Finite Group Databasebeta

← #33091052

Subgroups — #33091052

Order 18688.

1290 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989123C211yesmaximal, characteristic
2C2×#17988637C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic73C211yesmaximal, characteristic, complemented
4#22610885C212yesmaximal, complemented
5#22610860C212yesmaximal, complemented
6#22610248C211yesmaximal, characteristic
7#22610246C211yesmaximal, characteristic
8#22610239C211yesmaximal, characteristic, complemented
9#103×Dic73C212yesmaximal, complemented
10#22609745C212yesmaximal, complemented
11C73×#1872C211yesmaximal, characteristic, complemented
12#17989123—24no—
13#17989123C418yescomplemented
14C2×#14556586—21no—
15C2×#14556582C2211yescharacteristic
16C2×#14556580C2211yescharacteristic
17C2×#14556575—21no—
18C2×#14556565C2211yescharacteristic
19C2×Q8×Dic73C2211yescharacteristic
20C2×#14556553C2211yescharacteristic
21C2×#14556551—21no—
22C2×#14556549—21no—
23C22×C4⋊Dic73C2211yescharacteristic
24C22×C4×Dic73—21no—
25C22×C4×Dic73C2211yescharacteristic
26#17988637—24no—
27C2×#14556520C2212yes—
28C2×#14556515C2212yes—
29C2×#14556513C2212yes—
30C2×#14556511C2212yes—
31C2×C22⋊#10596831C2211yescharacteristic
32C2×C8×Dic73C2212yes—
33C4⋊C4×Dic73—21no—
34C4⋊C4×Dic73—21no—
35C4⋊C4×Dic73C2212yes—
36C4⋊C4×Dic73C2212yes—
37C2×#14556498—21no—
38#17988350C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×C22⋊Q8×C73C2211yescharacteristic
42C22×C4⋊C4×C73C2211yescharacteristic
43C2×#103×C73C2212yes—
44C2×C22⋊C8×C73C2211yescharacteristic
45#340×C73C2212yes—
46C22×D292C2311yescharacteristic
47C22×C22⋊D73—21no—
48C23×Dic73—21no—
49C23×Dic73D411yescharacteristic
50C23×Dic73D411yescharacteristic
51C22×C4×D73—21no—
52C22×#10596831C2312yes—
53#14556586—28no—
54#14556582—22no—
55#14556582C2×C414yescomplemented
56#14556580—22no—
57#14556580C2×C414yescomplemented
58#14556575—28no—
59#14556565—22no—
60#14556565C2×C414yes—
61Q8×Dic73—22no—
62Q8×Dic73C2×C414yescomplemented
63#14556553—22no—
64#14556553C2×C414yescomplemented
65#14556551—28no—
66#14556549—28no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—24no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2312yes—
73C2×#11924007—21no—
74C2×#11924005—21no—
75C2×#11924005—21no—
76C2×#11924005—21no—
77C2×#11924005—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—22no—
80C2×C4×Dic73—22no—
81C2×C4×Dic73—44no—
82C2×C4×Dic73C2×C414yes—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73D414yes—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556520—24no—
89#14556515—24no—
90#14556513—24no—
91#14556511—24no—
92C22⋊#10596831—22no—
93C8×Dic73—24no—
94#14556498—22no—
95#14556498—22no—
96C22×Q8×C73C2311yescharacteristic
97C23×C4×C73—21no—
98C23×C4×C73C2311yescharacteristic
99C22×C8×C73C2312yes—
100C22⋊Q8×C73—22no—
101C22⋊Q8×C73C2×C414yes—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—24no—
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2312yes—
107C2×C22⋊C4×C73—21no—
108#103×C73—24no—
109C22⋊C8×C73—22no—
110C23×D73—21no—
111C2×D292—22no—
112C2×D292—22no—
113C2×D292—21no—
114C2×D292C22×C412yes—
115C2×C22⋊D73—24no—
116C2×C22⋊D73—42no—
117C2×C22⋊D73—42no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—24no—
123C22×Dic73—22no—
124C22×Dic73—21no—
125C22×Dic73—42no—
126C22×Dic73—42no—
127C22×Dic73C22⋊C414yes—
128C22×Dic73C22⋊C414yes—
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C2×C4×D73—24no—
134C2×C4×D73—42no—
135C2×C4×D73—42no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C4⋊Dic73—24no—
140C4⋊Dic73—21no—
141C4⋊Dic73—22no—
142C4⋊Dic73—24no—
143C4⋊Dic73—24no—
144C4⋊Dic73C22×C412yes—
145C4⋊Dic73#4114yes—
146#11924007—24no—
147#11924005—24no—
148#11924005—24no—
149#11924005—24no—
150#11924005—24no—
151C4×Dic73—24no—
152C4×Dic73—44no—
153C4×Dic73—44no—
154C4×Dic73C22⋊C414yes—
155C4×Dic73C22⋊C414yes—
156C4×Dic73C22×C412yes—
157C4×Dic73C22×C412yes—
158C4×Dic73C2×D412yes—
159C4×Dic73C2×D412yes—
160C24×C73C2×D411yescharacteristic
161C2×Q8×C73—21no—
162C2×Q8×C73—22no—
163C2×Q8×C73—22no—
