Finite Group Databasebeta

← #32993195

Subgroups — #32993195

Order 18688.

938 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988605C211yesmaximal, characteristic
2C22×#14556519C211yesmaximal, characteristic
3C2×#17988560C211yesmaximal, characteristic
4C2×#17988357C211yesmaximal, characteristic, complemented
5C2×#17988347C211yesmaximal, characteristic, complemented
6C2×#17988343C211yesmaximal, characteristic, complemented
7C73×#2296C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9C22×#11924012C2211yescharacteristic
10C22×#11924012C2211yescharacteristic
11C2×#14556550C2211yescharacteristic
12#17988605—22no—
13#17988605—22no—
14C2×#14556519—22no—
15C2×#14556519—22no—
16C2×#14556519—22no—
17C2×#14556519—22no—
18#17988560—22no—
19#17988560C414yescomplemented
20C2×#14556505—22no—
21#17988357—22no—
22#17988347—22no—
23#17988343—22no—
24C2×#117×C73C2211yescharacteristic
25C2×#117×C73C2211yescharacteristic
26C22×C42×C73C2211yescharacteristic
27#389×C73—22no—
28#385×C73—22no—
29C4×C4⋊C4×C73—22no—
30C2×#11924028—21no—
31C2×#11924028—21no—
32C2×#11924028—21no—
33C2×#11924028—21no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028C2×C412yes—
41C2×#11924028D412yes—
42C22×Dic146Q811yescharacteristic
43C22×D292Q811yescharacteristic
44C22×C22⋊D73—21no—
45C22×C4×D73—21no—
46C2×#11924012—21no—
47C2×#11924012—21no—
48C2×#11924012—21no—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53C2×#11924012—21no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924012D412yes—
57C2×#11924012D412yes—
58C22×#10596831—21no—
59C22×#10596831—21no—
60#14556550—21no—
61#14556550—21no—
62#14556550—21no—
63#14556550C2×C412yes—
64C2×C4⋊Dic73—21no—
65C2×C22⋊Dic73—21no—
66C2×C4×Dic73—21no—
67#14556519—28no—
68#14556519—28no—
69#14556505—28no—
70C23×C4×C73—21no—
71C23×C4×C73C2311yescharacteristic
72#117×C73—21no—
73#117×C73—21no—
74#117×C73—21no—
75#117×C73—21no—
76#117×C73—21no—
77#117×C73—21no—
78#117×C73C2×C412yes—
79#117×C73D412yes—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—24no—
82C2×C4⋊C4×C73—21no—
83C2×C22⋊C4×C73—21no—
84C2×C22⋊C4×C73D412yes—
85C2×C42×C73—21no—
86C2×C42×C73—21no—
87C2×C42×C73—21no—
88C2×C42×C73—21no—
89C2×C42×C73—21no—
90C2×C42×C73—21no—
91C2×C42×C73D412yes—
92C2×C42×C73D412yes—
93#95×C73—24no—
94C23×D73—21no—
95#11924028—22no—
96#11924028—22no—
97#11924028—22no—
98#11924028—22no—
99#11924028—22no—
100#11924028—22no—
101#11924028—22no—
102#11924028—22no—
103#11924028—41no—
104#11924028—41no—
105#11924028—41no—
106#11924028—41no—
107#11924028—41no—
108#11924028—41no—
109#11924028—41no—
110#11924028—41no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146C4⋊C412yescomplemented
117C2×D292—21no—
118C2×D292—21no—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292C4⋊C412yescomplemented
123C2×C22⋊D73—22no—
124C2×C22⋊D73—22no—
125C2×C22⋊D73—41no—
126C2×C22⋊D73—41no—
127C2×C22⋊D73—41no—
128C2×C22⋊D73—41no—
129C22×Dic73—21no—
130C22×Dic73—41no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—41no—
134C2×C4×D73—41no—
135C2×C4×D73—41no—
136C2×C4×D73—41no—
137#11924012—22no—
138#11924012—22no—
139#11924012—22no—
140#11924012—22no—
141#11924012—41no—
142#11924012—41no—
143#11924012—41no—
144#11924012—41no—
145C2×#10596831—22no—
146C2×#10596831—22no—
147C2×#10596831—41no—
148C2×#10596831—41no—
149C2×#10596831—41no—
150C2×#10596831—41no—
151C4⋊Dic73—22no—
152C4⋊Dic73—41no—
153C22⋊Dic73—22no—
