14556569C811yescharacteristic
117#14556569C811yescharacteristic
118#14556564C811yescharacteristic
119#14556564C811yescharacteristic
120#14556561C811yescharacteristic
121#14556559C811yescharacteristic
122#14556553C2×C411yescharacteristic
123#14556553C2×C411yescharacteristic
124#14556551C2×C411yescharacteristic
125#14556551C2×C411yescharacteristic
126#14556547C2×C411yescharacteristic
127#123×C73C811yescharacteristic
128#11924040—21no—
129#11924040—21no—
130#11924040—21no—
131#11924040—21no—
132#11924040—21no—
133#11924040—21no—
134#11924040C2×D412yes—
135#11924040C2×D412yes—
136#11924038—22no—
137#11924037—22no—
138#11924037—22no—
139#11924037—22no—
140#11924036—22no—
141#11924036—22no—
142#11924036—21no—
143#11924036—21no—
144#11924036—21no—
145#11924036—21no—
146#11924036—21no—
147#11924036—21no—
148C2×#10596833—22no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833C22×C411yescharacteristic
152C2×#10596833C22×C411yescharacteristic
153C2×#10596833C22×C411yescharacteristic
154C2×#10596833C22×C411yescharacteristic
155C2×#10596833C22×C411yescharacteristic
156C2×#10596833C22×C411yescharacteristic
157C2×#10596833C22×C411yescharacteristic
158C2×#10596833C2×D412yes—
159C2×#10596833#4112yes—
160C2×#10596833C2411yescharacteristic
161C23×D73C2×C811yescharacteristic
162#11924027—22no—
163#11924027—22no—
164C2×Dic146C2×C811yescharacteristic
165C2×D292C2×C811yescharacteristic
166C2×D292C2×C811yescharacteristic
167C2×C22⋊D73C2×C811yescharacteristic
168C2×C22⋊D73C2×C811yescharacteristic
169C2×C22⋊D73C2×C811yescharacteristic
170C2×C22⋊D73C2×C811yescharacteristic
171C22×Dic73C22×C411yescharacteristic
172C22×Dic73C22×C411yescharacteristic
173C2×C4×D73—22no—
174C2×C4×D73—22no—
175C2×C4×D73C2×C811yescharacteristic
176C2×C4×D73C2×C811yescharacteristic
177C2×C4×D73C22×C411yescharacteristic
178C2×C4×D73C22×C411yescharacteristic
179C2×C4×D73C22×C411yescharacteristic
180#11924007C2×C811yescharacteristic
181#11924007C2×C811yescharacteristic
182#11924007C2×C811yescharacteristic
183#11924007C2×C811yescharacteristic
184C22⋊Dic73C2×C811yescharacteristic
185#11924005C2×C811yescharacteristic
186#11924005C2×C811yescharacteristic
187#11924005C2×C811yescharacteristic
188#11924005C2×C811yescharacteristic
189#11924005C2×C811yescharacteristic
190#11924005C2×C811yescharacteristic
191C4×Dic73C22×C411yescharacteristic
192C4×Dic73C22×C411yescharacteristic
193C2×D4×C73C2×C811yescharacteristic
194C22×C4×C73C2×C811yescharacteristic
195C4⋊C4×C73C2×C811yescharacteristic
196C4⋊C4×C73C2×C811yescharacteristic
197C22⋊C4×C73C2×C811yescharacteristic
198C22⋊C4×C73C2×C811yescharacteristic
199C22⋊C4×C73C2×C811yescharacteristic
200C23×C73C22×C811yescharacteristic
201C23×C73C22×C811yescharacteristic
202C22×D73—22no—
203C22×D73—22no—
204C22×D73—41no—
205C22×D73C22×C811yescharacteristic
206C22×D73C22×C811yescharacteristic
207C22×D73C22×C811yescharacteristic
208D4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73C22×C811yescharacteristic
211C2×C4×C73C22×C811yescharacteristic
212C2×C4×C73C22×C811yescharacteristic
213C2×C4×C73C22×C811yescharacteristic
214C2×C4×C73C22×C811yescharacteristic
215C22⋊D73—22no—
216C22⋊D73—22no—
217C22⋊D73—22no—
218C22⋊D73—41no—
219C2×Dic73—21no—
220C2×Dic73—21no—
221C2×Dic73C4×D412yes—
222C2×Dic73C22×C811yescharacteristic
223C2×Dic73C22×C811yescharacteristic
224C2×Dic73C22×C811yescharacteristic
225C2×Dic73C22×C811yescharacteristic
226C2×Dic73#13112yes—
227C2×Dic73C23×C411yescharacteristic
228D292—22no—
229D292—41no—
230C4×D73—21no—
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—21no—
