Finite Group Databasebeta

← #33158712

Subgroups — #33158712

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612934C211yesmaximal, characteristic
2#22611350C211yesmaximal, characteristic, complemented
3#22611323C211yesmaximal, characteristic, complemented
4#22611134C211yesmaximal, characteristic
5#22611127C211yesmaximal, characteristic
6#22610697C211yesmaximal, characteristic, complemented
7#22610661C211yesmaximal, characteristic, complemented
8#22610539C211yesmaximal, characteristic
9#22610516C211yesmaximal, characteristic, complemented
10#22610499C211yesmaximal, characteristic, complemented
11#22610184C211yesmaximal, characteristic, complemented
12#22610118C211yesmaximal, characteristic
13#22610015C211yesmaximal, characteristic
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2495C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988484—21no—
34#17988474—21no—
35#17988459—21no—
36C4×C4⋊Dic73C2211yescharacteristic
37C22⋊C4×Dic73—21no—
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#400×C73—21no—
49#397×C73—21no—
50#389×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#388×C73—21no—
53#383×C73—21no—
54#382×C73C2211yescharacteristic
55#380×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556570—22no—
66#14556570—22no—
67C4×Dic146—22no—
68C4×Dic146—41no—
69C2×C4⋊Dic73—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—41no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81#14556524—22no—
82#14556524—22no—
83#14556524—41no—
84#14556522—22no—
85C4⋊#10596831—22no—
86#14556514—22no—
87#14556514—22no—
88#14556514—22no—
89#14556514—41no—
90C22⋊#10596831—22no—
91C22⋊#10596831—41no—
92C4×#10596831—41no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C23×C4×C73C2311yescharacteristic
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73D411yescharacteristic
107C2×C22⋊C4×C73D411yescharacteristic
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111#95×C73—21no—
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146#4112yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—42no—
131C22×Dic73—42no—
132C22×Dic73—81no—
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—21no—
143C2×#10596831—21no—
144C2×#10596831#4112yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—42no—
150C4⋊Dic73#4112yes—
151C4⋊Dic73#4112yes—
152#11924007—24no—
153#11924007—42no—
154#11924007—81no—
155C22⋊Dic73—24no—
156C4×Dic73—42no—
157C4×Dic73—81no—
158C24×C73C2×D411yescharacteristic
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
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×C73C2411yescharacteristic
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—81no—
184C4⋊C4×C73—81no—
185C22⋊C4×C73—22no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—42no—
188C22⋊C4×C73—81no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—42no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73#13512yes—
201C23×C73C22×D411yescharacteristic
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
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—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
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—41no—
231C2×C4×C73—41no—
232C2×C4×C73#13712yes—
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73H(5,𝔽2)11yescharacteristic
238C2×C4×C732−1+411yescharacteristic
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—41no—
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—84no—
252Dic146—22no—
253Dic146—22no—
254Dic146—41no—
255Dic146—41no—
256#10596831—22no—
257#10596831—41no—
258#10596831—41no—
259#10596831—41no—
260C22×C73—22no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73#52511yescharacteristic
268C22×C73#54611yescharacteristic
269C22×C73#54711yescharacteristic
270C22×C73C2×#13511yescharacteristic
271C22×C73C2×#13711yescharacteristic
272C22×C73#57611yescharacteristic
273C22×C73#57811yescharacteristic
274C4×C73—22no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73—81no—
282C4×C73—81no—
283C4×C73#52911yescharacteristic
284C4×C73#53911yescharacteristic
285C4×C73#55311yescharacteristic
286C4×C73#55611yescharacteristic
287Dic73—42no—
288Dic73—42no—
289Dic73—42no—
290Dic73—81no—
291#27035—731nomaximal, Sylow
292C2×C73—42no—
293C2×C73#246611yescharacteristic
294C2×C73#310811yescharacteristic
295C2×C73#312411yescharacteristic
296C2×C73#329711yescharacteristic
297C2×C73#331211yescharacteristic
298C2×C73#333611yescharacteristic
299C2×C73#335711yescharacteristic
300#2495D7311yescharacteristic
301#2360—731no—
302#2074—731no—
303#2068—731no—
304#2036—731no—
305#2035—731no—
306#1949—731no—
307#1942—731no—