164C2×Q8×C73C22×C412yes—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—24no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—22no—
171C22×C4×C73—42no—
172C22×C4×C73—42no—
173C22×C4×C73C22×C412yes—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2411yescharacteristic
177C2×C8×C73—22no—
178C2×C8×C73—22no—
179C2×C8×C73—22no—
180C4⋊C4×C73—24no—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73C22×C412yes—
185C4⋊C4×C73#4114yes—
186C22⋊C4×C73—24no—
187C23×C73—21no—
188C23×C73—21no—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73C2×C22⋊C412yes—
192C23×C73#13512yes—
193C23×C73#13512yes—
194C23×C73C22×D411yescharacteristic
195C22×D73—22no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202Q8×C73—24no—
203Q8×C73—42no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—24no—
211C2×C4×C73—48no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73C2×C22⋊C412yes—
219C2×C4×C73C23×C411yescharacteristic
220C2×C4×C73C22×D411yescharacteristic
221C2×C4×C73C2×#4112yes—
222C2×C4×C73C2×#4112yes—
223C22⋊D73—48no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—28no—
231C2×Dic73—28no—
232C2×Dic73—22no—
233C2×Dic73—24no—
234C2×Dic73—48no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—42no—
239C2×Dic73C2×C22⋊C412yes—
240C2×Dic73C2×C22⋊C412yes—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243C2×Dic73AGL(1,ℤ/8)12yes—
244C2×Dic73#13712yes—
245D292—24no—
246D292—42no—
247C4×D73—48no—
248C8×C73—42no—
249#10596831—42no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73#41614yes—
257C22×C73C22×C22⋊C411yescharacteristic
258C22×C73#52511yescharacteristic
259C22×C73#52511yescharacteristic
260C22×C73C2×#13511yescharacteristic
261C22×C73C2×#13511yescharacteristic
262C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C73C2×#13711yescharacteristic
264C2×D73—42no—
265C2×D73—42no—
266C2×D73—42no—
267C2×D73—81no—
268C2×D73—81no—
269C2×D73—81no—
270C2×D73—81no—
271C4×C73—22no—
272C4×C73—44no—
273C4×C73—42no—
274C4×C73—42no—
275C4×C73C4×#4112yes—
276C4×C73#53212yes—
277Dic73—22no—
278Dic73—22no—
279Dic73—22no—
280Dic73—44no—
281Dic73—42no—
282Dic73#41714yes—
283#21388—731nomaximal, Sylow
284C2×C73—24no—
285C2×C73#229711yescharacteristic
286C2×C73C2×#41611yescharacteristic
287C2×C73C2×#41711yescharacteristic
288C2×C73#310511yescharacteristic
289C2×C73#310511yescharacteristic
290C2×C73#310811yescharacteristic
291C2×C73#310811yescharacteristic
292D73—82no—
293C2×#529—731no—
294C2×#431—731no—
295C2×C4×C4⋊C4—731no—
296#1989—732no—
297#1983—732no—
298#1874—731no—
299#1874—731no—
300#1872—731no—
301#1872D7311yescharacteristic, complemented
302C4×#103—732no—
303C4×#102—732no—
304C73#2138811yescharacteristic, complemented, Sylow
305#529—738no—
306#529—1464no—
307C2×#126—1461no—
308C2×#125—731no—
309C2×#124—731no—
310C2×#123—1461no—
311C2×C22⋊Q8C2×D7311yescharacteristic
312C2×#121—731no—
313C2×C4×Q8—731no—
314C2×C4×D4—731no—
315C2×C4×D4—1461no—
316C2×#117—1461no—
317C22×C4⋊C4—731no—
318C22×C4⋊C4C2×D7311yescharacteristic
319C22×C42—731no—
320C22×C42—1461no—
321#431—1464no—
322C2×C4⋊C8—732no—
323C2×#103—732no—
324C2×#103C2×D7312yes—
325C2×#102—732no—
326C2×#102—732no—
327C2×C22⋊C8—731no—
328C2×C22⋊C8C2×D7311yescharacteristic
329C2×C4×C8—732no—
330C4×C4⋊C4—732no—
331C4×C4⋊C4—732no—
332C4×C4⋊C4—1461no—
333C4×C4⋊C4—1461no—
334C2×#95—1461no—
335#340—732no—
336#340—732no—
337#340—732no—
338#340C2×D7312yes—
339C22×Q8C22×D7311yescharacteristic
340C22×D4—731no—
341C22×D4—1461no—
342C23×C4—21no—
343C23×C4—731no—
344C23×C4—731no—
345C23×C4—1461no—
346C23×C4—1461no—
347C23×C4C22×D7311yescharacteristic
348C22×C8—732no—
349C22×C8C22×D7312yes—
350#126—1468no—
351#125—734no—
352#125—1462no—
353#124—734no—
354#124—1462no—
355#123—1468no—
356C22⋊Q8—22no—
357C22⋊Q8C4×D7314yes—
358#121—734no—
359#121—1462no—
360C4×Q8—734no—
361C4×Q8—1462no—
362C4×D4—734no—
363C4×D4—1462no—
364C4×D4—1468no—
365#117—1468no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—732no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7312yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C42—2924no—
