Finite Group Databasebeta

← #33154116

Subgroups — #33154116

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612934C211yesmaximal, characteristic, complemented
2#22611248C211yesmaximal, characteristic, complemented
3#22611134C211yesmaximal, characteristic, complemented
4#22611079C211yesmaximal, characteristic, complemented
5#22610934C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610617C211yesmaximal, characteristic, complemented
8#22610548C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610523C211yesmaximal, characteristic, complemented
11#22610289C211yesmaximal, characteristic, complemented
12#22610249C211yesmaximal, characteristic
13#22610015C211yesmaximal, characteristic, complemented
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2490C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988484—21no—
34#17988474—21no—
35#17988459—21no—
36C4×C4⋊Dic73C2211yescharacteristic
37C22⋊C4×Dic73—21no—
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#125×C73C2211yescharacteristic
49C2×#121×C73C2211yescharacteristic
50#397×C73—21no—
51#394×C73—21no—
52#390×C73—21no—
53#390×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#388×C73—21no—
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C23×Dic73—21no—
61C2×#11924016—22no—
62C2×#11924016—41no—
63C2×#11924016—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556570—22no—
69#14556570—22no—
70C4×Dic146—22no—
71C4×Dic146—41no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84#14556524—22no—
85C4⋊#10596831—41no—
86#14556514—22no—
87#14556514—22no—
88#14556514—41no—
89#14556512—22no—
90#14556512—22no—
91#14556512—41no—
92C22⋊#10596831—41no—
93C22⋊#10596831—41no—
94C4×#10596831—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C22×D4×C73—21no—
100C22×D4×C73C2311yescharacteristic
101C23×C4×C73C2311yescharacteristic
102#125×C73—22no—
103#125×C73—41no—
104#121×C73—22no—
105#121×C73—41no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146#4112yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—81no—
139#11924016—44no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73#4112yes—
158C4⋊Dic73#4112yes—
159#11924007—24no—
160#11924007—42no—
161#11924007—81no—
162C22⋊Dic73—24no—
163C4×Dic73—42no—
164C4×Dic73—81no—
165C24×C73—21no—
166C24×C73C2×D411yescharacteristic
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—42no—
174C2×D4×C73—42no—
175C2×D4×C73—81no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73#4112yes—
199C22⋊C4×C73—22no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—81no—
204C22⋊C4×C73—81no—
205C22⋊C4×C73—81no—
206C22⋊C4×C73—81no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—81no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73#13512yes—
226C23×C73C22×D411yescharacteristic
227D4×C73—41no—
228D4×C73—41no—
229D4×C73—41no—
230D4×C73—41no—
231D4×C73—84no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73#13712yes—
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73C2×#4111yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73H(5,𝔽2)11yescharacteristic
265C2×C4×C73H(5,𝔽2)11yescharacteristic
266C2×Dic73—22no—
267C2×Dic73—22no—
268C2×Dic73—22no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—84no—
279Dic146—22no—
280Dic146—22no—
281Dic146—41no—
282Dic146—41no—
283#10596831—41no—
284#10596831—41no—
285#10596831—41no—
286#10596831—41no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—81no—
295C22×C73—81no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73#52511yescharacteristic
302C22×C73#54611yescharacteristic
303C22×C73#54611yescharacteristic
304C22×C73C2×#13511yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57611yescharacteristic
307C22×C73#57711yescharacteristic
308C4×C73—22no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—41no—
312C4×C73—42no—
313C4×C73—41no—
314C4×C73—81no—
315C4×C73—81no—
316C4×C73#52911yescharacteristic
317C4×C73#53811yescharacteristic
318C4×C73#55511yescharacteristic
319C4×C73#55911yescharacteristic
320Dic73—42no—
321Dic73—42no—
322Dic73—42no—
323Dic73—81no—
324#26727—731nomaximal, Sylow
325C2×C73—42no—
326C2×C73—161no—
327C2×C73#245911yescharacteristic
328C2×C73#310811yescharacteristic
329C2×C73#311911yescharacteristic
330C2×C73#329611yescharacteristic
331C2×C73#329711yescharacteristic
332C2×C73#332011yescharacteristic
333C2×C73#333311yescharacteristic
334#2490D7311yescharacteristic
335#2360—731no—
336#2055—731no—
337#2036—731no—
338#2026—731no—
339#2000—731no—
340#1989—731no—
341#1934—731no—
342#1925—731no—
343#1923—731no—
344#1921—731no—
345#1880—731no—
346#1874—731no—
347#1823—731no—
348C4×#103—731no—
349C73#2672711yescharacteristic, complemented, Sylow
350C22×SD16—731no—
351C2×#125C2×D7311yescharacteristic
352C2×C22⋊Q8—731no—
353C2×#121C2×D7311yescharacteristic
354C2×C4×Q8—731no—
355C22×C4⋊C4—731no—
356C2×#107—731no—
