Finite Group Databasebeta

← #32992292

Subgroups — #32992292

Order 18688.

1118 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556518C211yesmaximal, characteristic
2C2×M16×Dic73C211yesmaximal, characteristic, complemented
3#22612422C211yesmaximal, characteristic
4C2×#17988348C211yesmaximal, characteristic, complemented
5C2×#17988347C211yesmaximal, characteristic, complemented
6C4×#14556498C211yesmaximal, characteristic, complemented
7C22×#104×C73C211yesmaximal, characteristic
8C22×#11924012C2211yescharacteristic
9C2×#14556550C2211yescharacteristic
10C22×C4×Dic73C2211yescharacteristic
11C22×C4×Dic73C2211yescharacteristic
12C2×#14556518—22no—
13C2×#14556518—22no—
14C2×#14556518C414yescomplemented
15M16×Dic73—22no—
16M16×Dic73—22no—
17C2×#14556501—21no—
18C2×C8×Dic73—21no—
19#17988494C412yescomplemented
20#17988490C412yescomplemented
21#17988467—22no—
22C4×C22⋊Dic73—21no—
23C4×C22⋊Dic73—21no—
24C2×#14556498—22no—
25#17988348—22no—
26#17988347—22no—
27C22×#41×C73C2211yescharacteristic
28C22×M16×C73C2211yescharacteristic
29C22×C42×C73C2211yescharacteristic
30C2×#104×C73—22no—
31C2×#104×C73—22no—
32C2×#104×C73C414yescomplemented
33C23×Dic73—21no—
34C23×Dic73D412yes—
35C2×#11924012—21no—
36C2×#11924012—21no—
37C2×#11924012—21no—
38C2×#11924012C2×C412yes—
39C2×#11924012D412yes—
40C2×#11924012D412yes—
41C22×#10596831—21no—
42#14556550—21no—
43#14556550—21no—
44#14556550—21no—
45#14556550—21no—
46C2×C4⋊Dic73C2×C412yes—
47C2×C22⋊Dic73—22no—
48C2×C22⋊Dic73—22no—
49C2×C22⋊Dic73—21no—
50C2×C4×Dic73—21no—
51C2×C4×Dic73—21no—
52C2×C4×Dic73—21no—
53C2×C4×Dic73—21no—
54C2×C4×Dic73—21no—
55C2×C4×Dic73—21no—
56C2×C4×Dic73—21no—
57C2×C4×Dic73—21no—
58C2×C4×Dic73—21no—
59C2×C4×Dic73—21no—
60C2×C4×Dic73C2×C412yes—
61C2×C4×Dic73C2×C412yes—
62C2×C4×Dic73D411yescharacteristic
63C2×C4×Dic73D411yescharacteristic
64#14556518—28no—
65#14556501—24no—
66C8×Dic73—24no—
67#14556498—28no—
68#14556498—22no—
69#14556498—22no—
70C2×#41×C73—21no—
71C2×#41×C73—21no—
72C2×#41×C73—21no—
73C2×#41×C73—21no—
74C2×#41×C73—21no—
75C2×#41×C73—21no—
76C2×#41×C73—21no—
77C2×#41×C73—21no—
78C2×#41×C73—21no—
79C2×#41×C73—21no—
80C2×#41×C73C2×C412yes—
81C2×#41×C73C2×C412yes—
82C22×Q8×C73C2×C411yescharacteristic
83C22×D4×C73—21no—
84C22×D4×C73C2×C411yescharacteristic
85C23×C4×C73—21no—
86C23×C4×C73—21no—
87C23×C4×C73C2311yescharacteristic
88C2×M16×C73—21no—
89C2×M16×C73—21no—
90C2×M16×C73—21no—
91C2×M16×C73C2×C412yes—
92C2×M16×C73D412yes—
93C2×M16×C73D412yes—
94C22×C8×C73—21no—
95C2×C42×C73—21no—
96C2×C42×C73—21no—
97C2×C42×C73—21no—
98C2×C42×C73—21no—
99C2×C42×C73—21no—
100C2×C42×C73C2×C412yes—
101#104×C73—28no—
102C22×Dic73—24no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—24no—
107C22×Dic73—22no—
108C22×Dic73—42no—
109C22×Dic73—42no—
110C22×Dic73—41no—
111C22×Dic73—41no—
112C22×Dic73C22⋊C412yes—
113C22×Dic73C22⋊C412yes—
114#11924012—22no—
115#11924012—22no—
116#11924012—22no—
117#11924012C22⋊C414yescomplemented
