Finite Group Databasebeta

← #33053607

Subgroups — #33053607

Order 18688.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618589C211yesmaximal, characteristic, complemented
2#22612816C211yesmaximal, characteristic, complemented
3#22608503C211yesmaximal, characteristic, complemented
4#22608495C212yesmaximal, complemented
5#22608474C211yesmaximal, characteristic
6#22608469C212yesmaximal, complemented
7#22608451C211yesmaximal, characteristic, complemented
8#22608443C212yesmaximal, complemented
9#22608422C211yesmaximal, characteristic
10#22608417C212yesmaximal, complemented
11C73×#3495C211yesmaximal, characteristic
12#17988648—22no—
13#17988644C2212yes—
14#17988640C2212yes—
15#17988614—21no—
16#17988551C2211yescharacteristic
17C2×C22⋊#10596831C2211yescharacteristic
18#17988543—21no—
19#17988486—21no—
20#17988480—22no—
21#17988453C2212yes—
22#17988447C2211yescharacteristic
23C4⋊C4×Dic73—21no—
24C4×C22⋊Dic73C2211yescharacteristic
25#17988433C2212yes—
26#17988324C2212yes—
27#17988324C2212yes—
28#17988324C2211yescharacteristic
29#17988324C2211yescharacteristic
30#17988324C2212yes—
31#17988324C2212yes—
32#17988324C2212yes—
33#17988321C2211yescharacteristic
34#17988321C2212yes—
35#17988321C2211yescharacteristic
36#556×C73—21no—
37#554×C73—21no—
38#552×C73—22no—
39#552×C73—21no—
40#552×C73—21no—
41#551×C73—22no—
42#544×C73C411yescharacteristic, complemented
43#542×C73C412yescomplemented
44#541×C73C412yescomplemented
45#536×C73C411yescharacteristic, complemented
46#533×C73C2212yes—
47#529×C73C2212yes—
48#525×C73C2211yescharacteristic
49C2×C22⋊Q8×C73C2211yescharacteristic
50#520×C73C411yescharacteristic, complemented
51#519×C73C411yescharacteristic, complemented
52C2×#117×C73C2211yescharacteristic
53C2×#11924012—21no—
54C22×#10596831—21no—
55C2×C4⋊Dic73—21no—
56C2×C4⋊Dic73—22no—
57C2×C22⋊Dic73—21no—
58C2×C22⋊Dic73—21no—
59C2×C4×Dic73—22no—
60C2×C4×Dic73—21no—
61C4⋊#10596831—22no—
62C4⋊#10596831—22no—
63C22⋊#10596831—21no—
64C22⋊#10596831—21no—
65C22⋊#10596831C2311yescharacteristic
66C22⋊#10596831C2312yes—
67C22⋊#10596831C2311yescharacteristic
68#14556502—22no—
69C4×#10596831—22no—
70#14556498—22no—
71#14556498—22no—
72#14556498C2311yescharacteristic
73#14556498C2312yes—
74#14556498C2311yescharacteristic
75C2×#41×C73C2×C411yescharacteristic
76C22×Q8×C73C2×C411yescharacteristic
77C23×C4×C73C2311yescharacteristic
78C4⋊Q8×C73—21no—
79C4⋊Q8×C73C2×C412yes—
80C4⋊Q8×C73C2×C412yes—
81#126×C73—22no—
82#126×C73—22no—
83#126×C73—22no—
84#126×C73—22no—
85#125×C73—22no—
86#125×C73—22no—
87#125×C73—21no—
88#125×C73—22no—
89#125×C73C2×C412yes—
90#124×C73—22no—
91#124×C73C2×C412yes—
92#123×C73—21no—
93#123×C73—22no—
94#123×C73—22no—
95#123×C73—22no—
96C22⋊Q8×C73—21no—
97C22⋊Q8×C73—21no—
98C22⋊Q8×C73—22no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—21no—
101C22⋊Q8×C73—22no—
102C22⋊Q8×C73—22no—
103C22⋊Q8×C73C2×C412yes—
104C22⋊Q8×C73C2×C411yescharacteristic
105C22⋊Q8×C73C2312yes—
106C22⋊Q8×C73C2311yescharacteristic
107#121×C73C2×C411yescharacteristic
108#121×C73C2311yescharacteristic
109#120×C73—21no—
110C4×Q8×C73—21no—
111C4×Q8×C73—21no—
112C4×Q8×C73—22no—
113C4×Q8×C73C2×C412yes—
114C4×Q8×C73C2×C412yes—
115C4×Q8×C73C2×C412yes—
116C4×D4×C73—21no—
117C4×D4×C73—21no—
118C4×D4×C73—22no—
119C4×D4×C73—22no—
120C4×D4×C73C2×C412yes—
121#117×C73—22no—
