Finite Group Databasebeta

← #33114937

Subgroups — #33114937

Order 18688.

888 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988636C211yesmaximal, characteristic, complemented
2#22618451C211yesmaximal, characteristic
3#22618448C211yesmaximal, characteristic
4C2×#17988569C211yesmaximal, characteristic
5#22611397C214yesmaximal, complemented
6#22611390C214yesmaximal, complemented
7C2×#17988358C211yesmaximal, characteristic, complemented
8C2×#17988343C211yesmaximal, characteristic, complemented
9C2×#344×C73C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13C2×#14556550C2211yescharacteristic
14#17988636—22no—
15#17988636C2214yes—
16C2×#14556527—22no—
17#17988597C2212yes—
18#17988595C2212yes—
19#17988593C2212yes—
20#17988591C2212yes—
21#17988589—22no—
22#17988586—22no—
23#17988584—22no—
24#17988581—22no—
25#17988569—22no—
26#17988569—22no—
27C2×#14556509C2212yes—
28C2×#14556507C2212yes—
29#17988358C2214yes—
30#17988343C2214yes—
31C22×M16×C73C2211yescharacteristic
32C22×M16×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34#344×C73C2214yes—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028—21no—
43C2×#11924028—21no—
44C2×#11924028—21no—
45C2×#11924028D412yes—
46C2×#11924028D412yes—
47C22×Dic146C2311yescharacteristic
48C22×D292C2311yescharacteristic
49C22×C22⋊D73—21no—
50C22×C4×D73—21no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53C2×#11924012—21no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924012—21no—
57C2×#11924012—21no—
58C2×#11924012—21no—
59C2×#11924012—21no—
60C2×#11924012—21no—
61C2×#11924012C2312yes—
62C2×#11924012C2312yes—
63C22×#10596831—21no—
64C22×#10596831—21no—
65#14556550—21no—
66#14556550—21no—
67#14556550—21no—
68#14556550C2312yes—
69C2×C4⋊Dic73—21no—
70C2×C22⋊Dic73—21no—
71C2×C4×Dic73—21no—
72#14556527—28no—
73#14556516—24no—
74#14556514—24no—
75#14556513—24no—
76#14556511—24no—
77#14556509—24no—
78#14556507—24no—
79C23×C4×C73C2311yescharacteristic
80C2×M16×C73—21no—
81C2×M16×C73—21no—
82C2×M16×C73—21no—
83C2×M16×C73—21no—
84C2×M16×C73—21no—
85C2×M16×C73—21no—
86C2×M16×C73—21no—
87C2×M16×C73—21no—
88C2×M16×C73—21no—
89C2×M16×C73D412yes—
90C2×M16×C73C2312yes—
91C2×M16×C73C2312yes—
92C22×C8×C73—21no—
93C22×C8×C73D412yes—
94#117×C73—21no—
95#117×C73—21no—
96#117×C73—21no—
97#117×C73C2312yes—
98C2×C4⋊C4×C73—21no—
99C2×C22⋊C4×C73—21no—
100C2×C42×C73—21no—
101C23×D73—21no—
102#11924028—22no—
103#11924028—22no—
104#11924028—22no—
105#11924028—22no—
106#11924028—22no—
107#11924028—22no—
108#11924028—22no—
109#11924028—22no—
110#11924028—41no—
111#11924028—41no—
112#11924028—41no—
113#11924028—41no—
114#11924028—41no—
115#11924028—41no—
116#11924028—41no—
117#11924028—41no—
118C2×Dic146—21no—
119C2×Dic146—21no—
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146C2×D412yes—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292C2×D412yes—
130C2×C22⋊D73—22no—
131C2×C22⋊D73—22no—
132C2×C22⋊D73—41no—
133C2×C22⋊D73—41no—
134C2×C22⋊D73—41no—
135C2×C22⋊D73—41no—
136C22×Dic73—21no—
137C22×Dic73—41no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144#11924012—22no—
145#11924012—22no—
146#11924012—22no—
147#11924012—22no—
148#11924012—41no—
149#11924012—41no—
150#11924012—41no—
151#11924012—41no—
152C2×#10596831—22no—
153C2×#10596831—22no—
154C2×#10596831—41no—
155C2×#10596831—41no—
156C2×#10596831—41no—
157C2×#10596831—41no—
158C4⋊Dic73—22no—
