Finite Group Databasebeta

← #33139343

Subgroups — #33139343

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988957C211yesmaximal, characteristic, complemented
2C2×#17988797C211yesmaximal, characteristic, complemented
3#22614884C211yesmaximal, characteristic
4#22610939C212yesmaximal, complemented
5#22610939C212yesmaximal, complemented
6#22610889C211yesmaximal, characteristic, complemented
7#22610730C211yesmaximal, characteristic
8#22610243C212yesmaximal, complemented
9#22610238C212yesmaximal, complemented
10#22609761C211yesmaximal, characteristic, complemented
11C2×#552×C73C211yesmaximal, characteristic
12C22×#11924018C2212yes—
13C2×#14556591C2211yescharacteristic
14C2×#14556570C2212yes—
15C2×#14556570C2212yes—
16#17988957—24no—
17#17988797—24no—
18C2×C4⋊#10596831C2211yescharacteristic
19C2×#14556515C2212yes—
20C2×#14556514C2212yes—
21C2×#14556514C2212yes—
22C2×#14556514C2212yes—
23C2×C22⋊#10596831C2212yes—
24C2×C22⋊#10596831C2212yes—
25C2×#14556502C2211yescharacteristic
26#17988523—21no—
27#17988523—21no—
28#17988490—21no—
29#17988456—21no—
30C4×C4⋊Dic73C2211yescharacteristic
31#17988349C2212yes—
32#17988349C2212yes—
33#17988349C2212yes—
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
47C22×Dic146C2312yes—
48C2×#11924018—24no—
49C2×#11924018—42no—
50C2×#11924018—42no—
51C22×#10596831C2312yes—
52C22×#10596831C2312yes—
53#14556591—22no—
54#14556591—22no—
55#14556570—24no—
56#14556570—24no—
57#14556570—42no—
58#14556570—42no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73—21no—
61C2×C4⋊Dic73C2312yes—
62C2×#11924007—22no—
63C2×#11924007—22no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C4×Dic73—21no—
67C4⋊#10596831—22no—
68#14556515—24no—
69#14556514—24no—
70#14556514—24no—
71#14556514—24no—
72C22⋊#10596831—42no—
73C22⋊#10596831—42no—
74#14556502—22no—
75#14556498—21no—
76C22×Q8×C73C2311yescharacteristic
77C22×D4×C73D411yescharacteristic
78C23×C4×C73—21no—
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2311yescharacteristic
81#126×C73—28no—
82#125×C73—28no—
83#125×C73—22no—
84#125×C73D414yes—
85#123×C73—28no—
86C22⋊Q8×C73—22no—
87C22⋊Q8×C73—22no—
88C22⋊Q8×C73—22no—
89C22⋊Q8×C73—22no—
90C4×Q8×C73—22no—
91C4×Q8×C73D414yes—
92C4×D4×C73—22no—
93C4×D4×C73—22no—
94#117×C73—28no—
95#117×C73—22no—
96#117×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—22no—
104C2×C4⋊C4×C73—22no—
105C2×C4⋊C4×C73D411yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×Dic146—22no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125#11924018—48no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—22no—
131C2×#10596831—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—42no—
134C4⋊Dic73—42no—
135C4⋊Dic73#4114yes—
136#11924007—44no—
137#11924007—44no—
138C22⋊Dic73—44no—
139C22⋊Dic73—44no—
140C4×Dic73—42no—
141C24×C73C2×D411yescharacteristic
142C24×C73C2×D411yescharacteristic
143C2×Q8×C73—21no—
144C2×Q8×C73—22no—
145C2×Q8×C73—22no—
146C2×Q8×C73C2×D412yes—
147C2×D4×C73—22no—
148C2×D4×C73—22no—
149C2×D4×C73—41no—
150C22×C4×C73—24no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—42no—
164C22×C4×C73—42no—
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2411yescharacteristic
170C4⋊C4×C73—24no—
171C4⋊C4×C73—24no—
172C4⋊C4×C73—24no—
173C4⋊C4×C73—24no—
174C4⋊C4×C73—24no—