234C4×D73—21no—
235C4×D73—21no—
236C4×D73—22no—
237C4×D73—22no—
238Dic146—22no—
239#10596833—21no—
240#10596833—21no—
241#10596833—21no—
242#10596833—21no—
243#10596833—21no—
244#10596833—21no—
245#10596833C4×D412yes—
246#10596833C22×D411yescharacteristic
247#10596833C2×#4111yescharacteristic
248C22×C73—21no—
249C22×C73—21no—
250C22×C73—21no—
251C22×C73—21no—
252C22×C73—21no—
253C22×C73C8×D412yes—
254C22×C73C23×C811yescharacteristic
255C2×D73—21no—
256C2×D73—21no—
257C2×D73—21no—
258C2×D73—21no—
259C2×D73—21no—
260C2×D73—21no—
261C2×D73—21no—
262C2×D73—21no—
263C2×D73—44no—
264C2×D73#50412yescomplemented
265C4×C73—21no—
266C4×C73—21no—
267C4×C73—21no—
268C4×C73—21no—
269C4×C73—21no—
270Dic73—21no—
271Dic73—21no—
272Dic73—21no—
273Dic73C8×D412yes—
274Dic73C2×C4×D411yescharacteristic
275Dic73C2×#13111yescharacteristic
276#33436—731nomaximal, Sylow
277C2×C73—22no—
278C2×C73—41no—
279C2×C73C2×C8×D411yescharacteristic
280C2×C73C2×#50411yescharacteristic
281C2×C73#340711yescharacteristic
282D73—22no—
283D73—41no—
284D73—41no—
285C2×#504—731no—
286C22×M32—731no—
287#2929—731no—
288#2169—731no—
289#2169—731no—
290#2168—731no—
291#2168—731no—
292#2168—731no—
293#2168—731no—
294D4×C16—731no—
295D4×C16—731no—
296#2151—731no—
297#2113—731no—
298C2×C22⋊C16—731no—
299#2108—731no—
300C73#3343611yescharacteristic, complemented, Sylow
301C2×#131—731no—
302C23×C8—731no—
303#525—731no—
304#504—1464no—
305C2×M32—731no—
306C2×M32—731no—
307C2×M32—731no—
308C2×M32—731no—
309C2×M32—731no—
310C2×M32—731no—
311C2×M32—1464no—
312C22×C16—731no—
313C22×C16—731no—
314C22×C16—731no—
315C22×C16—731no—
316C22×C16—731no—
317C22×C16—731no—
318#435—731no—
319#435—731no—
320#435—731no—
321#435—731no—
322C8×D4—731no—
323C8×D4—731no—
324C8×D4—731no—
325C8×D4—731no—
326#424—731no—
327#408—731no—
328#407—731no—
329#405—731no—
330C4⋊C16—731no—
331C4⋊C16—731no—
332C4⋊C16—731no—
333C4⋊C16—731no—
334C22⋊C16—731no—
335C22⋊C16—731no—
336C22⋊C16—731no—
337C22⋊C16—731no—
338C22⋊C16—731no—
339C22⋊C16—731no—
340C22⋊C16—731no—
341C22⋊C16—731no—
342#346—731no—
343#346—731no—
344C4×C16—731no—
345C4×C16—731no—
346C2×#41—731no—
347C23×C4—731no—
348#131—1462no—
349#131—1462no—
350C2×M16—731no—
351C2×M16—731no—
352C2×M16—731no—
353C2×M16—731no—
354C22×C8—731no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—1462no—
363C22×C8—1462no—
364#123—731no—
365#123#1059684211yescharacteristic
366C22⋊Q8—731no—
367C22⋊Q8—731no—
368#121—731no—
369#121—731no—
370#120—731no—
371#120—731no—
372C4×D4—731no—
373C4×D4—731no—
374C4×D4—731no—
375C4×D4—731no—
376#117—731no—
377M32—734no—
378M32—1462no—
379M32—1462no—
380M32—1462no—
381M32—1462no—
382M32—1462no—
383C2×C16—731no—
384C2×C16—731no—
385C2×C16—731no—
386C2×C16—731no—
387C2×C16—731no—
388C2×C16—731no—
389C2×C16—731no—
390C2×C16—731no—
391C2×C16—1462no—
392C2×C16—1462no—
393C2×C16—1462no—
394C2×C16—1462no—
395C2×C16—1462no—
396C2×C16—1462no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—731no—
401C22⋊C8—731no—
402C22⋊C8—731no—
403C22⋊C8—731no—
404C22⋊C8—731no—
405C22⋊C8—731no—
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—731no—
409#97—731no—
410#97—731no—
411C4×C8—731no—
412C4×C8—731no—
413C24—731no—
414#41—1462no—
415#41—1462no—
416C2×Q8—731no—
417C2×D4—731no—
418C2×D4—731no—
419C2×D4—731no—
420C2×D4—731no—
421C2×D4—731no—
422C2×D4—731no—
423C2×D4C2×#1059684211yescharacteristic
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4C2×#1059684211yescharacteristic
434M16—1462no—
435M16—1462no—
436M16—1462no—
437M16—1462no—
438C2×C8—731no—
439C2×C8—732no—
