Finite Group Databasebeta

← #33245096

Subgroups — #33245096

Order 18688.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989191C211yesmaximal, characteristic
2#22612722C214yesmaximal, complemented
3#22612697C214yesmaximal, complemented
4#22612672C211yesmaximal, characteristic, complemented
5#22612618C212yesmaximal, complemented
6#22612568C212yesmaximal, complemented
7#22612541C211yesmaximal, characteristic, complemented
8#127×Dic73C211yesmaximal, characteristic, complemented
9#22612486C211yesmaximal, characteristic, complemented
10#22612456C214yesmaximal, complemented
11#22612447C212yesmaximal, complemented
12#22612441C214yesmaximal, complemented
13C4×D4×Dic73C212yesmaximal, complemented
14#22612270C211yesmaximal, characteristic, complemented
15C73×#2328C211yesmaximal, characteristic, complemented
16#17989191C4116yescomplemented
17C2×#14556583C2211yescharacteristic
18C2×#14556572C2214yes—
19C2×#14556563C2214yes—
20C2×#14556562C2212yes—
21C2×D4×Dic73C2212yes—
22C2×D4×Dic73C2214yes—
23C2×D4×Dic73C2212yes—
24C2×#14556551C2214yes—
25C2×#14556548C2211yescharacteristic
26C2×#14556548C2211yescharacteristic
27C22×C4×Dic73C2214yes—
28C2×C42×D73C2211yescharacteristic
29#17988490C2212yes—
30#17988490C2214yes—
31#17988487C2214yes—
32#17988485C2212yes—
33#17988484C2214yes—
34#17988483C2212yes—
35#17988481C2214yes—
36#17988481C2214yes—
37#17988481C2214yes—
38#17988479C2214yes—
39#17988475C2214yes—
40#17988474C2214yes—
41#17988474C2214yes—
42#17988469C2214yes—
43#17988466C2214yes—
44#17988463C2214yes—
45#17988460C2214yes—
46#17988457C2211yescharacteristic
47#17988448C2214yes—
48#17988445C2214yes—
49#17988444C2212yes—
50C4⋊C4×Dic73C2212yes—
51C4⋊C4×Dic73C2212yes—
52C4×C4⋊Dic73C2212yes—
53#17988438C2214yes—
54#17988438C2212yes—
55#17988438C2212yes—
56C22⋊C4×Dic73C2214yes—
57C22⋊C4×Dic73C2214yes—
58C22⋊C4×Dic73C2214yes—
59C4×C22⋊Dic73C2214yes—
60#17988434C2212yes—
61#17988434C2212yes—
62#17988433C2211yescharacteristic
63C42×Dic73C2211yescharacteristic
64C42×Dic73C2211yescharacteristic
65C2×#127×C73C2211yescharacteristic
66C2×C4×D4×C73C2212yes—
67#390×C73C2212yes—
68#386×C73C2214yes—
69#386×C73C2214yes—
70#382×C73C2211yescharacteristic
71#376×C73C2211yescharacteristic
72C22×C22⋊D73C2314yes—
73C23×Dic73C2314yes—
74C23×Dic73C2314yes—
75C22×C4×D73C2311yescharacteristic
76C22×C4×D73C2311yescharacteristic
77C22×C4×D73C2311yescharacteristic
78#14556583C2×C418yescomplemented
79#14556572C2×C4132yescomplemented
80#14556563C2×C4132yescomplemented
81#14556562C2×C4116yescomplemented
82D4×Dic73—28no—
83D4×Dic73—48no—
84D4×Dic73C2×C4116yescomplemented
85#14556551—48no—
86#14556548—24no—
87#14556548C2×C418yescomplemented
88C2×C4⋊Dic73C2312yes—
89C2×C4⋊Dic73C2314yes—
90C2×C4⋊Dic73C2312yes—
91C2×#11924007C2312yes—
92C2×#11924007C2314yes—
93C2×#11924007C2314yes—
94C2×#11924007C2312yes—
