Finite Group Databasebeta

← #33017338

Subgroups — #33017338

Order 18688.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556584C211yesmaximal, characteristic
2C2×#17988454C211yesmaximal, characteristic, complemented
3C2×C42×Dic73C211yesmaximal, characteristic, complemented
4#22608168C212yesmaximal
5#22608168C211yesmaximal, characteristic
6#22608168C212yesmaximal
7#22608166C211yesmaximal, characteristic, complemented
8#22608165C212yesmaximal, complemented
9#22608164C211yesmaximal, characteristic, complemented
10#22608162C212yesmaximal, complemented
11C73×#1437C211yesmaximal, characteristic
12C2×#14556584—24no—
13C2×#14556584—24no—
14C2×#14556584C418yescomplemented
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×C22⋊Dic73C2212yes—
19C22×C22⋊Dic73C2211yescharacteristic
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2212yes—
22C22×C4×Dic73C2212yes—
23C22×C4×Dic73C2211yescharacteristic
24C22×C4×Dic73C2211yescharacteristic
25#17988454—24no—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C42×Dic73—24no—
35C22×C22⋊C4×C73C2212yes—
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C42×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2211yescharacteristic
40C23×Dic73C2312yes—
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2312yes—
46#14556584—232no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73C2×C414yes—
50C2×#11924007—22no—
51C2×#11924007—22no—
52C2×#11924007—22no—
53C2×#11924007—22no—
54C2×#11924007C2×C414yes—
55C2×#11924007C2×C414yes—
56C2×C22⋊Dic73—22no—
57C2×C22⋊Dic73—22no—
58C2×C22⋊Dic73—22no—
59C2×C22⋊Dic73—22no—
60C2×C22⋊Dic73—24no—
61C2×C22⋊Dic73—24no—
62C2×C22⋊Dic73—24no—
63C2×C22⋊Dic73C2×C414yes—
64C2×C22⋊Dic73C2×C414yes—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73—24no—
72C2×C4×Dic73—24no—
73C2×C4×Dic73C2×C414yes—
74C2×C4×Dic73D414yes—
75C2×C4×Dic73D414yes—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—24no—
83#14556498—24no—
84C25×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2312yes—
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—24no—
92C2×C22⋊C4×C73—24no—
93C2×C22⋊C4×C73—24no—
94C2×C22⋊C4×C73C2×C414yes—
95C2×C42×C73—22no—
96C2×C42×C73—24no—
97#95×C73—22no—
98#95×C73—24no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—21no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73C22×C412yes—
123C22×Dic73C22×C412yes—
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C412yes—
126C22×Dic73C2×D412yes—
127C22×Dic73C2×D412yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D412yes—
130C22×Dic73#4114yes—
131C22×Dic73#4114yes—
132C22×Dic73#4114yes—
133C4⋊Dic73—28no—
134#11924007—28no—
135#11924007—28no—
136C22⋊Dic73—28no—
137C22⋊Dic73—28no—
138C22⋊Dic73—48no—
139C4×Dic73—216no—
140C4×Dic73—216no—
141C4×Dic73—28no—
142C4×Dic73C22⋊C4116yescomplemented
143C24×C73—21no—
144C24×C73—21no—
145C24×C73—22no—
146C24×C73—22no—
147C24×C73—22no—
148C24×C73—22no—
149C24×C73—22no—
150C24×C73—22no—
151C24×C73C22×C412yes—
152C24×C73C2411yescharacteristic
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
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×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×D412yes—
174C22×C4×C73#4114yes—
175C22×C4×C73#4114yes—
176C22×C4×C73#4114yes—
177C22⋊C4×C73—28no—
178C22⋊C4×C73—48no—
179C42×C73—28no—
180C23×C73—22no—
181C23×C73—24no—
182C23×C73—24no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—41no—
186C23×C73—41no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73C23×C411yescharacteristic