122#117×C73—21no—
123#117×C73—22no—
124#117×C73—22no—
125#117×C73C2×C411yescharacteristic
126#117×C73C2×C411yescharacteristic
127#117×C73C2×C411yescharacteristic
128#117×C73C2×C412yes—
129#117×C73D411yescharacteristic
130#117×C73D412yes—
131#117×C73D411yescharacteristic
132C2×C4⋊C4×C73—21no—
133C2×C4⋊C4×C73—21no—
134C2×C4⋊C4×C73C2×C411yescharacteristic
135C2×C4⋊C4×C73D411yescharacteristic
136C2×C4⋊C4×C73D411yescharacteristic
137C2×C22⋊C4×C73—21no—
138C2×C22⋊C4×C73D412yes—
139C2×C42×C73C2312yes—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—22no—
143#11924012—42no—
144C2×#10596831—21no—
145C2×#10596831—22no—
146C2×#10596831—21no—
147C2×#10596831—42no—
148C4⋊Dic73—42no—
149C4⋊Dic73—82no—
150C22⋊Dic73—42no—
151C22⋊Dic73—81no—
152C4×Dic73—44no—
153C4×Dic73—42no—
154C24×C73C2×D411yescharacteristic
155#41×C73—42no—
156C2×Q8×C73—21no—
157C2×Q8×C73—22no—
158C2×Q8×C73—42no—
159C2×Q8×C73C22×C412yes—
160C2×Q8×C73C22×C411yescharacteristic
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73C22×C411yescharacteristic
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—22no—
169C22×C4×C73—42no—
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—21no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73C22⋊C411yescharacteristic
192C4⋊C4×C73C22⋊C411yescharacteristic
193C4⋊C4×C73C22⋊C412yes—
194C4⋊C4×C73C22⋊C411yescharacteristic
195C4⋊C4×C73C22⋊C411yescharacteristic
196C4⋊C4×C73C22⋊C412yes—
197C4⋊C4×C73C22×C412yes—
198C4⋊C4×C73C22×C411yescharacteristic
199C4⋊C4×C73C22×C411yescharacteristic
200C22⋊C4×C73—22no—
201C22⋊C4×C73—21no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—21no—
205C22⋊C4×C73—22no—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—21no—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—21no—
210C22⋊C4×C73C22⋊C412yes—
211C22⋊C4×C73C22⋊C412yes—
212C22⋊C4×C73C22⋊C412yes—
213C22⋊C4×C73C22⋊C412yes—
214C42×C73—22no—
215C42×C73—22no—
216C42×C73—22no—
217C42×C73—21no—
218C42×C73C22×C412yes—
219C42×C73C22×C412yes—
220C42×C73C2×D412yes—
221C42×C73C2×D412yes—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73C2×C22⋊C411yescharacteristic
228C23×C73C2×C22⋊C411yescharacteristic
229C23×C73C22×D411yescharacteristic
230Q8×C73—42no—
231Q8×C73—41no—
232Q8×C73—42no—
233Q8×C73—82no—
234D4×C73—41no—
235D4×C73—42no—
236D4×C73—41no—
237D4×C73—41no—
238D4×C73—81no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73—42no—
255C2×C4×C73—82no—
256C2×C4×C73—81no—
257C2×C4×C73C2×C22⋊C412yes—
258C2×C4×C73C2×C22⋊C412yes—
259C2×C4×C73C2×C22⋊C412yes—
260C2×C4×C73C2×C22⋊C412yes—
261C2×C4×C73C2×C22⋊C411yescharacteristic
262C2×C4×C73C2×C22⋊C411yescharacteristic
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C22×D411yescharacteristic
265C2×Dic73—22no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—42no—
270C2×Dic73—41no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—41no—
274#10596831—41no—
275#10596831—41no—
276#10596831—42no—
277C22×C73—21no—
278C22×C73—21no—
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73C22×C22⋊C411yescharacteristic
291C22×C73#57711yescharacteristic
292C22×C73#57811yescharacteristic
293C4×C73—21no—
294C4×C73—21no—
295C4×C73—21no—
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—41no—
301C4×C73—42no—
302C4×C73—41no—
303C4×C73—42no—
304C4×C73—41no—
305C4×C73#41012yes—
