Finite Group Databasebeta

← #33008517

Subgroups — #33008517

Order 18688.

1862 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556548C211yesmaximal, characteristic
2C2×#17988446C211yesmaximal, characteristic
3C2×#17988434C211yesmaximal, characteristic
4#22608179C212yesmaximal
5#22608179C212yesmaximal
6#22608159C212yesmaximal, complemented
7#22608159C212yesmaximal, complemented
8#22608158C211yesmaximal, characteristic, complemented
9#22608155C211yesmaximal, characteristic, complemented
10#22608155C211yesmaximal, characteristic, complemented
11C73×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C2×#14556548—24no—
14C2×#14556548—24no—
15C2×#14556548C418yescomplemented
16C22×#11924007C2212yes—
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×#11924005C2212yes—
20C22×#11924005C2211yescharacteristic
21C22×#11924005C2211yescharacteristic
22C22×#11924005C2212yes—
23C22×C4×Dic73C2212yes—
24C22×C4×Dic73C2211yescharacteristic
25#17988446—24no—
26#17988434—24no—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C42×C73C2211yescharacteristic
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2212yes—
40C24×D73C2311yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2312yes—
45C23×Dic73C2311yescharacteristic
46C22×C4×D73—24no—
47C22×C4×D73—24no—
48C22×C4×D73C2×C414yes—
49C22×C4×D73D414yes—
50C22×C4×D73D414yes—
51#14556548—232no—
52C2×#11924007—22no—
53C2×#11924007—22no—
54C2×#11924007—22no—
55C2×#11924007—22no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007C2×C414yes—
60C2×#11924007C2×C414yes—
61C2×#11924005—24no—
62C2×#11924005—22no—
63C2×#11924005—22no—
64C2×#11924005—22no—
65C2×#11924005—22no—
66C2×#11924005—24no—
67C2×#11924005—24no—
68C2×#11924005—24no—
69C2×#11924005—42no—
70C2×#11924005—42no—
71C2×#11924005C2×C414yes—
72C2×#11924005C2×C414yes—
73C2×C4×Dic73—24no—
74C2×C4×Dic73—24no—
75C2×C4×Dic73—24no—
76C2×C4×Dic73—22no—
77C2×C4×Dic73—22no—
78C2×C4×Dic73C2×C414yes—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—22no—
83#14556498—24no—
84#14556498—42no—
85#14556498—42no—
86#14556498—42no—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2312yes—
91C23×C4×C73C2311yescharacteristic
92C2×C42×C73—22no—
93C2×C42×C73—22no—
94C2×C42×C73C2×C414yes—
95#95×C73—22no—
96#95×C73—22no—
97#95×C73—24no—
98#95×C73—42no—
99C23×D73—22no—
100C23×D73—22no—
101C23×D73—22no—
102C23×D73—22no—
103C23×D73—21no—
104C23×D73—21no—
105C23×D73—21no—
106C23×D73—21no—
107C23×D73C22×C412yes—
108C23×D73C2×D412yes—
109C23×D73C2×D412yes—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—22no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73C22×C412yes—
134C22×Dic73C22×C412yes—
135C22×Dic73C22×C412yes—
136C22×Dic73C2×D412yes—
137C22×Dic73C2×D412yes—
138C22×Dic73#4114yes—
139C22×Dic73#4114yes—
140C22×Dic73#4114yes—
141C2×C4×D73—28no—
142C2×C4×D73—28no—
143C2×C4×D73—216no—
144C2×C4×D73—216no—
145C2×C4×D73C22⋊C4116yes—
146#11924007—28no—
147#11924007—28no—
148#11924007—48no—
149#11924005—216no—
150#11924005—28no—
151#11924005—28no—
152#11924005—48no—
153C4×Dic73—28no—
154C4×Dic73—48no—
155C24×C73C2411yescharacteristic
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—21no—
169C22×C4×C73—22no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—22no—
173C22×C4×C73—21no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73C22×C412yes—
182C22×C4×C73C22×C412yes—
183C22×C4×C73C22×C412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D412yes—
186C22×C4×C73#4114yes—
187C42×C73—28no—
188C23×C73C23×C411yescharacteristic
189C23×C73C22×D411yescharacteristic
190C23×C73C22×D411yescharacteristic
191C23×C73C2×#4112yes—
192C23×C73C2×#4112yes—
193C23×C73C2×#4112yes—
194C23×C73C2×#4112yes—
195C23×C73H(5,𝔽2)11yescharacteristic
196C23×C73H(5,𝔽2)11yescharacteristic
197C23×C732−1+411yescharacteristic
198C23×C732−1+411yescharacteristic
199C22×D73—22no—
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—24no—
206C22×D73—24no—
