Finite Group Databasebeta

← #33189078

Subgroups — #33189078

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618313C211yesmaximal, characteristic, complemented
2#22611433C211yesmaximal, characteristic, complemented
3#22611323C211yesmaximal, characteristic, complemented
4#22611319C211yesmaximal, characteristic
5#22611248C211yesmaximal, characteristic
6#22610784C211yesmaximal, characteristic
7#22610661C211yesmaximal, characteristic, complemented
8#22610617C211yesmaximal, characteristic
9#22610606C211yesmaximal, characteristic, complemented
10#22610572C211yesmaximal, characteristic, complemented
11#22610509C211yesmaximal, characteristic, complemented
12#22610265C211yesmaximal, characteristic
13#22610178C211yesmaximal, characteristic
14C4×C4⋊#10596831C211yesmaximal, characteristic, complemented
15C73×#2293C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556524C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988493C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988471—21no—
37#17988471—21no—
38#17988471—21no—
39#17988455—21no—
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73C2211yescharacteristic
49C2×#117×C73—21no—
50#390×C73C2211yescharacteristic
51#390×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#388×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73—21no—
58C43×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924016—22no—
61C2×#11924016—22no—
62C2×#11924016—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73#14556524—22no—
74#14556524—41no—
75#14556522—22no—
76C4⋊#10596831—22no—
77C4⋊#10596831—22no—
78#14556514—22no—
79#14556514—22no—
80#14556514—41no—
81#14556512—22no—
82#14556512—22no—
83#14556512—41no—
84C4×#10596831—41no—
85C4×#10596831—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95C22×D4×C73C2311yescharacteristic
96C23×C4×C73—21no—
97C4×D4×C73—22no—
98C4×D4×C73—22no—
99#117×C73—44no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C42×C73—21no—
111C2×C42×C73D411yescharacteristic
112C2×C42×C73D411yescharacteristic
113C2×C42×C73D411yescharacteristic
114C2×C42×C73D411yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—21no—
126C2×Dic146—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131#11924016—44no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831#4112yes—
143C2×#10596831#4112yes—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150#11924007—42no—
151#11924007—42no—
152#11924007—42no—
153C24×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73#4112yes—
160C22×C4×C73—24no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—42no—
165C22×C4×C73—42no—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—42no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73#4112yes—
181C22⋊C4×C73—24no—
182C22⋊C4×C73—42no—
183C22⋊C4×C73—42no—
184C22⋊C4×C73—42no—
185C22⋊C4×C73—42no—
186C42×C73—22no—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73—22no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—42no—
197C23×C73—22no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73C22×D411yescharacteristic
205D4×C73—22no—
206D4×C73—22no—
207D4×C73—41no—
208D4×C73—41no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—48no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73#13512yes—
234C2×C4×C73#13512yes—
235C2×C4×C73AGL(1,ℤ/8)12yes—
236C2×C4×C73#13712yes—
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73H(5,𝔽2)11yescharacteristic
242C2×C4×C732−1+411yescharacteristic
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255Dic146—41no—
256Dic146—41no—
257Dic146—41no—
258Dic146—41no—
259#10596831—22no—
260#10596831—22no—
261#10596831—41no—
262#10596831—41no—
263C22×C73—42no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73#52511yescharacteristic
271C22×C73#53211yescharacteristic
272C22×C73#53711yescharacteristic
273C22×C73C2×#13511yescharacteristic
274C22×C73C2×#13511yescharacteristic
275C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C73C2×#13711yescharacteristic
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—42no—
282C4×C73—41no—
283C4×C73—41no—
284C4×C73—81no—
285C4×C73—81no—
286C4×C73#55111yescharacteristic
287C4×C73#55211yescharacteristic
288C4×C73#55311yescharacteristic
289C4×C73#55611yescharacteristic
290Dic73—81no—
291Dic73—81no—
292Dic73—81no—
293Dic73—81no—
294#29433—731nomaximal, Sylow
295C2×C73—82no—
296C2×C73#260111yescharacteristic
297C2×C73#310511yescharacteristic
298C2×C73#310711yescharacteristic
299C2×C73#316811yescharacteristic
