Finite Group Databasebeta

← #33189095

Subgroups — #33189095

Order 18688.

724 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
1#22612408C211yesmaximal, characteristic, complemented
2#22611432C211yesmaximal, characteristic, complemented
3#22611309C211yesmaximal, characteristic, complemented
4#22611305C211yesmaximal, characteristic
5#22611248C211yesmaximal, characteristic
6#22610784C211yesmaximal, characteristic
7#22610656C211yesmaximal, characteristic, complemented
8#22610617C211yesmaximal, characteristic
9#22610606C211yesmaximal, characteristic, complemented
10#22610555C211yesmaximal, characteristic
11#22610506C211yesmaximal, characteristic, complemented
12#22610289C211yesmaximal, characteristic
13#22610198C211yesmaximal, characteristic, complemented
14C4×C4⋊#10596831C211yesmaximal, characteristic, complemented
15C73×#2860C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988471—21no—
36#17988471—21no—
37C4×#11924007—21no—
38C4×C4⋊Dic73C2211yescharacteristic
39#17988433—21no—
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#126×C73—21no—
49#390×C73C2211yescharacteristic
50#390×C73C2211yescharacteristic
51#390×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#388×C73—21no—
56#388×C73—21no—
57#388×C73—21no—
58C43×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924016—22no—
61C2×#11924016—22no—
62C2×#11924016—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C4×Dic146—22no—
68C4×Dic146—22no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73—21no—
77#14556522—22no—
78#14556522—22no—
79#14556522—41no—
80C4⋊#10596831—22no—
81C4⋊#10596831—22no—
82#14556514—22no—
83#14556514—22no—
84#14556514—41no—
85#14556512—22no—
86#14556512—22no—
87#14556512—41no—
88C4×#10596831—41no—
89C4×#10596831—41no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×D4×C73C2311yescharacteristic
98#126×C73—44no—
99C2×C4⋊C4×C73—21no—
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×C22⋊C4×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#4112yes—
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#4112yes—
150#11924007—24no—
151#11924007—42no—
152#11924007—42no—
153C4×Dic73—24no—
154C4×Dic73—42no—
155C24×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2411yescharacteristic
172C4⋊C4×C73—21no—
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—42no—
181C22⋊C4×C73—42no—
182C22⋊C4×C73—42no—
183C22⋊C4×C73—42no—
184C22⋊C4×C73—42no—
185C22⋊C4×C73—42no—
186C22⋊C4×C73—42no—
187C42×C73—22no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—41no—
195C42×C73—41no—
196C42×C73SD1614yescomplemented
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73C22×D411yescharacteristic
205D4×C73—41no—
206D4×C73—41no—
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—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
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—41no—
231C2×C4×C73—41no—
232C2×C4×C73C2×SD1612yes—
233C2×C4×C73C2×SD1612yes—
234C2×C4×C73C2×SD1612yes—
235C2×C4×C73#13512yes—
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73H(5,𝔽2)11yescharacteristic
241C2×C4×C732−1+411yescharacteristic
242C2×Dic73—22no—
243C2×Dic73—22no—
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—
254Dic146—22no—
255Dic146—22no—
256Dic146—41no—
257Dic146—41no—
258#10596831—22no—
259#10596831—22no—
260#10596831—41no—
261#10596831—41no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73#52511yescharacteristic
270C22×C73#53211yescharacteristic
271C22×C73#53711yescharacteristic
272C22×C73C22×SD1611yescharacteristic
273C22×C73C2×#13511yescharacteristic
274C22×C73#57511yescharacteristic
275C22×C73#57511yescharacteristic
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73#55111yescharacteristic
286C4×C73#55211yescharacteristic
287C4×C73#55311yescharacteristic
288C4×C73#55611yescharacteristic
289Dic73—42no—
290Dic73—42no—
291Dic73—81no—
292Dic73—81no—
293#29434—731nomaximal, Sylow
294C2×C73—161no—
295C2×C73#260111yescharacteristic
296C2×C73#310211yescharacteristic
297C2×C73#310311yescharacteristic
298C2×C73#316711yescharacteristic
299C2×C73#318311yescharacteristic
300C2×C73#324611yescharacteristic
301C2×C73#325011yescharacteristic
302#2860D7311yescharacteristic
303#2295—731no—
304#2088—731no—
305#2066—731no—
306#2065—731no—
307#2055—731no—
308#1968—731no—
309#1941—731no—
310#1934—731no—
311#1933—731no—
312#1926—731no—
313#1919—731no—
314#1880—731no—
315#1865—731no—
