Finite Group Databasebeta

← #33105155

Subgroups — #33105155

Order 18688.

858 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×#17988678C211yesmaximal, characteristic
2#22612892C211yesmaximal, characteristic, complemented
3#22612836C211yesmaximal, characteristic
4#22610621C211yesmaximal, characteristic, complemented
5#22610619C211yesmaximal, characteristic, complemented
6#22610509C211yesmaximal, characteristic, complemented
7#22610507C211yesmaximal, characteristic, complemented
8#22610265C211yesmaximal, characteristic
9#22610263C211yesmaximal, characteristic
10#22610060C211yesmaximal, characteristic, complemented
11#22609945C211yesmaximal, characteristic, complemented
12#22609939C211yesmaximal, characteristic
13#22609799C211yesmaximal, characteristic, complemented
14#22609758C211yesmaximal, characteristic
15C73×#1762C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20#17988678C418yescomplemented
21C2×#14556522C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988493C2211yescharacteristic
30#17988492C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988471—21no—
34#17988469—21no—
35#17988460—21no—
36#17988459—21no—
37#17988454—21no—
38#17988452C2211yescharacteristic
39#17988452C2211yescharacteristic
40#17988445—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007C2211yescharacteristic
43#17988433—21no—
44C2×#14556498—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C23×Dic73—21no—
62C23×Dic73—21no—
63C2×#11924016—22no—
64C2×#11924016—22no—
65C2×#11924016C2×C414yes—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556556—22no—
69#14556556C2×C414yescomplemented
70D4×Dic73—22no—
71D4×Dic73—22no—
72D4×Dic73—41no—
73D4×Dic73C2×C414yes—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556522C2×C414yescomplemented
89#14556520—22no—
90#14556516—41no—
91#14556516C2×C414yescomplemented
92#14556514—41no—
93#14556512—22no—
94#14556501—41no—
95C8×Dic73C2×C414yescomplemented
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C23×C4×C73—21no—
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×D4×C73—22no—
114C4×D4×C73—41no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#102×C73—22no—
122#102×C73C2×C414yes—
123#97×C73—41no—
124#95×C73—21no—
125C2×Dic146—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146C22×C412yes—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—24no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—81no—
139C22×Dic73—81no—
140C22×Dic73—81no—
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145#11924016—28no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C22×C412yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73C22×C412yes—
158#11924007—24no—
159#11924007—41no—
160#11924007—41no—
161#11924007—81no—
162#11924007—81no—
163C22⋊Dic73—24no—
164C22⋊Dic73—42no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—24no—
168C4×Dic73—42no—
169C4×Dic73C22×C412yes—
170C4×Dic73#4112yes—
171C24×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73C22×C412yes—
176C2×D4×C73#4112yes—
177C2×D4×C73#4112yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—42no—
182C22×C4×C73—42no—
183C22×C4×C73—81no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C22×C412yes—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73C22×C412yes—
199C22⋊C4×C73—42no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—42no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73C22×D411yescharacteristic
211D4×C73—22no—
212D4×C73—22no—
213D4×C73—22no—
214D4×C73#11714yes—
215C2×C4×C73—22no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—84no—
231C2×C4×C73C23×C411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73H(5,𝔽2)11yescharacteristic
236C2×C4×C732−1+411yescharacteristic
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—84no—
260C2×Dic73—84no—
261C2×Dic73C4×D412yes—
262C2×Dic73C4×D412yes—
263Dic146—22no—
264Dic146—22no—
265Dic146—41no—
266Dic146—41no—
267C8×C73—22no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—41no—
271C22×C73—24no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73C2×C4×D411yescharacteristic
279C22×C73#54611yescharacteristic
280C22×C73#54811yescharacteristic
281C22×C73#57511yescharacteristic
282C22×C73#57511yescharacteristic
283C22×C73#57711yescharacteristic
284C22×C73#57711yescharacteristic
285C4×C73—42no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73—81no—
289C4×C73—81no—
290C4×C73C2×#11711yescharacteristic
291C4×C73#52011yescharacteristic
292C4×C73#55211yescharacteristic
293C4×C73#55311yescharacteristic
294Dic73—22no—
295Dic73—22no—
296Dic73—42no—
297Dic73—42no—
298Dic73—41no—
299Dic73—41no—
300Dic73—81no—
301Dic73—81no—
302#22760—731nomaximal, Sylow
303C2×C73—44no—
304C2×C73#235211yescharacteristic
305C2×C73#294311yescharacteristic
306C2×C73#294911yescharacteristic
307C2×C73#332311yescharacteristic
308C2×C73#332311yescharacteristic
309C2×C73#337611yescharacteristic
310C2×C73#337611yescharacteristic
311C2×C4×SD16—731no—
312#2356—731no—
313#2349—731no—
314#1935—731no—
