Finite Group Databasebeta

← #33246917

Subgroups — #33246917

Order 18688.

2858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989063C211yesmaximal, characteristic
2C2×#17989063C211yesmaximal, characteristic
3C2×C4×Q8×D73C211yesmaximal, characteristic
4#22612850C213yesmaximal, complemented
5#22612849C211yesmaximal, characteristic, complemented
6#22612848C213yesmaximal, complemented
7Q8×C4⋊Dic73C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic146C213yesmaximal, complemented
9#22612845C213yesmaximal, complemented
10Q8×#11924007C211yesmaximal, characteristic, complemented
11#22612556C213yesmaximal
12#22612553C213yesmaximal
13#22612412C213yesmaximal
14#22612411C213yesmaximal
15C73×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D73C2211yescharacteristic
17#17989063—28no—
18#17989063C4116yescomplemented
19C2×Q8×Dic73C2211yescharacteristic
20C2×Q8×Dic73C2211yescharacteristic
21C2×Q8×Dic73C2211yescharacteristic
22C2×C4×Dic146C2213yes—
23C2×C4×Dic146C2213yes—
24C2×C4×Dic146C2213yes—
25C2×#14556556C2213yes—
26C2×#14556556C2213yes—
27C2×#14556556C2213yes—
28C4×Q8×D73—28no—
29C2×#14556549C2213yes—
30C2×#14556549C2213yes—
31C2×#14556548C2213yes—
32C2×#14556548C2213yes—
33C2×C4⋊C4×D73C2213yes—
34C2×C42×D73C2213yes—
35#17988486C2213yes—
36#17988486C2213yes—
37#17988485C2213yes—
38#17988485C2213yes—
39#17988484C2216yes—
40#17988484C2216yes—
41#17988484C2213yes—
42#17988484C2213yes—
43#17988483C2213yes—
44#17988483C2213yes—
45#17988482C2213yes—
46#17988482C2213yes—
47#17988461C2216yes—
48#17988461C2213yes—
49#17988461C2211yescharacteristic
50#17988458C2213yes—
51#17988458C2213yes—
52#17988457C2213yes—
53#17988456C2216yes—
54#17988456C2213yes—
55#17988452C2213yes—
56#17988451C2216yes—
57C4⋊C4×Dic73C2216yes—
58C4×#11924007C2213yes—
59C4×#11924007C2213yes—
60C4×C4⋊Dic73C2213yes—
61#17988438C2216yes—
62C2×C4×Q8×C73C2211yescharacteristic
63C2×C4×Q8×C73C2211yescharacteristic
64C2×C4×Q8×C73C2211yescharacteristic
65#389×C73C2213yes—
66#389×C73C2213yes—
67#382×C73C2213yes—
68C4×C4⋊C4×C73C2213yes—
69C22×Dic146C2313yes—
70C2×Q8×D73C2×C418yes—
71C2×Q8×D73D418yes—
72C2×Q8×D73Q818yes—
73C22×C4×D73C2311yescharacteristic
74C22×C4×D73C2313yes—
75C22×C4×D73C2313yes—
76Q8×Dic73—24no—
77Q8×Dic73—24no—
78Q8×Dic73C2×C418yescomplemented
79C4×Dic146—212no—
80C4×Dic146—212no—
81C4×Dic146C2×C4124yescomplemented
82#14556556—212no—
83#14556556—212no—
84#14556556C2×C4124yescomplemented
85#14556549—212no—
86#14556549C2×C4124yes—
87#14556548—212no—
88#14556548C2×C4124yes—
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2313yes—
91C2×C4⋊Dic73C2313yes—
92C2×C4⋊Dic73C2313yes—
93C2×C4⋊Dic73C2316yes—
94C2×C4⋊Dic73C2313yes—
95C2×#11924007C2313yes—
96C2×#11924007C2311yescharacteristic
97C2×#11924007C2313yes—