357C2×C4⋊C8—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×#103—731no—
361C2×#102—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×C4×C8—731no—
367#397—21no—
368#394—21no—
369#391—731no—
370#390—21no—
371#390C2×D7311yescharacteristic
372#390C2×D7311yescharacteristic
373#389—731no—
374#389—1461no—
375#388—21no—
376#387—1461no—
377#385—1461no—
378#380C2×D7311yescharacteristic
379C4×C4⋊C4—731no—
380C4×C4⋊C4C2×D7311yescharacteristic
381C4×C22⋊C4—1461no—
382C4×C22⋊C4C2×D7311yescharacteristic
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#336—731no—
392#336—731no—
393C22×Q8—731no—
394C22×D4—21no—
395C22×D4C22×D7311yescharacteristic
396C23×C4—1461no—
397C23×C4C22×D7311yescharacteristic
398C2×SD16—2921no—
399C2×SD16—2921no—
400C2×SD16—1462no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405#125—22no—
406#125—41no—
407C22⋊Q8—1462no—
408C22⋊Q8—1462no—
409#121—22no—
410#121—41no—
411C4×Q8—2921no—
412C4×Q8—1462no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—2921no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1462no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4C22×D7311yescharacteristic
425C2×C4⋊C4C22×D7311yescharacteristic
426C2×C4⋊C4C22×D7311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4C22⋊D7311yescharacteristic
435C2×C22⋊C4C22⋊D7311yescharacteristic
436C2×C42—21no—
437C2×C42—1461no—
438C2×C42—1461no—
439C2×C42C22×D7311yescharacteristic
440C2×C42C22×D7311yescharacteristic
441#107—1462no—
442C4⋊C8—2921no—
443#103—2921no—
444#103—1462no—
445#103—1462no—
446#102—2921no—
447#102—1462no—
448#102—1462no—
449C22⋊C8—2921no—
450C22⋊C8—2921no—
451C4×C8—2921no—
452#95—21no—
453#95—21no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95C22⋊D7311yescharacteristic
459#95C22⋊D7311yescharacteristic
460C24—21no—
461C24C2×C22⋊D7311yescharacteristic
462C2×Q8—732no—
463C2×Q8—1461no—
464C2×Q8—1461no—
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×Q8—1461no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—81no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—5841no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4C2×C22⋊D7311yescharacteristic
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C23×D7311yescharacteristic
499SD16—2924no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—2922no—
520C4⋊C4—2922no—
521C4⋊C4—5841no—
522C4⋊C4—732no—
523C4⋊C4—732no—
524C4⋊C4—1464no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4#1192402712yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C22⋊C4—1464no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—81no—
544C42—2922no—
545C42—5841no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#1455661112yes—
560C23C22×C22⋊D7311yescharacteristic
561Q8—2921no—
562Q8—2921no—
563Q8—1462no—
564Q8—1462no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—5844no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4#1455662112yes—
611C2×C4#41×D7311yescharacteristic
612C2×C4#41×D7311yescharacteristic
613C2×C4#41×D7311yescharacteristic
614C2×C4C2×#1192402711yescharacteristic
615C2×C4#1455663611yescharacteristic
616C2×C4#1455663711yescharacteristic
617C8—2921no—
618C8—2921no—
619C8—2921no—
620C8—2921no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#1798909411yescharacteristic
636C22#1798922511yescharacteristic
637C22#1798923111yescharacteristic
638C22C2×#1455661111yescharacteristic
639C22C2×#1455662111yescharacteristic
640C22#1798939011yescharacteristic
641C22#1798939811yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—2922no—
651C4—2922no—
652C4—2922no—
653C4—5841no—
654C4#1798911611yescharacteristic
655C4#1798917111yescharacteristic
656C4#1798929911yescharacteristic
657C4#1798931911yescharacteristic
658C2—42no—
659C2—161no—
660C2#2261386411yescharacteristic
661C2#2262000711yescharacteristic
662C2#2262012411yescharacteristic
663C2#2262176611yescharacteristic
664C2#2262177511yescharacteristic
665C2#2262209611yescharacteristic
666C2#2262226911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2490C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2490D73
hypocenterC73#26727
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2490

Complements

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

NH ≅ G/Ntype
1. #22612934C2semidirect
2. #22611248C2semidirect
3. #22611134C2semidirect
4. #22611079C2semidirect
6. #22610889C2semidirect
7. #22610617C2semidirect
8. #22610548C2semidirect
9. #22610539C2semidirect
10. #22610523C2semidirect
11. #22610289C2semidirect
13. #22610015C2semidirect
14. C4×#14556514C2semidirect
349. C73#26727semidirect

Maximal subgroups

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

subgroupindex
#226129342
#226112482
#226111342
#226110792
#226109342
#226108892
#226106172
#226105482
#226105392
#226105232
#226102892
#226102492
#226100152
C4×#145565142
C73×#24902
#2672773