Finite Group Databasebeta

← #33095697

Subgroups — #33095697

Order 18688.

1106 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989141C211yesmaximal, characteristic
2C2×#17988639C211yesmaximal, characteristic
3#22612337C211yesmaximal, characteristic
4#22610890C212yesmaximal, complemented
5#22610731C212yesmaximal, complemented
6#22610242C212yesmaximal
7#22610240C212yesmaximal
8#22609947C211yesmaximal, characteristic, complemented
9#22609946C211yesmaximal, characteristic, complemented
10#22609943C211yesmaximal, characteristic, complemented
11C73×#1805C211yesmaximal, characteristic, complemented
12#17989141—24no—
13#17989141C418yescomplemented
14C2×#14556590C2211yescharacteristic
15C2×#14556589C2211yescharacteristic
16C2×#14556581—21no—
17C2×#14556571C2211yescharacteristic
18C2×#14556569—21no—
19C2×#14556567C2211yescharacteristic
20C2×C4×Dic146—21no—
21C2×C4×Dic146C2211yescharacteristic
22C2×#14556549C2212yes—
23C2×#14556548—21no—
24C2×C42×D73C2211yescharacteristic
25#17988639—24no—
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556516C2212yes—
28C2×#14556515C2212yes—
29C2×#14556514C2212yes—
30C2×#14556503C2212yes—
31C2×C8×Dic73C2212yes—
32#17988456C2212yes—
33#17988453C2211yescharacteristic
34#17988446—21no—
35#17988446—21no—
36#17988438—21no—
37#17988438—21no—
38#17988350C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4⋊C8×C73C2211yescharacteristic
43C2×#103×C73C2212yes—
44#382×C73C2211yescharacteristic
45#340×C73C2212yes—
46C22×Dic146—21no—
47C22×Dic146C2311yescharacteristic
48C22×C4×D73D411yescharacteristic
49C22×C4×D73D411yescharacteristic
50C22×C4×D73C2311yescharacteristic
51C22×#10596831C2312yes—
52#14556590—22no—
53#14556590C2×C414yes—
54#14556589—22no—
55#14556589C2×C414yes—
56#14556581—28no—
57#14556571—22no—
58#14556571C2×C414yes—
59#14556569—28no—
60#14556567—22no—
61#14556567C2×C414yes—
62C4×Dic146—28no—
63C4×Dic146—22no—
64C4×Dic146C2×C414yes—
65#14556549—42no—
66#14556549—42no—
67#14556548—28no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2312yes—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—22no—
76C2×#11924007—22no—
77C2×#11924007—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—22no—
81C2×C4×Dic73—22no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C42×D73C2×C414yes—
87C42×D73D414yes—
88C4⋊#10596831—22no—
89#14556516—42no—
90#14556515—42no—
91#14556514—42no—
92#14556503—24no—
93C8×Dic73—24no—
94#14556498—21no—
95#14556498—21no—
96C22×Q8×C73C2311yescharacteristic
97C22×C8×C73C2312yes—
98C4×Q8×C73—22no—
99C4×Q8×C73C2×C414yes—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73C2312yes—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C42×C73—21no—
104C2×C42×C73C2311yescharacteristic
105C4⋊C8×C73—22no—
106#103×C73—42no—
107#95×C73—21no—
108#95×C73—21no—
109C23×D73C2×D411yescharacteristic
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—24no—
