Finite Group Databasebeta

← #33189108

Subgroups — #33189108

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#22618298C211yesmaximal, characteristic
2#22611432C211yesmaximal, characteristic, complemented
3#22611326C211yesmaximal, characteristic, complemented
4#22611316C211yesmaximal, characteristic
5#22611246C211yesmaximal, characteristic
6#22610788C211yesmaximal, characteristic
7#22610661C211yesmaximal, characteristic, complemented
8#22610613C211yesmaximal, characteristic
9#22610610C211yesmaximal, characteristic, complemented
10#22610551C211yesmaximal, characteristic
11#22610516C211yesmaximal, characteristic, complemented
12#22610286C211yesmaximal, characteristic
13#22610185C211yesmaximal, characteristic, complemented
14C4×C4⋊#10596831C211yesmaximal, characteristic, complemented
15C73×#2295C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×#14556585—21no—
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988489C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988489C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988478—21no—
38#17988478—21no—
39#17988478—21no—
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×Q8×C73C2211yescharacteristic
49#391×C73C2211yescharacteristic
50#391×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#387×C73—21no—
54#387×C73—21no—
55C4×C4⋊C4×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57#376×C73—21no—
58C43×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C2×#11924015—22no—
61C2×#11924015—22no—
62C2×#11924015—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556585—44no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924005—21no—
75C2×#11924005—21no—
76C2×#11924005—21no—
77C2×#11924005—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80#14556524—22no—
81#14556524—41no—
82#14556522—22no—
83C4⋊#10596831—22no—
84C4⋊#10596831—22no—
85#14556515—22no—
86#14556515—22no—
87#14556515—41no—
88#14556511—22no—
89#14556511—22no—
90#14556511—41no—
91C4×#10596831—41no—
92C4×#10596831—41no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96C22×Q8×C73C2311yescharacteristic
97C4×Q8×C73—22no—
98C4×Q8×C73—22no—
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×C42×C73—21no—
106C2×C42×C73—21no—
107C2×C42×C73D411yescharacteristic
108C2×C42×C73D411yescharacteristic
109C2×C42×C73D411yescharacteristic
110C2×C42×C73D411yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121C23×D73—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131#11924015—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—
153#11924005—42no—
154#11924005—42no—
155#11924005—42no—
156#11924005—42no—
157#11924005—42no—
158#11924005—42no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73#4112yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2411yescharacteristic
176C4⋊C4×C73—24no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—42no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73#4112yes—
185C42×C73—24no—
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×C73C22×D411yescharacteristic
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—41no—
205Q8×C73—22no—
206Q8×C73—22no—
207Q8×C73—41no—
208Q8×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—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
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—41no—
234C2×C4×C73—41no—
235C2×C4×C73#13512yes—
236C2×C4×C73#13512yes—
237C2×C4×C73AGL(1,ℤ/8)12yes—
238C2×C4×C73#13712yes—
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73H(5,𝔽2)11yescharacteristic
244C2×C4×C732−1+411yescharacteristic
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—
254D292—41no—
255D292—41no—
256D292—41no—
257D292—41no—
258#10596831—22no—
259#10596831—22no—
260#10596831—41no—
261#10596831—41no—
262C22×C73#52511yescharacteristic
263C22×C73#53211yescharacteristic
264C22×C73#53711yescharacteristic
265C22×C73C2×#13511yescharacteristic
266C22×C73C2×#13511yescharacteristic
267C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C73C2×#13711yescharacteristic
269C2×D73—81no—
270C2×D73—81no—
271C2×D73—81no—
272C2×D73—81no—
273C2×D73—81no—
274C2×D73—81no—
275C2×D73—81no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—41no—
281C4×C73—42no—
282C4×C73—42no—
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—
293#29435—731nomaximal, Sylow
294C2×C73#260111yescharacteristic
295C2×C73#310611yescharacteristic
296C2×C73#310811yescharacteristic
297C2×C73#316811yescharacteristic
298C2×C73#318211yescharacteristic
299C2×C73#324711yescharacteristic
300C2×C73#324811yescharacteristic
301D73—161no—
302#2860—731no—
303#2295D7311yescharacteristic, complemented
