Finite Group Databasebeta

← #33249167

Subgroups — #33249167

Order 18688.

1998 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2#22613095C214yesmaximal
3#22613020C211yesmaximal, characteristic, complemented
4#22613020C211yesmaximal, characteristic, complemented
5#22612991C211yesmaximal, characteristic, complemented
6#22612895C212yesmaximal, complemented
7#22612849C212yesmaximal, complemented
8#22612820C212yesmaximal
9#22612815C212yesmaximal
10#22612671C212yesmaximal, complemented
11#22612668C212yesmaximal, complemented
12#22612616C212yesmaximal
13#22612607C212yesmaximal
14#22612580C214yesmaximal, complemented
15#22612335C211yesmaximal, characteristic
16#22612327C211yesmaximal, characteristic
17C73×#2365C211yesmaximal, characteristic, complemented
18#17989142C4116yescomplemented
19C2×#14556591C2211yescharacteristic
20C2×#14556582C2212yes—
21C2×#14556571C2212yes—
22C2×#14556567C2212yes—
23C2×Q8×Dic73C2212yes—
24C2×Q8×Dic73C2212yes—
25C2×#14556556C2212yes—
26C2×#14556549C2212yes—
27C2×#14556548C2212yes—
28C2×C42×D73C2211yescharacteristic
29#17988494C2212yes—
30#17988491C2214yes—
31#17988491C2214yes—
32#17988486C2212yes—
33#17988485C2212yes—
34#17988485C2212yes—
35#17988485C2214yes—
36#17988484C2214yes—
37#17988483C2212yes—
38#17988483C2212yes—
39#17988483C2214yes—
40#17988482C2214yes—
41#17988479C2214yes—
42#17988478C2214yes—
43#17988477C2214yes—
44#17988476C2214yes—
45#17988473C2214yes—
46#17988472C2214yes—
47#17988471C2214yes—
48#17988470C2214yes—
49#17988469C2214yes—
50#17988456C2211yescharacteristic
51#17988456C2211yescharacteristic
52#17988454C2214yes—
53#17988454C2212yes—
54#17988452C2214yes—
55#17988452C2211yescharacteristic
56#17988452C2211yescharacteristic
57#17988451C2212yes—
58#17988449C2212yes—
59#17988446C2212yes—
60#17988444C2211yescharacteristic
61#17988444C2211yescharacteristic
62#17988444C2211yescharacteristic
63#17988444C2211yescharacteristic
64C4⋊C4×Dic73C2212yes—
65C4⋊C4×Dic73C2212yes—
66C4⋊C4×Dic73C2212yes—
67C4×#11924007C2214yes—
68C4×#11924007C2212yes—
69C4×C4⋊Dic73C2212yes—
70C4×C4⋊Dic73C2211yescharacteristic
71C4×C4⋊Dic73C2211yescharacteristic
72C4×C4⋊Dic73C2212yes—
73C4×#11924005C2212yes—
74#17988433C2212yes—
75#17988432C2214yes—
76C2×C4⋊Q8×C73C2211yescharacteristic
77C2×C4×Q8×C73C2212yes—
78#391×C73C2212yes—
79#389×C73C2212yes—
80#387×C73C2214yes—
81C4×C4⋊C4×C73C2211yescharacteristic
82C4×C4⋊C4×C73C2211yescharacteristic
83C4×C4⋊C4×C73C2212yes—
84C22×Dic146C2312yes—
85C22×C4×D73C2311yescharacteristic
86C22×C4×D73C2311yescharacteristic
87C22×C4×D73C2311yescharacteristic
88#14556591C2×C418yes—
89#14556582C2×C4116yes—
90#14556571C2×C4116yes—
91#14556567C2×C4116yes—
92Q8×Dic73—44no—
93Q8×Dic73C2×C4116yes—
94#14556556C2×C4116yes—
95#14556549C2×C4116yes—
96#14556548—44no—
97C2×C4⋊Dic73C2311yescharacteristic
98C2×C4⋊Dic73C2311yescharacteristic
99C2×C4⋊Dic73C2311yescharacteristic
