Finite Group Databasebeta

← #33054532

Subgroups — #33054532

Order 18688.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2#22618561C211yesmaximal, characteristic
3#432×D73C211yesmaximal, characteristic
4#22608660C214yesmaximal, complemented
5#22608656C211yesmaximal, characteristic, complemented
6#22608656C211yesmaximal, characteristic, complemented
7#22608648C211yesmaximal, characteristic, complemented
8#22608587C214yesmaximal, complemented
9C73×#1556C211yesmaximal, characteristic, complemented
10#17989142—21no—
11#17989142—21no—
12#17989142—21no—
13#17989142—21no—
14#17989142C412yes—
15#17989142C412yes—
16#17989142C2212yes—
17#17989142C2212yes—
18C2×#14556591C2211yescharacteristic
19C2×#14556582—21no—
20C2×#14556571C412yes—
21C2×#14556567—21no—
22C2×Q8×Dic73—21no—
23C2×#14556556C412yes—
24C2×#14556549—21no—
25C2×C42×D73C2211yescharacteristic
26#17988644C2211yescharacteristic
27#17988644C2211yescharacteristic
28#17988643C2212yes—
29#17988642—21no—
30#17988642—21no—
31#17988641—21no—
32#17988639C2212yes—
33#17988637—21no—
34#17988609C2212yes—
35C4⋊C8×D73C2212yes—
36#17988549—21no—
37C22⋊C8×D73—21no—
38#17988539—21no—
39#97×D73—21no—
40#17988335C2214yescomplemented
41#17988335C2214yescomplemented
42#17988333C2214yescomplemented
43C2×C4⋊Q8×C73C2211yescharacteristic
44#432×C73C2211yescharacteristic
45#432×C73C2211yescharacteristic
46#332×C73C2214yescomplemented
47C22×Dic146C2×C412yes—
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73D411yescharacteristic
50C22×C4×D73D411yescharacteristic
51C2×#11924011—22no—
52C2×C8×D73—22no—
53#14556591—21no—
54#14556591—21no—
55#14556591C2×C412yes—
56#14556591C2312yes—
57#14556582—22no—
58#14556582—22no—
59#14556582—22no—
60#14556582—22no—
61#14556571—22no—
62#14556571—22no—
63#14556571C2×C414yes—
64#14556571C2×C414yes—
65#14556567—22no—
66#14556567—22no—
67#14556567—22no—
68#14556567—22no—
69Q8×Dic73—22no—
70Q8×Dic73—22no—
71Q8×Dic73—22no—
72Q8×Dic73—22no—
73#14556556—22no—
74#14556556—22no—
75#14556556C2×C414yes—
76#14556556C2×C414yes—
77#14556549—22no—
78#14556549—22no—
79#14556549—22no—
80#14556549—22no—
81C2×C4⋊Dic73—22no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×#11924007—22no—
85C2×#11924005—22no—
86C2×C4×Dic73—22no—
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73D411yescharacteristic
90C42×D73C2×C411yescharacteristic
91C42×D73C2×C411yescharacteristic
92C42×D73D411yescharacteristic
93C42×D73D411yescharacteristic
94C42×D73D411yescharacteristic
95C42×D73D411yescharacteristic
96C42×D73C2311yescharacteristic
97C42×D73C2311yescharacteristic
98C4⋊#10596831C2312yes—
99C4⋊#10596831C2312yes—
100#14556520—22no—
101#14556520—22no—
102C22⋊#10596831—21no—
103C22⋊#10596831—21no—
104#14556503—22no—
105#14556503—22no—
106#14556502—21no—
107#14556502—21no—
108#14556501—22no—
109C8×Dic73—22no—
110C22×Q8×C73—21no—
111C4⋊Q8×C73—21no—
112C4⋊Q8×C73—21no—
113C4⋊Q8×C73C2×C412yes—
114C4⋊Q8×C73C2312yes—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2×C412yes—
117C2×C42×C73C2311yescharacteristic
118C4⋊C8×C73C2312yes—
119C4⋊C8×C73C2312yes—
120C22⋊C8×C73—21no—
121C22⋊C8×C73—21no—