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C4⋊Dic73—22no—
122C4⋊Dic73—22no—
123C22⋊Dic73—44no—
124C22⋊Dic73—41no—
125C22⋊Dic73—41no—
126C22⋊Dic73—44no—
127C4×Dic73—22no—
128C4×Dic73—22no—
129C4×Dic73—21no—
130C4×Dic73—21no—
131C4×Dic73—21no—
132C4×Dic73—21no—
133C4×Dic73—22no—
134C4×Dic73—22no—
135C4×Dic73—41no—
136C4×Dic73—41no—
137C4×Dic73—41no—
138C4×Dic73—41no—
139C24×C73—21no—
140C24×C73C2×D411yescharacteristic
141#41×C73—22no—
142#41×C73—22no—
143#41×C73—22no—
144#41×C73—22no—
145#41×C73—22no—
146#41×C73—22no—
147#41×C73—22no—
148#41×C73—22no—
149#41×C73—41no—
150#41×C73—41no—
151#41×C73—41no—
152#41×C73—41no—
153#41×C73—41no—
154#41×C73—41no—
155#41×C73—41no—
156#41×C73—41no—
157C2×Q8×C73—21no—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73C4212yes—
163C2×D4×C73—22no—
164C2×D4×C73—22no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—41no—
171C2×D4×C73—41no—
172C2×D4×C73—41no—
173C2×D4×C73—41no—
174C2×D4×C73C4212yes—
175C22×C4×C73—24no—
176C22×C4×C73—21no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—42no—
181C22×C4×C73—41no—
182C22×C4×C73—41no—
183C22×C4×C73—41no—
184C22×C4×C73—41no—
185C22×C4×C73—41no—
186C22×C4×C73—41no—
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73#4111yescharacteristic
196C22×C4×C73#4111yescharacteristic
197C22×C4×C73#4111yescharacteristic
198C22×C4×C73#4111yescharacteristic
199C22×C4×C73#4111yescharacteristic
200C22×C4×C73#4111yescharacteristic
201M16×C73—22no—
202M16×C73—22no—
203M16×C73—22no—
204M16×C73C22⋊C414yescomplemented
205C2×C8×C73—22no—
206C2×C8×C73—22no—
207C2×C8×C73—22no—
208C42×C73—22no—
209C42×C73—22no—
210C42×C73—41no—
211C42×C73—41no—
212C23×C73—22no—
213C23×C73—22no—
214C23×C73—22no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—42no—
218C23×C73—42no—
219C23×C73C2×C22⋊C411yescharacteristic
220C23×C73C4×D411yescharacteristic
221C23×C73C4×D411yescharacteristic
222C23×C73#12111yescharacteristic
223C23×C73#12111yescharacteristic
224C23×C73#12411yescharacteristic
225C23×C73#12711yescharacteristic
226Q8×C73—22no—
227Q8×C73—22no—
228Q8×C73—41no—
229Q8×C73—41no—
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—42no—
233D4×C73—42no—
234D4×C73—42no—
235D4×C73—42no—
236D4×C73—41no—
237D4×C73—41no—
238C2×C4×C73—24no—
239C2×C4×C73—22no—
240C2×C4×C73—44no—
241C2×C4×C73—44no—
242C2×C4×C73—42no—
243C2×C4×C73—41no—
244C2×C4×C73—42no—
245C2×C4×C73—41no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73C2×C4211yescharacteristic
251C2×C4×C73C2×C22⋊C411yescharacteristic
252C2×C4×C73C2×C22⋊C411yescharacteristic
253C2×C4×C73#11711yescharacteristic
254C2×C4×C73#11711yescharacteristic
255C2×C4×C73#11711yescharacteristic
256C2×C4×C73C4×D411yescharacteristic
257C2×C4×C73C4×D411yescharacteristic
258C2×C4×C73C4×D411yescharacteristic
259C2×C4×C73C4×D411yescharacteristic
260C2×C4×C73C4×D411yescharacteristic
261C2×C4×C73C4×D411yescharacteristic
262C2×C4×C73C4×D411yescharacteristic
