Finite Group Databasebeta

← #33008306

Subgroups — #33008306

Order 18688.

1730 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
1C2×#17988477C212yesmaximal
2C2×#17988446C211yesmaximal, characteristic
3C2×C4×#11924005C211yesmaximal, characteristic
4#22608194C212yesmaximal, complemented
5#22608175C212yesmaximal
6#22608166C212yesmaximal, complemented
7#22608159C211yesmaximal, characteristic, complemented
8#22608158C211yesmaximal, characteristic, complemented
9#22608152C211yesmaximal, characteristic
10#95×Dic73C211yesmaximal, characteristic, complemented
11C2×#382×C73C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×#11924005C2212yes—
16C22×#11924005C2212yes—
17C22×#11924005C2211yescharacteristic
18C22×#11924005C2211yescharacteristic
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21#17988477—28no—
22#17988446—24no—
23C4×#11924005C418yescomplemented
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2211yescharacteristic
33C22×C4⋊C4×C73C2212yes—
34C22×C42×C73C2211yescharacteristic
35#382×C73—24no—
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C24×D73C2311yescharacteristic
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C22×C4×D73—24no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—42no—
49C22×C4×D73C2×C414yes—
50C2×#11924007—24no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—42no—
54C2×#11924007—42no—
55C2×#11924007—42no—
56C2×#11924005—24no—
57C2×#11924005—24no—
58C2×#11924005—24no—
59C2×#11924005—24no—
60C2×#11924005—24no—
61C2×#11924005—42no—
62C2×#11924005—42no—
63C2×#11924005C2×C414yes—
64C2×#11924005C2×C414yes—
65C2×#11924005D414yes—
66C2×#11924005Q814yes—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—42no—
70C2×C4×Dic73—42no—
71C2×C4×Dic73C2×C414yes—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—24no—
76#14556498—22no—
77#14556498—24no—
78#14556498—22no—
79#14556498C2×C414yes—
80#14556498C2×C414yes—
81C23×C4×C73C2312yes—
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73—24no—
87C2×C4⋊C4×C73—24no—
88C2×C4⋊C4×C73—42no—
89C2×C42×C73—22no—
90C2×C42×C73—22no—
91C2×C42×C73C2×C414yes—
92#95×C73—24no—
93#95×C73—22no—
94#95×C73D414yes—
95C23×D73—21no—
96C23×D73—22no—
97C23×D73—22no—
98C23×D73—22no—
99C23×D73—21no—
100C23×D73—21no—
101C23×D73—22no—
102C23×D73—21no—
103C23×D73C22×C412yes—
104C23×D73C2×D412yes—
105C23×D73C2×Q812yes—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—21no—
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×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73C22×C412yes—
128C22×Dic73C22×C412yes—
129C22×Dic73C22×C412yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×Q812yes—
132C22×Dic73#4114yes—
133C22×Dic73#4114yes—
134C22×Dic73#4114yes—
135C2×C4×D73—28no—
136C2×C4×D73—28no—
137C2×C4×D73—44no—
138C2×C4×D73—48no—
139C2×C4×D73—48no—
140#11924007—48no—
141#11924007—48no—
142#11924005—28no—
143#11924005—48no—
144#11924005—48no—
145#11924005C4⋊C4116yescomplemented
146C4×Dic73—44no—
147C4×Dic73—48no—
148C24×C73C2411yescharacteristic
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—22no—
157C22×C4×C73—21no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
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×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73C22×C412yes—
173C22×C4×C73C22×C412yes—
174C22×C4×C73C22×C412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D414yes—
177C22×C4×C73C2×D412yes—
178C22×C4×C73C2×Q812yes—
179C22×C4×C73#4112yes—
180C4⋊C4×C73—48no—
181C42×C73—28no—
182C23×C73C23×C411yescharacteristic
183C23×C73C22×D411yescharacteristic
184C23×C73C22×D411yescharacteristic
185C23×C73C22×Q811yescharacteristic
186C23×C73C2×#4112yes—
187C23×C73C2×#4112yes—
188C23×C73C2×#4112yes—
189C23×C73C2×#4111yescharacteristic
190C23×C73H(5,𝔽2)11yescharacteristic
191C23×C73H(5,𝔽2)12yes—
192C23×C732−1+411yescharacteristic
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—22no—
196C22×D73—22no—
197C22×D73—22no—
198C22×D73—22no—
199C22×D73—24no—
200C22×D73—24no—
201C22×D73—22no—