122#97×C73—21no—
123#97×C73—21no—
124C23×D73C22⋊C411yescharacteristic
125C2×Dic146—22no—
126C2×Dic146—44no—
127C2×Dic146C22×C414yes—
128C22×Dic73—22no—
129C22×Dic73C22⋊C411yescharacteristic
130C22×Dic73C22⋊C411yescharacteristic
131C22×Dic73C22⋊C411yescharacteristic
132C2×C4×D73—24no—
133C2×C4×D73—42no—
134C2×C4×D73—42no—
135C2×C4×D73C22⋊C411yescharacteristic
136C2×C4×D73C22⋊C411yescharacteristic
137C2×C4×D73C22⋊C411yescharacteristic
138C2×C4×D73C22⋊C411yescharacteristic
139C2×C4×D73C22⋊C411yescharacteristic
140C2×C4×D73C22⋊C411yescharacteristic
141C2×C4×D73C22×C411yescharacteristic
142C2×C4×D73C22×C411yescharacteristic
143C2×C4×D73C2×D411yescharacteristic
144C2×C4×D73C2×D411yescharacteristic
145C2×C4×D73C2×D411yescharacteristic
146C2×C4×D73C2×D411yescharacteristic
147#11924011—44no—
148C2×#10596831—22no—
149C2×#10596831—22no—
150C8×D73—44no—
151C4⋊Dic73—22no—
152C4⋊Dic73—22no—
153C4⋊Dic73—22no—
154C4⋊Dic73—22no—
155C4⋊Dic73—24no—
156C4⋊Dic73—24no—
157#11924007—24no—
158#11924007—24no—
159#11924005—24no—
160#11924005—24no—
161C4×Dic73—24no—
162C4×Dic73—24no—
163C4×Dic73C22⋊C411yescharacteristic
164C4×Dic73C22⋊C411yescharacteristic
165C4×Dic73C22⋊C411yescharacteristic
166C4×Dic73C22⋊C411yescharacteristic
167C4×Dic73C22⋊C411yescharacteristic
168C4×Dic73C22⋊C411yescharacteristic
169C4×Dic73C22×C411yescharacteristic
170C4×Dic73C22×C411yescharacteristic
171C4×Dic73C2×D411yescharacteristic
172C4×Dic73C2×D411yescharacteristic
173C4×Dic73C2×D411yescharacteristic
174C4×Dic73C2×D411yescharacteristic
175C2×Q8×C73—22no—
176C2×Q8×C73—22no—
177C2×Q8×C73—44no—
178C22×C4×C73—22no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C2×C8×C73—22no—
183C2×C8×C73—22no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73C22×C414yes—
188C42×C73C22×C411yescharacteristic
189C42×C73C2×D411yescharacteristic
190C42×C73C2×D411yescharacteristic
191C42×C73C2411yescharacteristic
192C23×C73C2×C22⋊C411yescharacteristic
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—41no—
196C22×D73C2×C22⋊C411yescharacteristic
197C22×D73C2×C22⋊C411yescharacteristic
198Q8×C73—44no—
199Q8×C73—44no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—24no—
204C2×C4×C73—44no—
205C2×C4×C73C2×C22⋊C411yescharacteristic
206C2×C4×C73C2×C22⋊C411yescharacteristic
207C2×C4×C73C2×C22⋊C411yescharacteristic
208C2×C4×C73C23×C411yescharacteristic
209C2×C4×C73C22×D411yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C2×Dic73—22no—
212C2×Dic73—22no—
213C2×Dic73—24no—
214C2×Dic73—41no—
215C2×Dic73—44no—
216C2×Dic73C2×C22⋊C411yescharacteristic
217C2×Dic73C2×C22⋊C411yescharacteristic
218C2×Dic73C2×C22⋊C411yescharacteristic
219C2×Dic73C2×C22⋊C411yescharacteristic
220C2×Dic73C2×C22⋊C411yescharacteristic
221C2×Dic73C2×C22⋊C411yescharacteristic
222C4×D73—22no—
223C4×D73—22no—
224C4×D73—22no—
225C4×D73—22no—
226C4×D73—22no—
227C4×D73—41no—
228C4×D73—41no—
229C4×D73—41no—
230C4×D73—41no—
231C4×D73—41no—
232C4×D73—41no—
233C4×D73#13714yes—
234Dic146—44no—
235Dic146—44no—
236C8×C73—42no—
237C8×C73—42no—
238#10596831—42no—
239#10596831—42no—
240C22×C73—21no—
241C22×C73—21no—
242C22×C73#41312yes—