98C2×#11924007C2316yes—
99C2×#11924007C2313yes—
100C2×#11924007C2316yes—
101C2×#11924007C2313yes—
102C2×#11924007C2313yes—
103C2×#11924007C2313yes—
104C2×#11924007C2313yes—
105C4⋊C4×D73—212no—
106C2×#11924005C2311yescharacteristic
107C2×#11924005C2313yes—
108C2×#11924005C2313yes—
109C2×#11924005C2311yescharacteristic
110C2×C4×Dic73C2313yes—
111C2×C4×Dic73C2316yes—
112C2×C4×Dic73C2313yes—
113C2×C4×Dic73C2313yes—
114C2×C4×Dic73C2313yes—
115C2×C4×Dic73C2316yes—
116C2×C4×Dic73C2313yes—
117C42×D73—212no—
118#14556498C2313yes—
119#14556498C2313yes—
120#14556498C2311yescharacteristic
121#14556498C2313yes—
122#14556498C2313yes—
123#14556498C2313yes—
124#14556498C2316yes—
125C22×Q8×C73C2311yescharacteristic
126C4×Q8×C73—24no—
127C4×Q8×C73—24no—
128C4×Q8×C73C2×C418yescomplemented
129C2×C4⋊C4×C73C2313yes—
130C2×C4⋊C4×C73C2313yes—
131C2×C4⋊C4×C73C2313yes—
132C2×C4⋊C4×C73C2313yes—
133C2×C4⋊C4×C73C2313yes—
134C2×C42×C73C2313yes—
135C2×C42×C73C2313yes—
136C2×C42×C73C2313yes—
137#95×C73C2311yescharacteristic
138#95×C73C2313yes—
139#95×C73C2316yes—
140C23×D73C2411yescharacteristic
141C2×Dic146C22×C4112yes—
142C2×Dic146C2×D4112yes—
143C2×Dic146C2×Q8112yes—
144Q8×D73C4⋊C4164yescomplemented
145C22×Dic73C2411yescharacteristic
146C22×Dic73C2413yes—
147C22×Dic73C2413yes—
148C22×Dic73C2411yescharacteristic
149C22×Dic73C2413yes—
150C22×Dic73C2411yescharacteristic
151C22×Dic73C2413yes—
152C2×C4×D73—22no—
153C2×C4×D73—22no—
154C2×C4×D73—22no—
155C2×C4×D73—26no—
156C2×C4×D73—43no—
157C2×C4×D73—43no—
158C2×C4×D73C22×C4112yes—
159C2×C4×D73C2×D4112yes—
160C2×C4×D73C2×Q8112yes—
161C4⋊Dic73—26no—
162C4⋊Dic73—26no—
163C4⋊Dic73—22no—
164C4⋊Dic73—43no—
165C4⋊Dic73—46no—
166C4⋊Dic73C22×C4112yes—
167#11924007—22no—
168#11924007—212no—
169#11924007—26no—
170#11924007—26no—
171#11924007—26no—
172#11924007—46no—
173#11924007—43no—
174#11924007—43no—
175#11924007C22×C4112yes—
176#11924007C22×C4112yes—
177#11924005—22no—
178#11924005—26no—
179#11924005C22×C414yes—
180#11924005C22×C4112yes—
181C4×Dic73—26no—
182C4×Dic73—26no—
183C4×Dic73—26no—
184C4×Dic73—26no—
185C4×Dic73—212no—
186C4×Dic73C22×C4112yes—
187C4×Dic73C22×C4124yes—
188C2×Q8×C73C22×C414yes—
189C2×Q8×C73C2×D414yes—
190C2×Q8×C73C2×Q814yes—
191C22×C4×C73C2413yes—
192C22×C4×C73C2413yes—
193C22×C4×C73C2411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C22×C4×C73C2413yes—
196C22×C4×C73C2411yescharacteristic
197C22×C4×C73C2413yes—
198C4⋊C4×C73—26no—
199C4⋊C4×C73—26no—
200C4⋊C4×C73—26no—
201C4⋊C4×C73—43no—
202C4⋊C4×C73C22×C4112yes—
203C42×C73—26no—
204C42×C73—26no—
205C42×C73C22×C4112yes—
206C23×C73C2511yescharacteristic
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—21no—
210C22×D73—21no—
211C22×D73C23×C412yes—
212C22×D73C22×D412yes—
213C22×D73C22×Q812yes—
