Finite Group Databasebeta

← #33096263

Subgroups — #33096263

Order 18688.

1234 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988549C211yesmaximal, characteristic
2C2×#17988444C211yesmaximal, characteristic
3#380×D73C211yesmaximal, characteristic
4#22610124C211yesmaximal, characteristic, complemented
5#22610122C212yesmaximal, complemented
6#22610121C211yesmaximal, characteristic, complemented
7#22610120C212yesmaximal, complemented
8#22610118C211yesmaximal, characteristic, complemented
9#22609963C212yesmaximal, complemented
10#22609961C212yesmaximal, complemented
11C73×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924011C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C2×C4⋊C4×D73C2211yescharacteristic
16C2×C4⋊C4×D73C2211yescharacteristic
17C22×#11924005—21no—
18C2×C22⋊C4×D73—21no—
19C2×#14556525C2212yescomplemented
20C2×#14556524C2212yescomplemented
21C2×#14556522C2212yescomplemented
22#17988549C418yescomplemented
23C2×C22⋊#10596831C2211yescharacteristic
24C2×#14556503C2212yes—
25#17988447C2211yescharacteristic
26#17988446—21no—
27#17988445—21no—
28#17988444—21no—
29#17988444—21no—
30#17988444C2211yescharacteristic
31#17988444C2212yes—
32#17988444C2211yescharacteristic
33#17988444C2212yes—
34C2×#14556498—21no—
35#95×D73—21no—
36#17988350C2212yescomplemented
37#17988349C2212yescomplemented
38#17988349C2212yescomplemented
39C22×C4⋊C4×C73C2211yescharacteristic
40C2×#106×C73C2212yescomplemented
41C2×C22⋊C8×C73C2211yescharacteristic
42#380×C73C2211yescharacteristic
43#340×C73C2212yescomplemented
44C24×D73D411yescharacteristic
45C23×Dic73—21no—
46C23×Dic73D411yescharacteristic
47C22×C4×D73—21no—
48C22×C4×D73—21no—
49C22×C4×D73—42no—
50C22×C4×D73—42no—
51C22×C4×D73C2×C414yes—
52C22×C4×D73D411yescharacteristic
53C22×C4×D73D411yescharacteristic
54C22×C4×D73Q814yes—
55C22×C4×D73C2311yescharacteristic
56C22×C4×D73C2311yescharacteristic
57C2×#11924011—42no—
58C2×#11924011—42no—
59C2×#11924011C2×C418yescomplemented
60C22×#10596831C2312yes—
61C2×C4⋊Dic73—21no—
62C2×C4⋊Dic73—42no—
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2312yes—
65C2×C4⋊Dic73C2311yescharacteristic
66C2×#11924007—21no—
67C2×#11924007—21no—
68C4⋊C4×D73—42no—
69C4⋊C4×D73—42no—
70C2×C22⋊Dic73—21no—
71C2×#11924005—22no—
72C2×#11924005—22no—
73C2×#11924005—21no—
74C2×#11924005—21no—
75C2×#11924005—44no—
76C22⋊C4×D73—44no—
77#14556525—42no—
78#14556524—42no—
79#14556522—42no—
80C22⋊#10596831C2×C414yescomplemented
81#14556503C2×C418yescomplemented
82#14556498—21no—
83#14556498—21no—
84#14556498—22no—
85#14556498—22no—
86#14556498—21no—
87C23×C4×C73—21no—
88C23×C4×C73C2311yescharacteristic
89C22×C8×C73C2312yes—
90C2×C4⋊C4×C73—21no—
91C2×C4⋊C4×C73—42no—
92C2×C4⋊C4×C73C2311yescharacteristic
93C2×C4⋊C4×C73C2311yescharacteristic
94C2×C4⋊C4×C73C2312yes—
95C2×C22⋊C4×C73—21no—
96#106×C73—42no—
97C22⋊C8×C73C2×C414yescomplemented
98#95×C73—21no—
99C23×D73—22no—
100C23×D73—22no—
101C23×D73—22no—
102C23×D73—22no—
103C23×D73—41no—
104C23×D73—41no—
105C23×D73C4⋊C414yes—
106C23×D73C2×D411yescharacteristic
107C23×D73C2×D411yescharacteristic
108C22×Dic73—22no—
109C22×Dic73—21no—
110C22×Dic73—21no—
111C22×Dic73—41no—
112C22×Dic73—41no—
113C22×Dic73—42no—
114C22×Dic73—42no—
115C22×Dic73—42no—
116C22×Dic73C4⋊C414yes—
