Finite Group Databasebeta

← #33052305

Subgroups — #33052305

Order 18688.

916 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989309C211yesmaximal, characteristic
2#22619020C211yesmaximal, characteristic, complemented
3#22608301C212yesmaximal, complemented
4#22608298C212yesmaximal, complemented
5#22608284C211yesmaximal, characteristic
6#22608282C211yesmaximal, characteristic
7#22608274C211yesmaximal, characteristic, complemented
8#22608263C212yesmaximal, complemented
9#22608254C211yesmaximal, characteristic, complemented
10#22608243C212yesmaximal, complemented
11C73×#2984C211yesmaximal, characteristic, complemented
12#17989309—21no—
13#17989309—22no—
14#17989309—21no—
15#17989309—21no—
16#17989309—22no—
17#17989309—21no—
18C2×#14556589C411yescharacteristic
19C2×#14556586C412yes—
20C2×#14556585C2212yes—
21C2×#14556580C411yescharacteristic
22C2×#14556579C2211yescharacteristic
23C2×#14556569C412yes—
24C2×C4×Dic146C2211yescharacteristic
25C2×#14556556C411yescharacteristic
26C2×#14556553C411yescharacteristic
27C2×#14556548C2212yes—
28#17988718—22no—
29#17988718—21no—
30Q8×#10596831—21no—
31#17988644C2212yes—
32#17988644C2211yescharacteristic
33#17988640C2212yes—
34#17988640C2211yescharacteristic
35#17988312C2211yescharacteristic
36#17988312C2212yes—
37#17988312C2211yescharacteristic
38#17988312C2211yescharacteristic
39#17988312C2211yescharacteristic
40#17988312C2212yes—
41#17988310C2212yes—
42#17988310C2212yes—
43#17988310C2212yes—
44#17988310C2212yes—
45C2×C4×Q8×C73C2211yescharacteristic
46#446×C73—22no—
47#446×C73—21no—
48C8×Q8×C73—21no—
49#432×C73C2212yes—
50#432×C73C2211yescharacteristic
51#431×C73C2212yes—
52#431×C73C2211yescharacteristic
53C22×Dic146C2×C411yescharacteristic
54C22×Dic146C2×C411yescharacteristic
55C22×D292C2×C411yescharacteristic
56C22×C4×D73C2×C412yes—
57#14556589—21no—
58#14556589—21no—
59#14556589—22no—
60#14556586—22no—
61#14556586—22no—
62#14556586—22no—
63#14556586—22no—
64#14556585—22no—
65#14556585—22no—
66#14556585—22no—
67#14556585—22no—
68#14556580—21no—
69#14556580—21no—
70#14556580—22no—
71#14556579—21no—
72#14556579—21no—
73#14556579—21no—
74#14556579—21no—
75#14556569—22no—
76#14556569—22no—
77#14556569—22no—
78#14556569—22no—
79C4×Dic146—21no—
80C4×Dic146—21no—
81C4×Dic146—21no—
82C4×Dic146—21no—
83#14556556—22no—
84#14556556—21no—
85#14556556—21no—
86#14556553—22no—
87#14556553—21no—
88#14556553—21no—
89#14556548—22no—
90#14556548—22no—
91#14556548—22no—
92#14556548—22no—
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×#11924007C2×C412yes—
95C2×#11924007C2×C412yes—
96C2×#11924007C2×C412yes—
97C2×#11924007C2311yescharacteristic
98C2×#11924007C2311yescharacteristic
99C2×#11924005C2×C412yes—
100C2×#11924005C2×C412yes—
101C2×#11924005C2×C412yes—
102C2×#11924005C2×C412yes—
103C2×#11924005C2312yes—
104C2×C4×Dic73C2×C411yescharacteristic
105C2×C4×Dic73C2×C412yes—
106C2×C4×Dic73C2×C411yescharacteristic
107C4⋊#10596831—22no—
108C4⋊#10596831—21no—
109C4⋊#10596831—22no—
110C4⋊#10596831—21no—
111C22⋊#10596831C2311yescharacteristic
112C22⋊#10596831C2312yes—
113C22⋊#10596831C2311yescharacteristic
114#14556502—22no—
115#14556502—21no—
116C4×#10596831—22no—
117C4×#10596831—21no—
118C22×Q8×C73C2×C411yescharacteristic
119C4×Q8×C73D411yescharacteristic
120C4×Q8×C73D412yes—
121C4×Q8×C73D411yescharacteristic
122C4×Q8×C73D411yescharacteristic