100C2×C4⋊Dic73C2311yescharacteristic
101C2×C4⋊Dic73C2314yes—
102C2×C4⋊Dic73C2312yes—
103C2×C4⋊Dic73C2312yes—
104C2×C4⋊Dic73C2314yes—
105C2×#11924007C2312yes—
106C2×#11924007C2312yes—
107C2×#11924007C2314yes—
108C2×#11924007C2314yes—
109C2×#11924007C2312yes—
110C2×#11924007C2312yes—
111C2×#11924007C2314yes—
112C2×#11924005C2312yes—
113C2×#11924005C2312yes—
114C2×#11924005C2312yes—
115C2×#11924005C2312yes—
116C2×#11924005C2314yes—
117C2×C4×Dic73C2312yes—
118C2×C4×Dic73C2312yes—
119C2×C4×Dic73C2314yes—
120C2×C4×Dic73C2314yes—
121C2×C4×Dic73C2314yes—
122C2×C4×Dic73C2314yes—
123C2×C4×Dic73C2311yescharacteristic
124C2×C4×Dic73C2311yescharacteristic
125C2×C4×Dic73C2311yescharacteristic
126C42×D73C2×C418yes—
127#14556498C2312yes—
128#14556498C2312yes—
129#14556498C2314yes—
130#14556498C2312yes—
131#14556498C2312yes—
132#14556498C2312yes—
133#14556498C2312yes—
134#14556498C2312yes—
135#14556498C2312yes—
136#14556498C2312yes—
137#14556498C2314yes—
138#14556498C2314yes—
139#14556498C2312yes—
140#14556498C2314yes—
141#14556498C2314yes—
142#14556498C2314yes—
143C22×Q8×C73C2312yes—
144C4⋊Q8×C73C2×C418yes—
145C4×Q8×C73—44no—
146C2×C4⋊C4×C73C2311yescharacteristic
147C2×C4⋊C4×C73C2311yescharacteristic
148C2×C4⋊C4×C73C2311yescharacteristic
149C2×C4⋊C4×C73C2311yescharacteristic
150C2×C4⋊C4×C73C2314yes—
151C2×C4⋊C4×C73C2312yes—
152C2×C4⋊C4×C73C2312yes—
153C2×C42×C73C2312yes—
154C2×C42×C73C2312yes—
155C2×C42×C73C2314yes—
156C2×C42×C73C2311yescharacteristic
157#95×C73C2312yes—
158#95×C73C2312yes—
159#95×C73C2314yes—
160#95×C73C2314yes—
161C23×D73C2411yescharacteristic
162C2×Dic146—42no—
163C2×Dic146—42no—
164C2×Dic146C22×C418yes—
165C22×Dic73C2412yes—
166C22×Dic73C2412yes—
167C22×Dic73C2412yes—
168C22×Dic73C2412yes—
169C22×Dic73C2411yescharacteristic
170C22×Dic73C2414yes—
171C22×Dic73C2411yescharacteristic
172C22×Dic73C2411yescharacteristic
173C2×C4×D73—22no—
174C2×C4×D73—22no—
175C2×C4×D73—22no—
176C2×C4×D73—22no—
177C2×C4×D73—41no—
178C2×C4×D73—41no—
179C2×C4×D73C22×C414yes—
180C2×C4×D73C22×C414yes—
181C2×C4×D73C22×C414yes—
182C4⋊Dic73—22no—
183C4⋊Dic73—22no—
184C4⋊Dic73—22no—
185C4⋊Dic73—22no—
186C4⋊Dic73—44no—
187C4⋊Dic73C22×C418yes—
188C4⋊Dic73C22×C418yes—
189C4⋊Dic73C22×C4116yes—
190#11924007—44no—
191#11924007—44no—
192#11924007—42no—
193#11924007—42no—
194#11924007—42no—
195#11924007—42no—
196#11924007C22×C4116yes—
197#11924005—42no—
198#11924005—42no—
199#11924005—42no—
200#11924005—42no—
201#11924005C22×C4116yes—
202C4×Dic73—24no—
203C4×Dic73—24no—
204C4×Dic73—44no—
205C4×Dic73—44no—
206C4×Dic73—44no—
207C4×Dic73C22×C414yes—
208C4×Dic73C22×C414yes—
209C4×Dic73C22×C414yes—
210C4×Dic73C22×C4116yes—
211C2×Q8×C73—24no—
212C2×Q8×C73—24no—
213C2×Q8×C73C22×C418yes—
214C22×C4×C73C2412yes—
215C22×C4×C73C2412yes—
216C22×C4×C73C2412yes—
217C22×C4×C73C2412yes—
218C22×C4×C73C2411yescharacteristic
219C22×C4×C73C2414yes—
