Finite Group Databasebeta

← #33085677

Subgroups — #33085677

Order 18688.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989353C211yesmaximal, characteristic
2C2×#17988635C211yesmaximal, characteristic, complemented
3C2×#17988630C211yesmaximal, characteristic, complemented
4#22618448C211yesmaximal, characteristic
5C2×#17988589C211yesmaximal, characteristic
6#22609994C214yesmaximal, complemented
7#22609989C214yesmaximal, complemented
8C2×#17988355C211yesmaximal, characteristic, complemented
9C73×#1742C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11#17989353—24no—
12#17989353C418yescomplemented
13C2×#14556599C2214yes—
14C2×#14556597—22no—
15C22×#11924012C2211yescharacteristic
16C22×#11924012C2211yescharacteristic
17C2×M16×D73—22no—
18C22×C4⋊Dic73C2211yescharacteristic
19#17988635—24no—
20C2×#14556526C2214yescomplemented
21#17988630—22no—
22#17988630—22no—
23C2×#14556524C2212yescomplemented
24C2×#14556522C2212yescomplemented
25#17988595—21no—
26#17988591—21no—
27#17988589—22no—
28#17988589C2214yes—
29#17988586C2212yes—
30#17988586C2212yes—
31#17988581—22no—
32C2×#14556516—21no—
33C2×#14556513—21no—
34#17988355C2214yescomplemented
35C22×M16×C73C2211yescharacteristic
36C22×M16×C73C2211yescharacteristic
37C2×#117×C73C2211yescharacteristic
38#343×C73C2212yescomplemented
39#343×C73C2212yescomplemented
40C2×#11924028—21no—
41C2×#11924028—22no—
42C2×#11924028—21no—
43C2×#11924028—22no—
44C2×#11924028C2×C414yes—
45C2×#11924028D412yes—
46C2×#11924028Q814yes—
47C2×#11924028C2312yes—
48C22×Dic146D411yescharacteristic
49C22×D292D411yescharacteristic
50C22×C22⋊D73—21no—
51C23×Dic73—21no—
52C22×C4×D73—21no—
53#14556599—28no—
54#14556599C2×C4116yescomplemented
55#14556597—216no—
56C2×#11924012—21no—
57C2×#11924012—22no—
58C2×#11924012—22no—
59C2×#11924012C2×C414yes—
60C2×#11924012D412yes—
61C2×#11924012D412yes—
62C2×#11924012D414yes—
63C2×#11924012C2312yes—
64C2×#11924011—24no—
65M16×D73—216no—
66C22×#10596831—21no—
67C22×#10596831C2312yes—
68C2×C8×D73—24no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2312yes—
75#14556526—28no—
76#14556524—24no—
77#14556522—24no—
78#14556516—24no—
79#14556514—22no—
80#14556514—22no—
81#14556513—24no—
82#14556511—22no—
83#14556511—22no—
84C23×C4×C73C2311yescharacteristic
85C2×M16×C73—21no—
86C2×M16×C73—22no—
87C2×M16×C73—21no—
88C2×M16×C73—22no—
89C2×M16×C73—22no—
90C2×M16×C73C2×C414yes—
91C2×M16×C73C2312yes—
92C2×M16×C73C2312yes—
93C22×C8×C73—21no—
94C22×C8×C73—21no—
95#117×C73—21no—
96#117×C73—21no—
97#117×C73—21no—
98#117×C73—21no—
99C2×C4⋊C4×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C22⋊C4×C73—21no—
102C2×C42×C73—21no—
103C23×D73—21no—
104#11924028—24no—
105#11924028—24no—
106#11924028—24no—
107#11924028—24no—
108#11924028—24no—
109#11924028—22no—
110#11924028—22no—
111#11924028C4⋊C418yes—
112#11924028C22×C414yes—
113#11924028C2×Q814yes—
114C2×Dic146—22no—
115C2×Dic146—21no—
116C2×Dic146C4⋊C414yes—
117C2×Dic146C2×D412yes—
118C2×D292—22no—
119C2×D292—21no—
120C2×D292C4⋊C414yes—
121C2×D292C2×D412yes—
122C2×C22⋊D73—24no—
123C2×C22⋊D73—22no—
124C2×C22⋊D73—24no—
125C2×C22⋊D73—22no—
126C22×Dic73—21no—
