Finite Group Databasebeta

← #33022748

Subgroups — #33022748

Order 18688.

1710 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988515C212yesmaximal
2C2×#17988512C211yesmaximal, characteristic
3C2×#17988456C211yesmaximal, characteristic, complemented
4C2×#17988445C212yesmaximal
5C2×C22⋊C4×Dic73C211yesmaximal, characteristic
6#22608200C211yesmaximal, characteristic, complemented
7#22608199C211yesmaximal, characteristic, complemented
8#22608197C212yesmaximal, complemented
9#22608159C212yesmaximal, complemented
10#22608159C211yesmaximal, characteristic, complemented
11C2×#397×C73C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924007C2212yes—
17C22×C22⋊Dic73C2212yes—
18C22×C22⋊Dic73C2211yescharacteristic
19C22×C4×Dic73C2211yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21#17988515—28no—
22#17988512—24no—
23#17988456—24no—
24#17988445—28no—
25C22⋊C4×Dic73—24no—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2211yescharacteristic
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2212yes—
37C22×C22⋊C4×C73C2211yescharacteristic
38#397×C73—24no—
39C2×#95×C73C2212yes—
40C2×#95×C73C2211yescharacteristic
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—24no—
44C23×Dic73—24no—
45C23×Dic73D414yes—
46C23×Dic73C2312yes—
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—24no—
60C2×#11924007—24no—
61C2×#11924007—24no—
62C2×#11924007—24no—
63C2×#11924007—24no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—24no—
67C2×C22⋊Dic73—24no—
68C2×C22⋊Dic73—24no—
69C2×C22⋊Dic73D414yes—
70C2×C4×Dic73—24no—
71C2×C4×Dic73—22no—
72C2×C4×Dic73—24no—
73C2×C4×Dic73D414yes—
74#14556498—22no—
75#14556498—24no—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—22no—
81#14556498—24no—
82#14556498—22no—
83C25×C73C2311yescharacteristic
84C23×C4×C73C2312yes—
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73—22no—
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—22no—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—24no—
94C2×C22⋊C4×C73—24no—
95C2×C22⋊C4×C73—24no—
96C2×C22⋊C4×C73D414yes—
97#95×C73—22no—
98#95×C73—24no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—28no—
110C22×Dic73—216no—
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×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—44no—
126C22×Dic73—44no—
127C22×Dic73—48no—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D412yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×D412yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73#4112yes—
135C22×Dic73#4112yes—
136C22×Dic73#4114yes—
137C22×Dic73#4114yes—
138C4⋊Dic73—44no—
139#11924007—48no—
140#11924007—48no—
141#11924007—48no—
142C22⋊Dic73—28no—
143C22⋊Dic73—48no—
144C4×Dic73—28no—
145C4×Dic73—44no—
146C24×C73—21no—
147C24×C73—21no—
148C24×C73—21no—
149C24×C73—21no—
150C24×C73—22no—
151C24×C73—22no—
152C24×C73—22no—
153C24×C73—22no—
154C24×C73C2×D412yes—
155C24×C73C2×D412yes—
156C24×C73C2×D412yes—
157C24×C73C2411yescharacteristic
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
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×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2×D412yes—
180C22×C4×C73C2×D412yes—
181C22×C4×C73#4112yes—
182C4⋊C4×C73—44no—
183C22⋊C4×C73—28no—
184C22⋊C4×C73—48no—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—24no—
193C23×C73—24no—
194C23×C73—24no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—42no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73#12714yes—
208C23×C73C22×D411yescharacteristic
209C23×C73C22×D411yescharacteristic
210C23×C73C22×D411yescharacteristic
211C23×C73C22×D411yescharacteristic
212C23×C73C2×#4111yescharacteristic
213C23×C73C2×#4111yescharacteristic
214C23×C73C2×#4112yes—
215C23×C73C2×#4112yes—
216C23×C73C2×#4111yescharacteristic
217C23×C732−1+412yes—