207C22×D73—24no—
208C22×D73—24no—
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—22no—
213C22×D73—42no—
214C22×D73—42no—
215C22×D73—42no—
216C22×D73—42no—
217C22×D73—42no—
218C22×D73—42no—
219C22×D73—42no—
220C22×D73—42no—
221C22×D73C2×C22⋊C414yes—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—24no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73C2×C22⋊C414yes—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73—42no—
279C2×Dic73C2×C22⋊C414yes—
280C2×Dic73#11714yes—
281C2×Dic73#11714yes—
282C2×Dic73#12318yes—
283C2×Dic73#12318yes—
284C4×D73—232no—
285C22×C73C22×C22⋊C411yescharacteristic
286C22×C73C2×#11711yescharacteristic
287C22×C73C2×#11711yescharacteristic
288C22×C73C4×#4111yescharacteristic
289C22×C73C4×#4111yescharacteristic
290C22×C73#52011yescharacteristic
291C22×C73#52011yescharacteristic
292C22×C73C2×#12312yes—
293C22×C73C2×#12312yes—
294C22×C73#52512yes—
295C22×C73#53212yes—
296C22×C73#53511yescharacteristic
297C22×C73#53611yescharacteristic
298C22×C73#53711yescharacteristic
299C22×C73#53711yescharacteristic
300C22×C73#55212yes—
301C22×C73#55212yes—
302C22×C73#55312yes—
303C22×C73#55312yes—
304C22×C73#55512yes—
305C22×C73#55512yes—
306C22×C73#55612yes—
307C22×C73#55612yes—
308C2×D73—24no—
309C2×D73—24no—
310C2×D73—24no—
311C2×D73—42no—
312C2×D73—42no—
313C2×D73—42no—
314C2×D73—42no—
315C2×D73—44no—
316C2×D73—44no—
317C2×D73—44no—
318C2×D73—44no—
319C4×C73—24no—
320C4×C73—42no—
321C4×C73—44no—
322C4×C73—42no—
323C4×C73—82no—
324Dic73—42no—
325Dic73—44no—
326Dic73—42no—
327Dic73—82no—
328Dic73#38118yes—
329#13939—731nomaximal, Sylow
330C2×C73C2×#38111yescharacteristic
331C2×C73#229711yescharacteristic
332C2×C73#231111yescharacteristic
333C2×C73#234611yescharacteristic
334C2×C73#234611yescharacteristic
335C2×C73#234711yescharacteristic
336C2×C73#234711yescharacteristic
337C2×C73#248512yes—
338C2×C73#248512yes—
339C2×C73#249912yes—
340C2×C73#250012yes—
341D73—44no—
342C22×#117—731no—
343C2×#380—731no—
344C2×C4×C22⋊C4—731no—
345#1440—732no—
346#1440—732no—
347C4⋊#95—732no—
348C4⋊#95—732no—
349#1434—731no—
350#1434—731no—
351#1434—731no—
352#1434D7311yescharacteristic, complemented
353C73#1393911yescharacteristic, complemented, Sylow
354C24×C4—731no—
355C2×#117—738no—
356C2×#117—1464no—
357C2×#117—1464no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C22×C4⋊C4—732no—
361C22×C22⋊C4—732no—
362C22×C22⋊C4—731no—
363C22×C22⋊C4—731no—
364C22×C22⋊C4—732no—
365C22×C42—731no—
366C22×C42—732no—
367C22×C42C2×D7311yescharacteristic
368#380—1464no—
369C4×C22⋊C4—1464no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—732no—
373C2×#95—731no—
374C2×#95—731no—
375C2×#95—732no—
376C2×#95—732no—
377C2×#95—732no—
378C2×#95C2×D7311yescharacteristic
379C2×#95C2×D7311yescharacteristic
380C2×#95C2×D7312yes—
381C2×#95C2×D7312yes—
382C25—731no—
383C23×C4—731no—
384C23×C4—734no—
385C23×C4—734no—
386C23×C4—734no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—732no—
390C23×C4—732no—
391C23×C4—1464no—
392C23×C4—1464no—
393C23×C4C22×D7312yes—
394C23×C4C22×D7311yescharacteristic
395C23×C4C22×D7311yescharacteristic
396C23×C4C22×D7312yes—
397C23×C4C22×D7311yescharacteristic
398#117—14632no—
399C2×C4⋊C4—734no—
400C2×C4⋊C4—734no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4—1464no—
403C2×C4⋊C4—1464no—
404C2×C4⋊C4—1462no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1462no—
407C2×C4⋊C4—1462no—
408C2×C22⋊C4—2922no—
409C2×C22⋊C4—2922no—
410C2×C22⋊C4—734no—
411C2×C22⋊C4—734no—
412C2×C22⋊C4—1464no—
413C2×C22⋊C4—1464no—
414C2×C22⋊C4—1462no—
415C2×C22⋊C4—1462no—
416C2×C22⋊C4—1462no—
417C2×C22⋊C4—1462no—
418C2×C22⋊C4—1464no—
419C2×C22⋊C4—1464no—
420C2×C42—22no—
421C2×C42—22no—
422C2×C42—734no—
423C2×C42—1464no—
424C2×C42—1464no—
425C2×C42—1464no—
426C2×C42—1462no—
427C2×C42—1462no—
428C2×C42C4×D7314yes—
429#95—22no—
430#95—22no—
431#95—24no—
432#95—42no—
433#95—2922no—
434#95—2922no—
