Finite Group Databasebeta

← #33086964

Subgroups — #33086964

Order 18688.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11924028C211yesmaximal, characteristic
2C2×#17988629C211yesmaximal, characteristic, complemented
3C2×#17988589C211yesmaximal, characteristic
4C2×#17988589C211yesmaximal, characteristic
5C2×#17988485C211yesmaximal, characteristic, complemented
6#22610202C212yesmaximal, complemented
7#22610195C212yesmaximal, complemented
8#22610188C212yesmaximal, complemented
9#22610181C212yesmaximal, complemented
10#22609651C211yesmaximal, characteristic, complemented
11C73×#1728C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C4×#11924028—24no—
14C4×#11924028—24no—
15C2×C4×Dic146C2211yescharacteristic
16C2×C4×Dic146C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×C22⋊D73—21no—
20C2×C4×C22⋊D73—21no—
21C2×#14556550C2211yescharacteristic
22C2×#14556548—21no—
23C22×C4⋊Dic73C2211yescharacteristic
24C22×C4⋊Dic73C2211yescharacteristic
25C22×#11924007—21no—
26C2×C42×D73—21no—
27#17988629—24no—
28C2×#14556525C2212yes—
29C2×#14556523C2212yes—
30#17988589—24no—
31#17988589C418yescomplemented
32C2×#14556516C2212yes—
33C2×#14556516C2212yes—
34C2×#14556513C2212yes—
35C2×#14556513C2212yes—
36#17988485—21no—
37#17988485—21no—
38#17988485C2212yes—
39#17988485C2212yes—
40C2×#14556498—21no—
41#17988350C2212yes—
42#17988350C2212yes—
43C22×M16×C73C2211yescharacteristic
44C22×M16×C73C2211yescharacteristic
45C22×C42×C73C2211yescharacteristic
46#336×C73C2212yes—
47#336×C73C2212yes—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028—22no—
51C2×#11924028C2×C414yes—
52C2×#11924028D414yes—
53C2×#11924028Q814yes—
54C22×Dic146C2311yescharacteristic
55C22×D292C2311yescharacteristic
56C22×C22⋊D73—21no—
57C23×Dic73—21no—
58C23×Dic73—21no—
59C22×C4×D73—21no—
60C22×C4×D73—21no—
61C22×C4×D73—21no—
62C4×Dic146—22no—
63C4×Dic146—22no—
64C4×Dic146—22no—
65C4×Dic146—22no—
66C4×D292—22no—
67C4×D292—22no—
68C4×D292—22no—
69C4×D292—22no—
70C4×C22⋊D73—28no—
71C4×C22⋊D73—28no—
72#14556550—22no—
73#14556550D414yes—
74#14556548—28no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—22no—
77C2×C4⋊Dic73—22no—
78C2×C4⋊Dic73—21no—
79C2×C4⋊Dic73—22no—
80C2×C4⋊Dic73C2×C414yescomplemented
81C2×C4⋊Dic73C2312yes—
82C2×C4⋊Dic73C2312yes—
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—22no—
88C2×#11924007—22no—
89C2×#11924007—44no—
90C2×C22⋊Dic73—21no—
91C2×#11924005—21no—
92C2×#11924005—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C42×D73—28no—
97#14556525—24no—
98#14556523—24no—
99#14556516—24no—
100#14556516C2×C418yescomplemented
101#14556513—24no—
102#14556513C2×C418yescomplemented
103#14556498—22no—
104#14556498—22no—
105C23×C4×C73D411yescharacteristic
106C23×C4×C73D411yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C2×M16×C73—22no—
109C2×M16×C73—22no—
110C2×M16×C73—22no—
111C2×M16×C73—22no—
112C2×M16×C73—22no—
113C2×M16×C73C2×C414yescomplemented
114C22×C8×C73C2312yes—
115C22×C8×C73C2312yes—
116C2×C42×C73—22no—
117C2×C42×C73—22no—
118C2×C42×C73D411yescharacteristic
119C2×C42×C73D411yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C23×D73—21no—
123#11924028—28no—
124#11924028—28no—
125#11924028—28no—
126#11924028C4⋊C4116yescomplemented
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146C22×C412yes—
131C2×Dic146C2×D412yes—
132C2×Dic146C2×Q812yes—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292C22×C412yes—
137C2×D292C2×D412yes—
138C2×D292C2×Q812yes—