300C2×C73#318211yescharacteristic
301C2×C73#324711yescharacteristic
302C2×C73#324811yescharacteristic
303#2863—731no—
304#2293D7311yescharacteristic
305#2088—731no—
306#2068—731no—
307#2067—731no—
308#2055—731no—
309#1968—731no—
310#1942—731no—
311#1934—731no—
312#1933—731no—
313#1928—731no—
314#1919—731no—
315#1877—731no—
316#1862—731no—
317C4×C4⋊C8—731no—
318C73#2943311yescharacteristic, complemented, Sylow
319C22×SD16—731no—
320C2×C4×D4C2×D7311yescharacteristic
321C2×#117—21no—
322C2×#107—731no—
323C2×#107—731no—
324C2×#106—731no—
325C2×C4⋊C8—731no—
326C2×C4⋊C8—731no—
327C2×#103—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×#102—731no—
331C2×#102—731no—
332C2×#102—731no—
333C2×C4×C8—731no—
334C2×C4×C8—731no—
335#391—731no—
336#391—731no—
337#391—731no—
338#390C2×D7311yescharacteristic
339#390C2×D7311yescharacteristic
340#389—731no—
341#389—731no—
342#389—731no—
343#389C2×D7311yescharacteristic
344#389C2×D7311yescharacteristic
345#388—21no—
346#388—21no—
347#387—1461no—
348#387—1461no—
349#387—1461no—
350#383—1461no—
351C4×C4⋊C4C2×D7311yescharacteristic
352C4×C22⋊C4—21no—
353C43C2×D7311yescharacteristic
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#336—731no—
361#336—731no—
362C22×Q8—731no—
363C22×D4C22×D7311yescharacteristic
364C23×C4—21no—
365C2×SD16—2921no—
366C2×SD16—1462no—
367C2×SD16—1462no—
368C22×C8—731no—
369C22×C8—731no—
370C22×C8—731no—
371C22×C8—731no—
372C4×D4—22no—
373C4×D4—22no—
374#117—44no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42C22⋊D7311yescharacteristic
393C2×C42C22⋊D7311yescharacteristic
394C2×C42C22⋊D7311yescharacteristic
395C2×C42C22⋊D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7311yescharacteristic
399#107—2921no—
400#107—1462no—
401#106—1462no—
402C4⋊C8—1462no—
403C4⋊C8—1462no—
404#103—2921no—
405#103—1462no—
406#103—1462no—
407#102—2921no—
408#102—1462no—
409#102—1462no—
410C4×C8—2921no—
411C4×C8—2921no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424C24—21no—
425C2×Q8—1461no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8—1461no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4#1192402812yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×C22⋊D7311yescharacteristic
451C22×C4C2×C22⋊D7311yescharacteristic
452C22×C4C2×C22⋊D7311yescharacteristic
453C22×C4C23×D7311yescharacteristic
454SD16—2924no—
455C2×C8—732no—
456C2×C8—732no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4#1192402812yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D7311yescharacteristic
508Q8—2921no—
509Q8—2921no—
510Q8—2921no—
511Q8—2921no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4#1455661112yes—
553C2×C4#1455661112yes—
554C2×C4#1455661712yes—
555C2×C4#1455662112yes—
556C2×C4#41×D7311yescharacteristic
557C2×C4#41×D7311yescharacteristic
558C2×C4C2×#1192402811yescharacteristic
559C2×C4C2×#1192402811yescharacteristic
560C2×C4#1455663711yescharacteristic
561C2×C4#1455663811yescharacteristic
562C8—2921no—
563C8—2921no—
564C8—1462no—
565C8—1462no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1798909811yescharacteristic
574C22#1798913311yescharacteristic
575C22#1798916511yescharacteristic
576C22C2×#1455661111yescharacteristic
577C22C2×#1455661111yescharacteristic
578C22C2×#1455661711yescharacteristic
579C22C2×#1455662111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—5841no—
590C4—5841no—
591C4—5841no—
592C4—5841no—
593C4#1798926111yescharacteristic
594C4#1798927011yescharacteristic
595C4#1798928111yescharacteristic
596C4#1798930511yescharacteristic
597C2—82no—
598C2#2261552811yescharacteristic
599C2#2261997811yescharacteristic
600C2#2262000211yescharacteristic
601C2#2262049511yescharacteristic
602C2#2262064511yescharacteristic
603C2#2262125111yescharacteristic
604C2#2262126811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2293C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2293D73
hypocenterC73#29433
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #22618313C2semidirect
2. #22611433C2semidirect
3. #22611323C2semidirect
7. #22610661C2semidirect
9. #22610606C2semidirect
10. #22610572C2semidirect
11. #22610509C2semidirect
14. C4×C4⋊#10596831C2semidirect
318. C73#29433semidirect

Maximal subgroups

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

subgroupindex
#226183132
#226114332
#226113232
#226113192
#226112482
#226107842
#226106612
#226106172
#226106062
#226105722
#226105092
#226102652
#226101782
C4×C4⋊#105968312
C73×#22932
#2943373