Finite Group Databasebeta

← #33246156

Subgroups — #33246156

Order 18688.

2470 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic73C211yesmaximal, characteristic
2#22612887C212yesmaximal, complemented
3#22612858C214yesmaximal, complemented
4D4×#11924007C212yesmaximal, complemented
5#22612818C211yesmaximal, characteristic, complemented
6#22612814C211yesmaximal, characteristic, complemented
7#22612793C212yesmaximal, complemented
8#22612787C212yesmaximal, complemented
9#22612761C211yesmaximal, characteristic
10#22612576C212yesmaximal, complemented
11#22612557C214yesmaximal, complemented
12#22612333C211yesmaximal, characteristic, complemented
13#22612325C211yesmaximal, characteristic, complemented
14#22612322C212yesmaximal, complemented
15C2×#17988451C212yesmaximal, complemented
16#22612292C212yesmaximal, complemented
17C73×#2339C211yesmaximal, characteristic
18#41×Dic73C4116yescomplemented
19C2×Q8×Dic73C2211yescharacteristic
20C2×Q8×Dic73C2211yescharacteristic
21C2×D4×Dic73C2212yes—
22C2×D4×Dic73C2211yescharacteristic
23C2×D4×Dic73C2211yescharacteristic
24C2×D4×Dic73C2212yes—
25C2×D4×Dic73C2212yes—
26C2×#14556550C2212yes—
27C2×#14556550C2211yescharacteristic
28C2×#14556547C2212yes—
29C22×#11924007C2214yes—
30C22×C4×Dic73C2212yes—
31C22×C4×Dic73C2211yescharacteristic
32#17988494C2212yes—
33#17988485C2212yes—
34#17988483C2212yes—
35#17988481C2214yes—
36#17988475C2214yes—
37#17988472C2214yes—
38#17988472C2214yes—
39#17988470C2214yes—
40#17988470C2214yes—
41#17988467C2214yes—
42#17988460C2214yes—
43#17988459C2214yes—
44#17988456C2212yes—
45#17988454C2211yescharacteristic
46#17988454C2211yescharacteristic
47#17988454C2214yes—
48#17988451C2212yes—
49#17988451C2212yes—
50#17988451C2214yes—
51#17988451C2211yescharacteristic
52#17988451C2212yes—
53#17988451C2214yes—
54#17988451C2211yescharacteristic
55#17988450C2212yes—
56#17988448C2212yes—
57#17988447C2211yescharacteristic
58#17988445C2212yes—
59#17988445C2214yes—
60#17988445C2212yes—
61C4⋊C4×Dic73C2212yes—
62C4⋊C4×Dic73C2212yes—
63C4×#11924007C2212yes—
64C4×C4⋊Dic73C2212yes—
65#17988438C2212yes—
66#17988438C2214yes—
67C22⋊C4×Dic73C2214yes—
68C22⋊C4×Dic73C2214yes—
69C4×C22⋊Dic73C2211yescharacteristic
70C4×C22⋊Dic73C2214yes—
71#17988433C2212yes—
72#17988432C2212yes—
73#17988432C2212yes—
74C2×#14556498C2212yes—
75C2×#14556498C2214yes—
76C22×#41×C73C2211yescharacteristic
77C2×C4×D4×C73C2212yes—
78#391×C73C2212yes—
79#386×C73C2214yes—
80#381×C73C2212yes—
81#380×C73C2211yescharacteristic
82C4×C22⋊C4×C73C2211yescharacteristic
83C2×#95×C73C2212yes—
84C23×Dic73C2312yes—
85C23×Dic73C2311yescharacteristic
86C23×Dic73C2314yes—