159C4⋊Dic73—41no—
160C22⋊Dic73—22no—
161C22⋊Dic73—41no—
162C4×Dic73—22no—
163C4×Dic73—41no—
164C24×C73#4111yescharacteristic
165C22×C4×C73—41no—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73#4111yescharacteristic
171C22×C4×C73#4111yescharacteristic
172C22×C4×C73#4111yescharacteristic
173C22×C4×C73#4111yescharacteristic
174C22×C4×C73#4111yescharacteristic
175C22×C4×C73#4111yescharacteristic
176C22×C4×C73#4111yescharacteristic
177C22×C4×C73#4111yescharacteristic
178C22×C4×C73#4111yescharacteristic
179C22×C4×C73C2411yescharacteristic
180M16×C73—22no—
181M16×C73—22no—
182M16×C73—22no—
183M16×C73—22no—
184M16×C73—22no—
185M16×C73—22no—
186M16×C73—41no—
187M16×C73—41no—
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73—22no—
191C2×C8×C73—41no—
192C2×C8×C73—41no—
193C2×C8×C73C2×D414yes—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—41no—
196C22⋊C4×C73—22no—
197C22⋊C4×C73—41no—
198C42×C73—22no—
199C42×C73—41no—
200C23×C73—81no—
201C23×C73#12311yescharacteristic
202C23×C73#12311yescharacteristic
203C23×C73#12411yescharacteristic
204C23×C73#12411yescharacteristic
205C23×C73#12611yescharacteristic
206C23×C73#12611yescharacteristic
207C23×C73C2×#4111yescharacteristic
208C22×D73—22no—
209C22×D73—41no—
210C22×D73—41no—
211C22×D73—82no—
212C2×C4×C73—42no—
213C2×C4×C73—82no—
214C2×C4×C73#12111yescharacteristic
215C2×C4×C73#12111yescharacteristic
216C2×C4×C73#12111yescharacteristic
217C2×C4×C73#12111yescharacteristic
218C2×C4×C73#12311yescharacteristic
219C2×C4×C73#12311yescharacteristic
220C2×C4×C73#12311yescharacteristic
221C2×C4×C73#12311yescharacteristic
222C2×C4×C73#12311yescharacteristic
223C2×C4×C73#12311yescharacteristic
224C2×C4×C73#12311yescharacteristic
225C2×C4×C73#12311yescharacteristic
226C2×C4×C73#12311yescharacteristic
227C2×C4×C73#12311yescharacteristic
228C2×C4×C73#12411yescharacteristic
229C2×C4×C73#12411yescharacteristic
230C2×C4×C73#12611yescharacteristic
231C2×C4×C73#12611yescharacteristic
232C2×C4×C73#12611yescharacteristic
233C2×C4×C73#12611yescharacteristic
234C2×C4×C73#12611yescharacteristic
235C2×C4×C73#12611yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C22×D411yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C22⋊D73—42no—
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—42no—
246C2×Dic73—22no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—42no—
250C2×Dic73—82no—
251D292—22no—
252D292—22no—
253D292—41no—
254D292—41no—
255C4×D73—42no—
256C4×D73—42no—
257C4×D73—42no—
258C4×D73—42no—
259Dic146—22no—
260Dic146—22no—
261Dic146—41no—
262Dic146—41no—
263C8×C73—24no—
264C8×C73—42no—
265#10596831—42no—
266#10596831—42no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73#39711yescharacteristic
275C22×C73#39711yescharacteristic
276C22×C73#39711yescharacteristic
277C22×C73#39711yescharacteristic
278C22×C73C2×#12311yescharacteristic
279C22×C73C2×#12411yescharacteristic
280C22×C73C2×#12611yescharacteristic
281C2×D73—42no—
282C2×D73—82no—
283C2×D73—82no—
284C2×D73—82no—
285C4×C73—82no—
286C4×C73#39711yescharacteristic
287C4×C73#39711yescharacteristic
288C4×C73#39711yescharacteristic
289C4×C73#39711yescharacteristic
290C4×C73C2×#12111yescharacteristic
291C4×C73C2×#12311yescharacteristic
292C4×C73C2×#12311yescharacteristic
293C4×C73C2×#12611yescharacteristic
294Dic73—42no—
295Dic73—82no—
296#23652—731nomaximal, Sylow
297C2×C73—21no—