202C23×C73C22×D411yescharacteristic
203C23×C73C22×D411yescharacteristic
204C23×C73C2×#4112yes—
205C23×C73C2×#4112yes—
206C23×C73C2×#4112yes—
207C23×C73C2×#4112yes—
208C23×C73C2×#4112yes—
209C23×C73C2×#4112yes—
210C2×C4×C73—24no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—22no—
221C2×C4×C73—24no—
222C2×C4×C73—42no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73C2×C22⋊C414yes—
233C2×C4×C73#12618yescomplemented
234C2×Dic73—24no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—24no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73C2×C22⋊C414yes—
266C2×Dic73C2×C22⋊C414yes—
267C2×Dic73#12618yescomplemented
268C22×C73—44no—
269C22×C73—44no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—82no—
274C22×C73—82no—
275C22×C73—82no—
276C22×C73—82no—
277C22×C73C22×C22⋊C411yescharacteristic
278C22×C73C4×#4112yes—
279C22×C73C4×#4112yes—
280C22×C73C4×#4111yescharacteristic
281C22×C73C4×#4111yescharacteristic
282C22×C73#52511yescharacteristic
283C22×C73#52511yescharacteristic
284C22×C73#52512yes—
285C22×C73#52512yes—
286C22×C73C2×#12612yes—
287C22×C73C2×#12612yes—
288C22×C73#52912yes—
289C22×C73#52912yes—
290C22×C73#52912yes—
291C22×C73#52912yes—
292C22×C73#52912yes—
293C22×C73#52912yes—
294C22×C73#53212yes—
295C22×C73#53212yes—
296C22×C73#53211yescharacteristic
297C22×C73#53211yescharacteristic
298C4×C73—28no—
299C4×C73—24no—
300C4×C73—42no—
301C4×C73—82no—
302Dic73—28no—
303Dic73—24no—
304Dic73—24no—
305Dic73—42no—
306Dic73—42no—
307Dic73—82no—
308#14711—731nomaximal, Sylow
309C2×C73—82no—
310C2×C73#229711yescharacteristic
311C2×C73#229711yescharacteristic
312C2×C73#229711yescharacteristic
313C2×C73C4×#12612yes—
314C2×C73C4×#12612yes—
315C2×C73#285212yes—
316C2×C73#285212yes—
317C2×C73#286012yes—
318C2×C73#286012yes—
319C22×#126—731no—
320C2×#383—731no—
321C2×C43—731no—
322#1437—732no—
323#1437—732no—
324#1437—731no—
325#1437D7311yescharacteristic
326#1436—731no—
327#1436—732no—
328#1436—732no—
329#1436—731no—
330C73#1471111yescharacteristic, complemented, Sylow
331C2×#126—738no—
332C2×#126—1464no—
333C2×#126—1464no—
334C22×C4⋊C4—731no—
335C22×C4⋊C4—731no—
336C22×C4⋊C4—731no—
337C22×C22⋊C4—731no—
338C22×C22⋊C4—731no—
339C22×C22⋊C4—732no—
340C22×C22⋊C4C2×D7312yes—
341C22×C22⋊C4C2×D7311yescharacteristic
342C22×C42—731no—
343C22×C42—732no—
344C22×C42—732no—
345C22×C42—731no—
346C22×C42C2×D7311yescharacteristic
347#383—1464no—
348C2×#95—732no—
349C2×#95—732no—
350C2×#95—732no—
351C2×#95—732no—
352C2×#95—732no—
353C2×#95—732no—
354C2×#95—732no—
355C2×#95—731no—
356C2×#95C2×D7312yes—
357C2×#95C2×D7311yescharacteristic
358C43—1464no—
359C25C22×D7311yescharacteristic
360C23×C4—731no—
361C23×C4—731no—
362C23×C4—732no—
363C23×C4—731no—
364C23×C4—731no—
365C23×C4—732no—
366C23×C4C22×D7312yes—
367C23×C4C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C23×C4C22×D7312yes—
370#126—14632no—
371C2×C4⋊C4—734no—
372C2×C4⋊C4—734no—
373C2×C4⋊C4—734no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1462no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1462no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—734no—
386C2×C22⋊C4—734no—
387C2×C22⋊C4—1464no—
388C2×C22⋊C4—1464no—
389C2×C22⋊C4—1464no—
390C2×C22⋊C4—1462no—
391C2×C22⋊C4—1462no—
392C2×C22⋊C4—1462no—
393C2×C22⋊C4—1462no—
394C2×C22⋊C4C4×D7314yes—
395C2×C42—22no—
396C2×C42—24no—
397C2×C42—734no—
398C2×C42—734no—
399C2×C42—734no—
400C2×C42—1464no—
401C2×C42—1464no—
402C2×C42—1464no—
403C2×C42—1464no—
404C2×C42—1464no—
405C2×C42—1462no—
406C2×C42—1464no—