87C23×Dic73C2312yes—
88C23×Dic73C2311yescharacteristic
89Q8×Dic73C2×C418yescomplemented
90Q8×Dic73C2×C418yescomplemented
91D4×Dic73—28no—
92D4×Dic73C2×C418yes—
93D4×Dic73C2×C418yes—
94D4×Dic73C2×C4116yescomplemented
95D4×Dic73C2×C4116yescomplemented
96#14556550C2×C4116yescomplemented
97#14556550C2×C418yescomplemented
98#14556547—44no—
99C2×C4⋊Dic73C2312yes—
100C2×C4⋊Dic73C2311yescharacteristic
101C2×C4⋊Dic73C2312yes—
102C2×C4⋊Dic73C2312yes—
103C2×C4⋊Dic73C2311yescharacteristic
104C2×#11924007—48no—
105C2×#11924007C2312yes—
106C2×#11924007C2314yes—
107C2×#11924007C2312yes—
108C2×#11924007C2314yes—
109C2×#11924007C2314yes—
110C2×#11924007C2314yes—
111C2×C22⋊Dic73C2314yes—
112C2×C22⋊Dic73C2312yes—
113C2×C22⋊Dic73C2312yes—
114C2×C22⋊Dic73C2312yes—
115C2×C22⋊Dic73C2312yes—
116C2×C22⋊Dic73C2312yes—
117C2×C4×Dic73C2×C418yescomplemented
118C2×C4×Dic73C2×C4116yescomplemented
119C2×C4×Dic73C2312yes—
120C2×C4×Dic73C2314yes—
121C2×C4×Dic73C2314yes—
122C2×C4×Dic73C2314yes—
123C2×C4×Dic73C2312yes—
124C2×C4×Dic73C2311yescharacteristic
125C2×C4×Dic73C2312yes—
126C2×C4×Dic73C2311yescharacteristic
127C2×C4×Dic73C2311yescharacteristic
128C2×C4×Dic73C2311yescharacteristic
129C2×C4×Dic73C2312yes—
130#14556498C2312yes—
131#14556498C2311yescharacteristic
132#14556498C2312yes—
133#14556498C2314yes—
134#14556498C2314yes—
135#14556498C2314yes—
136#14556498C2314yes—
137#14556498C2311yescharacteristic
138#14556498C2314yes—
139#14556498C2314yes—
140#14556498C2312yes—
141#14556498C2312yes—
142#14556498C2312yes—
143#14556498C2314yes—
144#14556498C2312yes—
145#14556498C2312yes—
146C2×#41×C73—28no—
147C2×#41×C73C2×C418yes—
148C22×Q8×C73C2311yescharacteristic
149C22×D4×C73C2312yes—
150C22×D4×C73C2311yescharacteristic
151C23×C4×C73C2311yescharacteristic
152C23×C4×C73C2312yes—
153C23×C4×C73C2314yes—
154C4×D4×C73—28no—
155C2×C4⋊C4×C73C2312yes—
156C2×C22⋊C4×C73C2314yes—
157C2×C22⋊C4×C73C2312yes—
158C2×C22⋊C4×C73C2312yes—
159C2×C42×C73C2312yes—
160#95×C73C2311yescharacteristic
161#95×C73C2311yescharacteristic
162#95×C73C2312yes—
163#95×C73C2314yes—
164#95×C73C2314yes—
165C22×Dic73—24no—
166C22×Dic73—22no—
167C22×Dic73—24no—
168C22×Dic73—22no—
169C22×Dic73—28no—
170C22×Dic73—44no—
171C22×Dic73—42no—
172C22×Dic73—44no—
173C22×Dic73—44no—
174C22×Dic73C22×C418yes—
175C22×Dic73C22×C414yes—
176C22×Dic73C22×C418yes—
177C22×Dic73C22×C414yes—
178C22×Dic73C2412yes—
179C22×Dic73C2412yes—
180C22×Dic73C2414yes—
181C22×Dic73C2411yescharacteristic
182C22×Dic73C2411yescharacteristic
183C22×Dic73C2412yes—
184C22×Dic73C2412yes—
185C22×Dic73C2411yescharacteristic
186C22×Dic73C2411yescharacteristic
187C4⋊Dic73—24no—
