Finite Group Databasebeta

← #33144052

Subgroups — #33144052

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613649C211yesmaximal, characteristic
2#22612816C211yesmaximal, characteristic
3#22611502C211yesmaximal, characteristic, complemented
4#22611435C211yesmaximal, characteristic, complemented
5#22611431C211yesmaximal, characteristic, complemented
6#22611272C211yesmaximal, characteristic, complemented
7#22610885C211yesmaximal, characteristic, complemented
8#22610702C211yesmaximal, characteristic, complemented
9#22610512C211yesmaximal, characteristic, complemented
10#22610508C211yesmaximal, characteristic, complemented
11#22610124C211yesmaximal, characteristic
12#22610114C211yesmaximal, characteristic
13#22610026C211yesmaximal, characteristic
14#22609964C211yesmaximal, characteristic
15C73×#1736C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556523C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24#17988526—21no—
25#17988520—21no—
26#17988486C2211yescharacteristic
27#17988486C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988483C2211yescharacteristic
30#17988480—21no—
31#17988480—21no—
32#17988460—21no—
33#17988453—21no—
34#17988452C2211yescharacteristic
35#17988447—21no—
36#17988447C2211yescharacteristic
37#17988447C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×C22⋊Dic73C2211yescharacteristic
41#17988433—21no—
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×M16×C73C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54C22×C4⋊C4×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61#14556582—22no—
62#14556582—41no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×C22⋊Dic73—21no—
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73D411yescharacteristic
76C2×C22⋊Dic73D411yescharacteristic
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81#14556525—41no—
82#14556523—22no—
83#14556522—22no—
84#14556522—41no—
85#14556520—22no—
86#14556520—41no—
87C22⋊#10596831—22no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498D411yescharacteristic
99#14556498D411yescharacteristic
100C23×C4×C73—21no—
101C23×C4×C73C2311yescharacteristic
102C2×M16×C73—22no—
103C2×M16×C73—22no—
104C2×M16×C73—41no—
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107#117×C73—22no—
108#117×C73Q814yes—
109C2×C4⋊C4×C73—22no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—41no—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C42×C73—21no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73C2×D411yescharacteristic
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—42no—
141C4⋊Dic73—81no—
142C4⋊Dic73—81no—
143C4⋊Dic73—81no—
144#11924007—42no—
145#11924007—81no—
146C22⋊Dic73—41no—
147C22⋊Dic73—41no—
148C22⋊Dic73—42no—
149C22⋊Dic73—81no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—42no—
155C4×Dic73—42no—
156C24×C73C2×D411yescharacteristic
157C22×C4×C73—21no—
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—42no—
166C22×C4×C73—42no—
167C22×C4×C73—81no—
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×Q812yes—
170C22×C4×C73#4112yes—
171C22×C4×C73C2411yescharacteristic
172M16×C73—44no—
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73#4112yes—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—42no—
185C4⋊C4×C73C2×Q812yes—
186C4⋊C4×C73C2×Q812yes—
187C4⋊C4×C73#4112yes—
188C22⋊C4×C73—24no—
189C42×C73—24no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73C22⋊Q812yes—
197C23×C73C22×D411yescharacteristic
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
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—84no—
217C2×C4×C73C22⋊Q812yes—
218C2×C4×C73C22×Q811yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73H(5,𝔽2)11yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C8×C73—22no—
249C8×C73—41no—
250#10596831—41no—
251#10596831—41no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73C2×C22⋊Q811yescharacteristic
260C22×C73#54611yescharacteristic
261C22×C73#54711yescharacteristic
262C22×C73#57511yescharacteristic
263C22×C73#57511yescharacteristic
264C22×C73#57711yescharacteristic
265C22×C73#57711yescharacteristic
266C4×C73—22no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—42no—
270C4×C73—81no—
271C4×C73#53311yescharacteristic
272C4×C73#54411yescharacteristic
273C4×C73#55111yescharacteristic
274C4×C73#55211yescharacteristic
275Dic73—41no—
276Dic73—41no—
277Dic73—41no—
278Dic73—41no—
279Dic73—81no—
280Dic73—81no—
281Dic73—81no—
282Dic73—81no—
283#26019—731nomaximal, Sylow
284C2×C73—42no—
285C2×C73#243811yescharacteristic
286C2×C73#308011yescharacteristic
287C2×C73#308411yescharacteristic
288C2×C73#332111yescharacteristic
289C2×C73#332311yescharacteristic