154C22⋊Dic73—41no—
155C4×Dic73—22no—
156C4×Dic73—41no—
157C24×C73C2×D411yescharacteristic
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—44no—
162C22×C4×C73—41no—
163C22×C4×C73—41no—
164C22×C4×C73C22×C411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×Q811yescharacteristic
172C22×C4×C73#4111yescharacteristic
173C22×C4×C73#4111yescharacteristic
174C22×C4×C73#4111yescharacteristic
175C22×C4×C73#4111yescharacteristic
176C22×C4×C73#4111yescharacteristic
177C22×C4×C73#4111yescharacteristic
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—48no—
181C4⋊C4×C73—41no—
182C4⋊C4×C73—41no—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—41no—
185C22⋊C4×C73—41no—
186C22⋊C4×C73C22⋊C414yes—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73—22no—
190C42×C73—22no—
191C42×C73—22no—
192C42×C73—22no—
193C42×C73—41no—
194C42×C73—41no—
195C23×C73—42no—
196C23×C73C2×C22⋊C411yescharacteristic
197C23×C73#12011yescharacteristic
198C23×C73#12111yescharacteristic
199C23×C73C22⋊Q811yescharacteristic
200C23×C73#12311yescharacteristic
201C23×C73#12411yescharacteristic
202C23×C73#12711yescharacteristic
203C22×D73—22no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—82no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—44no—
210C2×C4×C73—42no—
211C2×C4×C73—44no—
212C2×C4×C73—44no—
213C2×C4×C73—42no—
214C2×C4×C73—44no—
215C2×C4×C73—42no—
216C2×C4×C73C2×C4⋊C411yescharacteristic
217C2×C4×C73#11711yescharacteristic
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C4×D411yescharacteristic
221C2×C4×C73C4×D411yescharacteristic
222C2×C4×C73#12011yescharacteristic
223C2×C4×C73#12011yescharacteristic
224C2×C4×C73#12111yescharacteristic
225C2×C4×C73#12111yescharacteristic
226C2×C4×C73#12111yescharacteristic
227C2×C4×C73#12111yescharacteristic
228C2×C4×C73#12111yescharacteristic
229C2×C4×C73#12111yescharacteristic
230C2×C4×C73#12111yescharacteristic
231C2×C4×C73C22⋊Q811yescharacteristic
232C2×C4×C73C22⋊Q811yescharacteristic
233C2×C4×C73C22⋊Q811yescharacteristic
234C2×C4×C73C22⋊Q811yescharacteristic
235C2×C4×C73C22⋊Q811yescharacteristic
236C2×C4×C73#12311yescharacteristic
237C2×C4×C73#12311yescharacteristic
238C2×C4×C73#12311yescharacteristic
239C2×C4×C73#12311yescharacteristic
240C2×C4×C73#12411yescharacteristic
241C2×C4×C73#12411yescharacteristic
242C2×C4×C73#12611yescharacteristic
243C2×C4×C73#12611yescharacteristic
244C22⋊D73—42no—
245C22⋊D73—42no—
246C22⋊D73—42no—
247C22⋊D73—42no—
248C2×Dic73—22no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—42no—
252C2×Dic73—82no—
253D292—22no—
254D292—22no—
255D292—41no—
256D292—41no—
257C4×D73—42no—
258C4×D73—42no—
259C4×D73—42no—
260C4×D73—42no—
261Dic146—22no—
262Dic146—22no—
263Dic146—41no—
264Dic146—41no—
265#10596831—42no—
266#10596831—42no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73#38011yescharacteristic
275C22×C73#38611yescharacteristic
276C22×C73#39011yescharacteristic
277C22×C73#39211yescharacteristic
278C22×C73#39311yescharacteristic
279C22×C73#39711yescharacteristic
280C22×C73#39911yescharacteristic
281C2×D73—42no—
282C2×D73—82no—
283C2×D73—82no—
284C2×D73—82no—
285C4×C73—44no—
286C4×C73—44no—
287C4×C73—82no—
288C4×C73#38511yescharacteristic
289C4×C73#38511yescharacteristic
290C4×C73#38811yescharacteristic
291C4×C73#38811yescharacteristic
292C4×C73#39411yescharacteristic
293C4×C73#39411yescharacteristic
294C4×C73#39611yescharacteristic
295C4×C73#39611yescharacteristic