188C4⋊Dic73C22×C414yes—
189C4⋊Dic73C22×C414yes—
190C4⋊Dic73C22×C418yes—
191C4⋊Dic73C22×C418yes—
192#11924007—24no—
193#11924007—28no—
194#11924007—24no—
195#11924007—44no—
196#11924007—44no—
197#11924007—44no—
198C22⋊Dic73—28no—
199C22⋊Dic73—42no—
200C22⋊Dic73—42no—
201C22⋊Dic73C22×C418yes—
202C22⋊Dic73C22×C418yes—
203C22⋊Dic73C22×C418yes—
204C4×Dic73—24no—
205C4×Dic73—44no—
206C4×Dic73—44no—
207C4×Dic73—44no—
208C4×Dic73C22×C414yes—
209C4×Dic73C22×C418yes—
210C4×Dic73C22×C414yes—
211C4×Dic73C22×C414yes—
212C4×Dic73C22×C414yes—
213C4×Dic73C22×C418yes—
214C4×Dic73C22×C418yes—
215C24×C73C2411yescharacteristic
216C24×C73C2412yes—
217#41×C73—232no—
218C2×Q8×C73—24no—
219C2×Q8×C73C22×C414yes—
220C2×D4×C73—28no—
221C2×D4×C73C22×C418yes—
222C2×D4×C73C22×C414yes—
223C2×D4×C73#4118yes—
224C22×C4×C73—28no—
225C22×C4×C73—24no—
226C22×C4×C73—28no—
227C22×C4×C73—44no—
228C22×C4×C73—44no—
229C22×C4×C73C22×C418yes—
230C22×C4×C73C22×C414yes—
231C22×C4×C73C2412yes—
232C22×C4×C73C2412yes—
233C22×C4×C73C2414yes—
234C22×C4×C73C2411yescharacteristic
235C22×C4×C73C2412yes—
236C22×C4×C73C2411yescharacteristic
237C4⋊C4×C73—24no—
238C22⋊C4×C73—28no—
239C22⋊C4×C73—42no—
240C22⋊C4×C73—42no—
241C42×C73—24no—
242C23×C73—22no—
243C23×C73—22no—
244C23×C73—24no—
245C23×C73—22no—
246C23×C73—21no—
247C23×C73—21no—
248C23×C73—22no—
249C23×C73C23×C412yes—
250C23×C73C23×C414yes—
251C23×C73C2×#4118yes—
252C23×C73C2511yescharacteristic
253Q8×C73—28no—
254D4×C73—216no—
255D4×C73#117116yescomplemented
256C2×C4×C73—216no—
257C2×C4×C73—22no—
258C2×C4×C73—24no—
259C2×C4×C73—28no—
260C2×C4×C73—22no—
261C2×C4×C73—22no—
262C2×C4×C73—24no—
263C2×C4×C73—24no—
264C2×C4×C73—22no—
265C2×C4×C73—22no—
266C2×C4×C73—22no—
267C2×C4×C73—22no—
268C2×C4×C73—24no—
269C2×C4×C73—88no—
270C2×C4×C73C23×C412yes—
271C2×C4×C73C23×C414yes—
272C2×C4×C73C23×C412yes—
273C2×C4×C73C2×#4114yes—
274C2×Dic73—21no—
275C2×Dic73—22no—
276C2×Dic73—21no—
277C2×Dic73—22no—
278C2×Dic73—24no—
279C2×Dic73—22no—
280C2×Dic73—22no—
281C2×Dic73—216no—
282C2×Dic73—28no—
283C2×Dic73—22no—
284C2×Dic73—22no—
285C2×Dic73—24no—
286C2×Dic73—24no—
287C2×Dic73—22no—
288C2×Dic73—22no—
289C2×Dic73—22no—
290C2×Dic73—24no—
291C2×Dic73—22no—
292C2×Dic73—48no—
293C2×Dic73—44no—
294C2×Dic73—88no—
295C2×Dic73C23×C412yes—
296C2×Dic73C23×C412yes—
297C2×Dic73C23×C414yes—
298C2×Dic73C23×C412yes—
299C2×Dic73C23×C414yes—
300C2×Dic73C23×C412yes—
301C2×Dic73H(5,𝔽2)12yes—
302C2×Dic732−1+412yes—
303C22×C73—24no—
304C22×C73—24no—
305C22×C73—28no—
306C22×C73—28no—
307C22×C73—22no—
308C22×C73—22no—
309C22×C73—22no—
310C22×C73—42no—
311C22×C73—42no—
312C22×C73C2×#11718yes—
313C22×C73C24×C411yescharacteristic