117C22×Dic73C2×D411yescharacteristic
118C22×Dic73C2×D411yescharacteristic
119C2×C4×D73—22no—
120C2×C4×D73—22no—
121C2×C4×D73—22no—
122C2×C4×D73—22no—
123C2×C4×D73—48no—
124C2×C4×D73—48no—
125C2×C4×D73—41no—
126C2×C4×D73—41no—
127C2×C4×D73—41no—
128C2×C4×D73—41no—
129C2×C4×D73—42no—
130C2×C4×D73—42no—
131C2×C4×D73—42no—
132C2×C4×D73—42no—
133C2×C4×D73—42no—
134C2×C4×D73—42no—
135C2×C4×D73C4⋊C414yes—
136C2×C4×D73C4⋊C414yes—
137C2×C4×D73C22×C412yes—
138C2×C4×D73C22×C412yes—
139C2×C4×D73C2×Q812yes—
140C2×C4×D73C2×Q812yes—
141#11924011—48no—
142C2×#10596831—22no—
143C2×#10596831—22no—
144C2×#10596831C22×C414yes—
145C4⋊Dic73—22no—
146C4⋊Dic73—22no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73—42no—
150#11924007—42no—
151#11924007—42no—
152C22⋊Dic73—42no—
153#11924005—42no—
154#11924005—42no—
155#11924005—44no—
156#11924005—44no—
157C24×C73C2×D411yescharacteristic
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73—42no—
164C22×C4×C73—42no—
165C22×C4×C73—42no—
166C22×C4×C73C22×C412yes—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×Q812yes—
169C22×C4×C73C2411yescharacteristic
170C2×C8×C73—22no—
171C2×C8×C73—22no—
172C2×C8×C73C22×C414yes—
173C4⋊C4×C73—22no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—42no—
178C22⋊C4×C73—42no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—21no—
183C23×C73C2×C4⋊C412yes—
184C23×C73AGL(1,ℤ/8)12yes—
185C23×C73#13712yes—
186C23×C73C22×D411yescharacteristic
187C22×D73—21no—
188C22×D73—21no—
189C22×D73—21no—
190C22×D73—21no—
191C22×D73—21no—
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—28no—
198C22×D73—28no—
199C22×D73—44no—
200C22×D73—44no—
201C22×D73—44no—
202C22×D73—44no—
203C22×D73—44no—
204C22×D73C2×C4⋊C412yes—
205C22×D73C2×C4⋊C412yes—
206C22×D73AGL(1,ℤ/8)12yes—
207C22×D73#13712yes—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—42no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—41no—
219C2×C4×C73—42no—
220C2×C4×C73—41no—
221C2×C4×C73—84no—
222C2×C4×C73C2×C4⋊C412yes—
223C2×C4×C73C23×C411yescharacteristic
224C2×C4×C73C22×Q811yescharacteristic
225C2×C4×C73H(5,𝔽2)12yes—
226C2×C4×C73H(5,𝔽2)12yes—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—44no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—84no—
242C2×Dic73C2×C4⋊C412yes—
243C2×Dic73C2×C4⋊C412yes—
244C4×D73—24no—
245C4×D73—24no—
246C4×D73—24no—
247C4×D73—24no—
248C4×D73—44no—
249C4×D73—44no—
250C4×D73—84no—
251C4×D73—84no—
252C8×C73—24no—
253#10596831—24no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73#42814yes—
261C22×C73C22×C4⋊C411yescharacteristic
262C22×C73#53411yescharacteristic
263C22×C73#53411yescharacteristic
264C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C73C2×#13711yescharacteristic
267C22×C73C2×#13711yescharacteristic
268C2×D73—22no—
269C2×D73—22no—
270C2×D73—22no—
271C2×D73—22no—
272C2×D73—22no—
273C2×D73—22no—
274C2×D73—22no—
275C2×D73—22no—
276C2×D73—22no—
277C2×D73—416no—
278C2×D73—41no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73#42814yes—
283C4×C73—22no—
284C4×C73—81no—
285C4×C73—81no—
286C4×C73—82no—
287C4×C73#51812yes—
288C4×C73#54312yes—
289Dic73—22no—
290Dic73—22no—
291Dic73—81no—
292Dic73—82no—
293Dic73—81no—
294#21942—731nomaximal, Sylow
295C2×C73—24no—
296C2×C73#231511yescharacteristic
297C2×C73C2×#42811yescharacteristic
