Finite Group Databasebeta

← #33085995

Subgroups — #33085995

Order 18688.

1370 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D73C211yesmaximal, characteristic
2C2×#17988639C211yesmaximal, characteristic
3#22612336C211yesmaximal, characteristic
4#22610020C212yesmaximal
5#22609937C211yesmaximal, characteristic, complemented
6#22609936C211yesmaximal, characteristic, complemented
7#22609933C211yesmaximal, characteristic, complemented
8#22609864C212yesmaximal
9#103×Dic73C212yesmaximal, complemented
10#22609750C212yesmaximal, complemented
11C73×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D73C2211yescharacteristic
13C2×Q8×Dic73C2211yescharacteristic
14C2×C4×Dic146—21no—
15C2×C4×Dic146C2211yescharacteristic
16C2×#14556556—21no—
17C2×#14556556C2211yescharacteristic
18C4×Q8×D73—24no—
19C4×Q8×D73C418yescomplemented
20C2×#14556549—21no—
21C2×C4⋊C4×D73—21no—
22C2×C4⋊C4×D73C2211yescharacteristic
23C2×C42×D73—21no—
24C2×C42×D73C2211yescharacteristic
25#17988639—24no—
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556516C2212yes—
28C2×#14556515C2212yes—
29C2×#14556514C2212yes—
30C2×#14556503C2212yes—
31C2×C8×Dic73C2212yes—
32#17988457C2211yescharacteristic
33#17988452—21no—
34#17988444—21no—
35#17988444C2212yes—
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73C2212yes—
38#17988350C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4⋊C8×C73C2211yescharacteristic
43C2×#103×C73C2212yes—
44#384×C73C2211yescharacteristic
45#340×C73C2212yes—
46C22×Dic146—21no—
47C22×Dic146C2311yescharacteristic
48C2×Q8×D73—22no—
49C2×Q8×D73—22no—
50C2×Q8×D73—22no—
51C2×Q8×D73—22no—
52C2×Q8×D73—22no—
53C2×Q8×D73C2×C414yes—
54C22×C4×D73—21no—
55C22×C4×D73—21no—
56C22×C4×D73D411yescharacteristic
57C22×C4×D73D411yescharacteristic
58C22×C4×D73C2311yescharacteristic
59C22×#10596831C2312yes—
60Q8×Dic73—22no—
61Q8×Dic73C2×C414yes—
62C4×Dic146—28no—
63C4×Dic146—22no—
64C4×Dic146C2×C414yes—
65#14556556—28no—
66#14556556—22no—
67#14556556C2×C414yes—
68#14556549—44no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2312yes—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C4⋊C4×D73—28no—
78C4⋊C4×D73—22no—
79C4⋊C4×D73C2×C414yes—
80C2×#11924005—22no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—22no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C42×D73—28no—
89C42×D73C2×C414yes—
90C42×D73D414yes—
91C4⋊#10596831—22no—
92#14556516—24no—
93#14556515—24no—
94#14556514—24no—
95#14556503—24no—
96C8×Dic73—24no—
97#14556498—22no—
98#14556498—22no—
99C22×Q8×C73C2311yescharacteristic
100C22×C8×C73C2312yes—
101C4×Q8×C73—22no—
102C4×Q8×C73C2×C414yes—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73C2312yes—
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C4⋊C8×C73—22no—
111#103×C73—24no—
112C23×D73C2×D411yescharacteristic
113C2×Dic146—24no—
114C2×Dic146—24no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146—24no—
119C2×Dic146C22×C412yes—
120C2×Dic146#4112yes—
121C2×Dic146#4112yes—
122Q8×D73—28no—
123Q8×D73—28no—
124Q8×D73—28no—
125Q8×D73—28no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—22no—
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C2×C4×D73—24no—
133C2×C4×D73—24no—
134C2×C4×D73—24no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—24no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73C22⋊C414yes—
145C2×C4×D73C22⋊C414yes—
146C2×C4×D73SD1614yes—
147C2×C4×D73Q1614yes—
148C2×C4×D73C22×C412yes—
149C2×C4×D73C2×D412yes—
150C2×#10596831—22no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C4⋊Dic73—21no—
154C4⋊Dic73—24no—
155C4⋊Dic73—22no—
156C4⋊Dic73—22no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73C22×C412yes—
160#11924007—24no—
161#11924007—24no—
162#11924007—24no—
163#11924005—44no—
164C4×Dic73—24no—
165C4×Dic73—24no—
166C4×Dic73—24no—
167C4×Dic73—44no—
168C4×Dic73C22⋊C414yes—
169C4×Dic73C22⋊C414yes—
170C4×Dic73C22×C412yes—
171C4×Dic73C2×D412yes—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73C22×C412yes—
176C2×Q8×C73#4112yes—
177C2×Q8×C73#4112yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—22no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184C2×C8×C73—22no—
