Finite Group Databasebeta

← #33114663

Subgroups — #33114663

Order 18688.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988630C211yesmaximal, characteristic, complemented
2C2×#17988614C211yesmaximal, characteristic, complemented
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988581C211yesmaximal, characteristic
5#22612519C211yesmaximal, characteristic
6#22611323C212yesmaximal, complemented
7#22611316C212yesmaximal, complemented
8#22611309C212yesmaximal, complemented
9#22611302C212yesmaximal, complemented
10C2×#17988349C211yesmaximal, characteristic, complemented
11C73×#2317C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14C23×#10596831C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16#17988630—24no—
17C2×#14556524C2212yes—
18C2×#14556522C2212yes—
19#17988614—24no—
20C2×C4⋊#10596831C2212yes—
21C2×C4⋊#10596831C2212yes—
22#17988586—24no—
23#17988581—24no—
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2212yes—
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556511C2212yes—
30#17988493C2212yes—
31#17988489C2212yes—
32#17988464—22no—
33#17988444—21no—
34#17988444—21no—
35#17988349C2212yes—
36#17988349C2212yes—
37C2×#117×C73C2211yescharacteristic
38C2×#117×C73C2211yescharacteristic
39C22×C4⋊C4×C73C2211yescharacteristic
40#389×C73C2212yes—
41#389×C73C2212yes—
42#385×C73—22no—
43#384×C73—21no—
44#382×C73—21no—
45C2×#11924028—22no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028Q814yes—
51C22×Dic146C2311yescharacteristic
52C22×D292C2311yescharacteristic
53C22×C22⋊D73—21no—
54C23×Dic73—21no—
55C22×C4×D73—21no—
56C2×#11924012—22no—
57C2×#11924012—22no—
58C2×#11924012D414yes—
59C22×#10596831—22no—
60C22×#10596831—22no—
61C22×#10596831—22no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2312yes—
65C2×C4⋊Dic73—22no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73C2312yes—
69C2×#11924005—22no—
70C2×#11924005—22no—
71#14556524—24no—
72#14556522—24no—
73C4⋊#10596831—24no—
74C4⋊#10596831—24no—
75#14556514—24no—
76#14556514—22no—
77#14556514—22no—
78#14556511—24no—
79#14556511—22no—
80#14556511—22no—
81#14556498—22no—
82#14556498—22no—
83C23×C4×C73—21no—
84C23×C4×C73C2311yescharacteristic
85#117×C73—22no—
86#117×C73—22no—
87#117×C73—22no—
88#117×C73D414yes—
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—22no—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73—22no—
93C2×C4⋊C4×C73—22no—
94C2×C4⋊C4×C73D411yescharacteristic
95C2×C4⋊C4×C73D411yescharacteristic
96C2×C4⋊C4×C73C2312yes—
97C2×C4⋊C4×C73C2311yescharacteristic
98C2×C4⋊C4×C73C2311yescharacteristic
99C2×C22⋊C4×C73—21no—
100C2×C22⋊C4×C73D412yes—
101C2×C42×C73—21no—
102C2×C42×C73C2312yes—
103#95×C73—22no—
104#95×C73—22no—
105C23×D73—21no—
106#11924028—28no—
107#11924028—28no—
108#11924028—44no—
109#11924028—44no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146C2×Q812yes—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292—21no—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292C2×Q812yes—
122C2×C22⋊D73—24no—
123C2×C22⋊D73—42no—
124C2×C22⋊D73—42no—
125C22×Dic73—21no—
126C22×Dic73—22no—
127C22×Dic73—42no—
128C22×Dic73—81no—
129C22×Dic73—81no—
130C2×C4×D73—24no—
131C2×C4×D73—42no—
132C2×C4×D73—42no—
133#11924012—28no—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—44no—
143C2×#10596831C2×D414yes—
144C4⋊Dic73—22no—
145C4⋊Dic73—22no—
146C4⋊Dic73—42no—
147#11924005—82no—
148#11924005—82no—
149C24×C73C2×D411yescharacteristic
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—22no—
153C22×C4×C73—42no—
154C22×C4×C73—81no—
155C22×C4×C73—81no—
156C22×C4×C73C2×D412yes—
157C22×C4×C73C2×D412yes—
158C22×C4×C73C2×D412yes—
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D412yes—
161C22×C4×C73C2×Q812yes—
162C22×C4×C73#4112yes—
163C22×C4×C73#4112yes—
164C22×C4×C73#4112yes—
165C22×C4×C73C2411yescharacteristic
166C4⋊C4×C73—22no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—42no—
