Finite Group Databasebeta

← #33012944

Subgroups — #33012944

Order 18688.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C211yesmaximal, characteristic
2C2×#17988485C211yesmaximal, characteristic, complemented
3#22608200C211yesmaximal, characteristic, complemented
4#22608187C211yesmaximal, characteristic, complemented
5#22608182C212yesmaximal, complemented
6#22608176C211yesmaximal, characteristic
7#22608174C212yesmaximal
8#22608164C212yesmaximal, complemented
9#22608158C211yesmaximal, characteristic, complemented
10#22608154C212yesmaximal
11C2×C4×C22⋊C4×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×C22⋊Dic73C2212yes—
17C22×C22⋊Dic73C2212yes—
18#17988523C418yescomplemented
19#17988485—24no—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2211yescharacteristic
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2211yescharacteristic
30C24×C4×C73C2211yescharacteristic
31C22×C22⋊C4×C73C2211yescharacteristic
32C22×C22⋊C4×C73C2211yescharacteristic
33C22×C42×C73C2211yescharacteristic
34C22×C42×C73C2211yescharacteristic
35C4×C22⋊C4×C73—24no—
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C23×Dic73C2312yes—
39C23×Dic73C2312yes—
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C2×C4⋊Dic73—22no—
43C2×C4⋊Dic73—22no—
44C2×C4⋊Dic73—22no—
45C2×C4⋊Dic73—42no—
46C2×C4⋊Dic73C2×C414yescomplemented
47C2×#11924007—24no—
48C2×#11924007—42no—
49C2×#11924007—42no—
50C2×#11924007—42no—
51C2×#11924007—42no—
52C2×#11924007C2×C418yescomplemented
53C2×C22⋊Dic73—24no—
54C2×C22⋊Dic73—42no—
55C2×C22⋊Dic73—42no—
56C2×C22⋊Dic73—42no—
57C2×C22⋊Dic73—42no—
58C2×C22⋊Dic73C2×C418yes—
59#14556498—24no—
60#14556498—24no—
61#14556498—22no—
62#14556498—24no—
63#14556498—42no—
64#14556498—42no—
65#14556498—42no—
66#14556498—42no—
67#14556498—42no—
68#14556498C2×C414yescomplemented
69C25×C73C2311yescharacteristic
70C23×C4×C73—22no—
71C23×C4×C73—22no—
72C23×C4×C73—42no—
73C23×C4×C73—42no—
74C23×C4×C73Q814yes—
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2311yescharacteristic
77C23×C4×C73C2311yescharacteristic
78C23×C4×C73C2311yescharacteristic
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2311yescharacteristic
81C2×C22⋊C4×C73—22no—
82C2×C22⋊C4×C73—22no—
83C2×C22⋊C4×C73—22no—
84C2×C22⋊C4×C73—22no—
85C2×C22⋊C4×C73—22no—
86C2×C22⋊C4×C73C2×C414yes—
87C2×C42×C73—22no—
88C2×C42×C73—42no—
89C2×C42×C73—42no—
90C2×C42×C73D414yes—
91#95×C73—22no—
92#95×C73—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×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73C22×C414yes—
120C22×Dic73C22×C414yes—
121C4⋊Dic73—28no—
122C4⋊Dic73—44no—
123#11924007—48no—
124#11924007—48no—
125C22⋊Dic73—48no—
126C22⋊Dic73—48no—
127C24×C73—21no—
128C24×C73—21no—
129C24×C73—21no—
130C24×C73—21no—
131C24×C73—22no—
132C24×C73—22no—
133C24×C73—22no—
134C24×C73—22no—
135C24×C73C22×C412yes—
136C24×C73C2×Q812yes—
137C24×C73#4112yes—
138C24×C73C2411yescharacteristic
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
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—28no—
160C22×C4×C73—48no—
161C22×C4×C73—44no—
162C22×C4×C73—44no—
163C22×C4×C73—48no—
164C22×C4×C73C22×C412yes—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C2×D412yes—
167C22×C4×C73C2×D412yes—
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×Q812yes—
170C22×C4×C73C2×Q812yes—
171C22×C4×C73#4112yes—
172C22×C4×C73#4112yes—
173C22×C4×C73#4112yes—
174C22⋊C4×C73—28no—
175C22⋊C4×C73—28no—
176C42×C73—44no—
177C42×C73—44no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—22no—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—24no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73C4×Q814yes—
