Finite Group Databasebeta

← #33088679

Subgroups — #33088679

Order 18688.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988589C211yesmaximal, characteristic
2C2×#17988584C211yesmaximal, characteristic
3#22612547C211yesmaximal, characteristic, complemented
4#22612519C211yesmaximal, characteristic
5#22610628C212yesmaximal, complemented
6#22610621C212yesmaximal, complemented
7#22610614C212yesmaximal, complemented
8#22610607C212yesmaximal, complemented
9#22609651C211yesmaximal, characteristic, complemented
10C2×#17988350C211yesmaximal, characteristic, complemented
11C2×#405×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C2×#14556550C2211yescharacteristic
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16#17988589C418yescomplemented
17#17988584—24no—
18C2×#14556516C2211yescharacteristic
19C2×#14556516C2211yescharacteristic
20C2×#14556516C2212yes—
21C2×#14556513C2211yescharacteristic
22C2×#14556513C2211yescharacteristic
23C2×#14556513C2212yes—
24#17988493C2212yes—
25#17988489C2212yes—
26#17988485C2212yes—
27#17988485C2212yes—
28#17988483—21no—
29#17988483—21no—
30#17988464—22no—
31#17988445—22no—
32#17988444—21no—
33#17988444—21no—
34#17988350C2212yes—
35#17988350C2212yes—
36#17988350C2212yes—
37#17988350C2212yes—
38C22×M16×C73C2211yescharacteristic
39C23×C8×C73C2211yescharacteristic
40C2×#117×C73C2211yescharacteristic
41#405×C73—24no—
42C2×#97×C73C2212yes—
43C2×C4×C8×C73C2212yes—
44C2×#11924028—22no—
45C2×#11924028—22no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028C2×C414yes—
50C22×Dic146C2311yescharacteristic
51C22×D292C2311yescharacteristic
52C22×C22⋊D73—21no—
53C23×Dic73—21no—
54C23×Dic73—21no—
55C22×C4×D73—21no—
56#14556550—22no—
57#14556550—22no—
58C2×C4⋊Dic73—21no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73—22no—
61C2×C4⋊Dic73—22no—
62C2×C4⋊Dic73—22no—
63C2×C4⋊Dic73C2×C414yes—
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2312yes—
67C2×C4⋊Dic73C2312yes—
68C2×#11924007—22no—
69C2×#11924007—22no—
70C2×C22⋊Dic73—21no—
71C2×#11924005—22no—
72C2×#11924005—22no—
73C2×C4×Dic73—21no—
74#14556516—22no—
75#14556516—22no—
76#14556516C2×C418yescomplemented
77#14556513—22no—
78#14556513—22no—
79#14556513C2×C418yescomplemented
80#14556498—22no—
81#14556498—22no—
82#14556498—22no—
83#14556498—22no—
84C23×C4×C73C2311yescharacteristic
85C2×M16×C73—22no—
86C2×M16×C73—22no—
87C2×M16×C73C2×C414yes—
88C22×C8×C73—22no—
89C22×C8×C73—22no—
90C22×C8×C73D414yes—
91C22×C8×C73C2312yes—
92C22×C8×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94#117×C73—22no—
95#117×C73—22no—
96C2×C4⋊C4×C73D411yescharacteristic
97C2×C4⋊C4×C73D411yescharacteristic
98C2×C22⋊C4×C73D412yes—
99C2×C42×C73C2312yes—
100#97×C73—24no—
101C4×C8×C73—24no—
102C23×D73—21no—
103#11924028—28no—
104#11924028—28no—
105#11924028—44no—
106#11924028—44no—
107C2×Dic146—21no—
108C2×Dic146—21no—
109C2×Dic146—21no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146C22×C412yes—
113C2×D292—21no—
114C2×D292—21no—
115C2×D292—21no—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292C22×C412yes—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—42no—
121C2×C22⋊D73—42no—
122C22×Dic73—21no—
123C22×Dic73—22no—