185C2×C8×C73—22no—
186C2×C8×C73—22no—
187C4⋊C4×C73—24no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—42no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73C22×C412yes—
194C42×C73—24no—
195C42×C73C22×C412yes—
196C42×C73C2×D412yes—
197C23×C73C22×D411yescharacteristic
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73C2×C22⋊C412yes—
203C22×D73C2×SD1612yes—
204C22×D73C2×Q1612yes—
205Q8×C73—22no—
206Q8×C73—22no—
207Q8×C73—22no—
208Q8×C73#11714yescomplemented
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73C2×C22⋊C412yes—
221C2×C4×C73C2×C22⋊C412yes—
222C2×C4×C73C2×SD1612yes—
223C2×C4×C73C2×Q1612yes—
224C2×C4×C73C23×C411yescharacteristic
225C2×C4×C73C22×D411yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—22no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73C2×C22⋊C412yes—
244C2×Dic73C2×C22⋊C412yes—
245C2×Dic73C2×C22⋊C412yes—
246C2×Dic73C2×SD1612yes—
247C2×Dic73C2×Q1612yes—
248C4×D73—216no—
249C4×D73—22no—
250C4×D73—22no—
251C4×D73—22no—
252C4×D73—22no—
253C4×D73—44no—
254C4×D73—48no—
255C4×D73#103116yescomplemented
256Dic146—216no—
257Dic146—22no—
258Dic146—22no—
259Dic146—22no—
260Dic146#11714yescomplemented
261C8×C73—42no—
262#10596831—42no—
263C22×C73C22×C22⋊C411yescharacteristic
264C22×C73#52511yescharacteristic
265C22×C73#52511yescharacteristic
266C22×C73C22×SD1611yescharacteristic
267C22×C73C22×Q1611yescharacteristic
268C22×C73#57511yescharacteristic
269C22×C73#57511yescharacteristic
270C2×D73—22no—
271C2×D73—22no—
272C2×D73—22no—
273C2×D73—22no—
274C2×D73—22no—
275C2×D73—22no—
276C2×D73C2×#10314yes—
277C4×C73—22no—
278C4×C73—24no—
279C4×C73—42no—
280C4×C73—82no—
281C4×C73C2×#10314yes—
282C4×C73C2×#11711yescharacteristic
283C4×C73C2×#11711yescharacteristic
284C4×C73#52511yescharacteristic
285C4×C73#52511yescharacteristic
286Dic73—22no—
287Dic73—22no—
288Dic73—24no—
289Dic73—42no—
290Dic73—82no—
291Dic73C2×#10314yes—
292#20734—731nomaximal, Sylow
293C2×C73#228011yescharacteristic
294C2×C73C22×#10311yescharacteristic
295C2×C73#289611yescharacteristic
296C2×C73#310311yescharacteristic
297C2×C73#310311yescharacteristic
298C2×C73#310411yescharacteristic
299C2×C73#310411yescharacteristic
300D73—24no—
301C22×C4×Q8—731no—
302C2×#431—731no—
303#2285—731no—
304#1824—732no—
305#1803—731no—
306#1803—731no—
307#1803—731no—
308#1803D7311yescharacteristic, complemented
309#1787—732no—
310C4×#103—732no—
311C4×#103—732no—
312C73#2073411yescharacteristic, complemented, Sylow
313C23×Q8—731no—
314C2×C4×Q8—731no—
315C2×C4×Q8—738no—
316C2×C4×Q8—731no—
317C2×C4×Q8—731no—
318C2×C4×Q8—1461no—
319C2×C4×Q8—1461no—
320C2×C4×Q8—1464no—
321C2×C4×Q8C2×D7311yescharacteristic
322C2×#117—1461no—
323C22×C4⋊C4—731no—
324C22×C4⋊C4—1461no—
325C22×C42—731no—
326C22×C42—1461no—
327#431—1464no—
328C2×C4⋊C8—731no—
329C2×C4⋊C8C2×D7311yescharacteristic
330C2×#103—732no—
331C2×#103—732no—
332C2×#103—732no—
333C2×#103C2×D7312yes—
334C2×C22⋊C8—732no—
335C2×C4×C8—732no—
336#384—731no—
337#384C2×D7311yescharacteristic
338#382—1461no—
339#380—732no—
340#380—1461no—
341C4×C4⋊C4—732no—
342C4×C4⋊C4—1461no—
343#340—732no—
344#340—732no—
345#340—732no—
346#340C2×D7312yes—
347C22×Q8—734no—
348C22×Q8—731no—
349C22×Q8—1461no—
350C22×Q8—1462no—
351C22×Q8—1462no—
352C22×Q8—1462no—
353C22×Q8—1462no—
354C22×Q8—1462no—
355C22×Q8C22×D7311yescharacteristic
356C23×C4—731no—
357C23×C4—731no—
358C23×C4—731no—
359C23×C4—1461no—
360C23×C4—1461no—
361C22×C8—732no—
362C22×C8C22×D7312yes—
363C4×Q8—22no—
364C4×Q8—734no—
365C4×Q8—734no—
366C4×Q8—734no—
367C4×Q8—1468no—
368C4×Q8—1468no—
369C4×Q8—1462no—
370C4×Q8—1462no—
371C4×Q8—1462no—
372C4×Q8C4×D7314yes—
373#117—2924no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—734no—
379C2×C4⋊C4—732no—
380C2×C4⋊C4—1462no—
381C2×C4⋊C4—1468no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C22×D7312yes—
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C22⋊C4—1462no—
391C2×C42—21no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—734no—
396C2×C42—734no—
397C2×C42—1468no—
398C2×C42—1462no—
399C2×C42—1461no—
400C2×C42—1461no—
401C2×C42—1461no—
402C2×C42C22×D7311yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—1462no—