243C22×C73C22×C22⋊C411yescharacteristic
244C2×D73—21no—
245C2×D73—21no—
246C2×D73—21no—
247C2×D73—21no—
248C2×D73—21no—
249C2×D73—44no—
250C2×D73#41912yes—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—21no—
254C4×C73—21no—
255C4×C73—44no—
256C4×C73C2×#13712yes—
257C4×C73#57512yes—
258Dic73—21no—
259Dic73—21no—
260Dic73—21no—
261Dic73—21no—
262Dic73—21no—
263Dic73—44no—
264Dic73#41912yes—
265#17734—731nomaximal, Sylow
266C2×C73—22no—
267C2×C73C2×#41311yescharacteristic
268C2×C73C2×#41911yescharacteristic
269C2×C73#289611yescharacteristic
270D73—22no—
271D73—41no—
272C2×#533—731no—
273#2920—731no—
274C2×#432—731no—
275#1559—734no—
276#1558—731no—
277#1558—731no—
278#1556—731no—
279#1556D7311yescharacteristic, complemented
280#1542—734no—
281C73#1773411yescharacteristic, complemented, Sylow
282#533—732no—
283#533—732no—
284#533—732no—
285#533—732no—
286#533—1461no—
287#533—1461no—
288#533—1461no—
289#533—1461no—
290C2×C4⋊Q8—731no—
291C2×C4⋊Q8C2×D7311yescharacteristic
292C2×#125—1461no—
293C2×C22⋊Q8—732no—
294C2×C22⋊Q8—1461no—
295C2×C4×Q8—732no—
296C2×C4×Q8—1461no—
297C2×#117—1461no—
298C22×C42—731no—
299#432—731no—
300#432—731no—
301#432—732no—
302#432—1461no—
303#432—1461no—
304#432—1461no—
305#432C2×D7311yescharacteristic
306#432C2×D7311yescharacteristic
307#431—732no—
308#431—1461no—
309#423—732no—
310C2×C4⋊C8—732no—
311#407—1461no—
312C2×C22⋊C8—1461no—
313C4×M16—1461no—
314C2×#97—1461no—
315#333—734no—
316#333—734no—
317#332—734no—
318#332C2×D7314yescomplemented
319C22×Q8—21no—
320C22×Q8—732no—
321C23×C4—731no—
322C23×C4—731no—
323C23×C4—731no—
324C2×M16—1462no—
325C22×C8—1462no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—732no—
329C4⋊Q8—732no—
330C4⋊Q8—1461no—
331C4⋊Q8—1461no—
332C4⋊Q8C4×D7312yes—
333C4⋊Q8C22×D7312yes—
334#125—1462no—
335#125—1462no—
336#125—1462no—
337#125—1462no—
338C22⋊Q8—734no—
339C22⋊Q8—734no—
340C22⋊Q8—1462no—
341C22⋊Q8—1462no—
342C22⋊Q8—1462no—
343C22⋊Q8—1462no—
344C22⋊Q8—1462no—
345C22⋊Q8—1462no—
346C4×Q8—734no—
347C4×Q8—734no—
348C4×Q8—1462no—
349C4×Q8—1462no—
350C4×Q8—1462no—
351C4×Q8—1462no—
352C4×Q8—1462no—
353C4×Q8—1462no—
354#117—1462no—
355#117—1462no—
356#117—1462no—
357#117—1462no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—1462no—
360C2×C4⋊C4—1462no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1461no—
363C2×C4⋊C4C4×D7312yes—
364C2×C22⋊C4—1462no—
365C2×C42—731no—
366C2×C42—731no—
367C2×C42—731no—
368C2×C42—731no—
369C2×C42—731no—
370C2×C42—731no—
371C2×C42—731no—
372C2×C42—731no—
373C2×C42—731no—
374C2×C42—731no—
375C2×C42—731no—
376C2×C42—1462no—
377C2×C42C22×D7311yescharacteristic
378C4⋊C8—732no—
379C4⋊C8—732no—
380C4⋊C8—1462no—
381C4⋊C8—1462no—
382C4⋊C8C22×D7312yes—
383C4⋊C8C22×D7312yes—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—1462no—
387C22⋊C8—1462no—
388C22⋊C8—1461no—
389C22⋊C8—1461no—
390#97—21no—
391#97—21no—
392#97—1462no—
393#97—1461no—
394#97—1461no—
395C4×C8—1462no—
396C24—731no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—2924no—
401C2×Q8—734no—
402C2×Q8—1462no—
403C22×C4—22no—
404C22×C4—2922no—
405C22×C4—2922no—
406C22×C4—731no—
407C22×C4—731no—