214Q8×C73C2×C4⋊C4116yes—
215C2×C4×C73—21no—
216C2×C4×C73—23no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—23no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—23no—
223C2×C4×C73—23no—
224C2×C4×C73—23no—
225C2×C4×C73—23no—
226C2×C4×C73—21no—
227C2×C4×C73—23no—
228C2×C4×C73—21no—
229C2×C4×C73—23no—
230C2×C4×C73C23×C412yes—
231C2×C4×C73C23×C416yes—
232C2×C4×C73C23×C416yes—
233C2×C4×C73C22×D416yes—
234C2×C4×C73C22×Q816yes—
235C2×C4×C732−1+412yes—
236C2×Dic73—21no—
237C2×Dic73—23no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—23no—
242C2×Dic73—21no—
243C2×Dic73—23no—
244C2×Dic73—23no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—23no—
248C2×Dic73—23no—
249C2×Dic73—23no—
250C2×Dic73—23no—
251C2×Dic73C23×C412yes—
252C2×Dic73C23×C416yes—
253C2×Dic73C23×C416yes—
254C2×Dic73C22×D416yes—
255C2×Dic73C22×Q816yes—
256C2×Dic732−1+412yes—
257C4×D73—28no—
258C4×D73—412no—
259C4×D73C2×C4⋊C4148yes—
260Dic146C2×C4⋊C4148yes—
261C22×C73C24×C411yescharacteristic
262C22×C73C23×D411yescharacteristic
263C22×C73C23×Q811yescharacteristic
264C22×C73C2×2−1+411yescharacteristic
265C22×C73C2×2−1+411yescharacteristic
266C22×C73#58511yescharacteristic
267C22×C73#58511yescharacteristic
268C2×D73—22no—
269C2×D73—22no—
270C2×D73—22no—
271C2×D73—22no—
272C2×D73—22no—
273C2×D73—22no—
274C2×D73C22×C4⋊C414yes—
275C4×C73—22no—
276C4×C73—26no—
277C4×C73—22no—
278C4×C73—43no—
279C4×C73—43no—
280C4×C73—41no—
281C4×C73C22×C4⋊C4112yes—
282Dic73—26no—
283Dic73—22no—
284Dic73—22no—
285Dic73—41no—
286Dic73—43no—
287Dic73—43no—
288Dic73C22×C4⋊C4112yes—
289#34249—731nomaximal, Sylow
290C2×C73C23×C4⋊C411yescharacteristic
291C2×C73C4×2−1+411yescharacteristic
292C2×C73C4×2−1+411yescharacteristic
293C2×C73#348611yescharacteristic
294C2×C73#348711yescharacteristic
295C2×C73#350211yescharacteristic
296C2×C73#350211yescharacteristic
297D73—24no—
298C2×#519—731no—
299C2×#519—731no—
300C22×C4×Q8—731no—
301#2351—733no—
302#2351—733no—
303#2351—731no—
304#2351D7311yescharacteristic, complemented
305Q8×C4⋊C4—733no—
306Q8×C4⋊C4—733no—
307Q8×C4⋊C4—731no—
308Q8×C4⋊C4—731no—
309#2317—733no—
310#2317—733no—
311#2296—733no—
312#2296—733no—
313C73#3424911yescharacteristic, complemented, Sylow
314C23×Q8—731no—
315#519—7316no—
316#519—1468no—
317C2×C4×Q8—731no—
318C2×C4×Q8—731no—
319C2×C4×Q8—731no—
320C2×C4×Q8—733no—
321C2×C4×Q8—733no—
322C2×C4×Q8—733no—
323C2×C4×Q8—733no—
324C2×C4×Q8—733no—
325C2×C4×Q8—733no—
326C2×C4×Q8—1468no—
327C2×C4×Q8C2×D7311yescharacteristic
328C2×C4×Q8C2×D7311yescharacteristic
329C2×C4×Q8C2×D7311yescharacteristic
330C2×#117—733no—
331C2×#117—733no—
332C2×#117—733no—
333C2×#117—733no—
334C22×C4⋊C4—733no—
335C22×C42—733no—
336#389—733no—
337#389—733no—
338#389—733no—
339#389—733no—
340#389—733no—
341#389—733no—
