Finite Group Databasebeta

← #33012282

Subgroups — #33012282

Order 18688.

1470 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
1C2×#17988512C211yesmaximal, characteristic
2C2×C4×C4⋊Dic73C211yesmaximal, characteristic, complemented
3#22608198C211yesmaximal, characteristic, complemented
4#22608187C211yesmaximal, characteristic, complemented
5#22608176C211yesmaximal, characteristic
6#22608168C212yesmaximal
7#22608168C212yesmaximal
8#22608163C212yesmaximal, complemented
9#22608157C212yesmaximal, complemented
10C4×#14556498C211yesmaximal, characteristic, complemented
11C2×#388×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2212yes—
16C22×C4×Dic73C2212yes—
17C22×C4×Dic73C2212yes—
18#17988512C418yescomplemented
19C4×C4⋊Dic73—24no—
20C2×#14556498C2212yes—
21C2×#14556498C2211yescharacteristic
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C22×C4⋊C4×C73C2211yescharacteristic
31C22×C22⋊C4×C73C2211yescharacteristic
32C22×C22⋊C4×C73C2211yescharacteristic
33C22×C22⋊C4×C73C2211yescharacteristic
34C22×C42×C73C2211yescharacteristic
35#388×C73—24no—
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C23×Dic73C2312yes—
39C23×Dic73C2312yes—
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C2×C4⋊Dic73—24no—
43C2×C4⋊Dic73—22no—
44C2×C4⋊Dic73—22no—
45C2×C4⋊Dic73—22no—
46C2×C4⋊Dic73C2×C414yes—
47C2×C22⋊Dic73—24no—
48C2×C22⋊Dic73—24no—
49C2×C22⋊Dic73—42no—
50C2×C22⋊Dic73—42no—
51C2×C22⋊Dic73—42no—
52C2×C22⋊Dic73C2×C418yes—
53C2×C4×Dic73—24no—
54C2×C4×Dic73—24no—
55C2×C4×Dic73—24no—
56C2×C4×Dic73—42no—
57C2×C4×Dic73—42no—
58C2×C4×Dic73—42no—
59#14556498—24no—
60#14556498—24no—
61#14556498—24no—
62#14556498—24no—
63#14556498—22no—
64#14556498—42no—
65#14556498—42no—
66#14556498—42no—
67#14556498C2×C418yes—
68#14556498C2×C414yes—
69C25×C73C2311yescharacteristic
70C23×C4×C73C2311yescharacteristic
71C23×C4×C73C2311yescharacteristic
72C23×C4×C73C2311yescharacteristic
73C23×C4×C73C2311yescharacteristic
74C23×C4×C73C2311yescharacteristic
75C23×C4×C73C2311yescharacteristic
76C2×C4⋊C4×C73—22no—
77C2×C4⋊C4×C73—24no—
78C2×C22⋊C4×C73—22no—
79C2×C22⋊C4×C73—22no—
80C2×C22⋊C4×C73—22no—
81C2×C22⋊C4×C73—22no—
82C2×C22⋊C4×C73—24no—
83C2×C22⋊C4×C73—24no—
84C2×C22⋊C4×C73C2×C414yes—
85C2×C42×C73—24no—
86C2×C42×C73D414yes—
87#95×C73—22no—
88#95×C73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
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—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73C22×C414yes—
114C22×Dic73C22×C414yes—
115C22×Dic73#4114yes—
116C22×Dic73#4114yes—
117C4⋊Dic73—28no—
118C4⋊Dic73—28no—
119C22⋊Dic73—48no—
120C22⋊Dic73—48no—
121C4×Dic73—48no—
122C4×Dic73—48no—
123C24×C73—21no—
124C24×C73—22no—
125C24×C73—22no—
126C24×C73—21no—
127C24×C73—22no—
128C24×C73—22no—
129C24×C73—21no—
130C24×C73—21no—
131C24×C73—21no—
132C24×C73—21no—
133C24×C73C22×C412yes—
134C24×C73C2411yescharacteristic
135C22×C4×C73—22no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73C22×C412yes—
158C22×C4×C73C22×C412yes—
159C22×C4×C73C2×D412yes—
160C22×C4×C73C2×D412yes—
161C22×C4×C73C2×D412yes—
162C22×C4×C73#4114yes—
163C22×C4×C73#4114yes—
164C22×C4×C73#4112yes—
165C4⋊C4×C73—44no—
166C22⋊C4×C73—28no—
167C22⋊C4×C73—44no—
168C22⋊C4×C73—44no—
169C42×C73—28no—
170C23×C73—22no—
171C23×C73—24no—
172C23×C73—24no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—41no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73—41no—
181C23×C73—41no—
182C23×C73—41no—
183C23×C73—41no—
184C23×C73—41no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73C23×C411yescharacteristic
194C23×C73C22×D411yescharacteristic