314C22×C73C22×#4112yes—
315C22×C73C2×H(5,𝔽2)11yescharacteristic
316C22×C73C2×2−1+411yescharacteristic
317C22×C73#58512yes—
318C4×C73—22no—
319C4×C73—24no—
320C4×C73—44no—
321C4×C73—42no—
322C4×C73—42no—
323C4×C73C2×#11714yes—
324Dic73—22no—
325Dic73—24no—
326Dic73—22no—
327Dic73—24no—
328Dic73—22no—
329Dic73—44no—
330Dic73—42no—
331Dic73—42no—
332Dic73#52014yes—
333#34191—731nomaximal, Sylow
334C2×C73—28no—
335C2×C73—42no—
336C2×C73C22×#11711yescharacteristic
337C2×C73C2×#52011yescharacteristic
338C2×C73#342811yescharacteristic
339C2×C73#351412yes—
340C2×C73#351512yes—
341C2×C4×#41—731no—
342#2355—732no—
343#2352—734no—
344D4×C4⋊C4—732no—
345#2346—731no—
346#2346—731no—
347#2343—732no—
348#2343—732no—
349#2339—731no—
350#2339D7311yescharacteristic
351#2320—732no—
352#2318—734no—
353#2285—731no—
354#2283—732no—
355#2283—731no—
356C2×#382—732no—
357#2275—732no—
358C73#3419111yescharacteristic, complemented, Sylow
359C22×#41C2×D7311yescharacteristic
360C4×#41—7316no—
361C2×C4×Q8—731no—
362C2×C4×Q8—731no—
363C2×C4×D4—732no—
364C2×C4×D4—731no—
365C2×C4×D4—732no—
366C2×C4×D4—731no—
367C2×C4×D4—732no—
368C2×C4×D4C2×D7312yes—
369C2×#117—732no—
370C2×#117—731no—
371C2×#117—732no—
372C22×C4⋊C4—734no—
373C22×C42—732no—
374C22×C42—731no—
375#391—732no—
376#391C2×D7312yes—
377#389—732no—
378#389—732no—
379#388—734no—
380#388—734no—
381#387—734no—
382#387—734no—
383#387—734no—
384#387—734no—
385#386—734no—
386#386C2×D7314yes—
387#385—734no—
388#385—734no—
389#384—732no—
390#383—734no—
391#383—731no—
392#383—731no—
393#382—734no—
394#382—732no—
395#382—732no—
396#382—734no—
397#382—731no—
398#382—732no—
399#382—731no—
400#381—732no—
401#381—732no—
402#381C2×D7312yes—
403#380—731no—
404#380—732no—
405#380—734no—
406#380—732no—
407#380C2×D7311yescharacteristic
408C4×C4⋊C4—732no—
409C4×C4⋊C4—732no—
410C4×C4⋊C4—732no—
411C4×C4⋊C4—732no—
412C4×C4⋊C4—732no—
413C4×C4⋊C4—734no—
414C4×C22⋊C4—731no—
415C4×C22⋊C4—734no—
416C4×C22⋊C4—734no—
417C4×C22⋊C4—734no—
418C4×C22⋊C4C2×D7311yescharacteristic
419#376—732no—
420#376—732no—
421#376—732no—
422C2×#95—734no—
423C2×#95—732no—
424C2×#95C2×D7312yes—
425C2×#41—28no—
426C2×#41C4×D7318yes—
427C22×Q8C22×D7311yescharacteristic
428C22×D4C22×D7312yes—
429C22×D4C22×D7311yescharacteristic
430C23×C4—732no—
431C23×C4—731no—
432C23×C4—734no—
433C23×C4—732no—
434C23×C4—731no—
435C23×C4C22×D7312yes—
436C23×C4C22×D7311yescharacteristic
437C23×C4C22×D7314yes—
438C4×Q8—738no—
439C4×Q8—738no—
440C4×D4—28no—
441C4×D4—738no—
442C4×D4—738no—
443C4×D4—7316no—
444C4×D4—7316no—
445C4×D4—1468no—
446#117—2924no—
447#117—7316no—
448#117—738no—
449C2×C4⋊C4—2928no—