342#389—733no—
343#389—733no—
344#389—736no—
345#389—736no—
346#389—733no—
347#389—733no—
348#389C2×D7313yes—
349#389C2×D7313yes—
350#385—733no—
351#385—733no—
352#385—736no—
353#385—733no—
354#385—731no—
355#384—733no—
356#384—736no—
357#384—733no—
358#382—733no—
359#382—736no—
360#382C2×D7313yes—
361C4×C4⋊C4—736no—
362C4×C4⋊C4—733no—
363C4×C4⋊C4—733no—
364C4×C4⋊C4—733no—
365C4×C4⋊C4—736no—
366C4×C4⋊C4C2×D7313yes—
367C22×Q8—738no—
368C22×Q8—738no—
369C22×Q8—738no—
370C22×Q8—733no—
371C22×Q8C22×D7311yescharacteristic
372C23×C4—733no—
373C23×C4—733no—
374C23×C4—731no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—738no—
378C4×Q8—7324no—
379C4×Q8—7324no—
380C4×Q8—14612no—
381C4×Q8—14612no—
382C4×Q8—14612no—
383C4×Q8—14612no—
384C4×Q8—1464no—
385C4×Q8—1464no—
386C4×Q8C4×D7318yescomplemented
387#117—7324no—
388#117—7324no—
389#117—14612no—
390#117—14612no—
391C2×C4⋊C4—733no—
392C2×C4⋊C4—733no—
393C2×C4⋊C4—736no—
394C2×C4⋊C4—736no—
395C2×C4⋊C4—733no—
396C2×C4⋊C4—733no—
397C2×C4⋊C4—733no—
398C2×C4⋊C4—733no—
399C2×C4⋊C4—736no—
400C2×C4⋊C4—733no—
401C2×C4⋊C4—733no—
402C2×C4⋊C4—733no—
403C2×C4⋊C4—733no—
404C2×C4⋊C4—733no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—14612no—
408C2×C4⋊C4C22×D7313yes—
409C2×C4⋊C4C22×D7313yes—
410C2×C4⋊C4C22×D7313yes—
411C2×C4⋊C4C22×D7313yes—
412C2×C4⋊C4C22×D7313yes—
413C2×C22⋊C4—733no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—733no—
417C2×C42—736no—
418C2×C42—733no—
419C2×C42—736no—
420C2×C42—733no—
421C2×C42—733no—
422C2×C42—733no—
423C2×C42—733no—
424C2×C42—14612no—
425C2×C42C22×D7313yes—
426C2×C42C22×D7313yes—
427C2×C42C22×D7313yes—
428#95—731no—
429#95—733no—
430#95—733no—
431#95—733no—
432#95—733no—
433#95—736no—
434#95—733no—
435#95C22×D7316yes—
436#95C22×D7311yescharacteristic
437#95C22×D7313yes—
438C24—731no—
439C2×Q8—7364no—
440C2×Q8—7312no—
441C2×Q8—7312no—
442C2×Q8—7312no—
443C2×Q8C2×C4×D7314yes—
444C2×Q8D4×D7314yes—
445C2×Q8Q8×D7314yes—
446C22×C4—2923no—
447C22×C4—2923no—
448C22×C4—733no—
449C22×C4—7312no—
450C22×C4—7312no—
451C22×C4—7312no—
452C22×C4—731no—
453C22×C4—733no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—733no—
457C22×C4—733no—
458C22×C4—1466no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4C23×D7313yes—
463C22×C4C23×D7313yes—
464C22×C4C23×D7311yescharacteristic
465C22×C4C23×D7313yes—
466C22×C4C23×D7311yescharacteristic
467C22×C4C23×D7311yescharacteristic
468C22×C4C23×D7313yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—26no—
472C4⋊C4—43no—
473C4⋊C4—2923no—
474C4⋊C4—2923no—
475C4⋊C4—2926no—
476C4⋊C4—2926no—
477C4⋊C4—2923no—
478C4⋊C4—7312no—
479C4⋊C4—7312no—
480C4⋊C4—7312no—
481C4⋊C4—1466no—
482C4⋊C4—14612no—
483C4⋊C4—1466no—
484C4⋊C4—1466no—
485C4⋊C4—1462no—
486C4⋊C4—1466no—
487C4⋊C4—1466no—