407C2×C42—1464no—
408#95—22no—
409#95—24no—
410#95—1464no—
411#95—1464no—
412#95—1464no—
413#95—1464no—
414#95—1464no—
415#95—1462no—
416#95—1464no—
417#95—1464no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24C2×C4×D7312yes—
427C24C23×D7311yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—732no—
446C22×C4—732no—
447C22×C4—732no—
448C22×C4—732no—
449C22×C4—732no—
450C22×C4—732no—
451C22×C4—734no—
452C22×C4—734no—
453C22×C4—734no—
454C22×C4—732no—
455C22×C4—732no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4C2×C4×D7312yes—
480C22×C4C2×C4×D7312yes—
481C22×C4D4×D7312yes—
482C22×C4D4×D7312yes—
483C22×C4#1192402714yes—
484C22×C4#1192402714yes—
485C22×C4#1192402814yes—
486C4⋊C4—1468no—
487C4⋊C4—1468no—
488C4⋊C4—1468no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—2928no—
492C22⋊C4—1468no—
493C22⋊C4—1468no—
494C42—28no—
495C42—7316no—
496C42—1468no—
497C42—14616no—
498C42—14616no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C22×C4×D7311yescharacteristic
521C23C2×D4×D7311yescharacteristic
522C23C2×D4×D7311yescharacteristic
523C23#41×D7312yes—
524C23#41×D7312yes—
525C23#41×D7312yes—
526C23C2×#1192402712yes—
527C23C2×#1192402712yes—
528C23C2×#1192402812yes—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2922no—
556C2×C4—2922no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—734no—
564C2×C4—734no—
565C2×C4—738no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1464no—
569C2×C4—1462no—
570C2×C4—1462no—
571C2×C4—1464no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1464no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1464no—
578C2×C4—1464no—
579C2×C4—1464no—
580C2×C4—1464no—
581C2×C4—1464no—
582C2×C4—1464no—
583C2×C4—1464no—
584C2×C4—1464no—
585C2×C4C22⋊C4×D7314yes—
586C2×C4#1455658418yescomplemented
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×C22⋊C4×D7311yescharacteristic
597C22C4×#1192402712yes—
598C22C4×#1192402712yes—
599C22#1798905511yescharacteristic
600C22C4×#1192402811yescharacteristic
601C22#1798909311yescharacteristic
602C22#1798909511yescharacteristic
603C22#1798909612yes—
604C22#1798909712yes—
605C22#126×D7312yes—
606C22C2×#1455658412yes—
607C22#1798911412yes—
608C22#1798911512yes—
609C22#1798911612yes—
610C22#1798911712yes—
611C22#1798911812yes—
612C22#1798911912yes—
613C22#1798913212yes—
614C22#1798913412yes—
615C22#1798913511yescharacteristic
616C22#1798913611yescharacteristic
617C4—28no—
618C4—24no—
619C4—42no—
620C4—82no—
621C4—2922no—
622C4—2922no—
623C4—5842no—
624C4—1468no—
625C4—1464no—
626C4—1464no—
627C2—82no—
628C2#2261241611yescharacteristic
629C2#2261241711yescharacteristic
630C2#2261241811yescharacteristic
631C2C4×#1455658412yes—
632C2#2261248512yes—
633C2#2261824412yes—
634C2#2261824512yes—
635C2#2261829512yes—
636C2#2261829712yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1437C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1437D73
hypocenterC73#14711
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#17988454C2semidirect
3. C2×C42×Dic73C2semidirect
7. #22608166C2semidirect
8. #22608165C2semidirect
9. #22608164C2semidirect
10. #22608162C2semidirect
14. C2×#14556584C4semidirect
142. C4×Dic73C22⋊C4semidirect
233. C2×C4×C73#126semidirect
267. C2×Dic73#126semidirect
330. C73#14711semidirect
586. C2×C4#14556584semidirect

Maximal subgroups

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

subgroupindex
C22×#145565842
C2×#179884542
C2×C42×Dic732
#226081682
#226081662
#226081652
#226081642
#226081622
C73×#14372
#1471173