384C2×C42—734no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—734no—
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—1462no—
391C2×C42—1462no—
392C2×C42—1461no—
393C4⋊C8—1464no—
394#103—24no—
395#103—1464no—
396#102—1464no—
397#102—1464no—
398C22⋊C8—22no—
399C22⋊C8—1462no—
400C4×C8—1464no—
401#95—1462no—
402#95—1462no—
403C24—1461no—
404C24D4×D7311yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×C4×D7312yes—
409C2×D4—2922no—
410C2×D4—2922no—
411C2×D4—732no—
412C2×D4—1464no—
413C2×D4—1461no—
414C2×D4—1462no—
415C2×D4—1462no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—2922no—
425C22×C4—2922no—
426C22×C4—2922no—
427C22×C4—2922no—
428C22×C4—731no—
429C22×C4—734no—
430C22×C4—734no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—1464no—
435C22×C4—1464no—
436C22×C4—1462no—
437C22×C4—1461no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4C2×C4×D7312yes—
443C22×C4D4×D7311yescharacteristic
444C22×C4D4×D7312yes—
445C22×C4C23×D7311yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—1462no—
450C2×C8—1462no—
451C2×C8—1462no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—732no—
457C4⋊C4—734no—
458C4⋊C4—1464no—
459C4⋊C4—1464no—
460C4⋊C4—1464no—
461C4⋊C4—1464no—
462C4⋊C4—1461no—
463C4⋊C4—1462no—
464C4⋊C4C2×C4×D7312yes—
465C4⋊C4#1192402714yes—
466C22⋊C4—24no—
467C22⋊C4—1464no—
468C22⋊C4—1464no—
469C22⋊C4—1464no—
470C22⋊C4—1464no—
471C42—2924no—
472C42—2924no—
473C42—734no—
474C42—734no—
475C42—732no—
476C42—732no—
477C42—732no—
478C42—732no—
479C42—1464no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—2921no—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—1462no—
491C23C22⋊C4×D7312yes—
492C23#1455661212yes—
493C23#1455661412yes—
494C23C2×D4×D7311yescharacteristic
495Q8—24no—
496Q8—42no—
497D4—2928no—
498D4—2922no—
499D4—1464no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—2928no—
515C2×C4—2928no—
516C2×C4—2921no—
517C2×C4—2922no—
518C2×C4—2921no—
519C2×C4—2922no—
520C2×C4—732no—
521C2×C4—732no—
522C2×C4—732no—
523C2×C4—732no—
524C2×C4—732no—
525C2×C4—732no—
526C2×C4—1462no—
527C2×C4—1464no—
528C2×C4—1468no—
529C2×C4—1461no—
530C2×C4—1468no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4C22⋊C4×D7312yes—
537C2×C4C22×C4×D7311yescharacteristic
538C2×C4C2×D4×D7311yescharacteristic
539C2×C4#41×D7312yes—
540C2×C4C2×#1192402712yes—
541C8—42no—
542C8—2922no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—2922no—
550C22—2922no—
551C22—2922no—
552C22—5841no—
553C22—5841no—
554C22—5841no—
555C22—5841no—
556C22#1798858214yes—
557C22C2×C22⋊C4×D7311yescharacteristic
558C22#1798909311yescharacteristic
559C22#1798909511yescharacteristic
560C22C2×#1455661211yescharacteristic
561C22C2×#1455661411yescharacteristic
562C22AGL(1,ℤ/8)×D7311yescharacteristic
563C22#137×D7311yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—2922no—
569C4—2924no—
570C4—734no—
571C4—1462no—
572C4—1462no—
573C4—1462no—
574C4C4×#1192402712yes—
575C4#1798913212yes—
576C2—24no—
577C2—5842no—
578C2#2261241611yescharacteristic
579C2C2×#1798858211yescharacteristic
580C2#417×D7311yescharacteristic
581C2#2261997911yescharacteristic
582C2#2261998111yescharacteristic
583C2#2262001211yescharacteristic
584C2#2262001411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1872C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
cosocleC22×C4×C73C24
hypercenter#1872D73
hypocenterC73#21388
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#17988637C2semidirect
3. C2×C4⋊C4×Dic73C2semidirect
4. #22610885C2semidirect
5. #22610860C2semidirect
8. #22610239C2semidirect
9. #103×Dic73C2semidirect
10. #22609745C2semidirect
11. C73×#1872C2semidirect
13. #17989123C4semidirect
55. #14556582C2×C4semidirect
57. #14556580C2×C4semidirect
62. Q8×Dic73C2×C4semidirect
64. #14556553C2×C4semidirect
301. #1872D73semidirect
304. C73#21388semidirect

Maximal subgroups

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

subgroupindex
C2×#179891232
C2×#179886372
C2×C4⋊C4×Dic732
#226108852
#226108602
#226102482
#226102462
#226102392
#103×Dic732
#226097452
C73×#18722
#2138873