173C4⋊C4×C73—82no—
174C4⋊C4×C73—82no—
175C4⋊C4×C73C2×D412yes—
176C4⋊C4×C73C2×D412yes—
177C22⋊C4×C73—24no—
178C22⋊C4×C73—24no—
179C42×C73—24no—
180C42×C73—22no—
181C42×C73—22no—
182C23×C73—21no—
183C23×C73—21no—
184C23×C73—21no—
185C23×C73—21no—
186C23×C73#12112yes—
187C23×C73#12112yes—
188C23×C73C22⋊Q812yes—
189C23×C73C22×D411yescharacteristic
190C22×D73—22no—
191C22×D73—41no—
192C22×D73—41no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—84no—
213C2×C4×C73#12112yes—
214C2×C4×C73#12112yes—
215C2×C4×C73#12114yes—
216C2×C4×C73C22⋊Q812yes—
217C2×C4×C73C22⋊Q814yes—
218C2×C4×C73C22⋊Q814yes—
219C2×C4×C73#12614yes—
220C2×C4×C73C22×D411yescharacteristic
221C2×C4×C73C22×D411yescharacteristic
222C2×C4×C73C22×Q811yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C22⋊D73—48no—
227C2×Dic73—22no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—84no—
234D292—22no—
235D292—22no—
236D292—41no—
237D292—41no—
238C4×D73—48no—
239Dic146—22no—
240Dic146—22no—
241Dic146—41no—
242Dic146—41no—
243#10596831—24no—
244#10596831—41no—
245#10596831—41no—
246C22×C73—22no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73#39614yes—
253C22×C73C2×#12111yescharacteristic
254C22×C73C2×#12111yescharacteristic
255C22×C73C2×C22⋊Q811yescharacteristic
256C22×C73#57611yescharacteristic
257C22×C73#57711yescharacteristic
258C22×C73#57711yescharacteristic
259C22×C73#57811yescharacteristic
260C2×D73—42no—
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—81no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—82no—
272C4×C73#39614yes—
273C4×C73C2×#12111yescharacteristic
274C4×C73C2×C22⋊Q811yescharacteristic
275C4×C73C2×C22⋊Q811yescharacteristic
276C4×C73C2×#12611yescharacteristic
277Dic73—42no—
278Dic73—82no—
279#23616—731nomaximal, Sylow
280C2×C73—24no—
281C2×C73C2×#39611yescharacteristic
282C2×C73#304511yescharacteristic
283C2×C73#304611yescharacteristic
284C2×C73#306211yescharacteristic
285C2×C73#306311yescharacteristic
286C2×C73#308311yescharacteristic
287C2×C73#308411yescharacteristic
288D73—82no—
289C2×#428—731no—
290C2×#424—731no—
291C2×#417—731no—
292C2×#416—731no—
293#2317D7311yescharacteristic, complemented
294#2311—731no—
295#2068—732no—
296#2067—732no—
297#2066—732no—
298#2065—732no—
299C2×#340—731no—
300C73#2361611yescharacteristic, complemented, Sylow
301C22×#41—731no—
302C22×M16—731no—
303C23×C8—731no—
304C2×#117C2×D7311yescharacteristic
305C2×#117C2×D7311yescharacteristic
306C22×C4⋊C4—731no—
307C22×C4⋊C4C2×D7311yescharacteristic
308#428—1464no—
309C2×#107—732no—
310C2×#106—732no—
311#424—1464no—
312C2×C4⋊C8—732no—
313C2×C4⋊C8—732no—
314#417—1464no—
315#416—1464no—
316C2×#103—731no—
317C2×#103—732no—
318C2×#103—731no—
319C2×#102—731no—
320C2×#102—732no—
321C2×#102—731no—
322#391—732no—
323#390—732no—
324#389C2×D7312yes—
325#389C2×D7312yes—
326#386—1462no—
327#385—22no—
328#384—21no—
329#382—21no—
330#380—1461no—
331#380—1461no—
332#340—732no—
333#340—732no—
334C2×#41—734no—
335C2×#41—1462no—
336C2×#41—1462no—
337C2×#41—1462no—
338C2×#41—1462no—
339C2×#41—1462no—
340C22×Q8—731no—
341C22×D4—731no—
342C22×D4—1461no—
343C23×C4—21no—
344C23×C4—1461no—
345C23×C4—1461no—
346C23×C4C22×D7311yescharacteristic
347C2×M16—734no—
348C2×M16—1462no—
349C2×M16—1462no—
350C22×C8—732no—
351C22×C8—731no—
352C22×C8—731no—
353C22×C8—1462no—
354C22×C8—1462no—
355C22×C8—1462no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117D29214yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4C22⋊D7311yescharacteristic
370C2×C4⋊C4C22⋊D7311yescharacteristic
371C2×C4⋊C4C22×D7312yes—
372C2×C4⋊C4C22×D7311yescharacteristic
373C2×C4⋊C4C22×D7311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1462no—
376C2×C22⋊C4—1462no—
377C2×C22⋊C4C22⋊D7312yes—
378C2×C42—21no—
379C2×C42C22×D7312yes—
380#107—1464no—
381#106—1464no—
382C4⋊C8—1464no—
383C4⋊C8—1464no—
384#103—1464no—
385#103—1462no—
