Finite Group Databasebeta

← #33085467

Subgroups — #33085467

Order 18688.

1450 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×C4×Q8×D73C211yesmaximal, characteristic
2C2×#17988609C211yesmaximal, characteristic
3C2×#17988594C212yesmaximal
4#22612336C211yesmaximal, characteristic
5#22609945C212yesmaximal, complemented
6#22609944C212yesmaximal, complemented
7#22609937C211yesmaximal, characteristic, complemented
8#22609936C211yesmaximal, characteristic, complemented
9#22609933C211yesmaximal, characteristic, complemented
10#22609649C212yesmaximal
11C73×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D73C2211yescharacteristic
13C22×#11924011C2212yes—
14C2×Q8×Dic73C2211yescharacteristic
15C2×C4×Dic146—21no—
16C2×C4×Dic146C2211yescharacteristic
17C2×#14556556—21no—
18C2×#14556556C2211yescharacteristic
19C4×Q8×D73—24no—
20C4×Q8×D73C418yescomplemented
21C2×#14556549C2212yes—
22C2×C4⋊C4×D73—21no—
23C2×C4⋊C4×D73C2211yescharacteristic
24C2×C42×D73—21no—
25C2×C42×D73C2211yescharacteristic
26#17988609—24no—
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556520C2212yes—
29#17988594—28no—
30C2×#14556516C2212yes—
31C2×#14556515C2212yes—
32C2×#14556514C2212yes—
33#17988457C2211yescharacteristic
34#17988452C2212yes—
35#17988444—21no—
36#17988444—21no—
37C4⋊C4×Dic73—21no—
38C4⋊C4×Dic73—21no—
39#17988350C2212yes—
40#17988349C2212yes—
41#17988349C2212yes—
42C2×C4×Q8×C73C2211yescharacteristic
43C2×C4⋊C8×C73C2211yescharacteristic
44C2×#103×C73C2212yes—
45#384×C73C2211yescharacteristic
46#340×C73C2212yes—
47C22×Dic146—21no—
48C22×Dic146C2311yescharacteristic
49C2×Q8×D73—22no—
50C2×Q8×D73—22no—
51C2×Q8×D73—22no—
52C2×Q8×D73C2×C414yes—
53C2×Q8×D73D414yes—
54C2×Q8×D73Q814yes—
55C22×C4×D73—21no—
56C22×C4×D73—21no—
57C22×C4×D73D411yescharacteristic
58C22×C4×D73D411yescharacteristic
59C22×C4×D73C2311yescharacteristic
60C2×#11924011—24no—
61C2×#11924011—24no—
62C2×#11924011—24no—
63C22×#10596831C2312yes—
64Q8×Dic73—22no—
65Q8×Dic73C2×C414yes—
66C4×Dic146—28no—
67C4×Dic146—22no—
68C4×Dic146C2×C414yes—
69#14556556—28no—
70#14556556—22no—
71#14556556C2×C414yes—
72#14556549—24no—
73#14556549—24no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2312yes—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C4⋊C4×D73—28no—
83C4⋊C4×D73—22no—
84C4⋊C4×D73C2×C414yes—
85C2×#11924005—22no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—22no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73D411yescharacteristic
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93C42×D73—28no—
94C42×D73C2×C414yes—
95C42×D73D414yes—
96C4⋊#10596831—22no—
97#14556520—24no—
98#14556516—24no—
99#14556515—24no—
100#14556514—24no—
101#14556498—22no—
102#14556498—22no—
103C22×Q8×C73C2311yescharacteristic
104C22×C8×C73C2312yes—
105C4×Q8×C73—22no—
106C4×Q8×C73C2×C414yes—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2312yes—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C42×C73—21no—
113C2×C42×C73C2311yescharacteristic
114C4⋊C8×C73—22no—
115#103×C73—24no—
116C23×D73C2×D411yescharacteristic
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146—21no—
120C2×Dic146—24no—
121C2×Dic146—24no—
122C2×Dic146—24no—
123C2×Dic146C22×C412yes—