298C2×C73C2×#42811yescharacteristic
299C2×C73#319211yescharacteristic
300C2×C73#319311yescharacteristic
301C2×C73#319311yescharacteristic
302C2×C73#319511yescharacteristic
303D73—24no—
304D73—42no—
305C2×#407—731no—
306C2×#380—731no—
307C2×#380—731no—
308#1853—732no—
309#1853—731no—
310#1853D7311yescharacteristic, complemented
311#1852—731no—
312#1852—731no—
313#1852—732no—
314#1809—732no—
315#1809—732no—
316C73#2194211yescharacteristic, complemented, Sylow
317C24×C4—731no—
318C22×M16—732no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—731no—
322C22×C4⋊C4C2×D7311yescharacteristic
323C22×C22⋊C4—1461no—
324C22×C22⋊C4—1461no—
325C2×#107—732no—
326C2×#107—732no—
327C2×#106—732no—
328C2×#106C2×D7312yescomplemented
329#407—738no—
330C2×C22⋊C8—731no—
331C2×C22⋊C8—732no—
332C2×C22⋊C8C2×D7311yescharacteristic
333#380—731no—
334#380—731no—
335#380—731no—
336#380—732no—
337#380—732no—
338#380—1461no—
339#380—1461no—
340#380—1461no—
341#380—1461no—
342#380C2×D7311yescharacteristic
343C2×#95—1461no—
344C2×#95—1461no—
345#340—732no—
346#340—732no—
347#340—732no—
348#340C2×D7312yescomplemented
349C25—731no—
350C23×C4—21no—
351C23×C4—2922no—
352C23×C4—2922no—
353C23×C4—731no—
354C23×C4—734no—
355C23×C4—731no—
356C23×C4—731no—
357C23×C4—734no—
358C23×C4—731no—
359C23×C4—731no—
360C23×C4—1461no—
361C23×C4—1461no—
362C23×C4—1461no—
363C23×C4C22×D7311yescharacteristic
364C2×M16—2922no—
365C2×M16—2922no—
366C2×M16—738no—
367C22×C8—732no—
368C22×C8C22×D7312yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—2922no—
372C2×C4⋊C4—2922no—
373C2×C4⋊C4—2922no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—732no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C4⋊C4C22×D7312yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—2924no—
385C2×C22⋊C4—2924no—
386C2×C22⋊C4—1462no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1462no—
390C2×C22⋊C4—1461no—
391#107—2922no—
392#107—2922no—
393#106—42no—
394#106—2922no—
395C22⋊C8—734no—
396C22⋊C8—738no—
397C22⋊C8C4×D7314yescomplemented
398#95—21no—
399#95—1461no—
400#95—1461no—
401#95—1462no—
402#95—1462no—
403#95—1461no—
404C24—2921no—
405C24—2921no—
406C24—734no—
407C24—731no—
408C24—731no—
409C24—1462no—
410C24—1462no—
411C24—1462no—
412C24—1462no—
413C24D4×D7311yescharacteristic
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—2921no—
423C22×C4—2921no—
424C22×C4—2921no—
425C22×C4—2921no—
426C22×C4—2928no—
427C22×C4—2928no—
428C22×C4—2922no—
429C22×C4—2922no—
430C22×C4—2922no—
431C22×C4—2922no—
432C22×C4—2922no—
433C22×C4—2922no—
434C22×C4—2921no—
435C22×C4—2921no—
436C22×C4—2922no—
437C22×C4—2922no—
438C22×C4—2922no—
439C22×C4—734no—
440C22×C4—734no—
441C22×C4—734no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—732no—
445C22×C4—732no—
446C22×C4—732no—
447C22×C4—732no—
448C22×C4—1461no—
449C22×C4—1462no—
450C22×C4—1461no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4C2×C4×D7312yes—
456C22×C4D4×D7311yescharacteristic
457C22×C4Q8×D7312yes—
458C22×C4C23×D7311yescharacteristic
459M16—2928no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—734no—
463C2×C8—1462no—
464C2×C8—1462no—
465C2×C8C2×C4×D7314yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—2922no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—1462no—
