Finite Group Databasebeta

← #33017037

Subgroups — #33017037

Order 18688.

1530 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556584C211yesmaximal, characteristic
2C2×#17988456C211yesmaximal, characteristic, complemented
3C2×#17988454C211yesmaximal, characteristic, complemented
4#22608185C211yesmaximal, characteristic, complemented
5#22608183C212yesmaximal, complemented
6#22608182C212yesmaximal, complemented
7#22608181C211yesmaximal, characteristic, complemented
8#22608176C211yesmaximal, characteristic
9#22608174C212yesmaximal
10#22608174C212yesmaximal
11C73×#1439C211yesmaximal, characteristic
12C2×#14556584—24no—
13C2×#14556584—24no—
14C2×#14556584C418yescomplemented
15C22×C4⋊Dic73C2211yescharacteristic
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×#11924007C2212yes—
20C22×#11924007C2212yes—
21C22×C22⋊Dic73C2211yescharacteristic
22C22×C22⋊Dic73C2211yescharacteristic
23C22×C22⋊Dic73C2212yes—
24C22×C4×Dic73C2211yescharacteristic
25#17988456—24no—
26#17988454—24no—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2311yescharacteristic
46#14556584—232no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—22no—
50C2×C4⋊Dic73—42no—
51C2×C4⋊Dic73C2×C414yes—
52C2×#11924007—22no—
53C2×#11924007—22no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—42no—
59C2×#11924007—42no—
60C2×#11924007—42no—
61C2×#11924007—42no—
62C2×#11924007C2×C414yes—
63C2×#11924007C2×C414yes—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73—22no—
68C2×C22⋊Dic73—42no—
69C2×C22⋊Dic73—42no—
70C2×C22⋊Dic73—42no—
71C2×C22⋊Dic73C2×C414yes—
72C2×C22⋊Dic73C2×C414yes—
73C2×C4×Dic73C2×C414yes—
74C2×C4×Dic73D414yes—
75C2×C4×Dic73D414yes—
76#14556498—24no—
77#14556498—24no—
78#14556498—22no—
79#14556498—42no—
80#14556498—42no—
81#14556498—42no—
82#14556498—42no—
83#14556498—42no—
84C25×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C23×C4×C73C2312yes—
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—42no—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—42no—
94C2×C22⋊C4×C73—42no—
95C2×C22⋊C4×C73—42no—
96C2×C22⋊C4×C73C2×C414yes—
97#95×C73—22no—
98#95×C73—42no—
99C22×Dic73—22no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—22no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C412yes—
132C22×Dic73C22×C412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C22×Dic73C2×D412yes—
137C4⋊Dic73—28no—
138C4⋊Dic73—44no—
139#11924007—28no—
140#11924007—28no—
141#11924007—48no—
142#11924007—48no—
143C22⋊Dic73—28no—
144C22⋊Dic73—28no—
145C22⋊Dic73—48no—
146C4×Dic73C22⋊C4116yescomplemented
147C24×C73—21no—
148C24×C73—21no—
149C24×C73—21no—
150C24×C73—22no—
151C24×C73—22no—
152C24×C73—21no—
153C24×C73—22no—
154C24×C73—22no—
155C24×C73—21no—
156C24×C73—21no—
157C24×C73C22×C412yes—
158C24×C73C2411yescharacteristic
159C22×C4×C73—22no—
160C22×C4×C73—22no—
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—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73C22×C412yes—
180C22×C4×C73C22×C412yes—
181C22×C4×C73C2×D412yes—
182C22×C4×C73C2×D412yes—
183C4⋊C4×C73—44no—
184C22⋊C4×C73—28no—
185C22⋊C4×C73—48no—
186C23×C73—22no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—42no—
198C23×C73—41no—
199C23×C73—42no—
200C23×C73—41no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73C23×C411yescharacteristic
210C23×C73C22×D411yescharacteristic
211C23×C73C22×D411yescharacteristic
212C23×C73H(5,𝔽2)12yes—
213C23×C73H(5,𝔽2)12yes—
214C23×C73H(5,𝔽2)11yescharacteristic
215C23×C73H(5,𝔽2)11yescharacteristic
216C23×C732−1+412yes—
217C23×C732−1+412yes—
218C23×C732−1+411yescharacteristic