139C2×C22⋊D73—24no—
140C2×C22⋊D73—24no—
141C2×C22⋊D73—24no—
142C22×Dic73—21no—
143C22×Dic73—24no—
144C22×Dic73—22no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—22no—
148C22×Dic73—42no—
149C22×Dic73—42no—
150C22×Dic73—42no—
151C22×Dic73—42no—
152C22×Dic73—42no—
153C2×C4×D73—24no—
154C2×C4×D73—24no—
155C2×C4×D73—24no—
156C2×C4×D73—24no—
157C2×C4×D73—24no—
158C2×C4×D73—42no—
159C2×C4×D73—42no—
160C2×C4×D73—42no—
161C2×C4×D73—42no—
162C4⋊Dic73—24no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—22no—
166C4⋊Dic73—22no—
167C4⋊Dic73—22no—
168C4⋊Dic73—42no—
169C4⋊Dic73C22×C414yes—
170C4⋊Dic73C2×D412yes—
171C4⋊Dic73C2×D412yes—
172#11924007—24no—
173#11924007—24no—
174#11924007—44no—
175#11924007—44no—
176C22⋊Dic73—24no—
177#11924005—24no—
178#11924005—24no—
179C4×Dic73—24no—
180C4×Dic73—24no—
181C4×Dic73—24no—
182C24×C73C2×D411yescharacteristic
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—22no—
186C22×C4×C73—22no—
187C22×C4×C73—41no—
188C22×C4×C73—41no—
189C22×C4×C73C22×C412yes—
190C22×C4×C73C2×D412yes—
191C22×C4×C73C2×D412yes—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×Q812yes—
198C22×C4×C73#4112yes—
199C22×C4×C73#4112yes—
200C22×C4×C73C2411yescharacteristic
201M16×C73—28no—
202M16×C73—44no—
203C2×C8×C73—22no—
204C2×C8×C73—22no—
205C2×C8×C73—22no—
206C2×C8×C73—22no—
207C2×C8×C73—22no—
208C2×C8×C73C22×C414yes—
209C42×C73—22no—
210C42×C73—22no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73C4×D412yes—
220C23×C73#12012yes—
221C23×C73C22⋊Q812yes—
222C23×C73C22×D411yescharacteristic
223C22×D73—22no—
224C22×D73—22no—
225C22×D73—22no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—44no—
241C2×C4×C73—44no—
242C2×C4×C73C2×C4⋊C414yes—
243C2×C4×C73C4×D414yes—
244C2×C4×C73C4×D412yes—
245C2×C4×C73#12012yes—
246C2×C4×C73C22⋊Q814yes—
247C2×C4×C73C22⋊Q812yes—
248C2×C4×C73#12314yes—
249C2×C4×C73AGL(1,ℤ/8)12yes—
250C2×C4×C73#13712yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C22⋊D73—216no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—24no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—48no—
265C2×Dic73—42no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—84no—
275D292—22no—
276D292—22no—
277D292—22no—
278D292C2×C4⋊C414yescomplemented
279C4×D73—216no—
280C4×D73—48no—
281C4×D73—48no—
282Dic146—22no—
283Dic146—22no—
284Dic146—22no—
285Dic146C2×C4⋊C414yescomplemented
286C8×C73—24no—
287C8×C73—42no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73#38514yes—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73C2×#12011yescharacteristic
297C22×C73C2×C22⋊Q811yescharacteristic
298C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C73C2×#13711yescharacteristic
300C22×C73#57511yescharacteristic
301C22×C73#57511yescharacteristic
302C2×D73—24no—
303C2×D73—42no—
304C2×D73—42no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—42no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—41no—
312C4×C73—41no—
313C4×C73#38514yes—
314C4×C73C22×C4⋊C411yescharacteristic
315C4×C73C2×C4×D411yescharacteristic
316C4×C73C2×C22⋊Q811yescharacteristic
317C4×C73C2×#12311yescharacteristic
318Dic73—24no—
319Dic73—44no—
320Dic73—42no—
321Dic73—42no—
322Dic73—82no—
323#20875—731nomaximal, Sylow
324C2×C73—24no—
325C2×C73C2×#38511yescharacteristic
326C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
327C2×C73C4×#13711yescharacteristic
328C2×C73#300611yescharacteristic
329C2×C73#300711yescharacteristic
330C2×C73#307911yescharacteristic
