Finite Group Databasebeta

← #33103753

Subgroups — #33103753

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988628C211yesmaximal, characteristic
2#22612820C211yesmaximal, characteristic
3#22612807C211yesmaximal, characteristic
4#22610673C211yesmaximal, characteristic, complemented
5#22610656C211yesmaximal, characteristic, complemented
6#22610515C211yesmaximal, characteristic, complemented
7#22610510C211yesmaximal, characteristic, complemented
8#22610504C211yesmaximal, characteristic, complemented
9#22610024C211yesmaximal, characteristic
10#22610016C211yesmaximal, characteristic
11#22609963C211yesmaximal, characteristic
12#22609827C211yesmaximal, characteristic, complemented
13C4×#14556524C211yesmaximal, characteristic, complemented
14#22609711C211yesmaximal, characteristic
15C73×#1921C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×#14556549C2211yescharacteristic
18C2×C4⋊C4×D73C2211yescharacteristic
19#17988628C418yescomplemented
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988483C2211yescharacteristic
31#17988478—21no—
32#17988476—21no—
33#17988474—21no—
34#17988468—21no—
35#17988454—21no—
36#17988444—21no—
37#17988444C2211yescharacteristic
38#17988444C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×#11924007—21no—
42C4×C4⋊Dic73C2211yescharacteristic
43C4×#11924005C2211yescharacteristic
44#17988434—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73C2311yescharacteristic
62C2×#11924011—22no—
63C2×#11924011—22no—
64C2×#11924011C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556549—22no—
68#14556549C2×C414yes—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C4⋊C4×D73—22no—
78C4⋊C4×D73C2×C414yes—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005D411yescharacteristic
83C2×#11924005D411yescharacteristic
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73C2311yescharacteristic
89#14556524—22no—
90#14556524C2×C414yes—
91#14556523C2×C414yes—
92#14556522—22no—
93#14556522C2×C414yes—
94#14556503—22no—
95#14556501—22no—
96C4×#10596831—22no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116#107×C73C2×C414yes—
117#95×C73—21no—
118#95×C73—21no—
119C23×D73C2×D411yescharacteristic
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C2×C4×D73—21no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—24no—
132C2×C4×D73—81no—
133C2×C4×D73—81no—
134C2×C4×D73C22×C412yes—
135C2×C4×D73#4112yes—
136#11924011—28no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831C22×C412yes—
141C2×#10596831#4112yes—
142C2×#10596831#4112yes—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—81no—
150C4⋊Dic73C22×C412yes—
151C4⋊Dic73C22×C412yes—
152#11924007—24no—
153#11924007—42no—
154#11924007—42no—
155#11924005—24no—
156#11924005—22no—
157#11924005—41no—
158#11924005—42no—
159#11924005—81no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—24no—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73—81no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73C22×C412yes—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—81no—
186C4⋊C4×C73C22×C412yes—
187C4⋊C4×C73C22×C412yes—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—81no—
191C23×C73C22×D411yescharacteristic
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73C4×D412yes—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73#13712yes—
217C2×C4×C73C23×C411yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C732−1+411yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73C4×D412yes—
244C4×D73—22no—
245C4×D73—22no—
246C4×D73—22no—
247C4×D73—22no—
248C4×D73—84no—
249C8×C73—22no—
250C8×C73—41no—
251#10596831—41no—
252#10596831#11714yes—
253C22×C73C2×C4×D411yescharacteristic
254C22×C73#54711yescharacteristic
255C22×C73#54711yescharacteristic
256C22×C73C2×#13711yescharacteristic
257C22×C73#57511yescharacteristic
258C22×C73#57611yescharacteristic
259C22×C73#57711yescharacteristic
260C2×D73—22no—
261C2×D73—22no—
262C2×D73—22no—
263C2×D73—41no—
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C4×C73—22no—
268C4×C73—41no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73C2×#11711yescharacteristic
274C4×C73#51911yescharacteristic
275C4×C73#55211yescharacteristic
276C4×C73#55211yescharacteristic
277Dic73—22no—
278Dic73—41no—
279Dic73—41no—
280Dic73—42no—
281Dic73—42no—
282Dic73—81no—
283Dic73—81no—
284#22623—731nomaximal, Sylow
285C2×C73#234411yescharacteristic
286C2×C73C4×#13711yescharacteristic
287C2×C73#294611yescharacteristic
288C2×C73#334611yescharacteristic
289C2×C73#335011yescharacteristic
290C2×C73#335511yescharacteristic
291C2×C73#335611yescharacteristic
292D73—42no—
293C2×#428—731no—
294#2347—731no—
295#2345—731no—
296#1944—731no—
297#1941—731no—
298#1921D7311yescharacteristic, complemented
299#1920—731no—
300#1919—731no—
301#1918—731no—
302#1825—731no—
303#1823—731no—
304#1809—731no—
305#1776—731no—
306C4×#107—731no—