220C22×C4×C73C2411yescharacteristic
221C22×C4×C73C2411yescharacteristic
222C4⋊C4×C73—22no—
223C4⋊C4×C73—22no—
224C4⋊C4×C73—22no—
225C4⋊C4×C73—22no—
226C4⋊C4×C73—44no—
227C4⋊C4×C73C22×C418yes—
228C4⋊C4×C73C22×C418yes—
229C42×C73—24no—
230C42×C73—24no—
231C42×C73—44no—
232C42×C73C22×C414yes—
233C23×C73C2511yescharacteristic
234C22×D73—21no—
235C22×D73—21no—
236C22×D73—21no—
237C22×D73—21no—
238C22×D73—21no—
239C22×D73—21no—
240C22×D73C23×C412yes—
241Q8×C73—216no—
242C2×C4×C73—24no—
243C2×C4×C73—24no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—22no—
257C2×C4×C73—22no—
258C2×C4×C73C23×C412yes—
259C2×C4×C73C23×C418yes—
260C2×C4×C73C23×C412yes—
261C2×C4×C73C23×C412yes—
262C2×C4×C73H(5,𝔽2)12yes—
263C2×C4×C732−1+412yes—
264C2×C4×C732−1+412yes—
265C2×C4×C732−1+412yes—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—24no—
271C2×Dic73—24no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—22no—
275C2×Dic73—22no—
276C2×Dic73—22no—
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—22no—
281C2×Dic73—22no—
282C2×Dic73—22no—
283C2×Dic73—22no—
284C2×Dic73—22no—
285C2×Dic73—22no—
286C2×Dic73C23×C412yes—
287C2×Dic73C23×C412yes—
288C2×Dic73C23×C412yes—
289C2×Dic73C23×C418yes—
290C4×D73—28no—
291C4×D73—28no—
292C4×D73—44no—
293Dic146—48no—
294C22×C73C24×C411yescharacteristic
295C22×C73C2×H(5,𝔽2)11yescharacteristic
296C22×C73C2×2−1+411yescharacteristic
297C22×C73C2×2−1+411yescharacteristic
298C22×C73C2×2−1+411yescharacteristic
299C22×C73#58511yescharacteristic
300C22×C73#58511yescharacteristic
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—41no—
305C2×D73—41no—
306C2×D73—41no—
307C2×D73—41no—
308C4×C73—28no—
309C4×C73—22no—
310C4×C73—42no—
311C4×C73—42no—
312C4×C73—42no—
313C4×C73—42no—
314C4×C73#51914yes—
315C4×C73#52014yes—
316Dic73—22no—
317Dic73—22no—
318Dic73—28no—
319Dic73—22no—
320Dic73—42no—
321Dic73—42no—
322Dic73—42no—
323Dic73—42no—
324#34394—731nomaximal, Sylow
325C2×C73C2×#51911yescharacteristic
326C2×C73C2×#52011yescharacteristic
327C2×C73#342811yescharacteristic
328C2×C73#354711yescharacteristic
329C2×C73#354711yescharacteristic
330C2×C73#355511yescharacteristic
331C2×C73#355511yescharacteristic
332D73—42no—
333C2×#533—731no—
334#2378—734no—
335#2369—731no—
336#2369—731no—
337#2365—731no—
338#2365D7311yescharacteristic, complemented
339#2356—732no—
340#2351—732no—
341#2347—732no—
342#2346—732no—
343#2330—732no—
344#2330—732no—
345#2323—732no—
346#2323—732no—
347#2320—734no—
348#2285—731no—
349#2283—731no—
350C73#3439411yescharacteristic, complemented, Sylow
351#533—7316no—
352C2×C4⋊Q8—731no—
353C2×C4⋊Q8C2×D7311yescharacteristic
354C2×#125—732no—
355C2×C22⋊Q8—732no—
356C2×C22⋊Q8—732no—
357C2×C4×Q8—732no—
358C2×C4×Q8—732no—
359C2×C4×Q8—732no—
360C2×C4×Q8C2×D7312yes—
361C2×#117—732no—
362C2×#117—732no—
363C22×C42—731no—
364#391—732no—
365#391—734no—
366#391—734no—
367#391C2×D7312yes—
368#389—734no—
369#389—732no—