296Dic73—42no—
297Dic73—82no—
298#11091—731nomaximal, Sylow
299C2×C73—21no—
300C2×C73—21no—
301C2×C73—21no—
302C2×C73—21no—
303C2×C73—21no—
304C2×C73—21no—
305C2×C73#143911yescharacteristic
306C2×C73#203112yes—
307D73—82no—
308C2×#421—731no—
309C22×#104—731no—
310C2×#410—731no—
311#2296D7311yescharacteristic, complemented
312C2×#344—731no—
313C2×#339—731no—
314C2×H(3,ℤ/4)—731no—
315C73#1109111yescharacteristic, complemented, Sylow
316C22×#41—731no—
317C22×M16—731no—
318C22×M16—731no—
319C2×#117—731no—
320C2×#117C2×D7311yescharacteristic
321C2×#117C2×D7311yescharacteristic
322C22×C42C2×D7311yescharacteristic
323#421—1462no—
324#421—1462no—
325C2×#104—1462no—
326C2×#104—1462no—
327C2×#104—1462no—
328C2×#104—1462no—
329#410—734no—
330#410—1462no—
331C2×#99—1462no—
332#389—22no—
333#385—22no—
334C4×C4⋊C4—22no—
335#344—1462no—
336#339—1462no—
337H(3,ℤ/4)—1462no—
338C2×#41—732no—
339C2×#41—732no—
340C2×#41—1461no—
341C2×#41—1461no—
342C2×#41—1461no—
343C2×#41—1461no—
344C2×#41—1461no—
345C2×#41—1461no—
346C2×#41—1461no—
347C2×#41—1461no—
348C2×#41—1461no—
349C2×#41—1461no—
350C22×Q8—731no—
351C22×D4—731no—
352C22×D4—1461no—
353C23×C4—21no—
354C23×C4—1461no—
355C23×C4C22×D7311yescharacteristic
356C2×M16—732no—
357C2×M16—732no—
358C2×M16—1461no—
359C2×M16—1461no—
360C2×M16—1461no—
361C2×M16—1461no—
362C2×M16—1461no—
363C2×M16—1461no—
364C2×M16—1461no—
365C2×M16—1461no—
366C2×M16—1461no—
367C2×M16—1461no—
368C22×C8—1461no—
369C22×C8—1461no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—732no—
377#117—1461no—
378#117—1461no—
379#117—1461no—
380#117C4×D7312yes—
381#117D29212yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—1461no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4C22⋊D7312yes—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—1461no—
396C2×C42C22⋊D7312yes—
397C2×C42C22⋊D7312yes—
398#104—1468no—
399#104—1468no—
400#99—1468no—
401#95—24no—
402C24—1461no—
403C24C2×C22⋊D7311yescharacteristic
404#41—2921no—
405#41—2921no—
406#41—2921no—
407#41—2921no—
408#41—2921no—
409#41—2921no—
410#41—2921no—
411#41—2921no—
412#41—1462no—
413#41—1462no—
414#41—1462no—
415#41—1462no—
416#41—1462no—
417#41—1462no—
418#41—1462no—
419#41—1462no—
420C2×Q8—732no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×Q8—1461no—
424C2×Q8—1461no—
425C2×Q8—1461no—
426C2×D4—2921no—
427C2×D4—2921no—
428C2×D4—2921no—
429C2×D4—2921no—
430C2×D4—732no—
431C2×D4—1462no—
432C2×D4—1462no—
433C2×D4—1461no—
434C2×D4—1461no—
435C2×D4—1461no—
436C2×D4—1461no—
437C2×D4—1461no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—2921no—
445C22×C4—2921no—
446C22×C4—2921no—
447C22×C4—2921no—
448C22×C4—2921no—
449C22×C4—1461no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4C2×C4×D7311yescharacteristic
453C22×C4D4×D7311yescharacteristic
454C22×C4D4×D7311yescharacteristic
455C22×C4D4×D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×D29211yescharacteristic
459C22×C4Q8×D7311yescharacteristic
460C22×C4#1192402711yescharacteristic
461C22×C4#1192402811yescharacteristic
462C22×C4#1192402811yescharacteristic
463C22×C4#1192402911yescharacteristic
464C22×C4#1192402911yescharacteristic
465C22×C4#1192402911yescharacteristic
466M16—2921no—
467M16—2921no—
468M16—2921no—
469M16—2921no—
470M16—1462no—
471M16—1462no—
472M16—1462no—
473M16—1462no—
474C2×C8—2921no—
475C2×C8—2921no—
476C2×C8—2921no—
477C2×C8—2921no—
478C2×C8—1462no—