263C2×C4×C73C4×D411yescharacteristic
264C2×C4×C73C4×D411yescharacteristic
265C2×C4×C73C4×D411yescharacteristic
266C2×C4×C73#12011yescharacteristic
267C2×C4×C73#12111yescharacteristic
268C2×C4×C73#12111yescharacteristic
269C2×C4×C73#12111yescharacteristic
270C2×C4×C73#12111yescharacteristic
271C2×C4×C73#12311yescharacteristic
272C2×C4×C73#12311yescharacteristic
273C2×C4×C73#12311yescharacteristic
274C2×C4×C73#12411yescharacteristic
275C2×C4×C73#12411yescharacteristic
276C2×C4×C73#12611yescharacteristic
277C2×C4×C73#12611yescharacteristic
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—24no—
281C2×Dic73—22no—
282C2×Dic73—22no—
283C2×Dic73—22no—
284C2×Dic73—44no—
285C2×Dic73—44no—
286C2×Dic73—44no—
287C2×Dic73—44no—
288C2×Dic73—42no—
289C2×Dic73—42no—
290C2×Dic73—42no—
291C2×Dic73—42no—
292C2×Dic73—42no—
293C2×Dic73#10414yescomplemented
294C8×C73—24no—
295#10596831—24no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73—21no—
301C22×C73—21no—
302C22×C73—21no—
303C22×C73—21no—
304C22×C73—42no—
305C22×C73—42no—
306C22×C73—42no—
307C22×C73—82no—
308C22×C73—82no—
309C22×C73C4×C22⋊C411yescharacteristic
310C22×C73#38611yescharacteristic
311C22×C73#38811yescharacteristic
312C22×C73#38811yescharacteristic
313C22×C73#39011yescharacteristic
314C22×C73#39011yescharacteristic
315C22×C73#39011yescharacteristic
316C22×C73C2×#10412yes—
317C22×C73#42112yes—
318C4×C73—44no—
319C4×C73—42no—
320C4×C73C4×C22⋊C411yescharacteristic
321C4×C73C4×C22⋊C411yescharacteristic
322C4×C73#38611yescharacteristic
323C4×C73#38611yescharacteristic
324C4×C73#38811yescharacteristic
325C4×C73#38811yescharacteristic
326C4×C73#38811yescharacteristic
327C4×C73#38811yescharacteristic
328Dic73—24no—
329Dic73—44no—
330Dic73—42no—
331Dic73—82no—
332#10933—731nomaximal, Sylow
333C2×C73—21no—
334C2×C73—21no—
335C2×C73—21no—
336C2×C73—21no—
337C2×C73—82no—
338C2×C73#143711yescharacteristic
339C2×C73C4×#10411yescharacteristic
340C2×C73#175911yescharacteristic
341C2×C73#187112yes—
342C2×C73#198012yes—
343C22×#104—731no—
344C22×#104D7311yescharacteristic
345C2×C4×M16—731no—
346#2297—731no—
347C2×#339—731no—
348C2×#339—731no—
349C4×#95—731no—
350C73#1093311yescharacteristic, complemented, Sylow
351C22×#41C2×D7311yescharacteristic
352C22×M16—731no—
353C22×M16C2×D7311yescharacteristic
354C2×#117—731no—
355C22×C42—731no—
356C22×C42—731no—
357C22×C42C2×D7311yescharacteristic
358C2×#104—22no—
359C2×#104—22no—
360C2×#104—734no—
361C2×#104—1462no—
362C2×#104—1462no—
363C2×#104Dic7314yescomplemented
364C4×M16—1462no—
365C4×M16—1462no—
366C2×#97—1461no—
367C2×C4×C8—1461no—
368#391—732no—
369#390—732no—
370#386—1462no—
371C4×C22⋊C4—1461no—
372C4×C22⋊C4—1461no—
373C2×#95—1462no—
374#339—1462no—
375#339—1462no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41C4×D7312yes—
387C2×#41C2×Dic7312yes—
388C22×Q8C4×D7311yescharacteristic
389C22×D4—21no—
390C22×D4C4×D7311yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—732no—
