Finite Group Databasebeta

← #33096589

Subgroups — #33096589

Order 18688.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988648C211yesmaximal, characteristic
2#22612554C211yesmaximal, characteristic
3#22610699C211yesmaximal, characteristic, complemented
4#22610699C211yesmaximal, characteristic, complemented
5#22610694C211yesmaximal, characteristic, complemented
6#22610689C211yesmaximal, characteristic, complemented
7#22610128C211yesmaximal, characteristic
8#22610128C211yesmaximal, characteristic
9#22610123C211yesmaximal, characteristic
10#22610118C211yesmaximal, characteristic
11#22610100C211yesmaximal, characteristic, complemented
12#22610093C211yesmaximal, characteristic, complemented
13#22610066C211yesmaximal, characteristic, complemented
14#22610058C211yesmaximal, characteristic, complemented
15C73×#2314C211yesmaximal, characteristic
16C2×#14556550C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18#17988648C418yescomplemented
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33#17988486—21no—
34#17988486—21no—
35#17988462—21no—
36#17988462—21no—
37#17988457C2211yescharacteristic
38#17988457C2211yescharacteristic
39#17988453C2211yescharacteristic
40#17988453C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49C2×#117×C73C2211yescharacteristic
50#388×C73—21no—
51#388×C73—21no—
52#387×C73—21no—
53#387×C73—21no—
54#384×C73C2211yescharacteristic
55#382×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556550—41no—
66#14556550—41no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—41no—
73C2×C4⋊Dic73—41no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—41no—
81#14556524—41no—
82#14556524—41no—
83#14556524—41no—
84#14556522—41no—
85#14556522—41no—
86#14556522—41no—
87#14556522—41no—
88C4⋊#10596831C2×C414yescomplemented
89C4⋊#10596831C2×C414yescomplemented
90C22⋊#10596831C2×C414yes—
91C22⋊#10596831C2×C414yes—
92#14556502C2×C414yescomplemented
93C4×#10596831C2×C414yescomplemented
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98C23×C4×C73C2311yescharacteristic
99#117×C73C2×C414yes—
100#117×C73Q814yes—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73—21no—
114C2×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—42no—
128C22×Dic73—81no—
129C22×Dic73—81no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831C22×C412yes—
139C2×#10596831C22×C412yes—
140C2×#10596831C22×C412yes—
141C2×#10596831C22×C412yes—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—42no—
151C4⋊Dic73—81no—
152C4⋊Dic73—81no—
153C4⋊Dic73—81no—
154C4⋊Dic73—81no—
155C22⋊Dic73—42no—
156C4×Dic73—42no—
157C24×C73C2×D411yescharacteristic
158C22×C4×C73—21no—
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×C73C22×C412yes—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×Q812yes—
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73—21no—
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—81no—
183C4⋊C4×C73—81no—
184C4⋊C4×C73C4⋊C414yes—
185C4⋊C4×C73C4⋊C414yes—
186C22⋊C4×C73—81no—
187C22⋊C4×C73—81no—
188C22⋊C4×C73C4⋊C414yes—
189C22⋊C4×C73C4⋊C414yes—
190C42×C73—81no—
191C42×C73—81no—
192C42×C73C22×C412yes—
193C42×C73C22×C412yes—
194C42×C73C2×Q812yes—
195C42×C73C2×Q812yes—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73C2×C4⋊C412yes—
203C23×C73C22×D411yescharacteristic
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—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
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×C73C2×C4⋊C412yes—
231C2×C4×C73C2×C4⋊C412yes—
232C2×C4×C73C2×C4⋊C412yes—
233C2×C4×C73C2×C4⋊C412yes—
234C2×C4×C73C2×C4⋊C412yes—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C22×Q811yescharacteristic
237C2×C4×C73H(5,𝔽2)11yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×C4×C732−1+411yescharacteristic
241C2×Dic73—41no—
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—41no—
252C2×Dic73—41no—
253C2×Dic73—84no—
254#10596831—22no—
255#10596831—22no—
256#10596831—22no—
257#10596831—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73C22×C4⋊C411yescharacteristic
266C22×C73#53511yescharacteristic
267C22×C73#53611yescharacteristic
268C22×C73#57611yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57811yescharacteristic
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—22no—
276C4×C73—22no—
277C4×C73—81no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73#52011yescharacteristic
282C4×C73#52011yescharacteristic
283C4×C73#54411yescharacteristic
284C4×C73#54411yescharacteristic
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289#21992—731nomaximal, Sylow
290C2×C73—42no—
291C2×C73#231711yescharacteristic
292C2×C73#291211yescharacteristic
293C2×C73#291311yescharacteristic
294C2×C73#322411yescharacteristic
295C2×C73#322511yescharacteristic
296C2×C73#324111yescharacteristic
297C2×C73#324211yescharacteristic
298C2×#433—731no—
299#2317—731no—
300#2314D7311yescharacteristic
301#1949—731no—
302#1949—731no—
303#1948—731no—
304#1947—731no—
305#1854—731no—
306#1854—731no—