306Dic73—42no—
307Dic73—81no—
308Dic73—82no—
309#17605—731nomaximal, Sylow
310C2×C73—21no—
311C2×C73—21no—
312C2×C73—41no—
313C2×C73—41no—
314C2×C73—81no—
315C2×C73C2×#41011yescharacteristic
316C2×C73#289811yescharacteristic
317C2×C73#290111yescharacteristic
318#3495D7311yescharacteristic
319#2924—731no—
320#2346—731no—
321#1520—731no—
322#1518—732no—
323#1513—731no—
324#1512—732no—
325#1508—731no—
326#1506—732no—
327#1501—731no—
328#1500—732no—
329C73#1760511yescharacteristic, complemented, Sylow
330#556—21no—
331#554—21no—
332#552—22no—
333#552—21no—
334#552—21no—
335#551—22no—
336#544Dic7311yescharacteristic, complemented
337#542Dic7312yescomplemented
338#541Dic7312yescomplemented
339#536Dic7311yescharacteristic, complemented
340#533C2×D7312yes—
341#529C2×D7312yes—
342#525C2×D7311yescharacteristic
343C2×C22⋊Q8C2×D7311yescharacteristic
344#520Dic7311yescharacteristic, complemented
345#519Dic7311yescharacteristic, complemented
346C2×#117C2×D7311yescharacteristic
347#433—1462no—
348#432—732no—
349#431—732no—
350#424—1461no—
351#407—731no—
352C2×C22⋊C8—731no—
353#405—1461no—
354#389—1461no—
355#388—1462no—
356#382—732no—
357#380—731no—
358C4×C4⋊C4—1461no—
359C4×C22⋊C4—731no—
360#376—732no—
361#328—732no—
362#328—732no—
363#328—731no—
364#328—731no—
365#328—732no—
366#328—732no—
367#328—732no—
368#327—731no—
369#327—732no—
370#327—731no—
371C2×#41C2×Dic7311yescharacteristic
372C22×Q8C2×Dic7311yescharacteristic
373C23×C4C22×D7311yescharacteristic
374C2×M16—1461no—
375C22×C8—1461no—
376C4⋊Q8—21no—
377C4⋊Q8C2×Dic7312yes—
378C4⋊Q8C2×Dic7312yes—
379#126—22no—
380#126—22no—
381#126—22no—
382#126—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125—22no—
387#125C2×Dic7312yes—
388#124—22no—
389#124C2×Dic7312yes—
390#123—21no—
391#123—22no—
392#123—22no—
393#123—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8C2×Dic7312yes—
402C22⋊Q8C2×Dic7311yescharacteristic
403C22⋊Q8C22×D7312yes—
404C22⋊Q8C22×D7311yescharacteristic
405#121C2×Dic7311yescharacteristic
406#121C22×D7311yescharacteristic
407#120—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—22no—
411C4×Q8C2×Dic7312yes—
412C4×Q8C2×Dic7312yes—
413C4×Q8C2×Dic7312yes—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4C2×Dic7312yes—
419#117—22no—
420#117—21no—
421#117—22no—
422#117—22no—
423#117C2×Dic7311yescharacteristic
424#117C2×Dic7311yescharacteristic
425#117C2×Dic7311yescharacteristic
426#117C2×Dic7312yes—
427#117C22⋊D7311yescharacteristic
428#117C22⋊D7312yes—
429#117C22⋊D7311yescharacteristic
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4C2×Dic7311yescharacteristic
435C2×C4⋊C4C22⋊D7311yescharacteristic
436C2×C4⋊C4C22⋊D7311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4C22⋊D7312yes—
441C2×C42—1462no—
442C2×C42—1461no—
443C2×C42C22×D7312yes—
444C4⋊C8—1462no—
445C4⋊C8—1462no—
446C22⋊C8—731no—
447C22⋊C8—731no—
448C22⋊C8—732no—
449C22⋊C8—1461no—
450C22⋊C8—1461no—
451#97—1462no—
452C4×C8—1462no—
453#95—731no—
454#95—732no—
455#95—731no—
456#95—1462no—
457#95—1462no—
458C24C2×C22⋊D7311yescharacteristic
459#41—42no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—42no—
463C2×Q8C22×Dic7312yes—
464C2×Q8C22×Dic7311yescharacteristic
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C22×Dic7311yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—1461no—