331C2×C73#308011yescharacteristic
332D73—44no—
333C2×C4×#41—731no—
334C2×#428—731no—
335C2×#417—731no—
336C2×#417—731no—
337C2×#389—731no—
338#1865—732no—
339#1864—732no—
340#1863—732no—
341#1862—732no—
342#1736—731no—
343#1728D7311yescharacteristic, complemented
344C73#2087511yescharacteristic, complemented, Sylow
345C22×#41—731no—
346C22×M16C2×D7311yescharacteristic
347C22×M16C2×D7311yescharacteristic
348C4×#41—1464no—
349C4×#41—1464no—
350C2×C4×Q8—731no—
351C2×C4×Q8—731no—
352C2×C4×D4—731no—
353C2×C4×D4—731no—
354C2×C4×D4—1461no—
355C2×C4×D4—1461no—
356C2×#117—731no—
357C2×#117—1461no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C22×C4⋊C4—1461no—
361C22×C42—1461no—
362C22×C42C2×D7311yescharacteristic
363#428—1464no—
364C2×#107—732no—
365C2×#106—732no—
366#417—738no—
367#417—1464no—
368C2×#103—732no—
369C2×#103—732no—
370C2×#102—732no—
371C2×#102—732no—
372#389—732no—
373#389—732no—
374#389—1461no—
375#389—1461no—
376C2×#95—1461no—
377#340—732no—
378#340—732no—
379#336C2×D7312yes—
380#336C2×D7312yes—
381C2×#41—734no—
382C2×#41—734no—
383C2×#41—734no—
384C2×#41—1462no—
385C2×#41—1462no—
386C2×#41—1462no—
387C22×Q8—731no—
388C22×D4—731no—
389C22×D4—1461no—
390C23×C4—1461no—
391C23×C4—1461no—
392C23×C4—1461no—
393C23×C4—1461no—
394C23×C4—1461no—
395C23×C4D29211yescharacteristic
396C23×C4D29211yescharacteristic
397C23×C4C22×D7311yescharacteristic
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16—22no—
402C2×M16—22no—
403C2×M16C4×D7314yescomplemented
404C22×C8C22×D7312yes—
405C22×C8C22×D7312yes—
406C4×Q8—1462no—
407C4×Q8—1462no—
408C4×Q8—1462no—
409C4×Q8—1462no—
410C4×D4—1462no—
411C4×D4—1462no—
412C4×D4—1462no—
413C4×D4—1462no—
414C4×D4—1468no—
415C4×D4—1468no—
416#117—734no—
417#117—1468no—
418#117—1462no—
419C2×C4⋊C4—2924no—
420C2×C4⋊C4—734no—
421C2×C4⋊C4—732no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—732no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1462no—
427C2×C4⋊C4—1462no—
428C2×C4⋊C4—1462no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1462no—
434C2×C22⋊C4—1461no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C42—22no—
438C2×C42—22no—
439C2×C42—1468no—
440C2×C42—1461no—
441C2×C42—1461no—
442C2×C42—1461no—
443C2×C42D29211yescharacteristic
444C2×C42D29211yescharacteristic
445C2×C42C22×D7311yescharacteristic
446C2×C42C22×D7311yescharacteristic
447#107—1464no—
448#106—1464no—
449#103—738no—
450#103—1464no—
451#102—738no—
452#102—1464no—
453#95—1462no—
454#95—1462no—
455C24—1461no—
456C24C2×D29211yescharacteristic
457#41—7316no—
458#41—1468no—
459#41—1468no—
460#41—1468no—
461C2×Q8—732no—
462C2×Q8—732no—
463C2×Q8—732no—
464C2×Q8—1461no—
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×D4—732no—
468C2×D4—732no—
469C2×D4—732no—
470C2×D4—1464no—
471C2×D4—1464no—
472C2×D4—1464no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4—1461no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—2922no—
490C22×C4—2922no—
491C22×C4—1464no—
492C22×C4—1464no—
493C22×C4—1462no—
494C22×C4—1464no—
495C22×C4—1462no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1464no—
500C22×C4—1464no—
501C22×C4—1464no—
502C22×C4C2×C4×D7312yes—
503C22×C4D4×D7312yes—
504C22×C4D4×D7312yes—
505C22×C4C2×D29211yescharacteristic
506C22×C4C2×D29211yescharacteristic
507C22×C4C2×D29211yescharacteristic
508C22×C4C2×D29211yescharacteristic
509C22×C4C2×D29211yescharacteristic
510C22×C4Q8×D7312yes—
511C22×C4#1192402712yes—
512C22×C4#1192402812yes—