202C22×D73—24no—
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—24no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73—42no—
210C22×D73—42no—
211C22×D73—42no—
212C22×D73—42no—
213C22×D73—42no—
214C22×D73—42no—
215C22×D73C2×C4⋊C414yes—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—24no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—24no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
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×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73C2×C4⋊C414yes—
246C2×C4×C73C4×D414yes—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73C2×C4⋊C414yes—
274C2×Dic73#11714yes—
275C2×Dic73#11714yes—
276C4×D73—416no—
277C22×C73C22×C4⋊C411yescharacteristic
278C22×C73C2×#11711yescharacteristic
279C22×C73C2×#11711yescharacteristic
280C22×C73C2×C4×D411yescharacteristic
281C22×C73C4×#4111yescharacteristic
282C22×C73#51811yescharacteristic
283C22×C73#52011yescharacteristic
284C22×C73#52912yes—
285C22×C73#53212yes—
286C22×C73#53212yes—
287C22×C73#53312yes—
288C22×C73#53812yes—
289C22×C73#53912yes—
290C22×C73D4211yescharacteristic
291C22×C73#54612yes—
292C22×C73#54612yes—
293C22×C73#54711yescharacteristic
294C22×C73D4×Q811yescharacteristic
295C22×C73#55112yes—
296C22×C73#55212yes—
297C22×C73#55411yescharacteristic
298C22×C73#55412yes—
299C22×C73#55512yes—
300C2×D73—24no—
301C2×D73—24no—
302C2×D73—24no—
303C2×D73—42no—
304C2×D73—44no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—44no—
309C2×D73—44no—
310C2×D73—44no—
311C4×C73—24no—
312C4×C73—24no—
313C4×C73—42no—
314C4×C73—44no—
315C4×C73—82no—
316Dic73—24no—
317Dic73—24no—
318Dic73—24no—
319Dic73—44no—
320Dic73—82no—
321#13915—731nomaximal, Sylow
322C2×C73#228511yescharacteristic
323C2×C73#229311yescharacteristic
324C2×C73#231811yescharacteristic
325C2×C73#234211yescharacteristic
326C2×C73#234611yescharacteristic
327C2×C73D4×C4⋊C411yescharacteristic
328C2×C73#234911yescharacteristic
329C2×C73#255312yes—
330C2×C73#255512yes—
331C2×C73#255612yes—
332C2×C73#257412yes—
333D73—44no—
334C2×#388—732no—
335C2×#382D7311yescharacteristic, complemented
336C2×#380—731no—
337C2×C4×C22⋊C4—731no—
338#1444—732no—
339#1439—732no—
340#1436—732no—
341C4⋊#95—731no—
342#1434—731no—
343C22⋊#95—731no—
344C4×#95—731no—
345C73#1391511yescharacteristic, complemented, Sylow
346C24×C4—731no—
347C22×C4⋊C4—732no—
348C22×C4⋊C4—732no—
349C22×C4⋊C4C2×D7312yes—
350C22×C22⋊C4—732no—
351C22×C22⋊C4—732no—
352C22×C22⋊C4—731no—
353C22×C22⋊C4—731no—
354C22×C42—731no—
355C22×C42—732no—
356C22×C42C2×D7311yescharacteristic
357#388—1468no—
358#382—24no—
359#380—1464no—
360C4×C22⋊C4—738no—
361C2×#95—731no—
362C2×#95—731no—
363C2×#95—731no—
364C2×#95—732no—
365C2×#95—732no—
366C2×#95—731no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—732no—
370C2×#95C2×D7311yescharacteristic
371C2×#95C2×D7312yes—
372C2×#95C2×D7311yescharacteristic
373C25—731no—
374C23×C4—2922no—
375C23×C4—734no—
376C23×C4—731no—
377C23×C4—731no—
378C23×C4—731no—
379C23×C4—732no—
380C23×C4—732no—
381C23×C4—1464no—
382C23×C4—1462no—
383C23×C4—1462no—
384C23×C4C22×D7312yes—
385C23×C4C22×D7311yescharacteristic
386C23×C4C22×D7312yes—
387C23×C4C22×D7311yescharacteristic
388C23×C4C22×D7311yescharacteristic
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—2922no—
393C2×C4⋊C4—2922no—
394C2×C4⋊C4—2922no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C22⋊C4—2922no—
399C2×C22⋊C4—2922no—
400C2×C22⋊C4—734no—
401C2×C22⋊C4—734no—
402C2×C22⋊C4—734no—
403C2×C22⋊C4—734no—
404C2×C22⋊C4—1464no—
405C2×C22⋊C4—1464no—
406C2×C22⋊C4—1464no—
407C2×C22⋊C4—1464no—
408C2×C22⋊C4—1464no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—2922no—
412C2×C42—2922no—
413C2×C42—734no—
414C2×C42—1464no—
415C2×C42—1464no—
416C2×C42C4×D7314yes—
417#95—24no—
418#95—22no—
419#95—734no—
420#95—734no—
421#95—1464no—
422#95—1464no—
423#95—1464no—
424#95—1464no—
425#95—1464no—
426#95—1462no—
427#95—1462no—
428#95C22⋊D7314yes—
429C24—732no—
430C24—732no—
431C24—732no—
432C24—1461no—