298C2×C73—21no—
299C2×C73—21no—
300C2×C73—21no—
301C2×C73—21no—
302C2×C73—21no—
303C2×C73—21no—
304C2×C73C2×#39711yescharacteristic
305D73—82no—
306C2×#430—731no—
307#2896—731no—
308#2896—731no—
309C2×#413—731no—
310#2082—734no—
311#2081—734no—
312C2×#344—731no—
313C2×#344D7311yescharacteristic, complemented
314C2×H(3,ℤ/4)—731no—
315C73#2365211yescharacteristic, complemented, Sylow
316C22×#41—731no—
317C22×M16—731no—
318C22×M16—731no—
319C22×M16C2×D7311yescharacteristic
320C22×M16C2×D7311yescharacteristic
321C2×#117—731no—
322C2×#117C2×D7311yescharacteristic
323#430—734no—
324#430—1462no—
325C2×#108—1462no—
326#419—732no—
327#419—732no—
328#418—732no—
329#418—732no—
330#417—1462no—
331#417—1462no—
332#416—1462no—
333#416—1462no—
334#413—1462no—
335#413—1462no—
336C2×#101—732no—
337C2×#100—732no—
338#344—734no—
339#344C2×D7314yes—
340H(3,ℤ/4)—734no—
341C2×#41—732no—
342C2×#41—732no—
343C2×#41—1461no—
344C2×#41—1461no—
345C2×#41—1461no—
346C2×#41—1461no—
347C2×#41—1461no—
348C2×#41—1461no—
349C2×#41—1461no—
350C2×#41—1461no—
351C2×#41—1461no—
352C2×#41—1461no—
353C22×Q8—731no—
354C22×D4—731no—
355C22×D4—1461no—
356C23×C4—1461no—
357C23×C4C22×D7311yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—732no—
368C2×M16—732no—
369C2×M16—1461no—
370C2×M16—1461no—
371C2×M16—1461no—
372C2×M16—1461no—
373C2×M16—1461no—
374C2×M16—1461no—
375C2×M16—1461no—
376C2×M16—1461no—
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16D29212yes—
380C2×M16C22×D7312yes—
381C2×M16C22×D7312yes—
382C22×C8—21no—
383C22×C8—1461no—
384C22×C8—1461no—
385C22×C8D29212yes—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—732no—
390#117—1461no—
391#117—1461no—
392#117—1461no—
393#117C22×D7312yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—1461no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1461no—
398C2×C42—21no—
399C2×C42—1461no—
400#108—1468no—
401#103—1464no—
402#103—1464no—
403#102—1464no—
404#102—1464no—
405#101—1464no—
406#100—1464no—
407C24—1461no—
408C24#1192402811yescharacteristic
409#41—2921no—
410#41—2921no—
411#41—2921no—
412#41—2921no—
413#41—2921no—
414#41—2921no—
415#41—2921no—
416#41—2921no—
417#41—1462no—
418#41—1462no—
419#41—1462no—
420#41—1462no—
421#41—1462no—
422#41—1462no—
423#41—1462no—
424#41—1462no—
425C2×Q8—732no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8—1461no—
431C2×D4—2921no—
432C2×D4—2921no—
433C2×D4—2921no—
434C2×D4—2921no—
435C2×D4—732no—
436C2×D4—1462no—
437C2×D4—1462no—
438C2×D4—1461no—
439C2×D4—1461no—
440C2×D4—1461no—
441C2×D4—1461no—
442C2×D4—1461no—
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×C4D4×D7311yescharacteristic
453C22×C4D4×D7311yescharacteristic
454C22×C4C2×D29211yescharacteristic
455C22×C4C2×D29211yescharacteristic
456C22×C4#1192402711yescharacteristic
457C22×C4#1192402711yescharacteristic
458C22×C4#1192402711yescharacteristic
459C22×C4#1192402711yescharacteristic
460C22×C4#1192402811yescharacteristic
461C22×C4#1192402811yescharacteristic
462C22×C4#1192402811yescharacteristic
463C22×C4#1192402911yescharacteristic
464C22×C4#1192402911yescharacteristic
465C22×C4C23×D7311yescharacteristic
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—2921no—
475M16—2921no—
476M16—2921no—
477M16—2921no—
478M16—1462no—
479M16—1462no—
480M16—1462no—
481M16—1462no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—2921no—
488C2×C8—2921no—