95C2×#11924007C2314yes—
96C2×C22⋊Dic73C2314yes—
97C2×C22⋊Dic73C2314yes—
98C2×C22⋊Dic73C2314yes—
99C2×C22⋊Dic73C2314yes—
100C2×C22⋊Dic73C2314yes—
101C2×#11924005C2312yes—
102C2×#11924005C2314yes—
103C2×#11924005C2312yes—
104C2×C4×Dic73—48no—
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2312yes—
107C2×C4×Dic73C2312yes—
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2314yes—
110C2×C4×Dic73C2311yescharacteristic
111C2×C4×Dic73C2312yes—
112C2×C4×Dic73C2312yes—
113C2×C4×Dic73C2314yes—
114C2×C4×Dic73C2314yes—
115C2×C4×Dic73C2311yescharacteristic
116C2×C4×Dic73C2311yescharacteristic
117C42×D73—24no—
118#14556498C2312yes—
119#14556498C2312yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2314yes—
124#14556498C2314yes—
125#14556498C2314yes—
126C22×D4×C73C2312yes—
127C22×D4×C73C2314yes—
128C23×C4×C73C2314yes—
129#127×C73C2×C418yescomplemented
130C4×D4×C73—28no—
131C2×C4⋊C4×C73C2312yes—
132C2×C22⋊C4×C73C2314yes—
133C2×C22⋊C4×C73C2314yes—
134C2×C22⋊C4×C73C2314yes—
135C2×C42×C73C2311yescharacteristic
136C2×C42×C73C2311yescharacteristic
137C2×C42×C73C2311yescharacteristic
138#95×C73C2314yes—
139#95×C73C2312yes—
140#95×C73C2312yes—
141C23×D73C2411yescharacteristic
142C2×C22⋊D73—44no—
143C2×C22⋊D73—44no—
144C2×C22⋊D73C22×C4116yes—
145C22×Dic73—28no—
146C22×Dic73—28no—
147C22×Dic73—28no—
148C22×Dic73—44no—
149C22×Dic73—44no—
150C22×Dic73C22×C4116yes—
151C22×Dic73C2412yes—
152C22×Dic73C2414yes—
153C22×Dic73C2412yes—
154C22×Dic73C2411yescharacteristic
155C22×Dic73C2414yes—
156C22×Dic73C2411yescharacteristic
157C22×Dic73C2411yescharacteristic
158C2×C4×D73—22no—
159C2×C4×D73—22no—
160C2×C4×D73—22no—
161C2×C4×D73—22no—
162C2×C4×D73—24no—
163C2×C4×D73—42no—
164C2×C4×D73C22×C414yes—
165C4⋊Dic73—24no—
166C4⋊Dic73—44no—
167C4⋊Dic73C22×C418yes—
168#11924007—24no—
169#11924007—44no—
170#11924007—44no—
171#11924007C22×C418yes—
172#11924007C22×C4116yes—
173C22⋊Dic73—28no—
174C22⋊Dic73—44no—
175C22⋊Dic73—44no—
176C22⋊Dic73C22×C4116yes—
177C22⋊Dic73C22×C4116yes—
178#11924005—24no—
179#11924005—44no—
180#11924005C22×C418yes—
181C4×Dic73—22no—
182C4×Dic73—22no—
183C4×Dic73—22no—
184C4×Dic73—24no—
185C4×Dic73—28no—
186C4×Dic73—24no—
187C4×Dic73—44no—
188C4×Dic73—44no—
189C4×Dic73—42no—
190C4×Dic73—42no—
191C4×Dic73C22×C414yes—
192C4×Dic73#4114yes—
193C24×C73C2414yes—
194C2×D4×C73—24no—
195C2×D4×C73—24no—
196C2×D4×C73—44no—
197C2×D4×C73—44no—
198C2×D4×C73C22×C418yes—
199C2×D4×C73C22×C4116yes—
200C22×C4×C73—28no—
201C22×C4×C73—44no—
202C22×C4×C73—44no—
203C22×C4×C73C2411yescharacteristic
204C22×C4×C73C2411yescharacteristic
205C22×C4×C73C2414yes—
206C22×C4×C73C2412yes—
207C22×C4×C73C2412yes—
208C22×C4×C73C2411yescharacteristic
209C4⋊C4×C73—24no—
210C22⋊C4×C73—28no—
211C22⋊C4×C73—44no—
212C22⋊C4×C73—44no—
213C42×C73—22no—
214C42×C73C22×C414yes—
215C42×C73#4114yes—
216C23×C73—24no—
217C23×C73—24no—