315#1935—731no—
316#1919—731no—
317#1919—731no—
318#1877—731no—
319#1877—731no—
320#1835—731no—
321#1805—731no—
322#1804—731no—
323#1770—731no—
324#1762—731no—
325#1762D7311yescharacteristic
326C73#2276011yescharacteristic, complemented, Sylow
327C22×SD16—731no—
328C2×C4×Q8—731no—
329C2×C4×D4—731no—
330C2×C4×D4—731no—
331C2×C4×D4C2×D7311yescharacteristic
332C4×SD16—738no—
333C2×#106—731no—
334C2×C4⋊C8—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#102—731no—
339C2×#102C2×D7311yescharacteristic
340C2×#102C2×D7311yescharacteristic
341C2×#97—731no—
342C2×#97C2×D7311yescharacteristic
343C2×C4×C8—731no—
344#391—731no—
345#391—731no—
346#389—731no—
347#389—731no—
348#387—1461no—
349#387—1461no—
350#385—1461no—
351#385—1461no—
352#383—1461no—
353#382—731no—
354#382—731no—
355#380—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4C2×D7311yescharacteristic
359#376—1461no—
360C2×#95—1461no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340C2×D7311yescharacteristic
368#340C2×D7311yescharacteristic
369#336—731no—
370#336—731no—
371C22×Q8—731no—
372C22×D4C22×D7311yescharacteristic
373C23×C4—21no—
374C23×C4—1461no—
375C23×C4—1461no—
376C2×SD16—734no—
377C2×SD16—1462no—
378C2×SD16—1462no—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C4×Q8—734no—
384C4×Q8—1462no—
385C4×D4—22no—
386C4×D4—41no—
387C4×D4—2921no—
388C4×D4—734no—
389C4×D4—1462no—
390C4×D4—1462no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1461no—
404C2×C22⋊C4—1461no—
405C2×C42—21no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411#106—734no—
412C4⋊C8—1462no—
413#103—2921no—
414#103—2921no—
415#103—734no—
416#102—22no—
417#102—1462no—
418#102C4×D7314yes—
419#97—41no—
420#97—2921no—
421C4×C8—734no—
422#95—21no—
423#95—731no—
424#95—731no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—21no—
437C2×Q8—732no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C2×Q8—1461no—
442C2×Q8—1461no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D7312yes—
447C2×D4#1192402812yes—
448C2×D4#1192402812yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—5841no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1464no—
469C22×C4D4×D7311yescharacteristic
470C22×C4D4×D7311yescharacteristic
471C22×C4C23×D7311yescharacteristic
472SD16—1468no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8C2×C4×D7312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—2921no—
491C4⋊C4—2921no—
492C4⋊C4—5841no—
493C4⋊C4—5841no—
494C4⋊C4—732no—
495C4⋊C4—1464no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4C2×C4×D7312yes—
502C22⋊C4—42no—
503C22⋊C4—2922no—
504C22⋊C4—1464no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—2922no—
509C42—732no—
510C42—732no—
511C42—1464no—
512C42—1461no—
513C42—1461no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D7311yescharacteristic
522Q8—2921no—
523Q8—2921no—
524Q8—1462no—
525Q8—1462no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1455654814yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—5844no—
557C2×C4—5844no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4C22×C4×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4C2×#1192402811yescharacteristic
575C2×C4C2×#1192402811yescharacteristic
576C2×C4#1455663611yescharacteristic
577C2×C4#1455663811yescharacteristic
578C8—22no—
579C8—41no—
580C8—2921no—
581C8—1462no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C4×D4×D7311yescharacteristic
590C22#1798922711yescharacteristic
591C22#1798924611yescharacteristic
592C22#1798938311yescharacteristic
593C22#1798938311yescharacteristic
594C22#1798939611yescharacteristic
595C22#1798939711yescharacteristic
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—2922no—
602C4—2922no—
603C4—2921no—
604C4—2921no—
605C4—5841no—
606C4—5841no—
607C4—1462no—
608C4—1462no—
609C4C2×#1455654811yescharacteristic
610C4#1798906611yescharacteristic
611C4#1798927011yescharacteristic
612C4#1798928611yescharacteristic
613C2—44no—
614C2#2261285411yescharacteristic
615C2#2261872811yescharacteristic
616C2#2261878711yescharacteristic
617C2#2262214311yescharacteristic
618C2#2262214811yescharacteristic
619C2#2262275911yescharacteristic
620C2#2262276311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1762C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1762D73
hypocenterC73#22760
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #22612892C2semidirect
4. #22610621C2semidirect
5. #22610619C2semidirect
6. #22610509C2semidirect
7. #22610507C2semidirect
10. #22610060C2semidirect
11. #22609945C2semidirect
13. #22609799C2semidirect
20. #17988678C4semidirect
69. #14556556C2×C4semidirect
88. #14556522C2×C4semidirect
91. #14556516C2×C4semidirect
95. C8×Dic73C2×C4semidirect
326. C73#22760semidirect

Maximal subgroups

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

subgroupindex
C2×#179886782
#226128922
#226128362
#226106212
#226106192
#226105092
#226105072
#226102652
#226102632
#226100602
#226099452
#226099392
#226097992
#226097582
C73×#17622
#2276073