219C23×C732−1+411yescharacteristic
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73C2×C22⋊C414yes—
244C2×Dic73—24no—
245C2×Dic73—22no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—24no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—24no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—42no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
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×Dic73—42no—
280C2×Dic73C2×C22⋊C414yes—
281C2×Dic73C2×C22⋊C414yes—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73—82no—
292C22×C73—82no—
293C22×C73C22×C22⋊C411yescharacteristic
294C22×C73#52012yes—
295C22×C73#52012yes—
296C22×C73#52011yescharacteristic
297C22×C73#52011yescharacteristic
298C22×C73#53511yescharacteristic
299C22×C73#53511yescharacteristic
300C22×C73#53511yescharacteristic
301C22×C73#53611yescharacteristic
302C22×C73#53611yescharacteristic
303C22×C73#53611yescharacteristic
304C22×C73#53712yes—
305C22×C73#53711yescharacteristic
306C22×C73#53712yes—
307C22×C73#53711yescharacteristic
308C22×C73#56012yes—
309C22×C73#56012yes—
310C22×C73#56312yes—
311C22×C73#56312yes—
312C22×C73#56312yes—
313C22×C73#56312yes—
314C22×C73#56312yes—
315C22×C73#56312yes—
316C4×C73—24no—
317C4×C73—42no—
318C4×C73—82no—
319C4×C73—82no—
320Dic73—24no—
321Dic73—24no—
322Dic73—42no—
323Dic73—42no—
324Dic73—82no—
325Dic73—82no—
326#14665—731nomaximal, Sylow
327C2×C73—82no—
328C2×C73#231111yescharacteristic
329C2×C73#231111yescharacteristic
330C2×C73#231111yescharacteristic
331C2×C73#238312yes—
332C2×C73#238312yes—
333C2×C73#281811yescharacteristic
334C2×C73#282111yescharacteristic
335C2×C73#282111yescharacteristic
336C2×C73#282111yescharacteristic
337C2×C73#282312yes—
338C2×C73#282312yes—
339C22×#126—731no—
340C2×#384—731no—
341C2×#383—731no—
342#1441—732no—
343#1441—732no—
344#1441—731no—
345#1441—731no—
346#1439—732no—
347#1439—732no—
348#1439—731no—
349#1439D7311yescharacteristic
350C73#1466511yescharacteristic, complemented, Sylow
351C2×#126—738no—
352C2×#126—1464no—
353C2×#126—1464no—
354C22×C4⋊C4—732no—
355C22×C4⋊C4—732no—
356C22×C4⋊C4—731no—
357C22×C4⋊C4—731no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C22×C4⋊C4C2×D7311yescharacteristic
361C22×C22⋊C4—732no—
362C22×C22⋊C4—731no—
363C22×C22⋊C4—731no—
364C22×C22⋊C4C2×D7311yescharacteristic
365C22×C22⋊C4C2×D7312yes—
366C22×C42—731no—
367#384—1464no—
368#383—1464no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95—732no—
375C2×#95—732no—
376C2×#95—731no—
377C2×#95C2×D7311yescharacteristic
378C2×#95C2×D7312yes—
379C25C22×D7311yescharacteristic
380C23×C4—731no—
381C23×C4—731no—
382C23×C4—731no—
383C23×C4—731no—
384C23×C4—732no—
385C23×C4—732no—
386C23×C4C22×D7312yes—
387C23×C4C22×D7312yes—
388C23×C4C22×D7311yescharacteristic
389C23×C4C22×D7311yescharacteristic
390#126—14632no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—2922no—
394C2×C4⋊C4—2922no—
395C2×C4⋊C4—2922no—
396C2×C4⋊C4—2922no—
397C2×C4⋊C4—2922no—
398C2×C4⋊C4—734no—
399C2×C4⋊C4—734no—
400C2×C4⋊C4—734no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4—1464no—
403C2×C4⋊C4—1462no—
404C2×C4⋊C4—1462no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1462no—
407C2×C4⋊C4—1462no—
408C2×C4⋊C4—1462no—
409C2×C4⋊C4—1462no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—2922no—
416C2×C22⋊C4—2922no—
417C2×C22⋊C4—2922no—
418C2×C22⋊C4—734no—
419C2×C22⋊C4—734no—
420C2×C22⋊C4—1462no—
421C2×C22⋊C4—1462no—
422C2×C22⋊C4—1462no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4C4×D7314yes—
425C2×C42—734no—
426C2×C42—734no—
427C2×C42—734no—
428#95—22no—
429#95—42no—
430#95—2922no—