435#95—2922no—
436#95—1464no—
437#95—1464no—
438#95—1462no—
439#95—1462no—
440#95—1464no—
441C24—732no—
442C24—732no—
443C24—732no—
444C24—1462no—
445C24—1462no—
446C24—1462no—
447C24—1462no—
448C24—1461no—
449C24—1461no—
450C24—1461no—
451C24—1461no—
452C24C23×D7311yescharacteristic
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—732no—
481C22×C4—734no—
482C22×C4—734no—
483C22×C4—7316no—
484C22×C4—732no—
485C22×C4—732no—
486C22×C4—734no—
487C22×C4—1468no—
488C22×C4—1468no—
489C22×C4—14616no—
490C22×C4—14616no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4—1462no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1462no—
514C22×C4C2×C4×D7312yes—
515C22×C4C2×C4×D7312yes—
516C22×C4C2×C4×D7312yes—
517C22×C4D4×D7312yes—
518C22×C4D4×D7312yes—
519C22×C4#1192402814yes—
520C4⋊C4—2928no—
521C4⋊C4—1468no—
522C4⋊C4—1468no—
523C22⋊C4—2928no—
524C22⋊C4—1468no—
525C22⋊C4—1468no—
526C22⋊C4—14616no—
527C42—28no—
528C42—2928no—
529C42—1468no—
530C23—2922no—
531C23—2922no—
532C23—2922no—
533C23—2922no—
534C23—2922no—
535C23—2922no—
536C23—2922no—
537C23—2922no—
538C23—734no—
539C23—1464no—
540C23—1462no—
541C23—1462no—
542C23—1462no—
543C23—1462no—
544C23—1462no—
545C23—1462no—
546C23—1462no—
547C23—1462no—
548C23—1462no—
549C23—1462no—
550C23—1464no—
551C23—1464no—
552C23—1464no—
553C23C22×C4×D7311yescharacteristic
554C23C2×D4×D7311yescharacteristic
555C23C2×D4×D7311yescharacteristic
556C23#41×D7312yes—
557C23#41×D7312yes—
558C23#41×D7312yes—
559C23C2×#1192402812yes—
560C23#1455663611yescharacteristic
561C23#1455663711yescharacteristic
562C23#1455663811yescharacteristic
563C23#1455663911yescharacteristic
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—2922no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—2922no—
606C2×C4—2922no—
607C2×C4—2922no—
608C2×C4—2922no—
609C2×C4—2922no—
610C2×C4—2922no—
611C2×C4—734no—
612C2×C4—734no—
613C2×C4—734no—
614C2×C4—738no—
615C2×C4—738no—
616C2×C4—14632no—
617C2×C4—1464no—
618C2×C4—1464no—
619C2×C4—1462no—
620C2×C4—1462no—
621C2×C4—1462no—
622C2×C4—1462no—
623C2×C4—1462no—
624C2×C4—1462no—
625C2×C4—1464no—
626C2×C4C22⋊C4×D7314yes—
627C22—2922no—
628C22—2922no—
629C22—2922no—
630C22—2922no—
631C22—2924no—
632C22—2924no—
633C22—2924no—
634C22—2924no—
635C22—1464no—
636C22—1464no—
637C22—1464no—
638C22C2×C22⋊C4×D7311yescharacteristic
639C22#117×D7311yescharacteristic
640C22#117×D7311yescharacteristic
641C22#1798905511yescharacteristic
642C22C4×#1192402811yescharacteristic
643C22#1798906711yescharacteristic
644C22#1798906811yescharacteristic
645C22#123×D7312yes—
646C22#123×D7312yes—
647C22#1798909612yes—
648C22#1798913412yes—
649C22#1798915311yescharacteristic
650C22#1798916011yescharacteristic
651C22#1798916611yescharacteristic
652C22#1798916711yescharacteristic
653C22#1798926912yes—
654C22#1798927012yes—
655C22#1798928612yes—
656C22#1798928812yes—
657C22#1798929912yes—
658C22#1798930012yes—
659C22#1798930412yes—
660C22#1798930912yes—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—2922no—
667C4—2922no—
668C4—2924no—
669C4—5842no—
670C4—738no—
671C2—2924no—
672C2#381×D7311yescharacteristic
673C2#2261241711yescharacteristic
674C2#2261252011yescharacteristic
675C2#2261281111yescharacteristic
676C2#2261281711yescharacteristic
677C2#2261281911yescharacteristic
678C2#2261282511yescharacteristic
679C2#2261422512yes—
680C2#2261422712yes—
681C2#2261438212yes—
682C2#2261439112yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1434D73
hypocenterC73#13939
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
6. #22608159C2semidirect
7. #22608159C2semidirect
8. #22608158C2semidirect
9. #22608155C2semidirect
10. #22608155C2semidirect
11. C73×#1434C2semidirect
15. C2×#14556548C4semidirect
352. #1434D73semidirect
353. C73#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#145565482
C2×#179884462
C2×#179884342
#226081792
#226081592
#226081582
#226081552
C73×#14342
#1393973