175C4⋊C4×C73—24no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73C2×D412yes—
181C4⋊C4×C73C2×D412yes—
182C4⋊C4×C73C2×D412yes—
183C22⋊C4×C73—24no—
184C22⋊C4×C73—24no—
185C22⋊C4×C73—24no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—22no—
188C42×C73—24no—
189C42×C73—24no—
190C42×C73C2×D412yes—
191C42×C73C2×D412yes—
192C23×C73—21no—
193C23×C73—22no—
194C23×C73—21no—
195C23×C73—22no—
196C23×C73—21no—
197C23×C73—22no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—22no—
201C23×C73#13712yes—
202C23×C73C22×D411yescharacteristic
203Q8×C73—22no—
204Q8×C73—22no—
205Q8×C73—42no—
206D4×C73—44no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—22no—
210C2×C4×C73—21no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—48no—
219C2×C4×C73—42no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—42no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73#12012yes—
232C2×C4×C73#12012yes—
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C2×#4112yes—
236C2×C4×C73H(5,𝔽2)12yes—
237C2×Dic73—24no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243Dic146—42no—
244Dic146—42no—
245Dic146—42no—
246Dic146—42no—
247#10596831—42no—
248#10596831—42no—
249C22×C73—22no—
250C22×C73—24no—
251C22×C73—41no—
252C22×C73—41no—
253C22×C73—42no—
254C22×C73—41no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73C2×#12011yescharacteristic
260C22×C73#54612yes—
261C22×C73C2×#13711yescharacteristic
262C22×C73#57511yescharacteristic
263C22×C73#57812yes—
264C4×C73—22no—
265C4×C73—22no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73—42no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73#53211yescharacteristic
273C4×C73#53511yescharacteristic
274C4×C73#55012yes—
275Dic73—44no—
276Dic73—82no—
277#25584—731nomaximal, Sylow
278C2×C73—42no—
279C2×C73—81no—
280C2×C73#243111yescharacteristic
281C2×C73#300711yescharacteristic
282C2×C73#301111yescharacteristic
283C2×C73#331212yes—
284C2×C73#331512yes—
285C2×#552D7311yescharacteristic
286C2×#491—731no—
287C2×#462—731no—
288#2542—731no—
289#2001—732no—
290#2001—732no—
291#1989—731no—
292#1955—731no—
293#1873—732no—
294#1872—732no—
295#1763—731no—
296C73#2558411yescharacteristic, complemented, Sylow
297C22×Q16—732no—
298#552—24no—
299#552—24no—
300C2×C4⋊Q8—731no—
301C2×#126—21no—
302C2×#125—21no—
303C2×#125C2×D7311yescharacteristic
304C2×#123—21no—
305C2×C22⋊Q8—732no—
306C2×C22⋊Q8—732no—
307C2×C22⋊Q8C2×D7311yescharacteristic
308C2×C22⋊Q8C2×D7311yescharacteristic
309C2×C4×Q8C2×D7311yescharacteristic
310C2×C4×D4C2×D7311yescharacteristic
311C2×#117—21no—
312C2×#117C2×D7311yescharacteristic
313C22×C4⋊C4C2×D7311yescharacteristic
314#491—1464no—
315#462—1464no—
316C2×C4⋊C8—731no—
317C2×#103—732no—
318C2×#103—732no—
319C2×#103—732no—
320C2×#103—732no—
321C2×C22⋊C8—732no—
322C2×C22⋊C8—732no—
323C2×#97—731no—
324#399—1461no—
325#399—1461no—
326#390—1461no—
327#384—1461no—
328C4×C4⋊C4—731no—
329#340—732no—
330#340—732no—
331#340—732no—
332C22×Q8—732no—
333C22×Q8C22×D7311yescharacteristic
334C22×D4C22⋊D7311yescharacteristic
335C23×C4—21no—
336C23×C4C22×D7311yescharacteristic
337C23×C4C22×D7311yescharacteristic
338C2×Q16—2922no—
339C2×Q16—2922no—
340C2×Q16—1464no—
341C22×C8—732no—
342C22×C8—732no—
343C4⋊Q8—1462no—
344C4⋊Q8—1462no—
345#126—28no—
346#125—28no—
347#125—22no—