116C2×Dic146—44no—
117C2×Dic146C22×C412yes—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—22no—
121C22×Dic73C2×D411yescharacteristic
122C22×Dic73C2×D411yescharacteristic
123C22×Dic73C2×D411yescharacteristic
124C2×C4×D73—21no—
125C2×C4×D73—21no—
126C2×C4×D73—21no—
127C2×C4×D73—21no—
128C2×C4×D73—24no—
129C2×C4×D73—24no—
130C2×C4×D73C22⋊C414yes—
131C2×C4×D73C22⋊C414yes—
132C2×C4×D73C22×C412yes—
133C2×C4×D73C2×D412yes—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C4⋊Dic73—21no—
138C4⋊Dic73—24no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141C4⋊Dic73C22×C412yes—
142#11924007—24no—
143#11924007—24no—
144#11924007—24no—
145#11924007—24no—
146#11924007—24no—
147#11924007—44no—
148#11924007—44no—
149#11924005—24no—
150#11924005—24no—
151#11924005—44no—
152C4×Dic73—24no—
153C4×Dic73—44no—
154C4×Dic73C22⋊C414yes—
155C4×Dic73C22⋊C414yes—
156C4×Dic73C22×C412yes—
157C4×Dic73C2×D412yes—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73C22×C412yes—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2411yescharacteristic
170C2×C8×C73—22no—
171C2×C8×C73—22no—
172C2×C8×C73—22no—
173C4⋊C4×C73—24no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73C22×C412yes—
178C42×C73—24no—
179C42×C73C22×C412yes—
180C42×C73C2×D412yes—
181C23×C73C22×D411yescharacteristic
182C22×D73—21no—
183C22×D73—21no—
184C22×D73—22no—
185C22×D73—22no—
186C22×D73C2×C22⋊C412yes—
187Q8×C73—22no—
188Q8×C73—22no—
189Q8×C73—41no—
190Q8×C73—41no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—42no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73C2×C22⋊C412yes—
202C2×C4×C73C2×C22⋊C412yes—
203C2×C4×C73C23×C411yescharacteristic
204C2×C4×C73C22×D411yescharacteristic
205C2×C4×C73H(5,𝔽2)11yescharacteristic
206C2×C4×C73H(5,𝔽2)11yescharacteristic
207C2×C4×C732−1+411yescharacteristic
208C2×C4×C732−1+411yescharacteristic
209C2×Dic73—21no—
210C2×Dic73—21no—
211C2×Dic73—22no—
212C2×Dic73—22no—
213C2×Dic73—41no—
214C2×Dic73—41no—
215C2×Dic73—42no—
216C2×Dic73—42no—
217C2×Dic73—42no—
218C2×Dic73—42no—
219C2×Dic73C2×C22⋊C412yes—
220C2×Dic73C2×C22⋊C412yes—
221C2×Dic73C2×C22⋊C412yes—
222C2×Dic73#13514yes—
223C2×Dic73#13714yes—
224C4×D73—22no—
225C4×D73—22no—
226C4×D73—22no—
227C4×D73—22no—
228C4×D73—28no—
229C4×D73—28no—
230Dic146—22no—
231Dic146—22no—
232Dic146—41no—
233Dic146—41no—
234Dic146—48no—
235C8×C73—42no—
236#10596831—42no—
237C22×C73C22×C22⋊C411yescharacteristic
238C22×C73#53511yescharacteristic
239C22×C73#53611yescharacteristic
240C22×C73C2×#13512yes—
241C22×C73C2×#13712yes—
242C2×D73—22no—
243C2×D73—22no—
244C2×D73—22no—
245C2×D73—42no—
246C2×D73—42no—
247C4×C73—22no—
248C4×C73—22no—
249C4×C73—42no—
250C4×C73—42no—
251C4×C73—82no—
252C4×C73#52011yescharacteristic
253C4×C73#52011yescharacteristic
254C4×C73#53711yescharacteristic
255C4×C73#53711yescharacteristic
256Dic73—22no—
257Dic73—42no—