123C4×Q8×C73D412yes—
124C4×Q8×C73D411yescharacteristic
125C2×C4⋊C4×C73C2×C411yescharacteristic
126C2×C4⋊C4×C73C2×C412yes—
127C2×C42×C73C2312yes—
128C2×C42×C73C2311yescharacteristic
129C4⋊C8×C73—22no—
130C4⋊C8×C73—21no—
131C4⋊C8×C73—22no—
132C4⋊C8×C73—21no—
133C22⋊C8×C73C2311yescharacteristic
134C22⋊C8×C73C2312yes—
135C22⋊C8×C73C2311yescharacteristic
136#97×C73—22no—
137#97×C73—21no—
138C4×C8×C73—22no—
139C4×C8×C73—21no—
140C23×D73C22×C412yes—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—22no—
146C2×Dic146—41no—
147C2×Dic146—42no—
148C2×D292—21no—
149C2×D292—21no—
150C2×D292—42no—
151C22×Dic73C22×C411yescharacteristic
152C22×Dic73C22×C411yescharacteristic
153C22×Dic73C22×C411yescharacteristic
154C22×Dic73C22×C411yescharacteristic
155C22×Dic73C22×C412yes—
156C2×C4×D73—22no—
157C2×C4×D73—22no—
158C2×C4×D73—24no—
159C2×C4×D73—42no—
160C2×#10596831—21no—
161C2×#10596831—22no—
162C2×#10596831—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165#11924007—22no—
166#11924007—22no—
167#11924007—22no—
168#11924007—22no—
169#11924007—22no—
170#11924007—22no—
171#11924007—21no—
172#11924007—21no—
173#11924007—21no—
174#11924007—21no—
175#11924005—22no—
176#11924005—22no—
177#11924005—22no—
178#11924005—22no—
179#11924005—22no—
180#11924005—22no—
181#11924005—22no—
182#11924005—22no—
183#11924005—22no—
184#11924005—22no—
185C4×Dic73—22no—
186C4×Dic73—22no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C2×Q8×C73—42no—
192C2×Q8×C73C22⋊C411yescharacteristic
193C2×Q8×C73C22⋊C411yescharacteristic
194C2×Q8×C73C22⋊C412yes—
195C22×C4×C73C22×C412yes—
196C22×C4×C73C22×C411yescharacteristic
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C22×C412yes—
199C22×C4×C73C2411yescharacteristic
200C2×C8×C73—21no—
201C2×C8×C73—22no—
202C2×C8×C73—21no—
203C4⋊C4×C73C22⋊C412yes—
204C4⋊C4×C73C22⋊C411yescharacteristic
205C4⋊C4×C73C22⋊C411yescharacteristic
206C4⋊C4×C73C22⋊C412yes—
207C4⋊C4×C73C22⋊C412yes—
208C4⋊C4×C73C22⋊C412yes—
209C4⋊C4×C73C22⋊C412yes—
210C42×C73C2×D412yes—
211C42×C73C2×D411yescharacteristic
212C42×C73C2×D412yes—
213C42×C73C2×D411yescharacteristic
214C42×C73C2×D412yes—
215C42×C73C2×D411yescharacteristic
216C42×C73C2×D412yes—
217C42×C73C2×D411yescharacteristic
218C23×C73C23×C411yescharacteristic
219C22×D73—22no—
220C22×D73—22no—
221C22×D73—22no—
222C22×D73—42no—
223C22×D73—42no—
224Q8×C73—22no—
225Q8×C73—22no—
226Q8×C73—42no—
227C2×C4×C73—22no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—42no—
233C2×C4×C73—41no—
234C2×C4×C73C2×C22⋊C412yes—
235C2×C4×C73C2×C22⋊C412yes—
236C2×C4×C73C2×C22⋊C411yescharacteristic
237C2×C4×C73C2×C22⋊C411yescharacteristic
238C2×C4×C73C2×C22⋊C411yescharacteristic
239C2×C4×C73C2×C22⋊C412yes—
240C2×C4×C73C2×C22⋊C412yes—
241C2×C4×C73C2×C22⋊C411yescharacteristic
242C2×C4×C73C22×D411yescharacteristic
243C2×C4×C73C22×D411yescharacteristic
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—22no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—42no—
255C2×Dic73#13112yes—
256D292—42no—
257D292—42no—
258C4×D73—44no—
259C4×D73—44no—
260Dic146—42no—
261Dic146—42no—
262Dic146—42no—
263Dic146—42no—
264C8×C73—41no—
265C8×C73—42no—
266C8×C73—41no—
267#10596831—41no—
268#10596831—42no—
269#10596831—41no—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73C22×C22⋊C411yescharacteristic
273C22×C73C2×#13111yescharacteristic
274C22×C73#56811yescharacteristic