124C2×Dic146C2×D412yes—
125C2×Dic146C2×Q812yes—
126Q8×D73—28no—
127Q8×D73—28no—
128Q8×D73—28no—
129Q8×D73C4⋊C4116yescomplemented
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—22no—
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×C4×D73—24no—
137C2×C4×D73—24no—
138C2×C4×D73—24no—
139C2×C4×D73—22no—
140C2×C4×D73—22no—
141C2×C4×D73—24no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146C2×C4×D73C22⋊C414yes—
147C2×C4×D73C22⋊C414yes—
148C2×C4×D73C22×C412yes—
149C2×C4×D73C2×D412yes—
150C2×C4×D73C2×D412yes—
151C2×C4×D73C2×Q812yes—
152C2×C4×D73#4112yes—
153C2×C4×D73#4112yes—
154#11924011—48no—
155C2×#10596831—22no—
156C2×#10596831—22no—
157C2×#10596831—22no—
158C4⋊Dic73—21no—
159C4⋊Dic73—24no—
160C4⋊Dic73—22no—
161C4⋊Dic73—22no—
162C4⋊Dic73—42no—
163C4⋊Dic73—42no—
164C4⋊Dic73C22×C412yes—
165#11924007—24no—
166#11924007—24no—
167#11924007—24no—
168#11924005—44no—
169C4×Dic73—24no—
170C4×Dic73—24no—
171C4×Dic73—24no—
172C4×Dic73—44no—
173C4×Dic73C22⋊C414yes—
174C4×Dic73C22⋊C414yes—
175C4×Dic73C22×C412yes—
176C4×Dic73C2×D412yes—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73C22×C412yes—
181C2×Q8×C73C2×D412yes—
182C2×Q8×C73C2×Q812yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—22no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73—22no—
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C4⋊C4×C73—24no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—22no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73C22×C412yes—
199C42×C73—24no—
200C42×C73C22×C412yes—
201C42×C73C2×D412yes—
202C23×C73C22×D411yescharacteristic
203C22×D73—21no—
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—21no—
207C22×D73C2×C22⋊C412yes—
208C22×D73#12112yes—
209C22×D73C22⋊Q812yes—
210Q8×C73—22no—
211Q8×C73—22no—
212Q8×C73—22no—
213Q8×C73C2×C4⋊C414yes—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73C2×C22⋊C412yes—
226C2×C4×C73C2×C22⋊C412yes—
227C2×C4×C73#13712yes—
228C2×C4×C73#13712yes—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C22×Q811yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—22no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73C2×C22⋊C412yes—
249C2×Dic73C2×C22⋊C412yes—
250C2×Dic73C2×C22⋊C412yes—
251C2×Dic73#12112yes—
252C2×Dic73C22⋊Q812yes—
253C4×D73—216no—
254C4×D73—28no—
255C4×D73—44no—
256C4×D73—48no—
257C4×D73C2×C4⋊C414yes—
258C4×D73#11714yes—
259C4×D73#12114yes—
260C4×D73C22⋊Q814yes—
261Dic146—22no—
262Dic146—22no—
263Dic146—22no—
264Dic146—216no—
265Dic146C2×C4⋊C414yes—
266C8×C73—42no—
267#10596831—42no—
268C22×C73C22×C22⋊C411yescharacteristic
269C22×C73C2×#12111yescharacteristic
270C22×C73C2×C22⋊Q811yescharacteristic
271C22×C73C2×#13711yescharacteristic
272C22×C73C2×#13711yescharacteristic
273C22×C73#57511yescharacteristic
274C22×C73#57511yescharacteristic
275C2×D73—22no—
276C2×D73—22no—
277C2×D73—22no—
278C2×D73—22no—
279C2×D73—22no—
280C2×D73—22no—
281C2×D73#38014yes—
282C4×C73—22no—
283C4×C73—24no—
284C4×C73—42no—
285C4×C73—82no—
286C4×C73#41914yescomplemented
287C4×C73C22×C4⋊C411yescharacteristic
288C4×C73C2×#11711yescharacteristic
289C4×C73C2×#12111yescharacteristic
290C4×C73C2×C22⋊Q811yescharacteristic
291Dic73—22no—
292Dic73—22no—
