Finite Group Databasebeta

← #33119828

Subgroups — #33119828

Order 18688.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988915C211yesmaximal, characteristic, complemented
2C2×#17988630C211yesmaximal, characteristic, complemented
3#22614248C211yesmaximal, characteristic
4#22611266C212yesmaximal, complemented
5#22611131C212yesmaximal, complemented
6#22610365C211yesmaximal, characteristic, complemented
7#22610229C211yesmaximal, characteristic
8#22610199C212yesmaximal, complemented
9#22610124C212yesmaximal, complemented
10#22610026C211yesmaximal, characteristic, complemented
11C2×#552×C73C211yesmaximal, characteristic
12C22×#11924012C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14#17988915—24no—
15#17988630—24no—
16C2×#14556524C2212yes—
17C2×#14556522C2212yes—
18C2×#14556522C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×C22⋊#10596831C2211yescharacteristic
22C2×C22⋊#10596831C2212yes—
23#17988526—21no—
24#17988523—21no—
25#17988485C2212yes—
26#17988467—21no—
27#17988459—21no—
28#17988447C2211yescharacteristic
29#17988447C2212yes—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#552×C73—24no—
35#552×C73—24no—
36C2×#126×C73—21no—
37C2×#125×C73—21no—
38C2×#125×C73C2211yescharacteristic
39C2×#123×C73—21no—
40C2×C22⋊Q8×C73C2211yescharacteristic
41C2×C22⋊Q8×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×C4×D4×C73C2211yescharacteristic
44C2×#117×C73—21no—
45C2×#117×C73C2211yescharacteristic
46C22×C4⋊C4×C73C2211yescharacteristic
47C23×Dic73—21no—
48C2×#11924012—24no—
49C2×#11924012—22no—
50C22×#10596831C2312yes—
51C22×#10596831C2312yes—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—21no—
55C2×C4⋊Dic73C2312yes—
56C2×C4⋊Dic73C2312yes—
57C2×#11924007—22no—
58C2×C22⋊Dic73—21no—
59C2×C22⋊Dic73—22no—
60#14556524—24no—
61#14556522—24no—
62#14556522—24no—
63#14556515—24no—
64#14556515—24no—
65C22⋊#10596831—24no—
66C22⋊#10596831—22no—
67#14556498—22no—
68#14556498—21no—
69#14556498—22no—
70C22×Q8×C73C2311yescharacteristic
71C22×D4×C73D411yescharacteristic
72C23×C4×C73—21no—
73C23×C4×C73C2311yescharacteristic
74C23×C4×C73C2311yescharacteristic
75#126×C73—28no—
76#125×C73—28no—
77#125×C73—22no—
78#125×C73—22no—
79#123×C73—28no—
80C22⋊Q8×C73—22no—
81C22⋊Q8×C73—22no—
82C22⋊Q8×C73—22no—
83C22⋊Q8×C73D414yes—
84C4×Q8×C73—22no—
85C4×Q8×C73—22no—
86C4×D4×C73—22no—
87C4×D4×C73—22no—
88#117×C73—28no—
89#117×C73—22no—
90#117×C73D414yes—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73D411yescharacteristic
107C2×C22⋊C4×C73D411yescharacteristic
108C2×C42×C73—21no—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—42no—
114C22×Dic73—82no—
115#11924012—28no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C2×#10596831#4114yes—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73—22no—
126C4⋊Dic73—42no—
127#11924007—82no—
128C22⋊Dic73—42no—
129C22⋊Dic73—82no—
130C24×C73C2×D411yescharacteristic
131C24×C73C2×D411yescharacteristic
132C2×Q8×C73—21no—
133C2×Q8×C73—22no—
134C2×Q8×C73—22no—
135C2×Q8×C73C2×D412yes—
136C2×D4×C73—22no—
137C2×D4×C73—22no—
138C2×D4×C73—41no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—24no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—42no—
151C22×C4×C73—42no—
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D412yes—
157C22×C4×C73C2×D412yes—
158C22×C4×C73C2411yescharacteristic