258Dic73—42no—
259Dic73—82no—
260Dic73#41614yes—
261Dic73#41914yes—
262#21864—731nomaximal, Sylow
263C2×C73#231111yescharacteristic
264C2×C73C2×#41611yescharacteristic
265C2×C73C2×#41911yescharacteristic
266C2×C73#321012yes—
267C2×C73#323112yes—
268D73—42no—
269C2×#533—731no—
270C2×#431—731no—
271#2285—731no—
272#1990—732no—
273#1956—732no—
274#1873—732no—
275#1873—732no—
276#1805—731no—
277#1805—731no—
278#1805—731no—
279#1805D7311yescharacteristic, complemented
280C73#2186411yescharacteristic, complemented, Sylow
281#533—738no—
282#533—1464no—
283C2×C4⋊Q8—731no—
284C2×C4⋊Q8—731no—
285C2×#125—1461no—
286C2×C22⋊Q8—731no—
287C2×C22⋊Q8—731no—
288C2×C22⋊Q8—1461no—
289C2×C4×Q8—731no—
290C2×C4×Q8—1461no—
291C2×C4×Q8C2×D7311yescharacteristic
292C2×#117—732no—
293C2×#117—1461no—
294C22×C42—731no—
295#431—1464no—
296C2×C4⋊C8—731no—
297C2×C4⋊C8C2×D7311yescharacteristic
298C2×#103—732no—
299C2×#103—732no—
300C2×#103—732no—
301C2×#103C2×D7312yes—
302C2×C22⋊C8—732no—
303C2×C4×C8—732no—
304#384—732no—
305#382—731no—
306#382C2×D7311yescharacteristic
307#380—1461no—
308#380—1461no—
309C4×C4⋊C4—1461no—
310C4×C4⋊C4—1461no—
311#340—732no—
312#340—732no—
313#340—732no—
314#340C2×D7312yes—
315C22×Q8—731no—
316C22×Q8—1461no—
317C22×Q8C22×D7311yescharacteristic
318C23×C4—731no—
319C23×C4—731no—
320C23×C4—731no—
321C22×C8—732no—
322C22×C8C22×D7312yes—
323C4⋊Q8—734no—
324C4⋊Q8—734no—
325C4⋊Q8—1462no—
326C4⋊Q8—1462no—
327#125—1468no—
328C22⋊Q8—734no—
329C22⋊Q8—734no—
330C22⋊Q8—1462no—
331C22⋊Q8—1468no—
332C22⋊Q8—1462no—
333C4×Q8—22no—
334C4×Q8—734no—
335C4×Q8—1468no—
336C4×Q8—1462no—
337C4×Q8C4×D7314yes—
338#117—2922no—
339#117—2922no—
340#117—1468no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—731no—
343C2×C4⋊C4—732no—
344C2×C4⋊C4—1462no—
345C2×C4⋊C4—1462no—
346C2×C4⋊C4—1461no—
347C2×C4⋊C4—1461no—
348C2×C4⋊C4—1461no—
349C2×C4⋊C4—1461no—
350C2×C4⋊C4—1461no—
351C2×C4⋊C4—1461no—
352C2×C4⋊C4C22×D7312yes—
353C2×C4⋊C4C22×D7311yescharacteristic
354C2×C22⋊C4—1462no—
355C2×C22⋊C4—1461no—
356C2×C22⋊C4—1461no—
357C2×C42—21no—
358C2×C42—731no—
359C2×C42—731no—
360C2×C42—731no—
361C2×C42—734no—
362C2×C42—734no—
363C2×C42—1462no—
364C2×C42—1461no—
365C2×C42C22×D7311yescharacteristic
366C4⋊C8—22no—
367C4⋊C8—1462no—
368#103—42no—
369#103—2922no—
370#103—2922no—
371#103—2922no—
372C22⋊C8—1464no—
373C4×C8—1464no—
374#95—21no—
375#95—21no—
376#95—1461no—
377#95—1461no—
378C24—731no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—2924no—
385C2×Q8—732no—
386C2×Q8—1464no—
387C2×Q8—1461no—
388C2×Q8—1461no—
389C2×Q8—1461no—
390C2×Q8—1461no—
391C2×Q8—1461no—
392C2×Q8C2×C4×D7312yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—731no—
397C22×C4—731no—
398C22×C4—731no—
399C22×C4—734no—
400C22×C4—734no—
401C22×C4—732no—
402C22×C4—732no—
403C22×C4—1464no—
404C22×C4—1464no—
405C22×C4—1462no—
406C22×C4—1461no—
407C22×C4—1461no—
408C22×C4—1461no—