127C22×Dic73—22no—
128C22×Dic73—41no—
129C22×Dic73—41no—
130C22×Dic73—82no—
131C2×C4×D73—24no—
132C2×C4×D73—22no—
133C2×C4×D73—24no—
134C2×C4×D73—22no—
135#11924012—22no—
136#11924012—22no—
137#11924012—42no—
138#11924012C22⋊C418yes—
139#11924012C22⋊C418yes—
140#11924011—216no—
141C2×#10596831—22no—
142C2×#10596831—42no—
143C2×#10596831C22×C414yes—
144C2×#10596831C2×D414yes—
145C2×#10596831C2×D414yes—
146C8×D73—216no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73—41no—
150C4⋊Dic73—41no—
151C24×C73#4111yescharacteristic
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73C22×C412yes—
155C22×C4×C73C2×D412yes—
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×Q812yes—
161C22×C4×C73#4111yescharacteristic
162C22×C4×C73#4112yes—
163C22×C4×C73C2411yescharacteristic
164M16×C73—24no—
165M16×C73—22no—
166M16×C73—22no—
167M16×C73—22no—
168M16×C73—42no—
169M16×C73C22×C414yes—
170C2×C8×C73—24no—
171C2×C8×C73—22no—
172C2×C8×C73—22no—
173C2×C8×C73—42no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C42×C73—41no—
181C42×C73—41no—
182C23×C73—41no—
183C23×C73—41no—
184C23×C73#11712yes—
185C23×C73#12111yescharacteristic
186C23×C73#12111yescharacteristic
187C23×C73C22⋊Q812yes—
188C23×C73C2×#4111yescharacteristic
189C22×D73—22no—
190C22×D73—24no—
191C22×D73—81no—
192C22×D73—81no—
193C2×C4×C73—41no—
194C2×C4×C73—41no—
195C2×C4×C73—81no—
196C2×C4×C73—81no—
197C2×C4×C73C2×C22⋊C412yes—
198C2×C4×C73C2×C22⋊C412yes—
199C2×C4×C73C2×C4⋊C412yes—
200C2×C4×C73C2×C4⋊C412yes—
201C2×C4×C73#11712yes—
202C2×C4×C73#12112yes—
203C2×C4×C73#12112yes—
204C2×C4×C73#12111yescharacteristic
205C2×C4×C73#12111yescharacteristic
206C2×C4×C73C22⋊Q812yes—
207C2×C4×C73C22⋊Q812yes—
208C2×C4×C73C22⋊Q812yes—
209C2×C4×C73C23×C411yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C22×D411yescharacteristic
212C2×C4×C73C22×D411yescharacteristic
213C2×C4×C73C22×Q811yescharacteristic
214C2×C4×C73C2×#4111yescharacteristic
215C22⋊D73—24no—
216C22⋊D73—24no—
217C22⋊D73—28no—
218C2×Dic73—22no—
219C2×Dic73—24no—
220C2×Dic73—42no—
221C2×Dic73—82no—
222C2×Dic73—82no—
223C2×Dic73—82no—
224D292—22no—
225D292—24no—
226D292C2×C4⋊C414yes—
227C4×D73—24no—
228C4×D73—24no—
229C4×D73—28no—
230Dic146—22no—
231Dic146—24no—
232Dic146C2×C4⋊C414yes—
233C8×C73—24no—
234C8×C73—42no—
235#10596831—42no—
236#10596831C2×C22⋊C418yes—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—21no—
242C22×C73—21no—
243C22×C73—42no—
244C22×C73—42no—
245C22×C73#38012yes—
246C22×C73#38012yes—
247C22×C73C2×#11711yescharacteristic
248C22×C73C2×#12111yescharacteristic
249C22×C73C2×C22⋊Q811yescharacteristic
250C2×D73—24no—
251C2×D73—42no—
252C2×D73—42no—
253C2×D73—84no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73#38012yes—
257C4×C73#38012yes—
258C4×C73C22×C22⋊C411yescharacteristic
259C4×C73C22×C4⋊C411yescharacteristic
260C4×C73C2×#12111yescharacteristic
261C4×C73C2×C22⋊Q811yescharacteristic
262Dic73—24no—
263Dic73—82no—
264#20679—731nomaximal, Sylow
265C2×C73—22no—
266C2×C73—21no—
267C2×C73—21no—
268C2×C73—22no—
269C2×C73C2×#38011yescharacteristic
270C2×C73#306811yescharacteristic
271C2×C73#306911yescharacteristic