124C22×Dic73—24no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—22no—
128C22×Dic73—42no—
129C22×Dic73—81no—
130C22×Dic73—81no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C2×C4×D73—24no—
134C2×C4×D73—42no—
135C2×C4×D73—42no—
136C4⋊Dic73—24no—
137C4⋊Dic73—22no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145C4⋊Dic73C22×C414yes—
146#11924007—82no—
147#11924007—82no—
148C22⋊Dic73—42no—
149#11924005—82no—
150#11924005—82no—
151C4×Dic73—42no—
152C24×C73C2×D411yescharacteristic
153C22×C4×C73C22×C412yes—
154C22×C4×C73C2×D412yes—
155C22×C4×C73C2×D412yes—
156C22×C4×C73C2×D412yes—
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73#4112yes—
159C22×C4×C73#4112yes—
160C22×C4×C73#4112yes—
161C22×C4×C73#4112yes—
162C22×C4×C73C2411yescharacteristic
163M16×C73—28no—
164C2×C8×C73—28no—
165C2×C8×C73—22no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73—21no—
170C2×C8×C73—22no—
171C2×C8×C73C22×C414yes—
172C2×C8×C73C2×D412yes—
173C2×C8×C73C2×D412yes—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C22⋊C4×C73—24no—
177C42×C73—22no—
178C42×C73—22no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—21no—
183C23×C73C4×D412yes—
184C23×C73#12112yes—
185C23×C73#12412yes—
186C23×C73C22×D411yescharacteristic
187C22×D73—22no—
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—41no—
191C22×D73—41no—
192C22×D73—41no—
193C22×D73—41no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—22no—
200C2×C4×C73#11714yes—
201C2×C4×C73C4×D414yes—
202C2×C4×C73C4×D412yes—
203C2×C4×C73#12112yes—
204C2×C4×C73#12314yes—
205C2×C4×C73#12412yes—
206C2×C4×C73#12614yes—
207C2×C4×C73C23×C411yescharacteristic
208C2×C4×C73C22×D411yescharacteristic
209C2×C4×C73C2×#4111yescharacteristic
210C2×C4×C73C2×#4111yescharacteristic
211C2×C4×C73C2×#4111yescharacteristic
212C2×C4×C73C2×#4111yescharacteristic
213C22⋊D73—48no—
214C2×Dic73—24no—
215C2×Dic73—22no—
216C2×Dic73—41no—
217C2×Dic73—41no—
218C2×Dic73—42no—
219C2×Dic73—42no—
220C2×Dic73—41no—
221C2×Dic73—41no—
222C2×Dic73—42no—
223C2×Dic73—42no—
224C2×Dic73—41no—
225C2×Dic73—42no—
226C2×Dic73—41no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—84no—
230C2×Dic73—84no—
231D292—22no—
232D292—22no—
233D292—41no—
234D292—41no—
235C4×D73—48no—
236Dic146—22no—
237Dic146—22no—
238Dic146—41no—
239Dic146—41no—
240C8×C73—24no—
241C8×C73—22no—
242C8×C73—22no—
243C22×C73—22no—
244C22×C73—22no—
245C22×C73—22no—
246C22×C73—22no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73#38814yes—
250C22×C73C2×C4×D411yescharacteristic
251C22×C73C2×#12111yescharacteristic
252C22×C73C2×#12411yescharacteristic
253C22×C73#57611yescharacteristic
254C22×C73#57711yescharacteristic
255C22×C73#57711yescharacteristic
256C22×C73#57811yescharacteristic
257C2×D73—42no—
258C2×D73—42no—
259C2×D73—42no—
260C2×D73—81no—
261C2×D73—81no—
262C2×D73—81no—
263C2×D73—81no—
264C4×C73—42no—
265C4×C73—42no—
266C4×C73#38814yes—
267C4×C73C2×#11711yescharacteristic
268C4×C73C2×C4×D411yescharacteristic
269C4×C73C2×#12311yescharacteristic
270C4×C73C2×#12611yescharacteristic
271Dic73—44no—
272Dic73—42no—
273Dic73—82no—
274Dic73—81no—
275Dic73—81no—
276#21092—731nomaximal, Sylow
277C2×C73—24no—
278C2×C73C2×#38811yescharacteristic