201C23×C73C23×C411yescharacteristic
202C23×C73C22×D411yescharacteristic
203C23×C73C22×Q811yescharacteristic
204C23×C73C2×#4111yescharacteristic
205C23×C73C2×#4111yescharacteristic
206C23×C73C2×#4111yescharacteristic
207C23×C73C2×#4111yescharacteristic
208C23×C73H(5,𝔽2)12yes—
209C23×C73H(5,𝔽2)12yes—
210C23×C73H(5,𝔽2)12yes—
211C23×C732−1+412yes—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—22no—
221C2×C4×C73—24no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—416no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73C4×D414yes—
242C2×C4×C73C22⋊Q814yes—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
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—
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—
271C22×C73—28no—
272C22×C73—24no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—44no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—44no—
281C22×C73—44no—
282C22×C73C2×C4×D411yescharacteristic
283C22×C73C2×C4×Q811yescharacteristic
284C22×C73C4×#4111yescharacteristic
285C22×C73#51812yes—
286C22×C73#52012yes—
287C22×C73C2×C22⋊Q811yescharacteristic
288C22×C73#52511yescharacteristic
289C22×C73#52911yescharacteristic
290C22×C73#53311yescharacteristic
291C22×C73#53411yescharacteristic
292C22×C73#53511yescharacteristic
293C22×C73#53511yescharacteristic
294C22×C73#53611yescharacteristic
295C22×C73#53812yes—
296C22×C73#53812yes—
297C22×C73#53912yes—
298C22×C73#53912yes—
299C22×C73#53912yes—
300C22×C73#54012yes—
301C22×C73#54012yes—
302C22×C73#54212yes—
303C22×C73#54312yes—
304C22×C73#54412yes—
305C4×C73—24no—
306C4×C73—24no—
307C4×C73—42no—
308C4×C73—42no—
309C4×C73—42no—
310C4×C73—42no—
311Dic73—44no—
312Dic73—44no—
313Dic73—82no—
314Dic73—82no—
315#14317—731nomaximal, Sylow
316C2×C73—44no—
317C2×C73C4×C22⋊Q811yescharacteristic
318C2×C73#232211yescharacteristic
319C2×C73#232811yescharacteristic
320C2×C73#232912yes—
321C2×C73#233512yes—
322C2×C73#262611yescharacteristic
323C2×C73#262811yescharacteristic
324C2×C73#263311yescharacteristic
325C2×C73#263511yescharacteristic
326C2×C73#264912yes—
327C2×C73#266712yes—
328C2×#399—731no—
329C2×#389—731no—
330C2×C4×C22⋊C4D7311yescharacteristic
331#1445—731no—
332#1442—731no—
333#1441—732no—
334#1439—732no—
335#1439—731no—
336#1436—732no—
337#1434—731no—
338C22⋊#95—732no—
339C73#1431711yescharacteristic, complemented, Sylow
340C24×C4C2×D7311yescharacteristic
341C22×C4⋊C4—732no—
342C22×C4⋊C4—732no—
343C22×C4⋊C4—731no—
344C22×C4⋊C4—731no—
345C22×C22⋊C4—732no—
346C22×C22⋊C4—732no—
347C22×C22⋊C4C2×D7311yescharacteristic
348C22×C22⋊C4C2×D7311yescharacteristic
349C22×C42C2×D7311yescharacteristic
350C22×C42C2×D7311yescharacteristic
351#399—738no—
352#389—1464no—
353C4×C22⋊C4—24no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—731no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95—731no—
363C2×#95—732no—
364C2×#95C2×D7311yescharacteristic
365C2×#95C2×D7311yescharacteristic
366C25C22×D7311yescharacteristic
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—42no—
371C23×C4—732no—
372C23×C4—732no—
373C23×C4—732no—
374C23×C4—732no—
375C23×C4Dic14614yes—
376C23×C4C22×D7311yescharacteristic
377C23×C4C22×D7311yescharacteristic
378C23×C4C22×D7311yescharacteristic
379C23×C4C22×D7311yescharacteristic
380C23×C4C22×D7311yescharacteristic
381C23×C4C22×D7311yescharacteristic
382C2×C4⋊C4—2922no—
383C2×C4⋊C4—2922no—
384C2×C4⋊C4—2922no—
385C2×C4⋊C4—2922no—
386C2×C4⋊C4—2922no—
387C2×C4⋊C4—738no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—1464no—
390C2×C4⋊C4—1462no—
391C2×C4⋊C4—1462no—
392C2×C4⋊C4—1462no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—2922no—
399C2×C22⋊C4—2922no—