307#1753—731no—
308C73#2262311yescharacteristic, complemented, Sylow
309C22×M16—731no—
310C2×#117—731no—
311C22×C4⋊C4—731no—
312#428—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107C2×D7311yescharacteristic
316C2×#106—731no—
317C2×#106—731no—
318C2×#106—731no—
319C2×C22⋊C8—731no—
320C2×#97—731no—
321C2×C4×C8—731no—
322#389—731no—
323#389—731no—
324#389—731no—
325#389C2×D7311yescharacteristic
326#388—1461no—
327#388—1461no—
328#387—1461no—
329#387—1461no—
330#383—1461no—
331#380—731no—
332#380—731no—
333#380—1461no—
334C4×C4⋊C4—731no—
335C4×C4⋊C4—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—1461no—
338C4×C4⋊C4C2×D7311yescharacteristic
339C4×C22⋊C4—731no—
340C4×C22⋊C4—1461no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340C2×D7311yescharacteristic
349#340C2×D7311yescharacteristic
350#340C2×D7311yescharacteristic
351#336—731no—
352#336C2×D7311yescharacteristic
353C23×C4—731no—
354C23×C4—1461no—
355C2×M16—734no—
356C2×M16—1462no—
357C2×M16—1462no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#117—734no—
363#117—1462no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—734no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C22⋊C4—731no—
380C2×C22⋊C4—731no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C42—21no—
385C2×C42—731no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42C22×D7311yescharacteristic
391#107—734no—
392#107—1462no—
393#107C4×D7314yes—
394#106—734no—
395#106—734no—
396#106—1462no—
397C22⋊C8—1462no—
398#97—1462no—
399C4×C8—1462no—
400#95—21no—
401#95—21no—
402#95—731no—
403#95—731no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412C24—731no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—5841no—
418C22×C4—5841no—
419C22×C4—731no—
420C22×C4—732no—
421C22×C4—732no—
422C22×C4—731no—
423C22×C4—731no—
424C22×C4—1464no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4D4×D7311yescharacteristic
434C22×C4D4×D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436M16—1468no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—732no—
443C2×C8—732no—
444C2×C8—732no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8C2×C4×D7312yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—81no—
456C4⋊C4—2922no—
457C4⋊C4—2922no—
458C4⋊C4—5841no—
459C4⋊C4—732no—
460C4⋊C4—732no—
461C4⋊C4—1464no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C2×C4×D7312yes—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—2921no—
471C22⋊C4—2922no—
472C22⋊C4—5841no—
473C22⋊C4—1464no—
474C22⋊C4—1462no—
475C42—21no—
476C42—21no—
477C42—81no—
478C42—5841no—
479C42—5841no—
480C42—5841no—
481C42—1464no—
482C42—1461no—
483C42—1461no—
484C23—732no—
485C23—1461no—
486C23—1461no—
487C23—1461no—
488C23—1461no—
489C23—1461no—
490C23—1461no—
491C23C2×D4×D7311yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—2921no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—5844no—
520C2×C4—732no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1462no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4#1455662312yes—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4#41×D7311yescharacteristic
538C2×C4#41×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C2×C4#1455663811yescharacteristic
541C2×C4#1455663911yescharacteristic
542C8—22no—
543C8—41no—
544C8—2921no—
545C8—734no—
546C22—2921no—
547C22—2921no—
548C22—2921no—
549C22—2921no—
550C22—1462no—
551C22—1462no—
552C22—1462no—
553C22C4×D4×D7311yescharacteristic
554C22#1798923511yescharacteristic
555C22#1798923911yescharacteristic
556C22C2×#1455662311yescharacteristic
557C22#1798938411yescharacteristic
558C22#1798939111yescharacteristic
559C22#1798939711yescharacteristic
560C4—22no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—2921no—
567C4—2921no—
568C4—2922no—
569C4—2922no—
570C4—5841no—
571C4—5841no—
572C4—1462no—
573C4#117×D7311yescharacteristic
574C4#1798906311yescharacteristic
575C4#1798926911yescharacteristic
576C4#1798927311yescharacteristic
577C2—2922no—
578C2#2261279711yescharacteristic
579C2C4×#1455662311yescharacteristic
580C2#2261876011yescharacteristic
581C2#2262242011yescharacteristic
582C2#2262248011yescharacteristic
583C2#2262253411yescharacteristic
584C2#2262254911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1921C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1921D73
hypocenterC73#22623
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #22610673C2semidirect
5. #22610656C2semidirect
6. #22610515C2semidirect
7. #22610510C2semidirect
8. #22610504C2semidirect
12. #22609827C2semidirect
13. C4×#14556524C2semidirect
15. C73×#1921C2semidirect
19. #17988628C4semidirect
298. #1921D73semidirect
308. C73#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
#226128202
#226128072
#226106732
#226106562
#226105152
#226105102
#226105042
#226100242
#226100162
#226099632
#226098272
C4×#145565242
#226097112
C73×#19212
#2262373