275C2×D73—42no—
276C2×D73—42no—
277C2×D73—42no—
278C2×D73—82no—
279C2×D73—82no—
280C4×C73—21no—
281C4×C73—21no—
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—22no—
288C4×C73—22no—
289C4×C73#41312yes—
290Dic73—22no—
291Dic73—41no—
292Dic73—41no—
293Dic73—41no—
294Dic73—42no—
295#17428—731nomaximal, Sylow
296C2×C73—21no—
297C2×C73—21no—
298C2×C73#288011yescharacteristic
299C2×C73#288411yescharacteristic
300C2×C73C2×#41311yescharacteristic
301D73—82no—
302C2×#556—731no—
303#2984—731no—
304#2984D7311yescharacteristic, complemented
305#1473—732no—
306#1473—732no—
307#1469—731no—
308#1469—731no—
309#1467—731no—
310#1464—732no—
311#1461—731no—
312#1458—732no—
313C73#1742811yescharacteristic, complemented, Sylow
314#556—1461no—
315#556—1462no—
316#556—1461no—
317#556—1461no—
318#556—1462no—
319#556—1461no—
320C2×C4⋊Q8—731no—
321C2×#126—732no—
322C2×#126—732no—
323C2×#124—731no—
324C2×#124—731no—
325C2×C22⋊Q8—732no—
326C2×C4×Q8—731no—
327C2×C4×Q8—731no—
328C2×C4×Q8C2×D7311yescharacteristic
329C2×C4×D4—731no—
330C2×#117—732no—
331#446—22no—
332#446—21no—
333#446—1462no—
334#446—1461no—
335C8×Q8—21no—
336C8×Q8—1461no—
337#432—732no—
338#432—731no—
339#432C2×D7312yes—
340#432C2×D7311yescharacteristic
341#431—732no—
342#431—731no—
343#431C2×D7312yes—
344#431C2×D7311yescharacteristic
345#324—731no—
346#324—732no—
347#324—731no—
348#324—731no—
349#324—731no—
350#324—732no—
351#323—732no—
352#323—732no—
353#323—732no—
354#323—732no—
355C22×Q8—731no—
356C22×Q8—731no—
357C22×Q8C4×D7311yescharacteristic
358C22×D4—731no—
359C23×C4—732no—
360C4⋊Q8—1461no—
361C4⋊Q8—1461no—
362C4⋊Q8—1462no—
363#126—1462no—
364#126—1462no—
365#126—1462no—
366#126—1462no—
367#126—1462no—
368#126—1462no—
369#126—1462no—
370#126—1462no—
371#124—1461no—
372#124—1461no—
373#124—1462no—
374#124—1461no—
375#124—1461no—
376#124—1461no—
377#124—1461no—
378C22⋊Q8—1462no—
379C22⋊Q8—1462no—
380C22⋊Q8—1462no—
381C22⋊Q8—1462no—
382C4×Q8—1462no—
383C4×Q8—1461no—
384C4×Q8—1461no—
385C4×Q8—1461no—
386C4×Q8—1461no—
387C4×Q8—1461no—
388C4×Q8—1461no—
389C4×Q8D29211yescharacteristic
390C4×Q8D29212yes—
391C4×Q8D29211yescharacteristic
392C4×Q8C22⋊D7311yescharacteristic
393C4×Q8C22⋊D7312yes—
394C4×Q8C22⋊D7311yescharacteristic
395C4×D4—1462no—
396C4×D4—1461no—
397C4×D4—1461no—
398#117—1462no—
399#117—1462no—
400#117—1462no—
401#117—1462no—
402C2×C4⋊C4—732no—
403C2×C4⋊C4—732no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—732no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4C4×D7311yescharacteristic
409C2×C4⋊C4C4×D7312yes—
410C2×C22⋊C4—732no—
411C2×C22⋊C4—732no—
412C2×C22⋊C4—732no—
413C2×C22⋊C4—732no—
414C2×C22⋊C4—732no—
415C2×C42—732no—
416C2×C42—731no—
417C2×C42—731no—
418C2×C42C22×D7312yes—
419C2×C42C22×D7311yescharacteristic
420C4⋊C8—22no—
421C4⋊C8—21no—
422C4⋊C8—22no—
423C4⋊C8—21no—
424C4⋊C8—1462no—
425C4⋊C8—1461no—
426C4⋊C8—1462no—
427C4⋊C8—1461no—
428C22⋊C8—731no—
429C22⋊C8—732no—
430C22⋊C8—731no—
431C22⋊C8C22×D7311yescharacteristic
432C22⋊C8C22×D7312yes—
433C22⋊C8C22×D7311yescharacteristic
434#97—22no—
435#97—21no—
436#97—1462no—
437#97—1461no—
438C4×C8—22no—
439C4×C8—21no—
440C4×C8—1462no—
441C4×C8—1461no—
442C24—732no—
443C2×Q8—42no—
444C2×Q8—2922no—
445C2×Q8—2921no—
446C2×Q8—1462no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8#1192400511yescharacteristic
452C2×Q8#1192400511yescharacteristic