386#103—1462no—
387#102—1464no—
388#102—1462no—
389#102—1462no—
390#95—22no—
391#95—22no—
392#95—1462no—
393#95—1462no—
394C24—1461no—
395C24C2×C22⋊D7311yescharacteristic
396#41—2924no—
397#41—2924no—
398#41—1468no—
399#41—1468no—
400C2×Q8—732no—
401C2×Q8—1461no—
402C2×Q8—1461no—
403C2×Q8—1461no—
404C2×Q8—1461no—
405C2×Q8—1461no—
406C2×D4—2922no—
407C2×D4—2922no—
408C2×D4—732no—
409C2×D4—1464no—
410C2×D4—1461no—
411C2×D4—1461no—
412C2×D4—1461no—
413C2×D4—1461no—
414C2×D4—1461no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—2922no—
422C22×C4—2922no—
423C22×C4—2922no—
424C22×C4—5841no—
425C22×C4—5841no—
426C22×C4—1462no—
427C22×C4—1461no—
428C22×C4—1464no—
429C22×C4D4×D7312yes—
430C22×C4C2×C22⋊D7312yes—
431C22×C4C2×C22⋊D7312yes—
432C22×C4C2×C22⋊D7311yescharacteristic
433C22×C4C2×D29212yes—
434C22×C4Q8×D7312yes—
435C22×C4#1192402712yes—
436C22×C4#1192402812yes—
437C22×C4#1192402912yes—
438C22×C4C23×D7311yescharacteristic
439M16—1468no—
440C2×C8—2924no—
441C2×C8—734no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1462no—
449C2×C8—1462no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—2922no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C4⋊C4C2×D29212yes—
463C4⋊C4C2×D29212yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—5842no—
467C22⋊C4—5842no—
468C42—24no—
469C42—22no—
470C42—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—2921no—
476C23—2921no—
477C23—2921no—
478C23—2921no—
479C23—2921no—
480C23—2921no—
481C23—1462no—
482C23#1455656312yes—
483C23#1455656512yes—
484C23#1455656712yes—
485C23C22×C22⋊D7311yescharacteristic
486Q8—2921no—
487Q8—2921no—
488Q8—1462no—
489Q8—1462no—
490D4—2928no—
491D4—2921no—
492D4—2921no—
493D4—1462no—
494D4—1462no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—2928no—
514C2×C4—2922no—
515C2×C4—2922no—
516C2×C4—2922no—
517C2×C4—5844no—
518C2×C4—1462no—
519C2×C4#1455656312yes—
520C2×C4#1455656512yes—
521C2×C4#1455656614yes—
522C2×C4#1455656712yes—
523C2×C4#1455656914yes—
524C2×C4#1455657114yes—
525C2×C4#1455658614yes—
526C2×C4C2×D4×D7311yescharacteristic
527C2×C4C22×D29211yescharacteristic
528C2×C4C2×Q8×D7311yescharacteristic
529C2×C4C2×#1192402711yescharacteristic
530C2×C4C2×#1192402811yescharacteristic
531C2×C4C2×#1192402911yescharacteristic
532C8—2921no—
533C8—2921no—
534C8—1464no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—2922no—
542C22—2922no—
543C22—2922no—
544C22—5841no—
545C22—5841no—
546C22—5841no—
547C22—5841no—
548C22#1798850914yes—
549C22C2×#1455656311yescharacteristic
550C22C2×#1455656511yescharacteristic
551C22C2×#1455656711yescharacteristic
552C22#1798939011yescharacteristic
553C22#1798939511yescharacteristic
554C22#1798939811yescharacteristic
555C22#1798940211yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—2922no—
562C4—5842no—
563C4#1798850914yes—
564C4C2×#1455656611yescharacteristic
565C4C2×#1455656911yescharacteristic
566C4C2×#1455657111yescharacteristic
567C4C2×#1455658611yescharacteristic
568C2—24no—
569C2—5842no—
570C2C2×#1798850911yescharacteristic
571C2#2261947011yescharacteristic
572C2#2261948111yescharacteristic
573C2#2261960711yescharacteristic
574C2#2261961811yescharacteristic
575C2#2261977811yescharacteristic
576C2#2261978911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2317C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
hypercenter#2317D73
hypocenterC73#23616
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#17988630C2semidirect
2. C2×#17988614C2semidirect
6. #22611323C2semidirect
7. #22611316C2semidirect
8. #22611309C2semidirect
9. #22611302C2semidirect
10. C2×#17988349C2semidirect
11. C73×#2317C2semidirect
293. #2317D73semidirect
300. C73#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#179886302
C2×#179886142
C2×#179885862
C2×#179885812
#226125192
#226113232
#226113162
#226113092
#226113022
C2×#179883492
C73×#23172
#2361673