218C23×C73—24no—
219C23×C73—24no—
220C23×C73—24no—
221C23×C73—24no—
222C23×C73C23×C418yes—
223C23×C73C2511yescharacteristic
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—22no—
227C22×D73—22no—
228C22×D73C23×C412yes—
229D4×C73—216no—
230D4×C73—416no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—24no—
238C2×C4×C73—22no—
239C2×C4×C73—24no—
240C2×C4×C73—22no—
241C2×C4×C73—24no—
242C2×C4×C73—416no—
243C2×C4×C73C23×C414yes—
244C2×C4×C73C23×C412yes—
245C2×C4×C73C2×#4112yes—
246C2×C4×C73C2×#4114yes—
247C2×C4×C73C2×#4112yes—
248C22⋊D73—416no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—22no—
252C2×Dic73—24no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—24no—
256C2×Dic73—22no—
257C2×Dic73—24no—
258C2×Dic73—22no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—22no—
262C2×Dic73—416no—
263C2×Dic73—416no—
264C2×Dic73C23×C414yes—
265C2×Dic73C23×C412yes—
266C2×Dic73C23×C418yes—
267C2×Dic73C2×#4112yes—
268C2×Dic73C2×#4112yes—
269C2×Dic73H(5,𝔽2)14yes—
270C4×D73—28no—
271C4×D73—44no—
272C4×D73—44no—
273C22×C73—28no—
274C22×C73—28no—
275C22×C73—28no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73—44no—
280C22×C73C24×C411yescharacteristic
281C22×C73C22×#4111yescharacteristic
282C22×C73C22×#4111yescharacteristic
283C22×C73C2×H(5,𝔽2)12yes—
284C22×C73#58512yes—
285C2×D73—22no—
286C2×D73—24no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—42no—
290C4×C73—24no—
291C4×C73—24no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—44no—
295C4×C73C4×#4114yes—
296Dic73—22no—
297Dic73—24no—
298Dic73—28no—
299Dic73—22no—
300Dic73—44no—
301Dic73—42no—
302Dic73#54014yescomplemented
303C4×#540—731nomaximal, Sylow
304C2×C73—48no—
305C2×C73C2×C4×#4111yescharacteristic
306C2×C73C4×H(5,𝔽2)12yes—
307C2×C73C2×#54011yescharacteristic
308C2×C73#345511yescharacteristic
309C2×C73#346012yes—
310D73—42no—
311C2×#540—731no—
312#2335—734no—
313#2333—734no—
314#2330—731no—
315#2328D7311yescharacteristic, complemented
316#2324—732no—
317#2319—732no—
318#2314—731no—
319C4×#127—731no—
320C4×#125—731no—
321C4×#123—734no—
322C4×#121—732no—
323C4×#121—734no—
324C42×D4—732no—
325C4×#117—731no—
326C73C4×#54011yescharacteristic, complemented, Sylow
327#540—7316no—
328C2×#127C2×D7311yescharacteristic
329C2×#125—731no—
330C2×#123—734no—
331C2×#121—732no—
332C2×#121—734no—
333C2×C4×D4—734no—
334C2×C4×D4—732no—
335C2×C4×D4—732no—
336C2×C4×D4—734no—
337C2×C4×D4C2×D7312yes—
338C2×#117—731no—
339C2×#117—731no—
340C22×C42—734no—
341C22×C42—731no—
342#390—732no—
343#390—734no—
344#390—734no—
345#390C2×D7312yes—
346#389—732no—
347#389—732no—
348#389—734no—
349#388—734no—
350#388—734no—
351#388—734no—
352#388—734no—
353#388—734no—
354#387—734no—
355#387—734no—
356#387—734no—
357#386—734no—
358#386—734no—
359#386C2×D7314yes—
360#386C2×D7314yes—
361#385—734no—
362#384—731no—
363#382C2×D7311yescharacteristic