348#125C22⋊D7314yes—
349#123—28no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—2922no—
355C22⋊Q8—2922no—
356C22⋊Q8—1464no—
357C22⋊Q8—1464no—
358C4×Q8—22no—
359C4×Q8C22⋊D7314yes—
360C4×D4—22no—
361C4×D4—22no—
362#117—28no—
363#117—22no—
364#117—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—732no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4C22⋊D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1462no—
386C2×C22⋊C4—1462no—
387C2×C22⋊C4C22⋊D7311yescharacteristic
388C2×C22⋊C4C22⋊D7311yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—1461no—
392C2×C42C22×D7311yescharacteristic
393C4⋊C8—1462no—
394#103—1464no—
395#103—1464no—
396#103—1464no—
397#103—1464no—
398C22⋊C8—2922no—
399C22⋊C8—2922no—
400#97—1462no—
401#95—1461no—
402C24C2×C22⋊D7311yescharacteristic
403C24C2×C22⋊D7311yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—1462no—
408C2×Q8—1462no—
409C2×Q8—1462no—
410C2×Q8—1462no—
411C2×Q8—1462no—
412C2×Q8—1462no—
413C2×Q8C2×C22⋊D7312yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4C2×C22⋊D7311yescharacteristic
435C22×C4C2×C22⋊D7311yescharacteristic
436C22×C4C2×C22⋊D7311yescharacteristic
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4C23×D7311yescharacteristic
439Q16—2928no—
440C2×C8—1462no—
441C2×C8—1462no—
442C2×C8—1462no—
443C2×C8—1462no—
444C2×C8—1462no—
445C2×C8—1462no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—2924no—
457C4⋊C4—2924no—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—734no—
461C4⋊C4—1462no—
462C4⋊C4C2×C22⋊D7312yes—
463C4⋊C4C2×C22⋊D7312yes—
464C4⋊C4C2×C22⋊D7312yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—2924no—
471C22⋊C4—2924no—
472C42—24no—
473C42—24no—
474C42—2922no—
475C42C2×C22⋊D7312yes—
476C42C2×C22⋊D7312yes—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23#1455662112yes—
487C23C22×C22⋊D7311yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—42no—
491Q8—2922no—
492Q8—2922no—
493Q8—2922no—
494Q8—2922no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—1464no—
526C2×C4#1455656012yes—
527C2×C4#1455656012yes—
528C2×C4C22×C22⋊D7311yescharacteristic
529C2×C4C22×C22⋊D7311yescharacteristic
530C2×C4#41×D7312yes—
531C2×C4#1455663712yes—
532C8—2922no—
533C8—2922no—
534C22—24no—
535C22—22no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22C2×#1455656011yescharacteristic
545C22#1798922512yes—
546C22C2×#1455662111yescharacteristic
547C22#1798938511yescharacteristic
548C22#1798940212yes—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—2924no—
558C4—5842no—
559C4#1798913311yescharacteristic
560C4#1798915111yescharacteristic
561C4#1798925612yes—
562C2—42no—
563C2—81no—
564C2#2261354611yescharacteristic
565C2#2261919111yescharacteristic
566C2#2261922611yescharacteristic
567C2#2262197612yes—
568C2#2262202112yes—
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#25584
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×#17988957C2semidirect
2. C2×#17988797C2semidirect
4. #22610939C2semidirect
5. #22610939C2semidirect
6. #22610889C2semidirect
8. #22610243C2semidirect
9. #22610238C2semidirect
10. #22609761C2semidirect
296. C73#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#179889572
C2×#179887972
#226148842
#226109392
#226108892
#226107302
#226102432
#226102382
#226097612
C2×#552×C732
#2558473