450C2×C4⋊C4—734no—
451C2×C4⋊C4—734no—
452C2×C4⋊C4—734no—
453C2×C4⋊C4—731no—
454C2×C4⋊C4—732no—
455C2×C4⋊C4—732no—
456C2×C4⋊C4—731no—
457C2×C4⋊C4—732no—
458C2×C4⋊C4—734no—
459C2×C4⋊C4—732no—
460C2×C4⋊C4—732no—
461C2×C4⋊C4C22×D7312yes—
462C2×C22⋊C4—732no—
463C2×C22⋊C4—732no—
464C2×C22⋊C4—732no—
465C2×C22⋊C4—732no—
466C2×C22⋊C4—732no—
467C2×C22⋊C4—734no—
468C2×C22⋊C4C22×D7314yes—
469C2×C22⋊C4C22×D7312yes—
470C2×C22⋊C4C22×D7312yes—
471C2×C42—734no—
472C2×C42—734no—
473C2×C42—734no—
474C2×C42—7316no—
475C2×C42—738no—
476C2×C42—732no—
477C2×C42—731no—
478C2×C42—731no—
479C2×C42—732no—
480C2×C42—731no—
481C2×C42—732no—
482C2×C42—731no—
483C2×C42—732no—
484C2×C42C22×D7312yes—
485#95—732no—
486#95—731no—
487#95—732no—
488#95—734no—
489#95—734no—
490#95—734no—
491#95—734no—
492#95—731no—
493#95—732no—
494#95—732no—
495#95—732no—
496#95—734no—
497#95—734no—
498#95—732no—
499#95—732no—
500#95—734no—
501#95C22×D7311yescharacteristic
502#95C22×D7312yes—
503#95C22×D7314yes—
504#95C22×D7311yescharacteristic
505#95C22×D7314yes—
506C24C23×D7311yescharacteristic
507C24C23×D7312yes—
508#41—232no—
509C2×Q8—24no—
510C2×Q8C2×C4×D7314yes—
511C2×D4—28no—
512C2×D4C2×C4×D7318yes—
513C2×D4C2×C4×D7314yes—
514C2×D4#1192402718yes—
515C22×C4—28no—
516C22×C4—24no—
517C22×C4—28no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—2924no—
521C22×C4—2922no—
522C22×C4—2924no—
523C22×C4—2924no—
524C22×C4—734no—
525C22×C4—738no—
526C22×C4—731no—
527C22×C4—734no—
528C22×C4—738no—
529C22×C4—731no—
530C22×C4—731no—
531C22×C4—732no—
532C22×C4—732no—
533C22×C4—731no—
534C22×C4—732no—
535C22×C4—732no—
536C22×C4—734no—
537C22×C4—1468no—
538C22×C4—1464no—
539C22×C4—1462no—
540C22×C4—1464no—
541C22×C4—1462no—
542C22×C4C2×C4×D7318yes—
543C22×C4C2×C4×D7314yes—
544C22×C4C23×D7312yes—
545C22×C4C23×D7312yes—
546C22×C4C23×D7314yes—
547C22×C4C23×D7311yescharacteristic
548C22×C4C23×D7312yes—
549C22×C4C23×D7311yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—2924no—
552C4⋊C4—2924no—
553C4⋊C4—2924no—
554C4⋊C4—738no—
555C4⋊C4—738no—
556C4⋊C4—734no—
557C4⋊C4—734no—
558C4⋊C4—1464no—
559C4⋊C4—1468no—
560C4⋊C4—1464no—
561C4⋊C4—1464no—
562C22⋊C4—28no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—2922no—
566C22⋊C4—2922no—
567C22⋊C4—738no—
568C22⋊C4—738no—
569C22⋊C4—738no—
570C22⋊C4—1468no—
571C42—24no—
572C42—2924no—
573C42—2924no—
574C42—2924no—
575C42—734no—
576C42—738no—
577C42—734no—
578C42—738no—
579C42—734no—
580C42—738no—
581C42—734no—
582C42—1464no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23C22×C4×D7312yes—
591C23C22×C4×D7314yes—
592C23C2×#1192402718yes—
593C23C24×D7311yescharacteristic