513C22×C4C23×D7311yescharacteristic
514M16—28no—
515M16—44no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8C2×C4×D7314yes—
522C4⋊C4—2924no—
523C4⋊C4—2922no—
524C4⋊C4—2924no—
525C4⋊C4—734no—
526C4⋊C4—732no—
527C4⋊C4—732no—
528C4⋊C4—1464no—
529C4⋊C4—1464no—
530C4⋊C4—1462no—
531C4⋊C4—1462no—
532C4⋊C4—1464no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1462no—
536C22⋊C4—1464no—
537C22⋊C4—1464no—
538C22⋊C4—1464no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1464no—
546C42—1464no—
547C42—1464no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—2921no—
553C23—2921no—
554C23—2921no—
555C23—2921no—
556C23—1462no—
557C23—1462no—
558C23—1462no—
559C23C4×D29212yes—
560C23#1455656112yes—
561C23#1455657112yes—
562C23C22×D29211yescharacteristic
563Q8—734no—
564Q8—1462no—
565Q8—1462no—
566Q8—1462no—
567D4—734no—
568D4—14616no—
569D4—1462no—
570D4—1462no—
571D4—1462no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—2928no—
585C2×C4—2928no—
586C2×C4—2928no—
587C2×C4—2922no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—5844no—
597C2×C4—1464no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1462no—
603C2×C4—14616no—
604C2×C4C4⋊C4×D7314yes—
605C2×C4#1455655114yes—
606C2×C4C4×D29212yes—
607C2×C4#1455656112yes—
608C2×C4#1455656914yes—
609C2×C4#1455657112yes—
610C2×C4#1455657314yes—
611C2×C4#1455661912yes—
612C2×C4#1455662412yes—
613C2×C4C22×C4×D7311yescharacteristic
614C2×C4C2×D4×D7311yescharacteristic
615C2×C4C2×D4×D7311yescharacteristic
616C2×C4C2×Q8×D7311yescharacteristic
617C2×C4C2×#1192402711yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C8—24no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—2922no—
628C22—2922no—
629C22—2922no—
630C22—2922no—
631C22—2922no—
632C22—2922no—
633C22—1464no—
634C22#1798846114yes—
635C22C2×C4×D29211yescharacteristic
636C22C2×#1455656111yescharacteristic
637C22C2×#1455657111yescharacteristic
638C22C2×#1455661911yescharacteristic
639C22C2×#1455662411yescharacteristic
640C22#1798938711yescharacteristic
641C22#1798938711yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—2922no—
647C4—2922no—
648C4—2924no—
649C4—5842no—
650C4—1464no—
651C4#1798846114yes—
652C4C2×C4⋊C4×D7311yescharacteristic
653C4C2×#1455655111yescharacteristic
654C4C2×#1455656911yescharacteristic
655C4C2×#1455657311yescharacteristic
656C2—24no—
657C2—2924no—
658C2C2×#1798846111yescharacteristic
659C2C4×#1455661911yescharacteristic
660C2C4×#1455662411yescharacteristic
661C2#2261918511yescharacteristic
662C2#2261919311yescharacteristic
663C2#2261974711yescharacteristic
664C2#2261975411yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1728C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenter#1728D73
hypocenterC73#20875
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
2. C2×#17988629C2semidirect
5. C2×#17988485C2semidirect
6. #22610202C2semidirect
7. #22610195C2semidirect
8. #22610188C2semidirect
9. #22610181C2semidirect
10. #22609651C2semidirect
11. C73×#1728C2semidirect
31. #17988589C4semidirect
80. C2×C4⋊Dic73C2×C4semidirect
100. #14556516C2×C4semidirect
102. #14556513C2×C4semidirect
113. C2×M16×C73C2×C4semidirect
126. #11924028C4⋊C4semidirect
278. D292C2×C4⋊C4semidirect
285. Dic146C2×C4⋊C4semidirect
343. #1728D73semidirect
344. C73#20875semidirect
403. C2×M16C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#119240282
C2×#179886292
C2×#179885892
C2×#179884852
#226102022
#226101952
#226101882
#226101812
#226096512
C73×#17282
#2087573