431#95—2922no—
432#95—2922no—
433#95—2922no—
434#95—2922no—
435#95—1464no—
436#95—1464no—
437#95—1462no—
438C24—21no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24—22no—
443C24—21no—
444C24—22no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24C2×C4×D7312yes—
449C24C23×D7311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—732no—
471C22×C4—732no—
472C22×C4—732no—
473C22×C4—732no—
474C22×C4—732no—
475C22×C4—732no—
476C22×C4—732no—
477C22×C4—732no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1462no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1462no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1462no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1462no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4C2×C4×D7312yes—
509C22×C4C2×C4×D7312yes—
510C22×C4D4×D7312yes—
511C22×C4D4×D7312yes—
512C4⋊C4—44no—
513C4⋊C4—2924no—
514C4⋊C4—2928no—
515C4⋊C4—2928no—
516C4⋊C4—1468no—
517C4⋊C4—1468no—
518C4⋊C4—1468no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—2928no—
522C22⋊C4—1468no—
523C22⋊C4—1468no—
524C42—7316no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C22×C4×D7311yescharacteristic
549C23C2×D4×D7311yescharacteristic
550C23C2×D4×D7311yescharacteristic
551C23#1455663612yes—
552C23#1455663612yes—
553C23#1455663611yescharacteristic
554C23#1455663711yescharacteristic
555C23#1455663812yes—
556C23#1455663812yes—
557C23#1455663811yescharacteristic
558C23#1455663911yescharacteristic
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2922no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—734no—
604C2×C4—734no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1464no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4—1464no—
616C2×C4—1462no—
617C2×C4—1464no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4C22⋊C4×D7314yes—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22C2×C22⋊C4×D7311yescharacteristic
633C22#1798906612yes—
634C22#1798906612yes—
635C22#1798906711yescharacteristic
636C22#1798906811yescharacteristic
637C22#1798915011yescharacteristic
638C22#1798915211yescharacteristic
639C22#1798915311yescharacteristic
640C22#1798915711yescharacteristic
641C22#1798915911yescharacteristic
642C22#1798916011yescharacteristic
643C22#1798916412yes—
644C22#1798916611yescharacteristic
645C22#1798916612yes—
646C22#1798916711yescharacteristic
647C22#1798932512yes—
648C22#1798932612yes—
649C22#1798933712yes—
650C22#1798933812yes—
651C22#1798933812yes—
652C22#1798933912yes—
653C22#1798934012yes—
654C22#1798934012yes—
655C4—24no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—2922no—
660C4—2922no—
661C4—5842no—
662C4—5842no—
663C4—1464no—
664C4—1464no—
665C2—82no—
666C2#2261251811yescharacteristic
667C2#2261252011yescharacteristic
668C2#2261252111yescharacteristic
669C2#2261311812yes—
670C2#2261311912yes—
671C2#2261797011yescharacteristic
672C2#2261798811yescharacteristic
673C2#2261799311yescharacteristic
674C2#2261799411yescharacteristic
675C2#2261800212yes—
676C2#2261800412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14665
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#17988456C2semidirect
3. C2×#17988454C2semidirect
4. #22608185C2semidirect
5. #22608183C2semidirect
6. #22608182C2semidirect
7. #22608181C2semidirect
14. C2×#14556584C4semidirect
146. C4×Dic73C22⋊C4semidirect
350. C73#14665semidirect

Maximal subgroups

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

subgroupindex
C22×#145565842
C2×#179884562
C2×#179884542
#226081852
#226081832
#226081822
#226081812
#226081762
#226081742
C73×#14392
#1466573