218C23×C732−1+412yes—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—22no—
224C2×C4×C73—24no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—42no—
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#12114yes—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—42no—
255C2×Dic73—41no—
256C2×Dic73—42no—
257C2×Dic73—416no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
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#12114yes—
273C2×Dic73#12114yes—
274C2×Dic73#12314yes—
275C2×Dic73#12318yes—
276C2×Dic73#12414yes—
277C2×Dic73#12618yes—
278C22×C73—24no—
279C22×C73—28no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—44no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73C2×#12111yescharacteristic
290C22×C73C2×#12111yescharacteristic
291C22×C73C2×#12111yescharacteristic
292C22×C73C2×#12311yescharacteristic
293C22×C73C2×#12312yes—
294C22×C73C2×#12411yescharacteristic
295C22×C73C2×#12612yes—
296C22×C73C2×#12711yescharacteristic
297C22×C73#53211yescharacteristic
298C22×C73#53211yescharacteristic
299C22×C73#53211yescharacteristic
300C22×C73#53611yescharacteristic
301C22×C73#53611yescharacteristic
302C22×C73#53612yes—
303C22×C73#54112yes—
304C22×C73#54712yes—
305C22×C73#54712yes—
306C22×C73#54712yes—
307C22×C73#54712yes—
308C22×C73#55212yes—
309C22×C73#55212yes—
310C22×C73#55612yes—
311C22×C73#55612yes—
312C4×C73—24no—
313C4×C73—42no—
314C4×C73—82no—
315C4×C73—82no—
316Dic73—24no—
317Dic73—42no—
318Dic73—42no—
319Dic73—82no—
320Dic73—82no—
321Dic73#39718yescomplemented
322#15150—731nomaximal, Sylow
323C2×C73—44no—
324C2×C73C2×#39711yescharacteristic
325C2×C73#248611yescharacteristic
326C2×C73#248611yescharacteristic
327C2×C73#249612yes—
328C2×C73#249612yes—
329C2×C73#249912yes—
330C2×C73#250112yes—
331C2×C73#254711yescharacteristic
332C2×C73#254711yescharacteristic
333C2×C73#262111yescharacteristic
334C2×C73#285611yescharacteristic
335C2×#397—731no—
336C2×#397—732no—
337C2×#397D7311yescharacteristic
338C2×#384—731no—
339C2×#380—732no—
340C2×C4×C22⋊C4—731no—
341#1445—732no—
342#1445—731no—
343#1445—731no—
344C4⋊#95—731no—
345C4⋊#95—732no—
346C73#1515011yescharacteristic, complemented, Sylow
347C24×C4—731no—
348C22×C4⋊C4—732no—
349C22×C4⋊C4—731no—
350C22×C4⋊C4—732no—
351C22×C4⋊C4—732no—
352C22×C4⋊C4C2×D7311yescharacteristic
353C22×C22⋊C4—731no—
354C22×C22⋊C4—732no—
355C22×C22⋊C4C2×D7312yes—
356C22×C22⋊C4C2×D7311yescharacteristic
357C22×C42—731no—
358C22×C42—731no—
359#397—24no—
360#397—1468no—
361#397—1464no—
362#384—1464no—
363#380—1468no—
364C4×C22⋊C4—1464no—
365C2×#95—732no—
366C2×#95—732no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—731no—
370C2×#95—731no—
371C2×#95—731no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95C2×D7312yes—
375C2×#95C2×D7311yescharacteristic
376C25C22×D7311yescharacteristic
377C23×C4—731no—
378C23×C4—731no—
379C23×C4—734no—
380C23×C4—731no—
381C23×C4—731no—
382C23×C4—732no—
383C23×C4—732no—
384C23×C4—1462no—
385C23×C4—1464no—
386C23×C4—1464no—
387C23×C4—1462no—
388C23×C4C22×D7312yes—
389C23×C4C22×D7312yes—
390C23×C4C22×D7311yescharacteristic
391C23×C4C22×D7311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1464no—
399C2×C4⋊C4—1464no—
400C2×C4⋊C4—1464no—
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×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—734no—
413C2×C22⋊C4—1464no—
414C2×C22⋊C4—1464no—
415C2×C22⋊C4—1464no—
416C2×C22⋊C4—1462no—
417C2×C22⋊C4—1462no—
418C2×C22⋊C4C22⋊D7314yes—
419C2×C42—734no—
420C2×C42—1462no—
421C2×C42—1464no—
422C2×C42—1464no—
423#95—22no—
424#95—24no—
425#95—1462no—
426#95—1464no—
427#95—1464no—
428#95—1464no—
429#95—1464no—
430#95—1464no—
431#95—1462no—
432#95—1462no—
433#95—1464no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24—21no—
440C24—22no—
441C24—22no—
442C24D4×D7312yes—