440C2×C8—732no—
441C2×C8—731no—
442C2×C8—731no—
443C2×C8—731no—
444C2×C8—731no—
445C2×C8—731no—
446C2×C8—731no—
447C2×C8—731no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1462no—
456C2×C8—1462no—
457C2×C8—1461no—
458C2×C8—1462no—
459C4⋊C4—731no—
460C4⋊C4—731no—
461C4⋊C4—731no—
462C4⋊C4—731no—
463C4⋊C4C2×#1059684211yescharacteristic
464C4⋊C4C2×#1059684211yescharacteristic
465C22⋊C4—731no—
466C22⋊C4—731no—
467C22⋊C4—731no—
468C22⋊C4—731no—
469C22⋊C4—731no—
470C22⋊C4—731no—
471C22⋊C4—731no—
472C22⋊C4C2×#1059684211yescharacteristic
473C22⋊C4C2×#1059684211yescharacteristic
474C22⋊C4C2×#1059684211yescharacteristic
475C42—731no—
476C42—731no—
477C16—732no—
478C16—732no—
479C16—1461no—
480C16—1461no—
481C16—1461no—
482C16—1461no—
483C16—1461no—
484C16—1461no—
485C23—2921no—
486C23—731no—
487C23—731no—
488C23—731no—
489C23—1462no—
490C23—1462no—
491C23C22×#1059684211yescharacteristic
492C23C22×#1059684211yescharacteristic
493Q8—1462no—
494D4—22no—
495D4—2921no—
496D4—2921no—
497D4—1462no—
498D4—1462no—
499D4—1462no—
500D4—1462no—
501C2×C4—22no—
502C2×C4—732no—
503C2×C4—732no—
504C2×C4—731no—
505C2×C4—731no—
506C2×C4—731no—
507C2×C4—731no—
508C2×C4—731no—
509C2×C4—1461no—
510C2×C4—1461no—
511C2×C4—1461no—
512C2×C4—1461no—
513C2×C4—1461no—
514C2×C4—1462no—
515C2×C4—1462no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4C22×#1059684211yescharacteristic
520C2×C4C22×#1059684211yescharacteristic
521C2×C4C22×#1059684211yescharacteristic
522C2×C4C22×#1059684211yescharacteristic
523C2×C4C22×#1059684211yescharacteristic
524C8—731no—
525C8—731no—
526C8—732no—
527C8—1461no—
528C8—1461no—
529C8—1461no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—2924no—
539C22—732no—
540C22—1461no—
541C22—1461no—
542C22—1461no—
543C22—1461no—
544C22—1461no—
545C22—1461no—
546C22—1461no—
547C22—1461no—
548C22D4×#1059684212yes—
549C22C23×#1059684211yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—731no—
556C4—731no—
557C4—732no—
558C4—1461no—
559C4—1461no—
560C4—1461no—
561C2—22no—
562C2—41no—
563C2—2921no—
564C2—2921no—
565C2—1462no—
566C2C2×D4×#1059684211yescharacteristic
567C2#2262718211yescharacteristic
568C2#2262718511yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10596842
commutator, socleC22×C73C23×C8
Fitting#123×C73C8
prime radical#17989095C4
cosocleC2×#10596833C24
hypercenter#123#10596842
hypocenterC73#33436
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#17989708C2semidirect
2. C2×#17989706C2semidirect
3. #22627146C2semidirect
4. #22627145C2semidirect
5. #22627144C2semidirect
6. #22627142C2semidirect
7. #22627142C2semidirect
8. #22627141C2semidirect
9. #22627140C2semidirect
10. #22627139C2semidirect
11. #22627122C2semidirect
12. #22627101C2semidirect
13. C2×#17989684C2semidirect
14. #22627092C2semidirect
19. C2×#14556688C22semidirect
23. C2×#14556687C22semidirect
26. C2×#14556686C22semidirect
29. C22×#11924040C22semidirect
30. #17989690C22semidirect
31. #17989690C22semidirect
32. C4⋊#11924040C22semidirect
33. C4⋊#11924040C22semidirect
38. #17989684C22semidirect
39. #17989684C22semidirect
40. C22⋊#11924040C22semidirect
41. C22⋊#11924040C22semidirect
42. #17989681C22semidirect
43. #17989681C22semidirect
44. C4×#11924040C22semidirect
45. C4×#11924040C22semidirect
66. #14556687D4semidirect
264. C2×D73#504semidirect
300. C73#33436semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179897082
C2×#179897062
#226271462
#226271452
#226271442
#226271422
#226271412
#226271402
#226271392
#226271222
#226271012
C2×#179896842
#226270922
#226266482
#3343673