394C23×C4—1461no—
395C23×C4C22×D7311yescharacteristic
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—21no—
399C2×M16—732no—
400C2×M16—732no—
401C2×M16—732no—
402C2×M16—1461no—
403C2×M16—1461no—
404C2×M16—1461no—
405C2×M16C2×Dic7312yes—
406C2×M16C22⋊D7312yes—
407C2×M16C22⋊D7312yes—
408C22×C8—21no—
409C22×C8—1461no—
410#117—1461no—
411#117—1461no—
412#117—1461no—
413#117—1461no—
414C2×C4⋊C4—732no—
415C2×C22⋊C4—1462no—
416C2×C22⋊C4—1462no—
417C2×C22⋊C4—1461no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—732no—
424C2×C42—731no—
425C2×C42—731no—
426C2×C42—732no—
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42—1461no—
435C2×C42—1461no—
436C2×C42—1461no—
437C2×C42C2×Dic7312yes—
438#104—28no—
439#104—1468no—
440#97—1464no—
441C4×C8—1464no—
442#95—1462no—
443#95—1462no—
444#95—1468no—
445C24—21no—
446C24D4×D7311yescharacteristic
447#41—22no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41—22no—
452#41—22no—
453#41—22no—
454#41—22no—
455#41—41no—
456#41—41no—
457#41—41no—
458#41—41no—
459#41—41no—
460#41—41no—
461#41—41no—
462#41—41no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C4×Dic7312yes—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4C4×Dic7312yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—2922no—
494C22×C4—2921no—
495C22×C4—2921no—
496C22×C4—2922no—
497C22×C4—732no—
498C22×C4—732no—
499C22×C4—1462no—
500C22×C4—1464no—
501C22×C4—1464no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4C2×C4×D7311yescharacteristic
506C22×C4C2×C4×D7311yescharacteristic
507C22×C4C22×Dic7311yescharacteristic
508C22×C4D4×D7311yescharacteristic
509C22×C4D4×D7311yescharacteristic
510C22×C4D4×D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C2×C22⋊D7311yescharacteristic
513C22×C4#1192402711yescharacteristic
514C22×C4#1192402711yescharacteristic
515C22×C4#1192402711yescharacteristic
516C22×C4#1192402711yescharacteristic
517C22×C4#1192402811yescharacteristic
518C22×C4#1192402811yescharacteristic
519M16—22no—
520M16—22no—
521M16—22no—
522M16—734no—
523M16—1462no—
524M16—1462no—
525M16—1462no—
526M16C22⋊Dic7314yescomplemented
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1462no—
531C2×C8—1462no—
532C2×C8—1462no—
533C4⋊C4—1462no—
534C4⋊C4—1462no—
535C22⋊C4—2921no—
536C22⋊C4—2921no—
537C22⋊C4—2924no—
538C22⋊C4—2924no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—2921no—
544C42—2921no—
545C42—2921no—
546C42—2921no—
547C42—1462no—
548C42—1462no—
549C42—1462no—
550C42—1462no—
551C42—1461no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23C22⋊C4×D7311yescharacteristic
563C23#1455655111yescharacteristic
564C23D4×Dic7311yescharacteristic
565C23#1455656211yescharacteristic
566C23#1455656411yescharacteristic
567C23#1455657711yescharacteristic
568C23#1455658711yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—41no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—2924no—
594C2×C4—2922no—
595C2×C4—2924no—
596C2×C4—2922no—