408C22×C4—731no—
409C22×C4—731no—
410C22×C4—731no—
411C22×C4—731no—
412C22×C4—731no—
413C22×C4—731no—
414C22×C4—731no—
415C22×C4—731no—
416C22×C4—731no—
417C22×C4—731no—
418C22×C4—731no—
419C22×C4—731no—
420C22×C4—731no—
421C22×C4—1464no—
422C22×C4—1462no—
423C22×C4C2×C4×D7311yescharacteristic
424C22×C4D4×D7311yescharacteristic
425C22×C4D4×D7311yescharacteristic
426M16—2924no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—2924no—
430C2×C8—1462no—
431C2×C8—1462no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—1464no—
436C4⋊C4—1464no—
437C4⋊C4—1464no—
438C4⋊C4—1464no—
439C4⋊C4—1462no—
440C4⋊C4—1462no—
441C4⋊C4—1462no—
442C4⋊C4—1462no—
443C4⋊C4C2×C4×D7314yes—
444C22⋊C4—1464no—
445C22⋊C4—1464no—
446C42—731no—
447C42—731no—
448C42—731no—
449C42—731no—
450C42—731no—
451C42—731no—
452C42—731no—
453C42—731no—
454C42—731no—
455C42—731no—
456C42—731no—
457C42—731no—
458C42—1464no—
459C42—1464no—
460C42C2×C4×D7311yescharacteristic
461C42D4×D7311yescharacteristic
462C42D4×D7311yescharacteristic
463C42C23×D7311yescharacteristic
464C23—2921no—
465C23—731no—
466C23—731no—
467C23—1462no—
468C23—1462no—
469C23C22⋊C4×D7311yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—2924no—
473Q8—2924no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—2921no—
480C2×C4—2921no—
481C2×C4—2921no—
482C2×C4—2921no—
483C2×C4—2921no—
484C2×C4—2921no—
485C2×C4—2924no—
486C2×C4—2921no—
487C2×C4—731no—
488C2×C4—734no—
489C2×C4—731no—
490C2×C4—731no—
491C2×C4—731no—
492C2×C4—731no—
493C2×C4—731no—
494C2×C4—1462no—
495C2×C4—1462no—
496C2×C4—1464no—
497C2×C4—1462no—
498C2×C4—1462no—
499C2×C4—1462no—
500C2×C4—1462no—
501C2×C4—1462no—
502C2×C4C22⋊C4×D7311yescharacteristic
503C2×C4C22⋊C4×D7311yescharacteristic
504C2×C4C22⋊C4×D7311yescharacteristic
505C2×C4C22×C4×D7311yescharacteristic
506C2×C4C2×D4×D7311yescharacteristic
507C2×C4C2×D4×D7311yescharacteristic
508C8—42no—
509C8—42no—
510C8—2922no—
511C8—2922no—
512C22—21no—
513C22—21no—
514C22—2924no—
515C22—732no—
516C22—1461no—
517C22—1461no—
518C22—1461no—
519C22—1461no—
520C22—1461no—
521C22#1798857012yes—
522C22C2×C22⋊C4×D7311yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—2924no—
529C4—732no—
530C4—1461no—
531C4—1461no—
532C4—1461no—
533C4—1461no—
534C4—1461no—
535C4#137×D7312yes—
536C4#1798938312yes—
537C2—22no—
538C2—2921no—
539C2—1462no—
540C2C2×#1798857011yescharacteristic
541C2#419×D7311yescharacteristic
542C2#2261844511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1556C2
socleC22×C73C22×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC42×C73C24
hypercenter#1556D73
hypocenterC73#17734
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. #22608660C2semidirect
5. #22608656C2semidirect
6. #22608656C2semidirect
7. #22608648C2semidirect
8. #22608587C2semidirect
9. C73×#1556C2semidirect
40. #17988335C22semidirect
41. #17988335C22semidirect
42. #17988333C22semidirect
46. #332×C73C22semidirect
279. #1556D73semidirect
281. C73#17734semidirect
318. #332C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226185612
#432×D732
#226086602
#226086562
#226086482
#226085872
C73×#15562
#1773473