489C2×C8—2921no—
490C2×C8—2921no—
491C2×C8—1462no—
492C2×C8—1462no—
493C2×C8C2×D29214yes—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—2921no—
497C4⋊C4—1462no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—2921no—
501C22⋊C4—1462no—
502C42—22no—
503C42—41no—
504C42—2921no—
505C42—1462no—
506C23—81no—
507C23—2921no—
508C23—2921no—
509C23—5842no—
510C23—1462no—
511C23#1455657311yescharacteristic
512C23#1455657311yescharacteristic
513C23#1455657911yescharacteristic
514C23#1455657911yescharacteristic
515C23#1455658611yescharacteristic
516C23#1455658611yescharacteristic
517C23C2×#1192402811yescharacteristic
518Q8—2921no—
519Q8—2921no—
520Q8—1462no—
521Q8—1462no—
522D4—2922no—
523D4—2922no—
524D4—2922no—
525D4—2922no—
526D4—2921no—
527D4—2921no—
528D4—1462no—
529D4—1462no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—5842no—
540C2×C4—1462no—
541C2×C4#1455656611yescharacteristic
542C2×C4#1455656611yescharacteristic
543C2×C4#1455656611yescharacteristic
544C2×C4#1455656611yescharacteristic
545C2×C4#1455657311yescharacteristic
546C2×C4#1455657311yescharacteristic
547C2×C4#1455657311yescharacteristic
548C2×C4#1455657311yescharacteristic
549C2×C4#1455657311yescharacteristic
550C2×C4#1455657311yescharacteristic
551C2×C4#1455657511yescharacteristic
552C2×C4#1455657511yescharacteristic
553C2×C4#1455657511yescharacteristic
554C2×C4#1455657511yescharacteristic
555C2×C4#1455657911yescharacteristic
556C2×C4#1455657911yescharacteristic
557C2×C4#1455658611yescharacteristic
558C2×C4#1455658611yescharacteristic
559C2×C4#1455658611yescharacteristic
560C2×C4#1455658611yescharacteristic
561C2×C4#1455658611yescharacteristic
562C2×C4#1455658611yescharacteristic
563C2×C4C2×D4×D7311yescharacteristic
564C2×C4C22×D29211yescharacteristic
565C2×C4C2×#1192402711yescharacteristic
566C2×C4C2×#1192402711yescharacteristic
567C2×C4C2×#1192402811yescharacteristic
568C2×C4C2×#1192402911yescharacteristic
569C8—24no—
570C8—42no—
571C8—2922no—
572C8—2922no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—2922no—
581C22—5842no—
582C22—5842no—
583C22—5842no—
584C22#1798851411yescharacteristic
585C22#1798851411yescharacteristic
586C22#1798851411yescharacteristic
587C22#1798851411yescharacteristic
588C22C2×#1455657311yescharacteristic
589C22C2×#1455657911yescharacteristic
590C22C2×#1455658611yescharacteristic
591C4—82no—
592C4—2922no—
593C4—5842no—
594C4#1798851411yescharacteristic
595C4#1798851411yescharacteristic
596C4#1798851411yescharacteristic
597C4#1798851411yescharacteristic
598C4C2×#1455656611yescharacteristic
599C4C2×#1455657311yescharacteristic
600C4C2×#1455657511yescharacteristic
601C4C2×#1455658611yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—5842no—
610C2C2×#1798851411yescharacteristic
subgroupG/H
centerC4#17988514
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#344×C73C2
socleC2×C73C2×#397
prime radicalC22×#11924028C22
hypercenterC2×#344D73
hypocenterC73#23652
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
1. C2×#17988636C2semidirect
5. #22611397C2semidirect
6. #22611390C2semidirect
7. C2×#17988358C2semidirect
8. C2×#17988343C2semidirect
9. C2×#344×C73C2semidirect
313. C2×#344D73semidirect
315. C73#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#179886362
#226184512
#226184482
C2×#179885692
#226113972
#226113902
C2×#179883582
C2×#179883432
C2×#344×C732
#2365273