443C24D4×D7312yes—
444C24C2×C22⋊D7312yes—
445C24C23×D7311yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
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—2924no—
467C22×C4—2924no—
468C22×C4—2928no—
469C22×C4—732no—
470C22×C4—732no—
471C22×C4—734no—
472C22×C4—734no—
473C22×C4—732no—
474C22×C4—732no—
475C22×C4—732no—
476C22×C4—732no—
477C22×C4—732no—
478C22×C4—732no—
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—1462no—
493C22×C4—1468no—
494C22×C4—14616no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1462no—
498C22×C4—1461no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1461no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4D4×D7312yes—
506C22×C4C2×C22⋊D7312yes—
507C22×C4C2×C22⋊D7312yes—
508C22×C4#1192402712yes—
509C4⋊C4—44no—
510C4⋊C4—2928no—
511C4⋊C4—2928no—
512C4⋊C4—2928no—
513C4⋊C4—2924no—
514C22⋊C4—28no—
515C22⋊C4—48no—
516C22⋊C4—2928no—
517C22⋊C4—1468no—
518C42—2924no—
519C42—1468no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23#1455658714yes—
543C23C2×D4×D7311yescharacteristic
544C23C2×D4×D7311yescharacteristic
545C23C2×D4×D7311yescharacteristic
546C23C22×C22⋊D7311yescharacteristic
547C23#41×D7311yescharacteristic
548C23#41×D7311yescharacteristic
549C23#41×D7312yes—
550C23#41×D7312yes—
551C23C2×#1192402711yescharacteristic
552C23#1455663812yes—
553C23#1455663812yes—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
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—2921no—
578C2×C4—2922no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2922no—
583C2×C4—29216no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
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—734no—
599C2×C4—734no—
600C2×C4—738no—
601C2×C4—734no—
602C2×C4—738no—
603C2×C4—734no—
604C2×C4—1464no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1464no—
608C2×C4—1462no—
609C2×C4—1464no—
610C2×C4—1464no—
611C2×C4—1462no—
612C2×C4#1455656314yes—
613C22—28no—
614C22—24no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22#121×D7311yescharacteristic
625C22#121×D7311yescharacteristic
626C22C2×#1455656311yescharacteristic
627C22#123×D7312yes—
628C22#123×D7311yescharacteristic
629C22#124×D7311yescharacteristic
630C22#126×D7312yes—
631C22C2×#1455658711yescharacteristic
632C22#1798913211yescharacteristic
633C22#1798913211yescharacteristic
634C22#1798913311yescharacteristic
635C22#1798915711yescharacteristic
636C22#1798915912yes—
637C22#1798915911yescharacteristic
638C22#1798919412yes—
639C22#1798923512yes—
640C22#1798923512yes—
641C22#1798923612yes—
642C22#1798923612yes—
643C22#1798927312yes—
644C22#1798927412yes—
645C22#1798930412yes—
646C22#1798930612yes—
647C4—24no—
648C4—42no—
649C4—82no—
650C4—82no—
651C4—2922no—
652C4—2922no—
653C4—5842no—
654C4—5842no—
655C4—738no—
656C4—1464no—
657C2—44no—
658C2#397×D7311yescharacteristic
659C2#2261423211yescharacteristic
660C2#2261423611yescharacteristic
661C2#2261434612yes—
662C2#2261435012yes—
663C2#2261437912yes—
664C2#2261440012yes—
665C2#2261492211yescharacteristic
666C2#2261492411yescharacteristic
667C2#2261569611yescharacteristic
668C2#2261827411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutator, socle, cosocleC24×C73C24
FittingC2×#397×C73C2
prime radicalC25×C73C23
hypercenterC2×#397D73
hypocenterC73#15150
LCSG ▷ C24×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. C2×#17988456C2semidirect
6. #22608200C2semidirect
7. #22608199C2semidirect
8. #22608197C2semidirect
9. #22608159C2semidirect
10. #22608159C2semidirect
321. Dic73#397semidirect
346. C73#15150semidirect

Maximal subgroups

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

subgroupindex
C2×#179885152
C2×#179885122
C2×#179884562
C2×#179884452
C2×C22⋊C4×Dic732
#226082002
#226081992
#226081972
#226081592
C2×#397×C732
#1515073