364#381—734no—
365#380—732no—
366#380—734no—
367C4×C4⋊C4—732no—
368C4×C4⋊C4—732no—
369C4×C4⋊C4—732no—
370C4×C4⋊C4—734no—
371C4×C4⋊C4—732no—
372C4×C4⋊C4—732no—
373C4×C22⋊C4—734no—
374C4×C22⋊C4—734no—
375C4×C22⋊C4—734no—
376C4×C22⋊C4—734no—
377C4×C22⋊C4—732no—
378C4×C22⋊C4—732no—
379#376—731no—
380#376C2×D7311yescharacteristic
381C43—731no—
382C43—731no—
383C22×D4—734no—
384C22×D4C22×D7312yes—
385C22×D4C22×D7314yes—
386C23×C4—734no—
387C23×C4—731no—
388C23×C4—731no—
389C23×C4—734no—
390C23×C4—731no—
391C23×C4C22×D7314yes—
392#127C4×D7318yescomplemented
393#125—738no—
394#123—7332no—
395#121—7316no—
396#121—7332no—
397C4×D4—28no—
398C4×D4—2928no—
399C4×D4—2928no—
400C4×D4—7316no—
401C4×D4—1468no—
402#117—738no—
403#117—1464no—
404C2×C4⋊C4—734no—
405C2×C4⋊C4—734no—
406C2×C4⋊C4—734no—
407C2×C4⋊C4—732no—
408C2×C4⋊C4—732no—
409C2×C4⋊C4—734no—
410C2×C4⋊C4—732no—
411C2×C4⋊C4—732no—
412C2×C4⋊C4C22×D7312yes—
413C2×C22⋊C4—734no—
414C2×C22⋊C4—734no—
415C2×C22⋊C4—734no—
416C2×C22⋊C4—734no—
417C2×C22⋊C4—734no—
418C2×C22⋊C4—732no—
419C2×C22⋊C4—734no—
420C2×C22⋊C4—732no—
421C2×C22⋊C4C22×D7314yes—
422C2×C22⋊C4C22×D7314yes—
423C2×C22⋊C4C22×D7314yes—
424C2×C42—2928no—
425C2×C42—732no—
426C2×C42—732no—
427C2×C42—734no—
428C2×C42—734no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—732no—
433C2×C42—734no—
434C2×C42—732no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—1464no—
438C2×C42C22×D7311yescharacteristic
439C2×C42C22×D7311yescharacteristic
440C2×C42C22×D7311yescharacteristic
441#95—732no—
442#95—732no—
443#95—734no—
444#95—734no—
445#95—734no—
446#95—734no—
447#95—734no—
448#95—734no—
449#95C22×D7312yes—
450#95C22×D7312yes—
451#95C22×D7314yes—
452C24—731no—
453C24C23×D7314yes—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—2924no—
459C2×D4—2924no—
460C2×D4—7316no—
461C2×D4C2×C4×D7318yes—
462C2×D4C2×C4×D73116yes—
463C22×C4—28no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—2922no—
467C22×C4—2924no—
468C22×C4—2924no—
469C22×C4—7316no—
470C22×C4—731no—
471C22×C4—731no—
472C22×C4—732no—
473C22×C4—731no—
474C22×C4—734no—
475C22×C4—734no—
476C22×C4—732no—
477C22×C4—734no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1464no—
481C22×C4—1468no—
482C22×C4—1468no—
483C22×C4—1468no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4C23×D7311yescharacteristic
487C22×C4C23×D7314yes—
488C22×C4C23×D7311yescharacteristic
489C22×C4C23×D7312yes—
490C22×C4C23×D7312yes—
491C22×C4C23×D7311yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—2924no—
494C4⋊C4—2924no—
495C4⋊C4—2924no—
496C4⋊C4—738no—
497C4⋊C4—7316no—
498C4⋊C4—738no—
499C4⋊C4—1464no—
500C4⋊C4—1464no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—2924no—
505C22⋊C4—2924no—
506C22⋊C4—2924no—
507C22⋊C4—7316no—
508C22⋊C4—7316no—
509C22⋊C4—738no—
510C22⋊C4—1468no—
511C22⋊C4—1464no—
512C42—22no—