316C4×C4⋊C8—731no—
317C73#2943411yescharacteristic, complemented, Sylow
318C22×SD16—731no—
319C2×#126—21no—
320C2×C4×Q8—731no—
321C2×#106—731no—
322C2×#106—731no—
323C2×#106—731no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8—731no—
326C2×#103—731no—
327C2×#103—731no—
328C2×#103—731no—
329C2×#102—731no—
330C2×#102—731no—
331C2×#102—731no—
332C2×C4×C8—731no—
333C2×C4×C8—731no—
334#391—731no—
335#391—731no—
336#390C2×D7311yescharacteristic
337#390C2×D7311yescharacteristic
338#390C2×D7311yescharacteristic
339#389—731no—
340#389—731no—
341#389C2×D7311yescharacteristic
342#389C2×D7311yescharacteristic
343#389C2×D7311yescharacteristic
344#388—21no—
345#388—21no—
346#388—21no—
347#387—1461no—
348#387—1461no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4—1461no—
351#376—1461no—
352C43C2×D7311yescharacteristic
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#336—731no—
360#336—731no—
361C22×Q8—731no—
362C22×D4C22×D7311yescharacteristic
363C2×SD16—2921no—
364C2×SD16—1462no—
365C2×SD16—1462no—
366C22×C8—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370#126—44no—
371C4×Q8—1462no—
372C4×Q8—1462no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42C22⋊D7311yescharacteristic
394C2×C42C22⋊D7311yescharacteristic
395C2×C42C22⋊D7311yescharacteristic
396C2×C42C22⋊D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7311yescharacteristic
399C2×C42C22×D7311yescharacteristic
400#106—2921no—
401#106—1462no—
402#106—1462no—
403C4⋊C8—1462no—
404C4⋊C8—1462no—
405#103—2921no—
406#103—1462no—
407#103—1462no—
408#102—2921no—
409#102—1462no—
410#102—1462no—
411C4×C8—2921no—
412C4×C8—2921no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423C24—21no—
424C2×Q8—732no—
425C2×Q8—1461no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7311yescharacteristic
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450SD16—2924no—
451C2×C8—732no—
452C2×C8—732no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—732no—
475C4⋊C4—1464no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—2922no—
497C42—1464no—
498C42#1192401614yescomplemented
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D7311yescharacteristic
507Q8—2921no—
508Q8—2921no—
509Q8—1462no—
510Q8—1462no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
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—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—2921no—
539C2×C4—2921no—
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—1462no—
549C2×C4—1462no—
550C2×C4C2×#1192401612yes—
551C2×C4C2×#1192401612yes—
552C2×C4C2×#1192401612yes—
553C2×C4#1455661112yes—
554C2×C4#41×D7311yescharacteristic
555C2×C4#41×D7311yescharacteristic
556C2×C4#41×D7311yescharacteristic
557C2×C4#41×D7311yescharacteristic
558C2×C4#1455663711yescharacteristic
559C2×C4#1455663811yescharacteristic
560C8—2921no—
561C8—2921no—
562C8—1462no—
563C8—1462no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1798909411yescharacteristic
572C22#1798913311yescharacteristic
573C22#1798916511yescharacteristic
574C22C22×#1192401611yescharacteristic
575C22C2×#1455661111yescharacteristic
576C22#1798938511yescharacteristic
577C22#1798938511yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—2922no—
588C4—2922no—
589C4—5841no—
590C4—5841no—
591C4#1798926211yescharacteristic
592C4#1798927311yescharacteristic
593C4#1798928811yescharacteristic
594C4#1798930411yescharacteristic
595C2—161no—
596C2#2261553011yescharacteristic
597C2#2261994311yescharacteristic
598C2#2261995811yescharacteristic
599C2#2262048611yescharacteristic
600C2#2262065411yescharacteristic
601C2#2262124711yescharacteristic
602C2#2262127811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2860C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2860D73
hypocenterC73#29434
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #22612408C2semidirect
2. #22611432C2semidirect
3. #22611309C2semidirect
7. #22610656C2semidirect
9. #22610606C2semidirect
11. #22610506C2semidirect
13. #22610198C2semidirect
14. C4×C4⋊#10596831C2semidirect
196. C42×C73SD16semidirect
317. C73#29434semidirect
498. C42#11924016semidirect

Maximal subgroups

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

subgroupindex
#226124082
#226114322
#226113092
#226113052
#226112482
#226107842
#226106562
#226106172
#226106062
#226105552
#226105062
#226102892
#226101982
C4×C4⋊#105968312
C73×#28602
#2943473