370#389—732no—
371#389—732no—
372#389—734no—
373#389—732no—
374#389—732no—
375#389—734no—
376#389—734no—
377#389C2×D7312yes—
378#388—734no—
379#388—734no—
380#388—734no—
381#388—734no—
382#387—734no—
383#387—734no—
384#387—734no—
385#387—734no—
386#387—734no—
387#387C2×D7314yes—
388#384—731no—
389#384—731no—
390#383—732no—
391#383—734no—
392#382—734no—
393#382—731no—
394#382—731no—
395#382—732no—
396#381—732no—
397#380—731no—
398#380—731no—
399#380—731no—
400#380—731no—
401#380—732no—
402C4×C4⋊C4—732no—
403C4×C4⋊C4—732no—
404C4×C4⋊C4—732no—
405C4×C4⋊C4—734no—
406C4×C4⋊C4—732no—
407C4×C4⋊C4—732no—
408C4×C4⋊C4—732no—
409C4×C4⋊C4—731no—
410C4×C4⋊C4—731no—
411C4×C4⋊C4C2×D7312yes—
412C4×C4⋊C4C2×D7311yescharacteristic
413C4×C4⋊C4C2×D7311yescharacteristic
414C4×C22⋊C4—732no—
415#376—732no—
416#376—734no—
417C22×Q8—732no—
418C22×Q8C22×D7312yes—
419C23×C4—731no—
420C23×C4—731no—
421C23×C4—731no—
422C4⋊Q8—738no—
423C4⋊Q8C4×D7318yes—
424#125—7316no—
425C22⋊Q8—7316no—
426C22⋊Q8—7316no—
427C4×Q8—44no—
428C4×Q8—2924no—
429C4×Q8—7316no—
430C4×Q8—7316no—
431#117—2924no—
432#117—7316no—
433C2×C4⋊C4—732no—
434C2×C4⋊C4—732no—
435C2×C4⋊C4—734no—
436C2×C4⋊C4—732no—
437C2×C4⋊C4—732no—
438C2×C4⋊C4—734no—
439C2×C4⋊C4—734no—
440C2×C4⋊C4—732no—
441C2×C4⋊C4—734no—
442C2×C4⋊C4—734no—
443C2×C4⋊C4—732no—
444C2×C4⋊C4—731no—
445C2×C4⋊C4—731no—
446C2×C4⋊C4—731no—
447C2×C4⋊C4—731no—
448C2×C4⋊C4C22×D7311yescharacteristic
449C2×C4⋊C4C22×D7311yescharacteristic
450C2×C4⋊C4C22×D7311yescharacteristic
451C2×C4⋊C4C22×D7311yescharacteristic
452C2×C4⋊C4C22×D7314yes—
453C2×C4⋊C4C22×D7312yes—
454C2×C4⋊C4C22×D7312yes—
455C2×C22⋊C4—732no—
456C2×C22⋊C4—732no—
457C2×C22⋊C4—732no—
458C2×C22⋊C4—732no—
459C2×C22⋊C4—734no—
460C2×C42—734no—
461C2×C42—734no—
462C2×C42—734no—
463C2×C42—734no—
464C2×C42—731no—
465C2×C42—731no—
466C2×C42—731no—
467C2×C42—738no—
468C2×C42—732no—
469C2×C42—732no—
470C2×C42C22×D7312yes—
471C2×C42C22×D7312yes—
472C2×C42C22×D7314yes—
473C2×C42C22×D7311yescharacteristic
474#95—732no—
475#95—732no—
476#95—732no—
477#95—732no—
478#95—732no—
479#95—732no—
480#95—734no—
481#95—732no—
482#95—732no—
483#95—732no—
484#95—732no—
485#95—734no—
486#95—734no—
487#95—734no—
488#95—734no—
489#95—734no—
490#95C22×D7314yes—
491#95C22×D7312yes—
492#95C22×D7312yes—
493#95C22×D7314yes—
494C24—731no—
495C2×Q8—24no—
496C2×Q8—24no—
497C2×Q8—2922no—
498C2×Q8—2922no—
499C2×Q8—738no—
500C2×Q8C2×C4×D7318yes—
501C22×C4—2921no—
502C22×C4—2921no—
503C22×C4—734no—
504C22×C4—731no—
505C22×C4—734no—
506C22×C4—731no—
507C22×C4—731no—
508C22×C4—734no—
509C22×C4—734no—
510C22×C4—732no—
511C22×C4—732no—
512C22×C4—732no—
513C22×C4—732no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1462no—
518C22×C4C23×D7312yes—
519C22×C4C23×D7312yes—
520C22×C4C23×D7312yes—
521C22×C4C23×D7312yes—
522C22×C4C23×D7311yescharacteristic
523C22×C4C23×D7314yes—
524C22×C4C23×D7311yescharacteristic