159C4⋊C4×C73—24no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73—24no—
163C4⋊C4×C73—24no—
164C4⋊C4×C73—24no—
165C4⋊C4×C73—22no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73C2×D412yes—
170C4⋊C4×C73C2×D412yes—
171C4⋊C4×C73C2×D412yes—
172C22⋊C4×C73—24no—
173C22⋊C4×C73—24no—
174C22⋊C4×C73—24no—
175C22⋊C4×C73—22no—
176C22⋊C4×C73—22no—
177C42×C73—24no—
178C42×C73—24no—
179C42×C73—21no—
180C42×C73—21no—
181C23×C73—21no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—21no—
185C23×C73—21no—
186C23×C73—21no—
187C23×C73—22no—
188C23×C73—22no—
189C23×C73#12012yes—
190C23×C73#13712yes—
191C23×C73C22×D411yescharacteristic
192Q8×C73—24no—
193Q8×C73—41no—
194Q8×C73—41no—
195D4×C73—44no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—24no—
200C2×C4×C73—21no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—22no—
211C2×C4×C73—48no—
212C2×C4×C73—42no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—42no—
221C2×C4×C73#12012yes—
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C2×#4112yes—
225C2×C4×C732−1+412yes—
226C2×Dic73—42no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—84no—
233#10596831—24no—
234#10596831—42no—
235C22×C73—24no—
236C22×C73—22no—
237C22×C73—22no—
238C22×C73—22no—
239C22×C73—22no—
240C22×C73—42no—
241C22×C73—42no—
242C22×C73—42no—
243C22×C73—41no—
244C22×C73—41no—
245C22×C73C2×#12011yescharacteristic
246C22×C73#54712yes—
247C22×C73C2×#13711yescharacteristic
248C22×C73#57511yescharacteristic
249C22×C73#57712yes—
250C4×C73—22no—
251C4×C73—41no—
252C4×C73—42no—
253C4×C73—41no—
254C4×C73—42no—
255C4×C73—42no—
256C4×C73—41no—
257C4×C73—42no—
258C4×C73#52511yescharacteristic
259C4×C73#53611yescharacteristic
260C4×C73#54712yes—
261Dic73—82no—
262Dic73—82no—
263#24111—731nomaximal, Sylow
264C2×C73—42no—
265C2×C73—42no—
266C2×C73#240611yescharacteristic
267C2×C73#300711yescharacteristic
268C2×C73#301011yescharacteristic
269C2×C73#334412yes—
270C2×C73#335112yes—
271C2×#552D7311yescharacteristic
272C2×#483—731no—
273C2×#428—731no—
274#2487—731no—
275#2059—732no—
276#2036—732no—
277#1894—731no—
278#1870—731no—
279#1865—732no—
280#1853—732no—
281#1825—731no—
282C73#2411111yescharacteristic, complemented, Sylow
283C22×M16—731no—
284#552—24no—
285#552—24no—
286C2×#126—21no—
287C2×#125—21no—
288C2×#125C2×D7311yescharacteristic
289C2×#123—21no—
290C2×C22⋊Q8C2×D7311yescharacteristic
291C2×C22⋊Q8C2×D7311yescharacteristic
292C2×C4×Q8C2×D7311yescharacteristic
293C2×C4×D4C2×D7311yescharacteristic
294C2×#117—21no—
295C2×#117C2×D7311yescharacteristic
296C22×C4⋊C4—731no—
297C22×C4⋊C4C2×D7311yescharacteristic
298#483—1464no—
299#428—1464no—
300C2×#107—732no—
301C2×#106—732no—
302C2×#106—732no—
303C2×#103—732no—
304C2×#103—732no—
305C2×C22⋊C8—731no—
306C2×C22⋊C8—732no—
307#400—1461no—
308#399—1461no—
309#389—732no—
310#386—1461no—
311#385—1461no—
312#380—731no—
313#380—732no—
314#340—732no—
315#340—732no—
316#340—732no—
317#336—732no—
318C22×Q8C22×D7311yescharacteristic
319C22×D4C22⋊D7311yescharacteristic
320C23×C4—21no—
321C23×C4—1461no—
322C23×C4C22×D7311yescharacteristic
323C23×C4C22×D7311yescharacteristic
324C2×M16—1464no—
325C2×M16—1462no—
326C22×C8—732no—
327C22×C8—732no—
328#126—28no—
329#125—28no—
330#125—22no—
331#125—22no—