594Q8—28no—
595D4—216no—
596D4#14556547116yescomplemented
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—216no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—88no—
611C2×C4—2928no—
612C2×C4—2924no—
613C2×C4—5848no—
614C2×C4—732no—
615C2×C4—732no—
616C2×C4—732no—
617C2×C4—732no—
618C2×C4—732no—
619C2×C4—734no—
620C2×C4—734no—
621C2×C4—732no—
622C2×C4—1462no—
623C2×C4—1462no—
624C2×C4—1464no—
625C2×C4—1464no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1464no—
630C2×C4—1462no—
631C2×C4—14616no—
632C2×C4—1468no—
633C2×C4—1461no—
634C2×C4—1462no—
635C2×C4—1461no—
636C2×C4—1462no—
637C2×C4—1464no—
638C2×C4—1462no—
639C2×C4—1462no—
640C2×C4C22×C4×D7312yes—
641C2×C4C22×C4×D7314yes—
642C2×C4C22×C4×D7312yes—
643C2×C4C2×#1192402714yes—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—42no—
652C22—42no—
653C22C2×#1455654718yes—
654C22C23×C4×D7311yescharacteristic
655C22C22×#1192402712yes—
656C22H(5,𝔽2)×D7311yescharacteristic
657C222−1+4×D7311yescharacteristic
658C22#1798942312yes—
659C4—22no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—2922no—
665C4—2924no—
666C4—2922no—
667C4—734no—
668C4—1462no—
669C4—1464no—
670C4—1462no—
671C4—1464no—
672C4—1462no—
673C4C2×#1455654714yes—
674C2—28no—
675C2—42no—
676C2C22×#1455654711yescharacteristic
677C2#520×D7311yescharacteristic
678C2#2262313011yescharacteristic
679C2#2262387912yes—
680C2#2262389612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2339C2
socle, cosocleC23×C73C25
prime radicalC22×#41×C73C22
hypercenter#2339D73
hypocenterC73#34191
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2339

Complements

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

NH ≅ G/Ntype
2. #22612887C2semidirect
3. #22612858C2semidirect
4. D4×#11924007C2semidirect
5. #22612818C2semidirect
6. #22612814C2semidirect
7. #22612793C2semidirect
8. #22612787C2semidirect
10. #22612576C2semidirect
11. #22612557C2semidirect
12. #22612333C2semidirect
13. #22612325C2semidirect
14. #22612322C2semidirect
15. C2×#17988451C2semidirect
16. #22612292C2semidirect
18. #41×Dic73C4semidirect
89. Q8×Dic73C2×C4semidirect
90. Q8×Dic73C2×C4semidirect
94. D4×Dic73C2×C4semidirect
95. D4×Dic73C2×C4semidirect
96. #14556550C2×C4semidirect
97. #14556550C2×C4semidirect
117. C2×C4×Dic73C2×C4semidirect
118. C2×C4×Dic73C2×C4semidirect
255. D4×C73#117semidirect
358. C73#34191semidirect
596. D4#14556547semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic732
#226128872
#226128582
D4×#119240072
#226128182
#226128142
#226127932
#226127872
#226127612
#226125762
#226125572
#226123332
#226123252
#226123222
C2×#179884512
#226122922
C73×#23392
#3419173