405#103—24no—
406#103—1464no—
407#103—1464no—
408#103—1464no—
409C22⋊C8—1464no—
410C4×C8—1464no—
411#95—1462no—
412#95—1462no—
413C24—731no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—732no—
418C2×Q8—732no—
419C2×Q8—732no—
420C2×Q8—1464no—
421C2×Q8—1464no—
422C2×Q8—1464no—
423C2×Q8—1468no—
424C2×Q8—1468no—
425C2×Q8—1468no—
426C2×Q8—1468no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8C2×C4×D7312yes—
431C2×Q8#1192402712yes—
432C2×Q8#1192402912yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—2921no—
437C22×C4—2921no—
438C22×C4—2922no—
439C22×C4—2922no—
440C22×C4—734no—
441C22×C4—731no—
442C22×C4—734no—
443C22×C4—734no—
444C22×C4—734no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—732no—
448C22×C4—732no—
449C22×C4—1464no—
450C22×C4—1464no—
451C22×C4—1464no—
452C22×C4—1464no—
453C22×C4—1462no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4D4×D7311yescharacteristic
461C22×C4D4×D7311yescharacteristic
462C22×C4C23×D7311yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1462no—
467C2×C8—1462no—
468C2×C8—1462no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—732no—
478C4⋊C4—1464no—
479C4⋊C4—1464no—
480C4⋊C4—1464no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4—1464no—
484C4⋊C4—1461no—
485C4⋊C4C2×C4×D7312yes—
486C22⋊C4—2924no—
487C42—24no—
488C42—2924no—
489C42—734no—
490C42—734no—
491C42—732no—
492C42—732no—
493C42—1464no—
494C42—1464no—
495C42—1464no—
496C42C2×C4×D7312yes—
497C42D4×D7312yes—
498C23—732no—
499C23—732no—
500C23—732no—
501C23—1461no—
502C23—1461no—
503C23—1461no—
504C23—1461no—
505C23C2×D4×D7311yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—734no—
510Q8—14616no—
511Q8—1462no—
512Q8—1462no—
513Q8—1462no—
514Q8#1455654914yescomplemented
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—2924no—
527C2×C4—2928no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—732no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—7316no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—14616no—
540C2×C4—1462no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1462no—
544C2×C4—1461no—
545C2×C4—1462no—
546C2×C4—1461no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4C22⋊C4×D7312yes—
553C2×C4C22⋊C4×D7312yes—
554C2×C4SD16×D7312yes—
555C2×C4Q16×D7312yes—
556C2×C4C22×C4×D7311yescharacteristic
557C2×C4C2×D4×D7311yescharacteristic
558C2×C4#41×D7311yescharacteristic
559C2×C4#41×D7311yescharacteristic
560C2×C4C2×#1192402711yescharacteristic
561C2×C4C2×#1192402911yescharacteristic
562C8—42no—
563C8—2922no—
564C22—734no—
565C22—1462no—
566C22—1462no—
567C22—1462no—
568C22—1462no—
569C22—1462no—
570C22—1462no—
571C22C2×C22⋊C4×D7311yescharacteristic
572C22#1798909711yescharacteristic
573C22#1798910011yescharacteristic
574C22C2×SD16×D7311yescharacteristic
575C22C2×Q16×D7311yescharacteristic
576C22#1798938311yescharacteristic
577C22#1798938611yescharacteristic
578C4—22no—
579C4—24no—
580C4—42no—
581C4—82no—
582C4—2922no—
583C4—5842no—
584C4—734no—
585C4—1462no—
586C4—1462no—
587C4—1464no—
588C4#103×D7314yes—
589C4#117×D7311yescharacteristic
590C4C2×#1455654911yescharacteristic
591C4#1798909711yescharacteristic
592C4#1798910011yescharacteristic
593C2—1464no—
594C2#2261231311yescharacteristic
595C2C2×#103×D7311yescharacteristic
596C2#2261844711yescharacteristic
597C2#2261996111yescharacteristic
598C2#2261996411yescharacteristic
599C2#2261996911yescharacteristic
600C2#2261997211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1803C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1803D73
hypocenterC73#20734
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #22609937C2semidirect
6. #22609936C2semidirect
7. #22609933C2semidirect
9. #103×Dic73C2semidirect
10. #22609750C2semidirect
11. C73×#1803C2semidirect
19. C4×Q8×D73C4semidirect
208. Q8×C73#117semidirect
255. C4×D73#103semidirect
260. Dic146#117semidirect
308. #1803D73semidirect
312. C73#20734semidirect
514. Q8#14556549semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D732
C2×#179886392
#226123362
#226100202
#226099372
#226099362
#226099332
#226098642
#103×Dic732
#226097502
C73×#18032
#2073473