513C42—2924no—
514C42—2924no—
515C42—2922no—
516C42—2922no—
517C42—734no—
518C42—734no—
519C42—1464no—
520C42—1468no—
521C42—1462no—
522C42—1462no—
523C42—1464no—
524C42—1462no—
525C42C2×C4×D7314yes—
526C42#1192402714yes—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—732no—
534C23—1461no—
535C23—1461no—
536C23—1462no—
537C23—1462no—
538C23C22×C4×D7318yes—
539C23C24×D7311yescharacteristic
540D4—216no—
541D4—416no—
542D4—29216no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—416no—
555C2×C4—2924no—
556C2×C4—2924no—
557C2×C4—29216no—
558C2×C4—29216no—
559C2×C4—732no—
560C2×C4—734no—
561C2×C4—734no—
562C2×C4—732no—
563C2×C4—738no—
564C2×C4—732no—
565C2×C4—1464no—
566C2×C4—1462no—
567C2×C4—1464no—
568C2×C4—1462no—
569C2×C4—1464no—
570C2×C4—1462no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1462no—
574C2×C4—1464no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1464no—
578C2×C4—1468no—
579C2×C4C22×C4×D7314yes—
580C2×C4C22×C4×D7312yes—
581C2×C4#41×D7312yes—
582C2×C4C2×#1192402714yes—
583C2×C4C2×#1192402712yes—
584C22—28no—
585C22—28no—
586C22—28no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—2921no—
592C22—2921no—
593C22—2922no—
594C22—1462no—
595C22—1464no—
596C22C23×C4×D7311yescharacteristic
597C22C2×#41×D7311yescharacteristic
598C22C22×#1192402711yescharacteristic
599C22H(5,𝔽2)×D7312yes—
600C22#1798942212yes—
601C4—24no—
602C4—24no—
603C4—41no—
604C4—41no—
605C4—44no—
606C4—2924no—
607C4—2922no—
608C4—734no—
609C4—1462no—
610C4—1462no—
611C4—1464no—
612C4—1468no—
613C4C4×#1192402714yes—
614C2—48no—
615C2—2922no—
616C2C2×C4×#1192402711yescharacteristic
617C2#2262313512yes—
618C2#540×D7311yescharacteristic
619C2#2262331511yescharacteristic
620C2#2262334412yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2328C2
socle, cosocleC23×C73C25
prime radicalC2×#17989191C2
hypercenter#2328D73
hypocenterC73C4×#540
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #22612722C2semidirect
3. #22612697C2semidirect
4. #22612672C2semidirect
5. #22612618C2semidirect
6. #22612568C2semidirect
7. #22612541C2semidirect
8. #127×Dic73C2semidirect
9. #22612486C2semidirect
10. #22612456C2semidirect
11. #22612447C2semidirect
12. #22612441C2semidirect
13. C4×D4×Dic73C2semidirect
14. #22612270C2semidirect
15. C73×#2328C2semidirect
16. #17989191C4semidirect
78. #14556583C2×C4semidirect
79. #14556572C2×C4semidirect
80. #14556563C2×C4semidirect
81. #14556562C2×C4semidirect
84. D4×Dic73C2×C4semidirect
87. #14556548C2×C4semidirect
129. #127×C73C2×C4semidirect
302. Dic73#540semidirect
315. #2328D73semidirect
326. C73C4×#540semidirect
392. #127C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891912
#226127222
#226126972
#226126722
#226126182
#226125682
#226125412
#127×Dic732
#226124862
#226124562
#226124472
#226124412
C4×D4×Dic732
#226122702
C73×#23282
C4×#54073