195C23×C73C2×#4112yes—
196C23×C73C2×#4112yes—
197C23×C73C2×#4112yes—
198C23×C73C2×#4112yes—
199C23×C73C2×#4111yescharacteristic
200C23×C73H(5,𝔽2)11yescharacteristic
201C23×C73H(5,𝔽2)11yescharacteristic
202C23×C73H(5,𝔽2)11yescharacteristic
203C23×C732−1+411yescharacteristic
204C2×C4×C73—24no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—24no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—41no—
219C2×C4×C73—42no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
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×C73C4×D414yes—
232C2×C4×C73#12314yes—
233C2×C4×C73#12414yes—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
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—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C22×C73—42no—
263C22×C73—44no—
264C22×C73—44no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73—82no—
272C22×C73—82no—
273C22×C73C2×C4×D411yescharacteristic
274C22×C73C4×#4112yes—
275C22×C73C4×#4112yes—
276C22×C73#51811yescharacteristic
277C22×C73#52011yescharacteristic
278C22×C73C2×#12311yescharacteristic
279C22×C73#52511yescharacteristic
280C22×C73C2×#12411yescharacteristic
281C22×C73#52912yes—
282C22×C73#52912yes—
283C22×C73#53211yescharacteristic
284C22×C73#53511yescharacteristic
285C22×C73#53711yescharacteristic
286C22×C73#54011yescharacteristic
287C22×C73#54211yescharacteristic
288C22×C73#55112yes—
289C22×C73#55112yes—
290C22×C73#55212yes—
291C22×C73#55312yes—
292C22×C73#55312yes—
293C22×C73#55312yes—
294C22×C73#55612yes—
295C22×C73#55912yes—
296C4×C73—24no—
297C4×C73—24no—
298C4×C73—24no—
299C4×C73—42no—
300C4×C73—81no—
301C4×C73—81no—
302Dic73—44no—
303Dic73—44no—
304Dic73—44no—
305Dic73—44no—
306#14249—731nomaximal, Sylow
307C2×C73—82no—
308C2×C73C4×#12311yescharacteristic
309C2×C73C4×#12411yescharacteristic
310C2×C73#232411yescharacteristic
311C2×C73#237312yes—
312C2×C73#237412yes—
313C2×C73#249811yescharacteristic
314C2×C73#250411yescharacteristic
315C2×C73#251612yes—
316C2×C73#252512yes—
317C2×C73#259211yescharacteristic
318C2×C73#260311yescharacteristic
319C2×#397—731no—
320C2×#388D7311yescharacteristic
321C2×C4×C4⋊C4—731no—
322#1445—731no—
323#1442—731no—
324#1439—731no—
325#1437—732no—
326#1437—732no—
327#1436—732no—
328#1434—732no—
329C4×#95—731no—
330C73#1424911yescharacteristic, complemented, Sylow
331C22×C4⋊C4—731no—
332C22×C4⋊C4—731no—
333C22×C4⋊C4C2×D7311yescharacteristic
334C22×C22⋊C4—732no—
335C22×C22⋊C4—732no—
336C22×C22⋊C4C2×D7311yescharacteristic
337C22×C22⋊C4C2×D7311yescharacteristic
338C22×C22⋊C4C2×D7311yescharacteristic
339C22×C42—732no—
340C22×C42—732no—
341C22×C42C2×D7311yescharacteristic
342#397—738no—
343#388—24no—
344C4×C4⋊C4—1464no—
345C2×#95—732no—
346C2×#95—732no—
347C2×#95—732no—
348C2×#95—731no—
349C2×#95—732no—
350C2×#95—732no—
351C2×#95—731no—
352C2×#95—732no—
353C2×#95—732no—
354C2×#95—732no—
355C2×#95C2×D7311yescharacteristic
356C2×#95C2×D7311yescharacteristic
357C25C22×D7311yescharacteristic
358C23×C4—732no—
359C23×C4—732no—
360C23×C4—732no—
361C23×C4—732no—
362C23×C4C22×D7311yescharacteristic
363C23×C4C22×D7311yescharacteristic
364C23×C4C22×D7311yescharacteristic
365C23×C4C22×D7311yescharacteristic
366C23×C4C22×D7311yescharacteristic
367C23×C4C22×D7311yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—734no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1462no—
373C2×C4⋊C4—1462no—
374C2×C4⋊C4—1462no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—2922no—
382C2×C22⋊C4—2922no—
383C2×C22⋊C4—2922no—
384C2×C22⋊C4—738no—
385C2×C22⋊C4—1464no—
386C2×C22⋊C4—1464no—
387C2×C22⋊C4C4×D7314yes—