433C24—1462no—
434C24—1462no—
435C24—1462no—
436C24—1461no—
437C24—1461no—
438C24—1462no—
439C24—1461no—
440C24C23×D7311yescharacteristic
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
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—2928no—
465C22×C4—2928no—
466C22×C4—2924no—
467C22×C4—732no—
468C22×C4—732no—
469C22×C4—732no—
470C22×C4—732no—
471C22×C4—732no—
472C22×C4—734no—
473C22×C4—734no—
474C22×C4—734no—
475C22×C4—1468no—
476C22×C4—1462no—
477C22×C4—1462no—
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—1468no—
490C22×C4—1461no—
491C22×C4—1462no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1461no—
498C22×C4C2×C4×D7312yes—
499C22×C4C2×C4×D7312yes—
500C22×C4C2×C4×D7312yes—
501C22×C4D4×D7312yes—
502C22×C4C2×C22⋊D7314yes—
503C22×C4C2×C22⋊D7312yes—
504C22×C4Q8×D7312yes—
505C22×C4#1192402812yes—
506C4⋊C4—48no—
507C4⋊C4—2928no—
508C4⋊C4—2928no—
509C22⋊C4—2928no—
510C22⋊C4—2928no—
511C22⋊C4—7316no—
512C22⋊C4—1468no—
513C42—28no—
514C42—2924no—
515C42—2928no—
516C23—2922no—
517C23—2922no—
518C23—2922no—
519C23—2922no—
520C23—2922no—
521C23—2922no—
522C23—2922no—
523C23—2922no—
524C23—734no—
525C23—1464no—
526C23—1464no—
527C23—1462no—
528C23—1464no—
529C23—1462no—
530C23—1462no—
531C23—1462no—
532C23—1464no—
533C23—1462no—
534C23—1462no—
535C23—1462no—
536C23—1462no—
537C23—1462no—
538C23—1462no—
539C23C22×C4×D7311yescharacteristic
540C23C2×D4×D7311yescharacteristic
541C23C22×C22⋊D7311yescharacteristic
542C23C2×Q8×D7311yescharacteristic
543C23#41×D7312yes—
544C23#41×D7312yes—
545C23#41×D7312yes—
546C23C2×#1192402811yescharacteristic
547C23#1455663611yescharacteristic
548C23#1455663712yes—
549C23#1455663911yescharacteristic
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
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—29216no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—734no—
592C2×C4—734no—
593C2×C4—734no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4C4⋊C4×D7314yes—
610C2×C4C4×C22⋊D7314yes—
611C22—2922no—
612C22—2922no—
613C22—2922no—
614C22—2924no—
615C22—2922no—
616C22—2924no—
617C22—2924no—
618C22—2924no—
619C22—1464no—
620C22—1464no—
621C22—1464no—
622C22C2×C4⋊C4×D7311yescharacteristic
623C22#117×D7311yescharacteristic
624C22#117×D7311yescharacteristic
625C22C2×C4×C22⋊D7311yescharacteristic
626C22#1798905511yescharacteristic
627C22#1798906111yescharacteristic
628C22#1798906811yescharacteristic
629C22#1798911512yes—
630C22#1798913312yes—
631C22#1798913612yes—
632C22#1798914312yes—
633C22#1798917112yes—
634C22#1798918512yes—
635C22D4×C22⋊D7311yescharacteristic
636C22#1798922512yes—
637C22#1798923012yes—
638C22#1798923811yescharacteristic
639C22Q8×C22⋊D7311yescharacteristic
640C22#1798926112yes—
641C22#1798926912yes—
642C22#1798929211yescharacteristic
643C22#1798929312yes—
644C22#1798929912yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—44no—
649C4—82no—
650C4—2924no—
651C4—5842no—
652C4—1464no—
653C4—1464no—
654C4—1464no—
655C2—2924no—
656C2#2261233711yescharacteristic
657C2#2261239011yescharacteristic
658C2#2261256711yescharacteristic
659C2#2261278111yescharacteristic
660C2#2261281111yescharacteristic
661C2C4⋊C4×C22⋊D7311yescharacteristic
662C2#2261283911yescharacteristic
663C2#2261498312yes—
664C2#2261500212yes—
665C2#2261501112yes—
666C2#2261520112yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#382×C73C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC2×#382D73
hypocenterC73#13915
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
4. #22608194C2semidirect
6. #22608166C2semidirect
7. #22608159C2semidirect
8. #22608158C2semidirect
10. #95×Dic73C2semidirect
11. C2×#382×C73C2semidirect
23. C4×#11924005C4semidirect
145. #11924005C4⋊C4semidirect
335. C2×#382D73semidirect
345. C73#13915semidirect

Maximal subgroups

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

subgroupindex
C2×#179884772
C2×#179884462
C2×C4×#119240052
#226081942
#226081752
#226081662
#226081592
#226081582
#226081522
#95×Dic732
C2×#382×C732
#1391573