272D73—44no—
273C2×#568—731no—
274C2×#430—731no—
275C2×#428—731no—
276#2896—731no—
277C2×#417—731no—
278#1816—734no—
279#1815—734no—
280#1742D7311yescharacteristic, complemented
281C2×#343—731no—
282C73#2067911yescharacteristic, complemented, Sylow
283C22×#41—731no—
284#568—738no—
285#568—1464no—
286C2×#131—734no—
287C2×#131—1462no—
288C22×M16—731no—
289C22×M16—731no—
290C22×M16—1462no—
291C22×M16C2×D7311yescharacteristic
292C22×M16C2×D7311yescharacteristic
293C2×#117C2×D7311yescharacteristic
294C22×C4⋊C4—731no—
295#430—1464no—
296C2×#108—734no—
297#428—1462no—
298#428—1462no—
299C2×#107—732no—
300C2×#106—732no—
301#419—1461no—
302#418—1461no—
303#417—734no—
304#417—732no—
305#417—732no—
306#417—1462no—
307#416—1462no—
308C2×#103—1461no—
309C2×#102—1461no—
310#343—734no—
311#343C2×D7312yescomplemented
312#343C2×D7312yescomplemented
313C2×#41—734no—
314C2×#41—732no—
315C2×#41—734no—
316C2×#41—732no—
317C2×#41—1461no—
318C2×#41—1462no—
319C2×#41—1461no—
320C2×#41—1462no—
321C22×Q8—731no—
322C22×D4—731no—
323C22×D4—1461no—
324C23×C4—1461no—
325C23×C4—1461no—
326C23×C4C22×D7311yescharacteristic
327#131—7316no—
328#131—14616no—
329#131—1468no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—21no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—734no—
336C2×M16—732no—
337C2×M16—732no—
338C2×M16—734no—
339C2×M16—732no—
340C2×M16—1464no—
341C2×M16—1462no—
342C2×M16—1462no—
343C2×M16—14616no—
344C2×M16—1461no—
345C2×M16C4×D7314yes—
346C2×M16C22×D7312yes—
347C2×M16C22×D7312yes—
348C22×C8—21no—
349C22×C8—21no—
350C22×C8—732no—
351C22×C8—1464no—
352C22×C8—1461no—
353#117—21no—
354#117—21no—
355#117—21no—
356#117—21no—
357C2×C4⋊C4—732no—
358C2×C4⋊C4—1461no—
359C2×C4⋊C4—1461no—
360C2×C4⋊C4—1461no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1461no—
363C2×C4⋊C4C22×D7311yescharacteristic
364C2×C4⋊C4C22×D7311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C42—21no—
367#108—1468no—
368#107—1464no—
369#106—1464no—
370#103—1462no—
371#103—1462no—
372#103—1464no—
373#102—1462no—
374#102—1462no—
375#102—1464no—
376C24—1461no—
377C24#1192402811yescharacteristic
378#41—738no—
379#41—734no—
380#41—734no—
381#41—1464no—
382#41—1464no—
383#41—1464no—
384#41—1464no—
385#41—1464no—
386#41—1462no—
387#41—1462no—
388C2×Q8—734no—
389C2×Q8—732no—
390C2×Q8—1462no—
391C2×Q8—1461no—
392C2×D4—734no—
393C2×D4—732no—
394C2×D4—1464no—
395C2×D4—1462no—
396C2×D4—1464no—
397C2×D4—1462no—
398C2×D4—1461no—
399C2×D4—1462no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—2921no—
403C22×C4—2921no—
404C22×C4—5842no—
405C22×C4—1462no—
406C22×C4—1461no—
407C22×C4—1464no—
408C22×C4—1462no—
409C22×C4—1464no—
410C22×C4—1462no—
411C22×C4C2×C4×D7312yes—
412C22×C4D4×D7312yes—
413C22×C4D4×D7311yescharacteristic
414C22×C4D4×D7311yescharacteristic
415C22×C4D4×D7311yescharacteristic
416C22×C4D4×D7311yescharacteristic
417C22×C4Q8×D7312yes—
418C22×C4#1192402811yescharacteristic
419C22×C4#1192402812yes—
420C22×C4C23×D7311yescharacteristic
421M16—24no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—42no—
426M16—2922no—
427M16—738no—
428M16—738no—
429M16—14616no—