388C2×C42—24no—
389C2×C42—2922no—
390C2×C42—2922no—
391C2×C42—2922no—
392C2×C42—1464no—
393C2×C42—1464no—
394C2×C42—1464no—
395C2×C42C22⋊D7314yes—
396#95—22no—
397#95—22no—
398#95—2922no—
399#95—2922no—
400#95—2922no—
401#95—734no—
402#95—738no—
403#95—1464no—
404#95—1464no—
405#95—1464no—
406#95—1464no—
407#95—1462no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24C2×C4×D7312yes—
419C24C23×D7311yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—734no—
443C22×C4—734no—
444C22×C4—734no—
445C22×C4—734no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4C2×C4×D7312yes—
471C22×C4C2×C4×D7312yes—
472C22×C4C2×C22⋊D7312yes—
473C22×C4C2×C22⋊D7312yes—
474C22×C4C2×C22⋊D7312yes—
475C22×C4#1192402714yes—
476C22×C4#1192402714yes—
477C22×C4#1192402812yes—
478C4⋊C4—44no—
479C4⋊C4—1468no—
480C4⋊C4—1468no—
481C22⋊C4—28no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—2928no—
485C22⋊C4—2928no—
486C42—28no—
487C42—2928no—
488C42—2928no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23C22×C4×D7311yescharacteristic
513C23C22×C22⋊D7311yescharacteristic
514C23#41×D7312yes—
515C23#41×D7312yes—
516C23C2×#1192402712yes—
517C23C2×#1192402712yes—
518C23C2×#1192402811yescharacteristic
519C23#1455663711yescharacteristic
520C23#1455663711yescharacteristic
521C23#1455663711yescharacteristic
522C23#1455663811yescharacteristic
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
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—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—1464no—
567C2×C4—1464no—
568C2×C4—1464no—
569C2×C4—1464no—
570C2×C4—1464no—
571C2×C4—1464no—
572C2×C4—1464no—
573C2×C4—1464no—
574C2×C4—1464no—
575C2×C4—1464no—
576C2×C4—1464no—
577C2×C4—1464no—
578C2×C4C4×C22⋊D7314yes—
579C2×C4#1455657214yes—
580C2×C4#1455657814yes—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22C2×C4×C22⋊D7311yescharacteristic
593C22C4×#1192402712yes—
594C22C4×#1192402712yes—
595C22#1798906111yescharacteristic
596C22#1798906711yescharacteristic
597C22C2×#1455657211yescharacteristic
598C22#1798909411yescharacteristic
599C22C2×#1455657811yescharacteristic
600C22#1798911712yes—
601C22#1798911912yes—
602C22#1798913311yescharacteristic
603C22#1798915111yescharacteristic
604C22#1798916511yescharacteristic
605C22#1798919211yescharacteristic
606C22#1798920811yescharacteristic
607C22#1798926012yes—
608C22#1798926212yes—
609C22#1798927412yes—
610C22#1798928812yes—
611C22#1798928912yes—
612C22#1798928912yes—
613C22#1798930612yes—
614C22#1798931912yes—
615C4—24no—
616C4—24no—
617C4—24no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—2924no—
622C4—2924no—
623C4—2924no—
624C4—2924no—
625C2—82no—
626C2C4×#1455657211yescharacteristic
627C2C4×#1455657811yescharacteristic
628C2#2261262411yescharacteristic
629C2#2261305812yes—
630C2#2261306112yes—
631C2#2261437711yescharacteristic
632C2#2261442411yescharacteristic
633C2#2261456712yes—
634C2#2261469012yes—
635C2#2261542011yescharacteristic
636C2#2261556011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#388×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#388D73
hypocenterC73#14249
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic73C2semidirect
3. #22608198C2semidirect
4. #22608187C2semidirect
8. #22608163C2semidirect
9. #22608157C2semidirect
10. C4×#14556498C2semidirect
18. #17988512C4semidirect
330. C73#14249semidirect

Maximal subgroups

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

subgroupindex
C2×#179885122
C2×C4×C4⋊Dic732
#226081982
#226081872
#226081762
#226081682
#226081632
#226081572
C4×#145564982
C2×#388×C732
#1424973