453C2×Q8#1192400512yes—
454C2×D4—2922no—
455C2×D4—1461no—
456C2×D4—1461no—
457C22×C4—2922no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—732no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1464no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4C2×C4×D7311yescharacteristic
467C22×C4C2×C4×D7312yes—
468C22×C4C2×C4×D7311yescharacteristic
469C22×C4C2×C4×D7312yes—
470C22×C4C23×D7311yescharacteristic
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—1461no—
475C2×C8—1462no—
476C2×C8—1461no—
477C4⋊C4—1462no—
478C4⋊C4—1462no—
479C4⋊C4—1462no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4#1192400512yes—
490C4⋊C4#1192400511yescharacteristic
491C4⋊C4#1192400511yescharacteristic
492C4⋊C4#1192400512yes—
493C4⋊C4#1192400512yes—
494C4⋊C4#1192400512yes—
495C4⋊C4#1192400512yes—
496C22⋊C4—1462no—
497C22⋊C4—1462no—
498C22⋊C4—1462no—
499C22⋊C4—1462no—
500C22⋊C4—1462no—
501C22⋊C4—1462no—
502C22⋊C4—1462no—
503C22⋊C4—1462no—
504C22⋊C4—1462no—
505C22⋊C4—1462no—
506C42—1462no—
507C42—1462no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C42C2×C22⋊D7312yes—
513C42C2×C22⋊D7311yescharacteristic
514C42C2×C22⋊D7312yes—
515C42C2×C22⋊D7311yescharacteristic
516C42C2×D29212yes—
517C42C2×D29211yescharacteristic
518C42C2×D29212yes—
519C42C2×D29211yescharacteristic
520C23—2922no—
521C23—2922no—
522C23—1462no—
523C23—1462no—
524C23—1462no—
525C23C22×C4×D7311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—2922no—
530Q8—2922no—
531Q8—2922no—
532Q8—2922no—
533D4—2922no—
534D4—2922no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—2924no—
543C2×C4—2924no—
544C2×C4—2922no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1462no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4C2×#1192400511yescharacteristic
557C2×C4C2×#1192400512yes—
558C2×C4C2×#1192400512yes—
559C2×C4C2×#1192400512yes—
560C2×C4C2×#1192400512yes—
561C2×C4C2×#1192400511yescharacteristic
562C2×C4C2×#1192400511yescharacteristic
563C2×C4C2×#1192400511yescharacteristic
564C2×C4C22×C22⋊D7311yescharacteristic
565C2×C4C22×D29211yescharacteristic
566C8—41no—
567C8—42no—
568C8—41no—
569C8—2921no—
570C8—2922no—
571C8—2921no—
572C22—21no—
573C22—21no—
574C22—2922no—
575C22—2922no—
576C22—2922no—
577C22—5842no—
578C22—5842no—
579C22C22×#1192400511yescharacteristic
580C22#131×D7311yescharacteristic
581C22#1798935211yescharacteristic
582C4—22no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—2921no—
592C4—2921no—
593C4—2921no—
594C4—2922no—
595C4—1462no—
596C4#1798856912yes—
597C2—21no—
598C2—21no—
599C2—5842no—
600C2#2261837811yescharacteristic
601C2#2261839411yescharacteristic
602C2C2×#1798856911yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2984C2
socleC22×C73C22×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC22×C4×C73C24
hypercenter#2984D73
hypocenterC73#17428
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #22619020C2semidirect
3. #22608301C2semidirect
4. #22608298C2semidirect
7. #22608274C2semidirect
8. #22608263C2semidirect
9. #22608254C2semidirect
10. #22608243C2semidirect
11. C73×#2984C2semidirect
304. #2984D73semidirect
313. C73#17428semidirect

Maximal subgroups

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

subgroupindex
C2×#179893092
#226190202
#226083012
#226082982
#226082842
#226082822
#226082742
#226082632
#226082542
#226082432
C73×#29842
#1742873