304#2088—731no—
305#2068—731no—
306#2067—731no—
307#2055—731no—
308#1968—731no—
309#1942—731no—
310#1934—731no—
311#1933—731no—
312#1925—731no—
313#1920—731no—
314#1880—731no—
315#1863—731no—
316C4×C4⋊C8—731no—
317C73#2943511yescharacteristic, complemented, Sylow
318C22×SD16—731no—
319C2×#126—1461no—
320C2×C4×Q8C2×D7311yescharacteristic
321C2×#107—731no—
322C2×#107—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#391C2×D7311yescharacteristic
335#391C2×D7311yescharacteristic
336#390—731no—
337#390—731no—
338#390—731no—
339#389—731no—
340#389—731no—
341#389—731no—
342#389C2×D7311yescharacteristic
343#389C2×D7311yescharacteristic
344#388—1461no—
345#388—1461no—
346#388—1461no—
347#387—21no—
348#387—21no—
349C4×C4⋊C4—21no—
350C4×C4⋊C4C2×D7311yescharacteristic
351#376—21no—
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×Q8C22×D7311yescharacteristic
362C22×D4—731no—
363C2×SD16—2921no—
364C2×SD16—1462no—
365C2×SD16—1462no—
366C22×C8—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370#126—2924no—
371C4×Q8—22no—
372C4×Q8—22no—
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—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4—1461no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C22⋊D7311yescharacteristic
394C2×C42C22⋊D7311yescharacteristic
395C2×C42C22⋊D7311yescharacteristic
396C2×C42C22⋊D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7311yescharacteristic
399C2×C42C22×D7311yescharacteristic
400#107—2921no—
401#107—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—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423C24—1461no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1192402812yes—
430C2×D4—1461no—
431C2×D4—1461no—
432C2×D4—1461no—
433C2×D4—1461no—
434C2×D4—1461no—
435C2×D4—1461no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
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—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—2922no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4#1192402812yes—
481C22⋊C4—2922no—
482C22⋊C4—2922no—
483C22⋊C4—2922no—
484C22⋊C4—2922no—
485C22⋊C4—2922no—
486C22⋊C4—2922no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—42no—
499C23—2921no—
500C23—2921no—
501C23—2921no—
502C23—2921no—
503C23—2921no—
504C23—2921no—
505C23—2921no—
506C23C22×C22⋊D7311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—2921no—
512D4—2921no—
513D4—2921no—
514D4—2921no—
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—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
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—41no—
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#1455661112yes—
551C2×C4#1455661112yes—
552C2×C4#1455661612yes—
553C2×C4#1455662212yes—
554C2×C4#41×D7311yescharacteristic
555C2×C4#41×D7311yescharacteristic
556C2×C4C2×#1192402811yescharacteristic
557C2×C4C2×#1192402811yescharacteristic
558C2×C4#1455663711yescharacteristic
559C2×C4#1455663811yescharacteristic
560C8—2921no—
561C8—2921no—
562C8—1462no—
563C8—1462no—
564C22—5841no—
565C22—5841no—
566C22—5841no—
567C22—5841no—
568C22—5841no—
569C22—5841no—
570C22—5841no—
571C22#1798909811yescharacteristic
572C22#1798913311yescharacteristic
573C22#1798916511yescharacteristic
574C22C2×#1455661111yescharacteristic
575C22C2×#1455661111yescharacteristic
576C22C2×#1455661611yescharacteristic
577C22C2×#1455662211yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—5841no—
589C4—5841no—
590C4—5841no—
591C4#1798926111yescharacteristic
592C4#1798927011yescharacteristic
593C4#1798928111yescharacteristic
594C4#1798930511yescharacteristic
595C2—11681no—
596C2#2261552811yescharacteristic
597C2#2261999111yescharacteristic
598C2#2262001111yescharacteristic
599C2#2262049511yescharacteristic
600C2#2262065011yescharacteristic
601C2#2262125711yescharacteristic
602C2#2262126211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2295C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenter#2295D73
hypocenterC73#29435
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

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

NH ≅ G/Ntype
2. #22611432C2semidirect
3. #22611326C2semidirect
7. #22610661C2semidirect
9. #22610610C2semidirect
11. #22610516C2semidirect
13. #22610185C2semidirect
14. C4×C4⋊#10596831C2semidirect
15. C73×#2295C2semidirect
303. #2295D73semidirect
317. C73#29435semidirect

Maximal subgroups

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

subgroupindex
#226182982
#226114322
#226113262
#226113162
#226112462
#226107882
#226106612
#226106132
#226106102
#226105512
#226105162
#226102862
#226101852
C4×C4⋊#105968312
C73×#22952
#2943573