290C2×C73#334811yescharacteristic
291C2×C73#335111yescharacteristic
292#2442—731no—
293#2346—731no—
294#2101—731no—
295#2088—731no—
296#2088—731no—
297#2059—731no—
298#1989—731no—
299#1949—731no—
300#1919—731no—
301#1919—731no—
302#1853—731no—
303#1851—731no—
304#1825—731no—
305#1809—731no—
306#1736D7311yescharacteristic
307C73#2601911yescharacteristic, complemented, Sylow
308C22×M16C2×D7311yescharacteristic
309C2×#125—731no—
310C2×#117C2×D7311yescharacteristic
311C22×C4⋊C4C2×D7311yescharacteristic
312C2×#107—731no—
313C2×#106—731no—
314C2×#106—731no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×C22⋊C8—731no—
319#400—1461no—
320#398—1461no—
321#389—731no—
322#389—731no—
323#389—731no—
324#389—731no—
325#388—1461no—
326#388—1461no—
327#385—1461no—
328#382—731no—
329#382—1461no—
330#380—731no—
331#380—731no—
332#380—1461no—
333C4×C4⋊C4—731no—
334C4×C4⋊C4—731no—
335C4×C22⋊C4—731no—
336#376—1461no—
337#340—731no—
338#340—731no—
339#340—731no—
340#340—731no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340C2×D7311yescharacteristic
346#340C2×D7311yescharacteristic
347#340C2×D7311yescharacteristic
348#340C2×D7311yescharacteristic
349#336—731no—
350#336—731no—
351C23×C4—21no—
352C23×C4C22×D7311yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—41no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8C22×D7311yescharacteristic
359C22×C8C22×D7311yescharacteristic
360#125—2921no—
361#125—1462no—
362#117—22no—
363#117Dic14614yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391#107—2921no—
392#106—2921no—
393#106—1462no—
394#106—1462no—
395C4⋊C8—2921no—
396C4⋊C8—1462no—
397C22⋊C8—1462no—
398#95—731no—
399#95—731no—
400#95—1461no—
401#95—1461no—
402#95—1461no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410C24D4×D7311yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—81no—
422C22×C4—731no—
423C22×C4—731no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4D4×D7311yescharacteristic
431C22×C4C2×Dic14612yes—
432C22×C4#1192402712yes—
433C22×C4C23×D7311yescharacteristic
434M16—44no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8#1192402812yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—2922no—
454C4⋊C4—2922no—
455C4⋊C4—5841no—
456C4⋊C4—5841no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4C2×Dic14612yes—
468C4⋊C4C2×Dic14612yes—
469C4⋊C4#1192402812yes—
470C22⋊C4—24no—
471C22⋊C4—2922no—
472C22⋊C4—2921no—
473C22⋊C4—2921no—
474C22⋊C4—5841no—
475C42—24no—
476C42—2922no—
477C42—2922no—
478C42—1461no—
479C42—1461no—
480C42—1461no—
481C42—1461no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1455656812yes—
489C23C2×D4×D7311yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
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—84no—
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—2921no—
520C2×C4—2921no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4#1455656812yes—
534C2×C4C22×Dic14611yescharacteristic
535C2×C4C2×#1192402711yescharacteristic
536C2×C4C2×#1192402811yescharacteristic
537C2×C4C2×#1192402811yescharacteristic
538C2×C4#1455663611yescharacteristic
539C2×C4#1455663911yescharacteristic
540C8—22no—
541C8—41no—
542C8—2921no—
543C8—2921no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1455656811yescharacteristic
552C22#1798922711yescharacteristic
553C22#1798923811yescharacteristic
554C22#1798938311yescharacteristic
555C22#1798938611yescharacteristic
556C22#1798939611yescharacteristic
557C22#1798939711yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—2921no—
564C4—2921no—
565C4—2921no—
566C4—2921no—
567C4—5841no—
568C4—5841no—
569C4—5841no—
570C4—5841no—
571C4#1798914411yescharacteristic
572C4#1798921811yescharacteristic
573C4#1798926311yescharacteristic
574C4#1798927211yescharacteristic
575C2—42no—
576C2#2261360711yescharacteristic
577C2#2261975111yescharacteristic
578C2#2261979711yescharacteristic
579C2#2262211811yescharacteristic
580C2#2262214311yescharacteristic
581C2#2262244711yescharacteristic
582C2#2262249211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1736C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1736D73
hypocenterC73#26019
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. #22611502C2semidirect
4. #22611435C2semidirect
5. #22611431C2semidirect
6. #22611272C2semidirect
7. #22610885C2semidirect
8. #22610702C2semidirect
9. #22610512C2semidirect
10. #22610508C2semidirect
307. C73#26019semidirect

Maximal subgroups

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

subgroupindex
#226136492
#226128162
#226115022
#226114352
#226114312
#226112722
#226108852
#226107022
#226105122
#226105082
#226101242
#226101142
#226100262
#226099642
C73×#17362
#2601973