525C22×C4C23×D7311yescharacteristic
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—2922no—
532C4⋊C4—2924no—
533C4⋊C4—2922no—
534C4⋊C4—2924no—
535C4⋊C4—2922no—
536C4⋊C4—2922no—
537C4⋊C4—2924no—
538C4⋊C4—7316no—
539C4⋊C4—738no—
540C4⋊C4—7316no—
541C4⋊C4—738no—
542C4⋊C4—1462no—
543C4⋊C4—1462no—
544C4⋊C4—1462no—
545C4⋊C4—1462no—
546C4⋊C4C2×C4×D7318yes—
547C4⋊C4C2×C4×D7318yes—
548C22⋊C4—2922no—
549C22⋊C4—2922no—
550C22⋊C4—2922no—
551C22⋊C4—2922no—
552C22⋊C4—7316no—
553C42—24no—
554C42—24no—
555C42—44no—
556C42—2924no—
557C42—2924no—
558C42—2924no—
559C42—734no—
560C42—734no—
561C42—734no—
562C42—7316no—
563C42—1464no—
564C42—1464no—
565C42C2×C4×D7314yes—
566C23—732no—
567C23—1461no—
568C23—1461no—
569C23—1461no—
570C23—1461no—
571C23—1461no—
572C23—1461no—
573C23C24×D7311yescharacteristic
574Q8—216no—
575Q8—2928no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—2924no—
593C2×C4—732no—
594C2×C4—732no—
595C2×C4—732no—
596C2×C4—738no—
597C2×C4—1462no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1464no—
614C2×C4—1464no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1468no—
618C2×C4—1468no—
619C2×C4C22×C4×D7312yes—
620C2×C4C22×C4×D7318yes—
621C2×C4C22×C4×D7312yes—
622C2×C4C22×C4×D7312yes—
623C2×C4#1455663612yes—
624C2×C4#1455663812yes—
625C2×C4#1455663812yes—
626C2×C4#1455663912yes—
627C22—2921no—
628C22—2921no—
629C22—2921no—
630C22—2921no—
631C22—1462no—
632C22—1462no—
633C22—1462no—
634C22C23×C4×D7311yescharacteristic
635C22C2×#1455663611yescharacteristic
636C22C2×#1455663811yescharacteristic
637C22C2×#1455663811yescharacteristic
638C22C2×#1455663911yescharacteristic
639C22#1798942411yescharacteristic
640C22#1798942411yescharacteristic
641C4—28no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—2922no—
648C4—2922no—
649C4—2922no—
650C4—2922no—
651C4—1462no—
652C4—1462no—
653C4—1462no—
654C4—1468no—
655C4#1798906314yes—
656C4#1798906614yes—
657C2—2922no—
658C2C2×#1798906311yescharacteristic
659C2C2×#1798906611yescharacteristic
660C2#2262313311yescharacteristic
661C2#2262430811yescharacteristic
662C2#2262431611yescharacteristic
663C2#2262440711yescharacteristic
664C2#2262441511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2365C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2365D73
hypocenterC73#34394
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
3. #22613020C2semidirect
4. #22613020C2semidirect
5. #22612991C2semidirect
6. #22612895C2semidirect
7. #22612849C2semidirect
10. #22612671C2semidirect
11. #22612668C2semidirect
14. #22612580C2semidirect
17. C73×#2365C2semidirect
18. #17989142C4semidirect
338. #2365D73semidirect
350. C73#34394semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226130952
#226130202
#226129912
#226128952
#226128492
#226128202
#226128152
#226126712
#226126682
#226126162
#226126072
#226125802
#226123352
#226123272
C73×#23652
#3439473