479C2×C8—1462no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—2921no—
486C4⋊C4—1462no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—2921no—
491C22⋊C4—1462no—
492C22⋊C4#1192400514yes—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—2921no—
502C42—1462no—
503C23—42no—
504C23—2921no—
505C23—2921no—
506C23—5842no—
507C23—1462no—
508C23C2×#1192400511yescharacteristic
509C23#1455656011yescharacteristic
510C23#1455656211yescharacteristic
511C23#1455656711yescharacteristic
512C23#1455657411yescharacteristic
513C23#1455658011yescharacteristic
514C23#1455658711yescharacteristic
515Q8—2921no—
516Q8—2921no—
517Q8—1462no—
518Q8—1462no—
519D4—2922no—
520D4—2922no—
521D4—2922no—
522D4—2922no—
523D4—2921no—
524D4—2921no—
525D4—1462no—
526D4—1462no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—5842no—
544C2×C4—1462no—
545C2×C4C4⋊C4×D7311yescharacteristic
546C2×C4#1455654911yescharacteristic
547C2×C4C4×C22⋊D7311yescharacteristic
548C2×C4C4×C22⋊D7311yescharacteristic
549C2×C4#1455655311yescharacteristic
550C2×C4#1455655311yescharacteristic
551C2×C4#1455655911yescharacteristic
552C2×C4#1455655911yescharacteristic
553C2×C4#1455656311yescharacteristic
554C2×C4#1455656311yescharacteristic
555C2×C4#1455656511yescharacteristic
556C2×C4#1455656511yescharacteristic
557C2×C4#1455656611yescharacteristic
558C2×C4#1455656611yescharacteristic
559C2×C4#1455656611yescharacteristic
560C2×C4#1455656711yescharacteristic
561C2×C4#1455656711yescharacteristic
562C2×C4#1455656911yescharacteristic
563C2×C4#1455656911yescharacteristic
564C2×C4#1455657111yescharacteristic
565C2×C4#1455657611yescharacteristic
566C2×C4#1455657611yescharacteristic
567C2×C4#1455657611yescharacteristic
568C2×C4#1455657611yescharacteristic
569C2×C4#1455658011yescharacteristic
570C2×C4#1455658011yescharacteristic
571C2×C4#1455658611yescharacteristic
572C2×C4#1455658611yescharacteristic
573C8—2922no—
574C8—2922no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—2922no—
583C22—5842no—
584C22—5842no—
585C22—5842no—
586C22#1798844411yescharacteristic
587C22#1798846611yescharacteristic
588C22#1798848811yescharacteristic
589C22#1798849511yescharacteristic
590C22#1798849811yescharacteristic
591C22#1798851311yescharacteristic
592C22#1798852211yescharacteristic
593C4—44no—
594C4—44no—
595C4—82no—
596C4—2922no—
597C4—5842no—
598C4#1798845811yescharacteristic
599C4#1798845811yescharacteristic
600C4#1798847711yescharacteristic
601C4#1798847711yescharacteristic
602C4#1798850411yescharacteristic
603C4#1798850411yescharacteristic
604C4#1798850911yescharacteristic
605C4#1798850911yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—5842no—
613C2#2260817311yescharacteristic
614C2#2261110012yes—
subgroupG/H
centerC2×C4#14556580
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2296C2
socleC22×C73#393
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenter#2296D73
hypocenterC73#11091
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
4. C2×#17988357C2semidirect
5. C2×#17988347C2semidirect
6. C2×#17988343C2semidirect
7. C73×#2296C2semidirect
19. #17988560C4semidirect
116. C2×Dic146C4⋊C4semidirect
122. C2×D292C4⋊C4semidirect
311. #2296D73semidirect
315. C73#11091semidirect

Maximal subgroups

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

subgroupindex
C2×#179886052
C22×#145565192
C2×#179885602
C2×#179883572
C2×#179883472
C2×#179883432
C73×#22962
#1109173