279C2×C73#294611yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#306011yescharacteristic
282C2×C73#306111yescharacteristic
283C2×C73#313611yescharacteristic
284C2×C73#313711yescharacteristic
285D73—82no—
286C2×#417—731no—
287C2×#416—731no—
288C2×#405D7311yescharacteristic, complemented
289#2316—731no—
290#2311—731no—
291#1936—732no—
292#1935—732no—
293#1934—732no—
294#1933—732no—
295#1736—731no—
296C2×#340—731no—
297C73#2109211yescharacteristic, complemented, Sylow
298C22×#41—731no—
299C22×M16C2×D7311yescharacteristic
300C23×C8C2×D7311yescharacteristic
301C2×#117—731no—
302C2×#117C2×D7311yescharacteristic
303C22×C4⋊C4—731no—
304C22×C4⋊C4—731no—
305#417—738no—
306#416—1464no—
307C2×#103—732no—
308C2×#103—731no—
309C2×#103—731no—
310C2×#102—732no—
311C2×#102—731no—
312C2×#102—731no—
313#405—24no—
314C2×#97C2×D7312yes—
315C2×C4×C8C2×D7312yes—
316#391—732no—
317#390—732no—
318#389—732no—
319#389—732no—
320#389—1461no—
321#389—1461no—
322#386—1462no—
323#380—1461no—
324#380—1461no—
325#380—1462no—
326#340—732no—
327#340—732no—
328#340—732no—
329#340—732no—
330C2×#41—734no—
331C2×#41—1462no—
332C2×#41—1462no—
333C2×#41—1462no—
334C2×#41—1462no—
335C2×#41—1462no—
336C22×Q8—731no—
337C22×D4—731no—
338C22×D4—1461no—
339C23×C4—1461no—
340C23×C4—1461no—
341C23×C4—1461no—
342C23×C4C22×D7311yescharacteristic
343C2×M16—22no—
344C2×M16—22no—
345C2×M16C4×D7314yes—
346C22×C8—22no—
347C22×C8—22no—
348C22×C8C22⋊D7314yes—
349C22×C8C22×D7312yes—
350C22×C8C22×D7311yescharacteristic
351C22×C8C22×D7311yescharacteristic
352#117—22no—
353#117—22no—
354#117—1462no—
355#117—1462no—
356C2×C4⋊C4—734no—
357C2×C4⋊C4—732no—
358C2×C4⋊C4—732no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—731no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1462no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4D29211yescharacteristic
369C2×C4⋊C4D29211yescharacteristic
370C2×C22⋊C4—1461no—
371C2×C22⋊C4—1462no—
372C2×C22⋊C4—1462no—
373C2×C22⋊C4D29212yes—
374C2×C42—1461no—
375C2×C42C22×D7312yes—
376#103—738no—
377#103—1462no—
378#103—1462no—
379#102—738no—
380#102—1462no—
381#102—1462no—
382#97—24no—
383C4×C8—24no—
384#95—1462no—
385#95—1462no—
386#95—1462no—
387#95—1462no—
388C24—1461no—
389C24C2×D29211yescharacteristic
390#41—2924no—
391#41—2924no—
392#41—1468no—
393#41—1468no—
394C2×Q8—732no—
395C2×Q8—1461no—
396C2×Q8—1461no—
397C2×Q8—1461no—
398C2×Q8—1461no—
399C2×Q8—1461no—
400C2×D4—2922no—
401C2×D4—2922no—
402C2×D4—732no—
403C2×D4—1464no—
404C2×D4—1461no—
405C2×D4—1461no—
406C2×D4—1461no—
407C2×D4—1461no—
408C2×D4—1461no—
409C22×C4—2922no—
410C22×C4—2922no—
411C22×C4—2922no—
412C22×C4—5841no—
413C22×C4—5841no—
414C22×C4—5841no—
415C22×C4—5841no—
416C22×C4—1461no—
417C22×C4—1461no—
418C22×C4—1462no—
419C22×C4—1461no—
420C22×C4—1462no—
421C22×C4—1464no—
422C22×C4—1464no—
423C22×C4C2×C4×D7312yes—
424C22×C4C2×C22⋊D7312yes—
425C22×C4C2×D29212yes—
426C22×C4C2×D29212yes—
427C22×C4C2×D29211yescharacteristic
428C22×C4#1192402812yes—
429C22×C4#1192402812yes—
430C22×C4#1192402812yes—
431C22×C4#1192402812yes—
432C22×C4C23×D7311yescharacteristic
433M16—28no—
434C2×C8—28no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—22no—
441C2×C8C2×C4×D7314yes—
442C2×C8C2×C22⋊D7312yes—