409C22×C4—1461no—
410C22×C4—1461no—
411C22×C4—1461no—
412C22×C4D4×D7311yescharacteristic
413C22×C4D4×D7311yescharacteristic
414C22×C4C23×D7311yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1462no—
419C2×C8—1462no—
420C2×C8—1462no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—2924no—
426C4⋊C4—2924no—
427C4⋊C4—732no—
428C4⋊C4—1464no—
429C4⋊C4—1464no—
430C4⋊C4—1464no—
431C4⋊C4—1464no—
432C4⋊C4—1464no—
433C4⋊C4—1462no—
434C4⋊C4—1462no—
435C4⋊C4—1464no—
436C4⋊C4—1461no—
437C4⋊C4C2×C4×D7312yes—
438C22⋊C4—2924no—
439C22⋊C4—1464no—
440C22⋊C4—1464no—
441C42—24no—
442C42—2924no—
443C42—734no—
444C42—734no—
445C42—732no—
446C42—732no—
447C42—1464no—
448C42C2×C4×D7312yes—
449C42D4×D7312yes—
450C23—732no—
451C23—1462no—
452C23—1461no—
453C23—1461no—
454C23—1462no—
455C23C2×D4×D7311yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—2928no—
461Q8—2921no—
462Q8—2921no—
463Q8—1462no—
464Q8—1462no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—2921no—
476C2×C4—2921no—
477C2×C4—2922no—
478C2×C4—2922no—
479C2×C4—2922no—
480C2×C4—2922no—
481C2×C4—734no—
482C2×C4—732no—
483C2×C4—732no—
484C2×C4—734no—
485C2×C4—732no—
486C2×C4—1468no—
487C2×C4—1468no—
488C2×C4—1462no—
489C2×C4—1462no—
490C2×C4—1461no—
491C2×C4—1461no—
492C2×C4—1462no—
493C2×C4—1462no—
494C2×C4—1462no—
495C2×C4—1462no—
496C2×C4C22⋊C4×D7312yes—
497C2×C4C22⋊C4×D7312yes—
498C2×C4C22×C4×D7311yescharacteristic
499C2×C4C2×D4×D7311yescharacteristic
500C2×C4#1455663611yescharacteristic
501C2×C4#1455663711yescharacteristic
502C2×C4#1455663811yescharacteristic
503C2×C4#1455663911yescharacteristic
504C8—42no—
505C8—2922no—
506C22—2922no—
507C22—2922no—
508C22—1462no—
509C22—1462no—
510C22—1462no—
511C22C2×C22⋊C4×D7311yescharacteristic
512C22#1798915311yescharacteristic
513C22#1798916011yescharacteristic
514C22#135×D7312yes—
515C22#137×D7312yes—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—2922no—
522C4—2922no—
523C4—5842no—
524C4—734no—
525C4—734no—
526C4—1462no—
527C4#1798906711yescharacteristic
528C4#1798906811yescharacteristic
529C4#1798916611yescharacteristic
530C4#1798916711yescharacteristic
531C2—2922no—
532C2#2261252011yescharacteristic
533C2#416×D7311yescharacteristic
534C2#419×D7311yescharacteristic
535C2#2262088612yes—
536C2#2262108912yes—
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1805C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1805D73
hypocenterC73#21864
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #22610890C2semidirect
5. #22610731C2semidirect
8. #22609947C2semidirect
9. #22609946C2semidirect
10. #22609943C2semidirect
11. C73×#1805C2semidirect
13. #17989141C4semidirect
279. #1805D73semidirect
280. C73#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#179891412
C2×#179886392
#226123372
#226108902
#226107312
#226102422
#226102402
#226099472
#226099462
#226099432
C73×#18052
#2186473