475C4⋊C4—1462no—
476C4⋊C4—1462no—
477C4⋊C4—1462no—
478C22⋊C4—42no—
479C22⋊C4—2924no—
480C22⋊C4—2924no—
481C22⋊C4—2922no—
482C22⋊C4—2922no—
483C22⋊C4—2922no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—2924no—
489C23—2924no—
490C23—2924no—
491C23—2924no—
492C23—2924no—
493C23—732no—
494C23—732no—
495C23—732no—
496C23—732no—
497C23—1468no—
498C23—1461no—
499C23—1461no—
500C23—1468no—
501C23—1461no—
502C23—1461no—
503C23—1461no—
504C23—1461no—
505C23—1461no—
506C23—1461no—
507C23—1461no—
508C23—1461no—
509C23C4⋊C4×D7312yes—
510C23#1455661812yes—
511C23#1455662012yes—
512C23C2×D4×D7311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—2924no—
528C2×C4—2924no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2922no—
532C2×C4—2921no—
533C2×C4—2922no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2922no—
538C2×C4—2924no—
539C2×C4—5844no—
540C2×C4—5844no—
541C2×C4—5844no—
542C2×C4—732no—
543C2×C4—732no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1464no—
549C2×C4—1464no—
550C2×C4—1464no—
551C2×C4—1464no—
552C2×C4C4⋊C4×D7312yes—
553C2×C4C22×C4×D7311yescharacteristic
554C2×C4C2×Q8×D7311yescharacteristic
555C2×C4#1455663612yes—
556C2×C4#1455663712yes—
557C8—24no—
558C8—1464no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—29216no—
566C22—2921no—
567C22—2921no—
568C22—2921no—
569C22—2921no—
570C22—734no—
571C22—1462no—
572C22—1462no—
573C22—1462no—
574C22—1462no—
575C22—1462no—
576C22—1462no—
577C22—1462no—
578C22—1462no—
579C22—1462no—
580C22#1798862614yes—
581C22C2×C4⋊C4×D7311yescharacteristic
582C22#1798914711yescharacteristic
583C22#1798914811yescharacteristic
584C22AGL(1,ℤ/8)×D7311yescharacteristic
585C22C2×#1455661811yescharacteristic
586C22#137×D7311yescharacteristic
587C22C2×#1455662011yescharacteristic
588C4—22no—
589C4—81no—
590C4—82no—
591C4—81no—
592C4—5842no—
593C4—5841no—
594C4—5841no—
595C4—1462no—
596C4—1462no—
597C4#1798905912yes—
598C4#1798921212yes—
599C2—24no—
600C2—2922no—
601C2—1464no—
602C2#2261254211yescharacteristic
603C2#428×D7311yescharacteristic
604C2C2×#1798862611yescharacteristic
605C2#2262073011yescharacteristic
606C2#2262073411yescharacteristic
607C2#2262073511yescharacteristic
608C2#2262075411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1853C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1853D73
hypocenterC73#21942
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #22610124C2semidirect
5. #22610122C2semidirect
6. #22610121C2semidirect
7. #22610120C2semidirect
8. #22610118C2semidirect
9. #22609963C2semidirect
10. #22609961C2semidirect
11. C73×#1853C2semidirect
19. C2×#14556525C22semidirect
20. C2×#14556524C22semidirect
21. C2×#14556522C22semidirect
22. #17988549C4semidirect
36. #17988350C22semidirect
37. #17988349C22semidirect
38. #17988349C22semidirect
40. C2×#106×C73C22semidirect
43. #340×C73C22semidirect
59. C2×#11924011C2×C4semidirect
80. C22⋊#10596831C2×C4semidirect
81. #14556503C2×C4semidirect
97. C22⋊C8×C73C2×C4semidirect
310. #1853D73semidirect
316. C73#21942semidirect
328. C2×#106C2×D73semidirect
348. #340C2×D73semidirect
397. C22⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179885492
C2×#179884442
#380×D732
#226101242
#226101222
#226101212
#226101202
#226101182
#226099632
#226099612
C73×#18532
#2194273