443C2×C8C2×C22⋊D7312yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—2922no—
447C4⋊C4—5842no—
448C4⋊C4—5842no—
449C4⋊C4—734no—
450C4⋊C4—1461no—
451C4⋊C4—1461no—
452C4⋊C4—1462no—
453C4⋊C4—1462no—
454C4⋊C4—1461no—
455C4⋊C4—1461no—
456C4⋊C4—1464no—
457C4⋊C4—1462no—
458C22⋊C4—24no—
459C22⋊C4—2922no—
460C22⋊C4—5842no—
461C22⋊C4—5842no—
462C42—22no—
463C42—22no—
464C42—2922no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—2921no—
470C23—2921no—
471C23—2921no—
472C23—2921no—
473C23—2921no—
474C23—2921no—
475C23—1462no—
476C23C4×D29212yes—
477C23#1455656512yes—
478C23#1455657912yes—
479C23C22×D29211yescharacteristic
480Q8—2921no—
481Q8—2921no—
482Q8—1462no—
483Q8—1462no—
484D4—2928no—
485D4—2921no—
486D4—2921no—
487D4—1462no—
488D4—1462no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—2922no—
496C2×C4—2922no—
497C2×C4—2921no—
498C2×C4—2921no—
499C2×C4—2928no—
500C2×C4—2922no—
501C2×C4—2921no—
502C2×C4—2922no—
503C2×C4—2921no—
504C2×C4—2921no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2922no—
509C2×C4—5844no—
510C2×C4—5844no—
511C2×C4—1464no—
512C2×C4—1462no—
513C2×C4#1455654814yes—
514C2×C4C4×C22⋊D7314yes—
515C2×C4C4×D29212yes—
516C2×C4#1455656512yes—
517C2×C4#1455657414yes—
518C2×C4#1455657912yes—
519C2×C4#1455658514yes—
520C2×C4C22×C4×D7311yescharacteristic
521C2×C4C22×C22⋊D7311yescharacteristic
522C2×C4C2×#1192402811yescharacteristic
523C2×C4C2×#1192402811yescharacteristic
524C2×C4C2×#1192402811yescharacteristic
525C2×C4C2×#1192402811yescharacteristic
526C8—24no—
527C8—22no—
528C8—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—2922no—
536C22—2922no—
537C22—2922no—
538C22—5841no—
539C22—5841no—
540C22—5841no—
541C22—5841no—
542C22#1798847814yes—
543C22C2×C4×D29211yescharacteristic
544C22C2×#1455656511yescharacteristic
545C22C2×#1455657911yescharacteristic
546C22#1798939211yescharacteristic
547C22#1798939911yescharacteristic
548C22#1798939911yescharacteristic
549C22#1798940411yescharacteristic
550C4—42no—
551C4—42no—
552C4—2922no—
553C4—2924no—
554C4—5842no—
555C4—5841no—
556C4—5841no—
557C4#1798847814yes—
558C4C2×#1455654811yescharacteristic
559C4C2×C4×C22⋊D7311yescharacteristic
560C4C2×#1455657411yescharacteristic
561C4C2×#1455658511yescharacteristic
562C2—24no—
563C2—5842no—
564C2C2×#1798847811yescharacteristic
565C2#2261876511yescharacteristic
566C2#2261877611yescharacteristic
567C2#2261959311yescharacteristic
568C2#2261960111yescharacteristic
569C2#2262028711yescharacteristic
570C2#2262029811yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#405×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#405D73
hypocenterC73#21092
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #22612547C2semidirect
5. #22610628C2semidirect
6. #22610621C2semidirect
7. #22610614C2semidirect
8. #22610607C2semidirect
9. #22609651C2semidirect
10. C2×#17988350C2semidirect
11. C2×#405×C73C2semidirect
16. #17988589C4semidirect
76. #14556516C2×C4semidirect
79. #14556513C2×C4semidirect
288. C2×#405D73semidirect
297. C73#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#179885892
C2×#179885842
#226125472
#226125192
#226106282
#226106212
#226106142
#226106072
#226096512
C2×#179883502
C2×#405×C732
#2109273