597C2×C4—2924no—
598C2×C4—2924no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—734no—
603C2×C4—1464no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4C2×C4×Dic7311yescharacteristic
610C2×C4C22⋊C4×D7311yescharacteristic
611C2×C4C2×C22⋊Dic7311yescharacteristic
612C2×C4#1455654711yescharacteristic
613C2×C4#1455654711yescharacteristic
614C2×C4#1455655011yescharacteristic
615C2×C4#1455655111yescharacteristic
616C2×C4#1455655111yescharacteristic
617C2×C4#1455655111yescharacteristic
618C2×C4C4×C22⋊D7311yescharacteristic
619C2×C4C4×C22⋊D7311yescharacteristic
620C2×C4C4×C22⋊D7311yescharacteristic
621C2×C4C4×C22⋊D7311yescharacteristic
622C2×C4D4×Dic7311yescharacteristic
623C2×C4D4×Dic7311yescharacteristic
624C2×C4D4×Dic7311yescharacteristic
625C2×C4#1455655911yescharacteristic
626C2×C4#1455656211yescharacteristic
627C2×C4#1455656311yescharacteristic
628C2×C4#1455656311yescharacteristic
629C2×C4#1455656411yescharacteristic
630C2×C4#1455657211yescharacteristic
631C2×C4#1455657311yescharacteristic
632C2×C4#1455657311yescharacteristic
633C2×C4#1455657711yescharacteristic
634C2×C4#1455657811yescharacteristic
635C2×C4#1455658411yescharacteristic
636C2×C4#1455658411yescharacteristic
637C8—24no—
638C8—1464no—
639C22—22no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22C22⋊C4×Dic7311yescharacteristic
653C22#1798846311yescharacteristic
654C22#1798847511yescharacteristic
655C22#1798848011yescharacteristic
656C22#1798848711yescharacteristic
657C22#1798848711yescharacteristic
658C22#1798849011yescharacteristic
659C22#104×D7312yes—
660C22#1798860212yes—
661C4—44no—
662C4—42no—
663C4—2924no—
664C4—2922no—
665C4—5842no—
666C4—1464no—
667C4C4×C22⋊Dic7311yescharacteristic
668C4C22⋊C4×Dic7311yescharacteristic
669C4#1798846311yescharacteristic
670C4#1798846711yescharacteristic
671C4#1798847511yescharacteristic
672C4#1798848011yescharacteristic
673C4#1798848111yescharacteristic
674C4#1798848111yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—82no—
680C2#2260816811yescharacteristic
681C2#104×Dic7311yescharacteristic
682C2#2260974311yescharacteristic
683C2#2261023012yes—
684C2#2261084812yes—
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C2×C42
FrattiniC23×C4C22×D73
FittingC22×#104×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC22×#41×C73C22
cosocleC23×C4×C73C23
hypercenterC22×#104D73
hypocenterC73#10933
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic73C2semidirect
4. C2×#17988348C2semidirect
5. C2×#17988347C2semidirect
6. C4×#14556498C2semidirect
14. C2×#14556518C4semidirect
19. #17988494C4semidirect
20. #17988490C4semidirect
32. C2×#104×C73C4semidirect
117. #11924012C22⋊C4semidirect
204. M16×C73C22⋊C4semidirect
293. C2×Dic73#104semidirect
350. C73#10933semidirect
363. C2×#104Dic73semidirect
526. M16C22⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565182
C2×M16×Dic732
#226124222
C2×#179883482
C2×#179883472
C4×#145564982
C22×#104×C732
#1093373