308#1923—731no—
309#1920—731no—
310#1918—731no—
311#1863—731no—
312#1852—731no—
313#1823—731no—
314C4×#103—731no—
315C73#2703511yescharacteristic, complemented, Sylow
316C2×C22⋊Q8—731no—
317C2×C4×Q8—731no—
318C22×C4⋊C4—731no—
319C2×#107—731no—
320C2×#107—731no—
321C2×#107—731no—
322C2×#106—731no—
323C2×C4⋊C8—731no—
324C2×#103—731no—
325C2×#103—731no—
326C2×#103—731no—
327C2×#103—731no—
328C2×C22⋊C8—731no—
329C2×C22⋊C8—731no—
330C2×C4×C8—731no—
331#400—21no—
332#397—21no—
333#391—731no—
334#389—731no—
335#389—1461no—
336#389C2×D7311yescharacteristic
337#389C2×D7311yescharacteristic
338#388—21no—
339#387—1461no—
340#385—1461no—
341#383—21no—
342#382C2×D7311yescharacteristic
343#380C2×D7311yescharacteristic
344#380C2×D7311yescharacteristic
345C4×C4⋊C4—731no—
346C4×C4⋊C4C2×D7311yescharacteristic
347C4×C22⋊C4—1461no—
348C4×C22⋊C4C2×D7311yescharacteristic
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#336—731no—
358#336—731no—
359C22×Q8—731no—
360C23×C4—1461no—
361C23×C4C22×D7311yescharacteristic
362C22×C8—731no—
363C22×C8—731no—
364C22×C8—731no—
365C22×C8—731no—
366C22⋊Q8—1462no—
367C22⋊Q8—1462no—
368C4×Q8—2921no—
369C4×Q8—1462no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—2921no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4C22⋊D7311yescharacteristic
389C2×C22⋊C4C22⋊D7311yescharacteristic
390C2×C42—21no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42C22×D7311yescharacteristic
394C2×C42C22×D7311yescharacteristic
395#107—2921no—
396#107—1462no—
397#107—1462no—
398#106—1462no—
399C4⋊C8—1462no—
400#103—2921no—
401#103—1462no—
402#103—1462no—
403#103—1462no—
404C22⋊C8—2921no—
405C22⋊C8—1462no—
406C4×C8—2921no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95C22⋊D7311yescharacteristic
419#95C22⋊D7311yescharacteristic
420C24C2×C22⋊D7311yescharacteristic
421C2×Q8—732no—
422C2×Q8—1461no—
423C2×Q8—1461no—
424C2×Q8—1461no—
425C2×Q8—1461no—
426C2×Q8—1461no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—2922no—
438C22×C4—2922no—
439C22×C4—5841no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7311yescharacteristic
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C2×C8—732no—
451C2×C8—1461no—
452C2×C8—1461no—
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—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—5841no—
475C4⋊C4—732no—
476C4⋊C4—732no—
477C4⋊C4—1464no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—1464no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—2922no—
493C42—5841no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1455661112yes—
501C23C22×C22⋊D7311yescharacteristic
502Q8—2921no—
503Q8—2921no—
504Q8—1462no—
505Q8—1462no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—5844no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4#1455662112yes—
550C2×C4#41×D7311yescharacteristic
551C2×C4#41×D7311yescharacteristic
552C2×C4#41×D7311yescharacteristic
553C2×C4#41×D7311yescharacteristic
554C2×C4#1455663711yescharacteristic
555C2×C4#1455663811yescharacteristic
556C8—2921no—
557C8—2921no—
558C8—2921no—
559C8—1462no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1798909411yescharacteristic
568C22#1798922511yescharacteristic
569C22#1798923611yescharacteristic
570C22C2×#1455661111yescharacteristic
571C22C2×#1455662111yescharacteristic
572C22#1798939011yescharacteristic
573C22#1798940211yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—2922no—
584C4—2922no—
585C4—2922no—
586C4—5841no—
587C4#1798912011yescharacteristic
588C4#1798917811yescharacteristic
589C4#1798928811yescharacteristic
590C4#1798930411yescharacteristic
591C2—42no—
592C2#2261396311yescharacteristic
593C2#2262000711yescharacteristic
594C2#2262018711yescharacteristic
595C2#2262177511yescharacteristic
596C2#2262197611yescharacteristic
597C2#2262229311yescharacteristic
598C2#2262255111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2495C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2495D73
hypocenterC73#27035
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2495

Complements

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

NH ≅ G/Ntype
2. #22611350C2semidirect
3. #22611323C2semidirect
6. #22610697C2semidirect
7. #22610661C2semidirect
9. #22610516C2semidirect
10. #22610499C2semidirect
11. #22610184C2semidirect
14. C4×#14556514C2semidirect
315. C73#27035semidirect

Maximal subgroups

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

subgroupindex
#226129342
#226113502
#226113232
#226111342
#226111272
#226106972
#226106612
#226105392
#226105162
#226104992
#226101842
#226101182
#226100152
C4×#145565142
C73×#24952
#2703573