293Dic73—24no—
294Dic73—42no—
295Dic73—82no—
296Dic73#38014yes—
297#20647—731nomaximal, Sylow
298C2×C73C2×#38011yescharacteristic
299C2×C73C2×#41911yescharacteristic
300C2×C73#289611yescharacteristic
301C2×C73#303711yescharacteristic
302C2×C73#303711yescharacteristic
303C2×C73#307811yescharacteristic
304C2×C73#307811yescharacteristic
305D73—24no—
306C22×C4×Q8—731no—
307C2×#423—731no—
308C2×#419—732no—
309#2285—731no—
310#1805—732no—
311#1805—732no—
312#1803—731no—
313#1803—731no—
314#1803—731no—
315#1803D7311yescharacteristic, complemented
316#1736—732no—
317C73#2064711yescharacteristic, complemented, Sylow
318C23×Q8—731no—
319C22×M16—732no—
320C2×C4×Q8—731no—
321C2×C4×Q8—738no—
322C2×C4×Q8—731no—
323C2×C4×Q8—731no—
324C2×C4×Q8—1461no—
325C2×C4×Q8—1461no—
326C2×C4×Q8—1464no—
327C2×C4×Q8C2×D7311yescharacteristic
328C2×#117—732no—
329C22×C4⋊C4—731no—
330C22×C4⋊C4—1461no—
331C22×C42—731no—
332C22×C42—1461no—
333#423—1464no—
334C2×C4⋊C8—732no—
335C2×C4⋊C8—731no—
336C2×C4⋊C8C2×D7311yescharacteristic
337#419—1468no—
338C2×#103—732no—
339C2×#103—732no—
340C2×#103—732no—
341C2×#103C2×D7312yes—
342#384—731no—
343#384C2×D7311yescharacteristic
344#382—732no—
345#380—1461no—
346#380—1461no—
347C4×C4⋊C4—1461no—
348C4×C4⋊C4—1461no—
349#340—732no—
350#340—732no—
351#340—732no—
352#340C2×D7312yes—
353C22×Q8—734no—
354C22×Q8—734no—
355C22×Q8—734no—
356C22×Q8—731no—
357C22×Q8—1461no—
358C22×Q8—1462no—
359C22×Q8—1462no—
360C22×Q8—1462no—
361C22×Q8C22×D7311yescharacteristic
362C23×C4—731no—
363C23×C4—731no—
364C23×C4—731no—
365C23×C4—1461no—
366C23×C4—1461no—
367C2×M16—1464no—
368C2×M16—1464no—
369C2×M16—1464no—
370C22×C8—732no—
371C22×C8C22×D7312yes—
372C4×Q8—22no—
373C4×Q8—734no—
374C4×Q8—734no—
375C4×Q8—734no—
376C4×Q8—1468no—
377C4×Q8—1468no—
378C4×Q8—1462no—
379C4×Q8—1462no—
380C4×Q8—1462no—
381C4×Q8C4×D7314yes—
382#117—1464no—
383#117—1464no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—732no—
390C2×C4⋊C4—1462no—
391C2×C4⋊C4—1468no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4C22×D7312yes—
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—1462no—
401C2×C42—21no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—734no—
406C2×C42—734no—
407C2×C42—1468no—
408C2×C42—1462no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42C22×D7311yescharacteristic
413C4⋊C8—22no—
414C4⋊C8—1464no—
415C4⋊C8—1462no—
416#103—24no—
417#103—1464no—
418#103—1464no—
419#103—1464no—
420#95—1462no—
421#95—1462no—
422C24—731no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—7316no—
427C2×Q8—732no—
428C2×Q8—732no—
429C2×Q8—732no—
430C2×Q8—1464no—
431C2×Q8—1464no—
432C2×Q8—1464no—
433C2×Q8—1468no—
434C2×Q8—1468no—
435C2×Q8—1468no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×Q8—1461no—
439C2×Q8C2×C4×D7312yes—
440C2×Q8D4×D7312yes—
441C2×Q8Q8×D7312yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—2921no—
446C22×C4—2921no—
447C22×C4—2922no—
448C22×C4—2922no—
449C22×C4—734no—
450C22×C4—731no—
451C22×C4—732no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—732no—
456C22×C4—732no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—734no—
460C22×C4—1464no—
461C22×C4—1464no—
462C22×C4—1464no—
463C22×C4—1464no—