400C2×C22⋊C4—2922no—
401C2×C22⋊C4—2922no—
402C2×C22⋊C4—738no—
403C2×C22⋊C4—1464no—
404C2×C22⋊C4C4×D7314yes—
405C2×C42—22no—
406C2×C42—42no—
407C2×C42—42no—
408C2×C42C22⋊D7314yes—
409#95—22no—
410#95—22no—
411#95—2922no—
412#95—2922no—
413#95—2922no—
414#95—2922no—
415#95—2922no—
416#95—734no—
417#95—1464no—
418#95—1462no—
419#95—1464no—
420#95—1464no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D7312yes—
430C24C2×Dic14612yes—
431C24#1192402812yes—
432C24C23×D7311yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—734no—
459C22×C4—734no—
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×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×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4C2×C4×D7312yes—
487C22×C4C2×C4×D7312yes—
488C22×C4C2×C22⋊D7312yes—
489C22×C4C2×C22⋊D7312yes—
490C22×C4C2×C22⋊D7312yes—
491C22×C4C2×Dic14612yes—
492C22×C4C2×Dic14612yes—
493C22×C4#1192402812yes—
494C22×C4#1192402812yes—
495C22×C4#1192402812yes—
496C4⋊C4—2928no—
497C4⋊C4—2924no—
498C4⋊C4—2928no—
499C4⋊C4—1468no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—2928no—
503C22⋊C4—2928no—
504C42—44no—
505C42—44no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C4×Dic14614yes—
529C23C22×C4×D7311yescharacteristic
530C23C22×C22⋊D7311yescharacteristic
531C23C22×Dic14611yescharacteristic
532C23C2×#1192402811yescharacteristic
533C23C2×#1192402811yescharacteristic
534C23C2×#1192402811yescharacteristic
535C23C2×#1192402811yescharacteristic
536C23#1455663612yes—
537C23#1455663612yes—
538C23#1455663712yes—
539C23#1455663812yes—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
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—1464no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C2×C4C4×C22⋊D7314yes—
598C2×C4#1455657014yes—
599C22—28no—
600C22—24no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22C2×C4×C22⋊D7311yescharacteristic
611C22C2×C4×Dic14611yescharacteristic
612C22C4×#1192402811yescharacteristic
613C22#1798905912yes—
614C22#1798906612yes—
615C22C2×#1455657011yescharacteristic
616C22#1798909811yescharacteristic
617C22#1798912111yescharacteristic
618C22#1798914411yescharacteristic
619C22#1798914611yescharacteristic
620C22#1798915111yescharacteristic
621C22#1798915411yescharacteristic
622C22#1798915811yescharacteristic
623C22#1798917012yes—
624C22#1798917512yes—
625C22#1798917912yes—
626C22#1798917912yes—
627C22#1798917912yes—
628C22#1798918812yes—
629C22#1798918812yes—
630C22#1798920812yes—
631C22#1798921312yes—
632C22#1798921512yes—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—2924no—
640C4—2924no—
641C4—5842no—
642C4—5842no—
643C2—44no—
644C2C4×#1455657011yescharacteristic
645C2#2261259611yescharacteristic
646C2#2261265311yescharacteristic
647C2#2261265912yes—
648C2#2261272312yes—
649C2#2261573611yescharacteristic
650C2#2261575211yescharacteristic
651C2#2261579911yescharacteristic
652C2#2261581511yescharacteristic
653C2#2261592812yes—
654C2#2261612512yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×C4×C22⋊C4×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×C4×C22⋊C4D73
hypocenterC73#14317
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17988485C2semidirect
3. #22608200C2semidirect
4. #22608187C2semidirect
5. #22608182C2semidirect
8. #22608164C2semidirect
9. #22608158C2semidirect
18. #17988523C4semidirect
46. C2×C4⋊Dic73C2×C4semidirect
52. C2×#11924007C2×C4semidirect
68. #14556498C2×C4semidirect
339. C73#14317semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×#179884852
#226082002
#226081872
#226081822
#226081762
#226081742
#226081642
#226081582
#226081542
C2×C4×C22⋊C4×C732
#1431773