488C4⋊C4—1462no—
489C4⋊C4C2×C4×D73112yes—
490C22⋊C4—7312no—
491C22⋊C4—734no—
492C22⋊C4—1462no—
493C22⋊C4—1466no—
494C42—26no—
495C42—26no—
496C42—7324no—
497C42—7312no—
498C42—1466no—
499C42—1466no—
500C42—14612no—
501C42—1466no—
502C42—1466no—
503C42C2×C4×D73112yes—
504C23—732no—
505C23—732no—
506C23—732no—
507C23—1461no—
508C23—1461no—
509C23—1461no—
510C23—1461no—
511C23C24×D7311yescharacteristic
512Q8—7348no—
513Q8C4⋊C4×D73116yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—21no—
526C2×C4—23no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—29212no—
530C2×C4—736no—
531C2×C4—736no—
532C2×C4—736no—
533C2×C4—7348no—
534C2×C4—736no—
535C2×C4—732no—
536C2×C4—732no—
537C2×C4—1468no—
538C2×C4—1461no—
539C2×C4—1463no—
540C2×C4—1463no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1463no—
544C2×C4—1463no—
545C2×C4—1463no—
546C2×C4—1463no—
547C2×C4—1461no—
548C2×C4—1463no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1463no—
553C2×C4C22×C4×D7316yes—
554C2×C4C22×C4×D7312yes—
555C2×C4C22×C4×D7316yes—
556C2×C4C2×D4×D7316yes—
557C2×C4C2×Q8×D7316yes—
558C2×C4#1455663812yes—
559C22—734no—
560C22—1462no—
561C22—1462no—
562C22—1462no—
563C22—1462no—
564C22—1462no—
565C22—1462no—
566C22C23×C4×D7311yescharacteristic
567C22C22×D4×D7311yescharacteristic
568C22C22×Q8×D7311yescharacteristic
569C222−1+4×D7311yescharacteristic
570C22C2×#1455663811yescharacteristic
571C22#1798942311yescharacteristic
572C22#1798942411yescharacteristic
573C4—22no—
574C4—22no—
575C4—26no—
576C4—43no—
577C4—43no—
578C4—41no—
579C4—2921no—
580C4—2923no—
581C4—2923no—
582C4—7312no—
583C4—1462no—
584C4—1466no—
585C4—1462no—
586C4C2×C4⋊C4×D73112yes—
587C2—1464no—
588C2C22×C4⋊C4×D7311yescharacteristic
589C2#2262314011yescharacteristic
590C2C4×#1455663811yescharacteristic
591C2#2262357711yescharacteristic
592C2#2262358811yescharacteristic
593C2#2262373511yescharacteristic
594C2#2262373811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2351C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2351D73
hypocenterC73#34249
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #22612850C2semidirect
5. #22612849C2semidirect
6. #22612848C2semidirect
7. Q8×C4⋊Dic73C2semidirect
8. C4⋊C4×Dic146C2semidirect
9. #22612845C2semidirect
10. Q8×#11924007C2semidirect
15. C73×#2351C2semidirect
18. #17989063C4semidirect
78. Q8×Dic73C2×C4semidirect
81. C4×Dic146C2×C4semidirect
84. #14556556C2×C4semidirect
128. C4×Q8×C73C2×C4semidirect
144. Q8×D73C4⋊C4semidirect
304. #2351D73semidirect
313. C73#34249semidirect
386. C4×Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890632
C2×C4×Q8×D732
#226128502
#226128492
#226128482
Q8×C4⋊Dic732
C4⋊C4×Dic1462
#226128452
Q8×#119240072
#226125562
#226125532
#226124122
#226124112
C73×#23512
#3424973