464C22×C4—1462no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4D4×D7311yescharacteristic
470C22×C4D4×D7311yescharacteristic
471C22×C4C23×D7311yescharacteristic
472M16—2928no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—1462no—
477C2×C8—1462no—
478C2×C8—1462no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—732no—
488C4⋊C4—1464no—
489C4⋊C4—1464no—
490C4⋊C4—1464no—
491C4⋊C4—1462no—
492C4⋊C4—1462no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4C2×C4×D7312yes—
496C22⋊C4—2924no—
497C42—24no—
498C42—2924no—
499C42—734no—
500C42—734no—
501C42—732no—
502C42—732no—
503C42—1464no—
504C42—1464no—
505C42—1464no—
506C42C2×C4×D7312yes—
507C42D4×D7312yes—
508C23—732no—
509C23—732no—
510C23—732no—
511C23—1461no—
512C23—1461no—
513C23—1461no—
514C23—1461no—
515C23C2×D4×D7311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—734no—
520Q8—14616no—
521Q8—1462no—
522Q8—1462no—
523Q8—1462no—
524Q8C4⋊C4×D7314yes—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—2924no—
537C2×C4—2928no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—732no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—734no—
547C2×C4—734no—
548C2×C4—734no—
549C2×C4—734no—
550C2×C4—732no—
551C2×C4—732no—
552C2×C4—14616no—
553C2×C4—1462no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1468no—
562C2×C4C22⋊C4×D7312yes—
563C2×C4C22⋊C4×D7312yes—
564C2×C4#1455662012yes—
565C2×C4#1455662312yes—
566C2×C4C22×C4×D7311yescharacteristic
567C2×C4C2×D4×D7311yescharacteristic
568C2×C4C2×D4×D7311yescharacteristic
569C2×C4C2×Q8×D7311yescharacteristic
570C2×C4#41×D7311yescharacteristic
571C2×C4#41×D7311yescharacteristic
572C8—42no—
573C8—2922no—
574C22—734no—
575C22—1462no—
576C22—1462no—
577C22—1462no—
578C22—1462no—
579C22—1462no—
580C22—1462no—
581C22C2×C22⋊C4×D7311yescharacteristic
582C22#121×D7311yescharacteristic
583C22C22⋊Q8×D7311yescharacteristic
584C22C2×#1455662011yescharacteristic
585C22C2×#1455662311yescharacteristic
586C22#1798938311yescharacteristic
587C22#1798938611yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—82no—
592C4—2922no—
593C4—5842no—
594C4—734no—
595C4—1462no—
596C4—1464no—
597C4—1462no—
598C4#1798859414yescomplemented
599C4C2×C4⋊C4×D7311yescharacteristic
600C4#117×D7311yescharacteristic
601C4#121×D7311yescharacteristic
602C4C22⋊Q8×D7311yescharacteristic
603C2—1464no—
604C2#380×D7311yescharacteristic
605C2C2×#1798859411yescharacteristic
606C2#2261844711yescharacteristic
607C2#2261939811yescharacteristic
608C2#2261940411yescharacteristic
609C2#2261973111yescharacteristic
610C2#2261973711yescharacteristic
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#20647
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. #22609945C2semidirect
6. #22609944C2semidirect
7. #22609937C2semidirect
8. #22609936C2semidirect
9. #22609933C2semidirect
11. C73×#1803C2semidirect
20. C4×Q8×D73C4semidirect
129. Q8×D73C4⋊C4semidirect
286. C4×C73#419semidirect
315. #1803D73semidirect
317. C73#20647semidirect
598. C4#17988594semidirect

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×#179886092
C2×#179885942
#226123362
#226099452
#226099442
#226099372
#226099362
#226099332
#226096492
C73×#18032
#2064773