332#123—28no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8C22⋊D7314yes—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—28no—
342#117—22no—
343#117C22⋊D7314yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—732no—
353C2×C4⋊C4—732no—
354C2×C4⋊C4—1461no—
355C2×C4⋊C4—1462no—
356C2×C4⋊C4—1462no—
357C2×C4⋊C4—1462no—
358C2×C4⋊C4C22⋊D7311yescharacteristic
359C2×C4⋊C4C22×D7311yescharacteristic
360C2×C4⋊C4C22×D7311yescharacteristic
361C2×C4⋊C4C22×D7311yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—1461no—
366C2×C22⋊C4—1462no—
367C2×C22⋊C4C22⋊D7311yescharacteristic
368C2×C22⋊C4C22⋊D7311yescharacteristic
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C22×D7311yescharacteristic
372#107—1464no—
373#106—1464no—
374#106—1464no—
375#103—1464no—
376#103—1464no—
377C22⋊C8—1464no—
378C22⋊C8—1462no—
379#95—1462no—
380#95—1461no—
381#95—1462no—
382C24C2×C22⋊D7311yescharacteristic
383C24C2×C22⋊D7311yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D7312yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—2922no—
405C22×C4—5842no—
406C22×C4—1462no—
407C22×C4—1462no—
408C22×C4C2×C22⋊D7311yescharacteristic
409C22×C4C2×C22⋊D7311yescharacteristic
410C22×C4C2×C22⋊D7311yescharacteristic
411C22×C4C2×C22⋊D7311yescharacteristic
412C22×C4C2×C22⋊D7312yes—
413C22×C4C2×C22⋊D7312yes—
414C22×C4C23×D7311yescharacteristic
415M16—1468no—
416C2×C8—734no—
417C2×C8—1462no—
418C2×C8—1462no—
419C2×C8—1462no—
420C2×C8—1462no—
421C2×C8—1462no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—2922no—
433C4⋊C4—5842no—
434C4⋊C4—1462no—
435C4⋊C4—1462no—
436C4⋊C4—1462no—
437C4⋊C4—1462no—
438C4⋊C4C2×C22⋊D7312yes—
439C4⋊C4C2×C22⋊D7312yes—
440C4⋊C4C2×C22⋊D7312yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—2922no—
447C22⋊C4—5842no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—21no—
460C23#1455656012yes—
461C23#1455662112yes—
462C23C22×C22⋊D7311yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—2922no—
493C2×C4—2922no—
494C2×C4—2922no—
495C2×C4—2922no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—5844no—
499C2×C4#1455656012yes—
500C2×C4C22×C22⋊D7311yescharacteristic
501C2×C4C22×C22⋊D7311yescharacteristic
502C2×C4#41×D7312yes—
503C2×C4#1455663812yes—
504C8—2922no—
505C8—1464no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22C2×#1455656011yescharacteristic
517C22#1798923612yes—
518C22C2×#1455662111yescharacteristic
519C22#1798938511yescharacteristic
520C22#1798939512yes—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—5842no—
530C4—5842no—
531C4#1798909411yescharacteristic
532C4#1798915811yescharacteristic
533C4#1798923612yes—
534C2—42no—
535C2—42no—
536C2#2261326211yescharacteristic
537C2#2261919111yescharacteristic
538C2#2261921711yescharacteristic
539C2#2262238612yes—
540C2#2262249112yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#552×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterC2×#552D73
hypocenterC73#24111
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#17988915C2semidirect
2. C2×#17988630C2semidirect
4. #22611266C2semidirect
5. #22611131C2semidirect
6. #22610365C2semidirect
8. #22610199C2semidirect
9. #22610124C2semidirect
10. #22610026C2semidirect
282. C73#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#179889152
C2×#179886302
#226142482
#226112662
#226111312
#226103652
#226102292
#226101992
#226101242
#226100262
C2×#552×C732
#2411173