307#1853—731no—
308#1852—731no—
309#1847—731no—
310#1846—731no—
311#1836—731no—
312#1834—731no—
313C73#2199211yescharacteristic, complemented, Sylow
314C2×#117—731no—
315C2×#117C2×D7311yescharacteristic
316C22×C4⋊C4—731no—
317#433—738no—
318C2×#107—731no—
319C2×#107—731no—
320C2×#107—731no—
321C2×#107—731no—
322C2×#106—731no—
323C2×#106—731no—
324C2×#106—731no—
325C2×#106—731no—
326C2×C4⋊C8—731no—
327C2×C4⋊C8—731no—
328C2×C22⋊C8—731no—
329C2×C22⋊C8—731no—
330C2×#97—731no—
331C2×C4×C8—731no—
332#389—1461no—
333#389—1461no—
334#388—21no—
335#388—21no—
336#387—21no—
337#387—21no—
338#385—1461no—
339#385—1461no—
340#384—731no—
341#384—731no—
342#384C2×D7311yescharacteristic
343#382—731no—
344#382—731no—
345#382C2×D7311yescharacteristic
346#380C2×D7311yescharacteristic
347#380C2×D7311yescharacteristic
348C4×C4⋊C4C2×D7311yescharacteristic
349C4×C4⋊C4C2×D7311yescharacteristic
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358C23×C4—1461no—
359C23×C4C22×D7311yescharacteristic
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8—731no—
364#117—2921no—
365#117—2921no—
366#117Dic14614yes—
367#117C4×D7314yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—2921no—
371C2×C4⋊C4—2921no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
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×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4C22⋊D7311yescharacteristic
391C2×C22⋊C4C22⋊D7311yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—1461no—
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397#107—2921no—
398#107—2921no—
399#107—2921no—
400#107—2921no—
401#106—2921no—
402#106—2921no—
403#106—2921no—
404#106—2921no—
405C4⋊C8—734no—
406C4⋊C8—734no—
407C22⋊C8—734no—
408C22⋊C8—734no—
409#97—734no—
410C4×C8—734no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421C24C2×C22⋊D7311yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—2922no—
431C22×C4—5841no—
432C22×C4—5841no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4C2×C4×D7312yes—
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C2×C22⋊D7311yescharacteristic
440C22×C4C2×C22⋊D7311yescharacteristic
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C2×Dic14612yes—
444C22×C4C23×D7311yescharacteristic
445C2×C8—732no—
446C2×C8—732no—
447C2×C8—732no—
448C2×C8—732no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—2922no—
468C4⋊C4—5841no—
469C4⋊C4—5841no—
470C4⋊C4—5841no—
471C4⋊C4—5841no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4#1192400714yes—
481C4⋊C4#1192400714yes—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—2922no—
485C22⋊C4#1192400714yes—
486C22⋊C4#1192400714yes—
487C42—81no—
488C42—81no—
489C42—2922no—
490C42C2×C4×D7312yes—
491C42C2×C4×D7312yes—
492C42C2×Dic14612yes—
493C42C2×Dic14612yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×#1192400712yes—
501C23C22×C22⋊D7311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
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—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—5844no—
541C2×C4C2×#1192400712yes—
542C2×C4C2×#1192400712yes—
543C2×C4C2×#1192400712yes—
544C2×C4C2×#1192400712yes—
545C2×C4C2×#1192400712yes—
546C2×C4C22×C4×D7311yescharacteristic
547C2×C4C22×Dic14611yescharacteristic
548C2×C4#1455663711yescharacteristic
549C2×C4#1455663711yescharacteristic
550C2×C4#1455663811yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—1462no—
553C8—1462no—
554C8—1462no—
555C8—1462no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×#1192400711yescharacteristic
564C22#1798915111yescharacteristic
565C22#1798915811yescharacteristic
566C22#1798939011yescharacteristic
567C22#1798939511yescharacteristic
568C22#1798939811yescharacteristic
569C22#1798940211yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—5841no—
580C4—5841no—
581C4—5841no—
582C4—5841no—
583C4#1798906711yescharacteristic
584C4#1798906711yescharacteristic
585C4#1798921511yescharacteristic
586C4#1798921511yescharacteristic
587C2—42no—
588C2#2261255111yescharacteristic
589C2#2261852411yescharacteristic
590C2#2261852911yescharacteristic
591C2#2262101911yescharacteristic
592C2#2262103411yescharacteristic
593C2#2262117811yescharacteristic
594C2#2262119311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2314C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2314D73
hypocenterC73#21992
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #22610699C2semidirect
4. #22610699C2semidirect
5. #22610694C2semidirect
6. #22610689C2semidirect
11. #22610100C2semidirect
12. #22610093C2semidirect
13. #22610066C2semidirect
14. #22610058C2semidirect
18. #17988648C4semidirect
88. C4⋊#10596831C2×C4semidirect
89. C4⋊#10596831C2×C4semidirect
92. #14556502C2×C4semidirect
93. C4×#10596831C2×C4semidirect
313. C73#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#179886482
#226125542
#226106992
#226106942
#226106892
#226101282
#226101232
#226101182
#226101002
#226100932
#226100662
#226100582
C73×#23142
#2199273