475C22×C4—1462no—
476C22×C4—1461no—
477C22×C4C22×Dic7311yescharacteristic
478C22×C4C22×Dic7311yescharacteristic
479C22×C4C2×C22⋊D7312yes—
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7312yes—
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C23×D7311yescharacteristic
485M16—2922no—
486C2×C8—2922no—
487C2×C8—1461no—
488C2×C8—1462no—
489C2×C8—1461no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—2922no—
504C4⋊C4—5842no—
505C4⋊C4C22⋊Dic7311yescharacteristic
506C4⋊C4C22⋊Dic7311yescharacteristic
507C4⋊C4C22⋊Dic7312yes—
508C4⋊C4C22⋊Dic7311yescharacteristic
509C4⋊C4C22⋊Dic7311yescharacteristic
510C4⋊C4C22⋊Dic7312yes—
511C4⋊C4C22×Dic7312yes—
512C4⋊C4C22×Dic7311yescharacteristic
513C4⋊C4C22×Dic7311yescharacteristic
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—2922no—
525C22⋊C4—5841no—
526C22⋊C4C22⋊Dic7312yes—
527C22⋊C4C22⋊Dic7312yes—
528C22⋊C4C22⋊Dic7312yes—
529C22⋊C4C22⋊Dic7312yes—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—2922no—
535C42—2924no—
536C42C22×Dic7312yes—
537C42C22×Dic7312yes—
538C42C2×C22⋊D7312yes—
539C42C2×C22⋊D7312yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23C2×C22⋊Dic7311yescharacteristic
546C23C2×C22⋊Dic7311yescharacteristic
547C23C22×C22⋊D7311yescharacteristic
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—82no—
552D4—41no—
553D4—42no—
554D4—41no—
555D4—41no—
556D4—81no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2922no—
579C2×C4—2921no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2921no—
583C2×C4—1462no—
584C2×C4C2×C22⋊Dic7312yes—
585C2×C4C2×C22⋊Dic7312yes—
586C2×C4C2×C22⋊Dic7312yes—
587C2×C4C2×C22⋊Dic7312yes—
588C2×C4C2×C22⋊Dic7311yescharacteristic
589C2×C4C2×C22⋊Dic7311yescharacteristic
590C2×C4C23×Dic7311yescharacteristic
591C2×C4C22×C22⋊D7311yescharacteristic
592C8—2921no—
593C8—2921no—
594C8—2922no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22C22×C22⋊Dic7311yescharacteristic
609C22#1798939511yescharacteristic
610C22#1798940211yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—2922no—
624C4—5841no—
625C4—5842no—
626C4#1798856112yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—81no—
632C2C2×#1798856111yescharacteristic
633C2#2261846411yescharacteristic
634C2#2261847811yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#3495C2
socleC22×C73C22×C22⋊C4
prime radical#525×C73C22
cosocleC22×C4×C73C24
hypercenter#3495D73
hypocenterC73#17605
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#117 ◁ #3495

Complements

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

NH ≅ G/Ntype
1. #22618589C2semidirect
2. #22612816C2semidirect
3. #22608503C2semidirect
4. #22608495C2semidirect
6. #22608469C2semidirect
7. #22608451C2semidirect
8. #22608443C2semidirect
10. #22608417C2semidirect
42. #544×C73C4semidirect
43. #542×C73C4semidirect
44. #541×C73C4semidirect
45. #536×C73C4semidirect
50. #520×C73C4semidirect
51. #519×C73C4semidirect
329. C73#17605semidirect
336. #544Dic73semidirect
337. #542Dic73semidirect
338. #541Dic73semidirect
339. #536Dic73semidirect
344. #520Dic73semidirect
345. #519Dic73semidirect

Maximal subgroups

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

subgroupindex
#226185892
#226128162
#226085032
#226084952
#226084742
#226084692
#226084512
#226084432
#226084222
#226084172
C73×#34952
#1760573