430M16—1462no—
431M16—1462no—
432M16C2×C4×D7314yes—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—2922no—
438C2×C8—734no—
439C2×C8—734no—
440C2×C8—734no—
441C2×C8—14616no—
442C2×C8—1462no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—2921no—
448C4⋊C4—2921no—
449C4⋊C4—1462no—
450C4⋊C4—1462no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—41no—
454C42—41no—
455C23—41no—
456C23—41no—
457C23—5841no—
458C23—5841no—
459C23—1462no—
460C23—1464no—
461C23#1455654812yes—
462C23#1455656411yescharacteristic
463C23#1455656411yescharacteristic
464C23#1455656912yes—
465C23C2×#1192402811yescharacteristic
466Q8—734no—
467Q8—1464no—
468Q8—1462no—
469D4—734no—
470D4—1468no—
471D4—1464no—
472D4—1464no—
473D4—1464no—
474D4—1462no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—2922no—
480C2×C4—5842no—
481C2×C4—5842no—
482C2×C4—5842no—
483C2×C4—1464no—
484C2×C4—1462no—
485C2×C4—1468no—
486C2×C4—1464no—
487C2×C4—1464no—
488C2×C4C22⋊C4×D7312yes—
489C2×C4C22⋊C4×D7312yes—
490C2×C4C4⋊C4×D7312yes—
491C2×C4C4⋊C4×D7312yes—
492C2×C4#1455654812yes—
493C2×C4#1455656411yescharacteristic
494C2×C4#1455656411yescharacteristic
495C2×C4#1455656412yes—
496C2×C4#1455656412yes—
497C2×C4#1455656912yes—
498C2×C4#1455656912yes—
499C2×C4#1455656912yes—
500C2×C4C22×C4×D7311yescharacteristic
501C2×C4C2×D4×D7311yescharacteristic
502C2×C4C2×D4×D7311yescharacteristic
503C2×C4C2×D4×D7311yescharacteristic
504C2×C4C2×Q8×D7311yescharacteristic
505C2×C4C2×#1192402811yescharacteristic
506C8—24no—
507C8—42no—
508C8—2922no—
509C8—738no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—2922no—
519C22—2922no—
520C22—5844no—
521C22—1464no—
522C22#1798844612yes—
523C22#1798844612yes—
524C22C2×#1455654811yescharacteristic
525C22C2×#1455656411yescharacteristic
526C22C2×#1455656911yescharacteristic
527C4—81no—
528C4—81no—
529C4—5842no—
530C4—1464no—
531C4#1798844612yes—
532C4#1798844612yes—
533C4C2×C22⋊C4×D7311yescharacteristic
534C4C2×C4⋊C4×D7311yescharacteristic
535C4C2×#1455656411yescharacteristic
536C4C2×#1455656911yescharacteristic
537C2—22no—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—2924no—
542C2C2×#1798844611yescharacteristic
543C2#2261966211yescharacteristic
544C2#2261967111yescharacteristic
subgroupG/H
centerC22C2×#14556564
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1742C2
socleC22×C73C2×#121
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenter#1742D73
hypocenterC73#20679
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C2×#17988635C2semidirect
3. C2×#17988630C2semidirect
6. #22609994C2semidirect
7. #22609989C2semidirect
8. C2×#17988355C2semidirect
9. C73×#1742C2semidirect
12. #17989353C4semidirect
20. C2×#14556526C22semidirect
23. C2×#14556524C22semidirect
24. C2×#14556522C22semidirect
34. #17988355C22semidirect
38. #343×C73C22semidirect
39. #343×C73C22semidirect
54. #14556599C2×C4semidirect
280. #1742D73semidirect
282. C73#20679semidirect
311. #343C2×D73semidirect
312. #343C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179893532
C2×#179886352
C2×#179886302
#226184482
C2×#179885892
#226099942
#226099892
C2×#179883552
C73×#17422
#2067973