Finite Group Databasebeta

← #33098244

Subgroups — #33098244

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988868C211yesmaximal, characteristic
2#22612677C211yesmaximal, characteristic, complemented
3#22612569C211yesmaximal, characteristic
4#22610879C211yesmaximal, characteristic, complemented
5#22610667C211yesmaximal, characteristic, complemented
6#22610662C211yesmaximal, characteristic, complemented
7#22610626C211yesmaximal, characteristic
8#22610612C211yesmaximal, characteristic
9#22610557C211yesmaximal, characteristic
10#22610550C211yesmaximal, characteristic
11#22610522C211yesmaximal, characteristic, complemented
12#22610501C211yesmaximal, characteristic, complemented
13#22609938C211yesmaximal, characteristic
14#22609790C211yesmaximal, characteristic, complemented
15C73×#1949C211yesmaximal, characteristic, complemented
16C2×#14556591C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18#17988868C418yescomplemented
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556513C2211yescharacteristic
23C2×#14556513C2211yescharacteristic
24C2×#14556511C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28#17988491—21no—
29#17988491—21no—
30#17988488C2211yescharacteristic
31#17988488C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988483C2211yescharacteristic
36#17988478—21no—
37#17988478—21no—
38#17988472—21no—
39#17988472—21no—
40#17988449—21no—
41#17988446—21no—
42C4⋊C4×Dic73C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44#17988433C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#106×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56#382×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556591—22no—
66#14556591C2×C414yes—
67C4×D292—22no—
68C4×D292C2×C414yes—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924005—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556525C2×C414yes—
87#14556525C2×C414yes—
88C4⋊#10596831—22no—
89#14556513C2×C414yes—
90#14556513C2×C414yes—
91#14556511—41no—
92#14556511—41no—
93#14556501—41no—
94C8×Dic73—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C42×C73—21no—
112C2×C42×C73C2311yescharacteristic
113#107×C73—41no—
114#106×C73—41no—
115C4⋊C8×C73C2×C414yes—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73—21no—
119C23×D73—21no—
120C2×Dic146—24no—
121C2×Dic146—81no—
122C2×Dic146—81no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292C22×C412yes—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×C4×D73—24no—
137C2×C4×D73—81no—
138C2×C4×D73—81no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C4⋊Dic73—24no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73C22×C412yes—
150C4⋊Dic73C22×C412yes—
151C4⋊Dic73C22×C412yes—
152#11924007—81no—
153#11924007—81no—
154#11924007—81no—
155#11924007—81no—
156#11924005—24no—
157#11924005—81no—
158#11924005—81no—
159#11924005—81no—
160#11924005—81no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—81no—
166C4×Dic73—81no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73C22×C412yes—
179C2×C8×C73C22×C412yes—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C42×C73—81no—
189C42×C73C22×C412yes—
190C42×C73#4112yes—
191C23×C73C22×D411yescharacteristic
192C22×D73—22no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73C4×D412yes—
216C2×C4×C73C4×D412yes—
217C2×C4×C73C23×C411yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73H(5,𝔽2)11yescharacteristic
220C2×C4×C732−1+411yescharacteristic
221C2×C4×C732−1+411yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244D292—22no—
245D292—22no—
246D292—41no—
247D292—41no—
248C4×D73—84no—
249Dic146—84no—
250C8×C73—22no—
251C8×C73—22no—
252#10596831—41no—
253#10596831—41no—
254C22×C73C2×C4×D411yescharacteristic
255C22×C73#53611yescharacteristic
256C22×C73#53711yescharacteristic
257C22×C73#57611yescharacteristic
258C22×C73#57611yescharacteristic
259C22×C73#57711yescharacteristic
260C22×C73#57811yescharacteristic
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C2×D73—81no—
268C4×C73—22no—
269C4×C73—22no—
270C4×C73—81no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73#51911yescharacteristic
275C4×C73#52011yescharacteristic
276C4×C73#54111yescharacteristic
277C4×C73#54211yescharacteristic
278Dic73—42no—
279Dic73—42no—
280Dic73—42no—
281Dic73—41no—
282Dic73—41no—
283Dic73—41no—
284Dic73—41no—
285#22127—731nomaximal, Sylow
286C2×C73#232311yescharacteristic
287C2×C73#294711yescharacteristic
288C2×C73#294811yescharacteristic
289C2×C73#324111yescharacteristic
290C2×C73#324311yescharacteristic
291C2×C73#325711yescharacteristic
292C2×C73#326211yescharacteristic
293D73—82no—
294C2×#474—731no—
295#2331—731no—
296#2319—731no—
297#1987—731no—
298#1949D7311yescharacteristic, complemented
299#1943—731no—
300#1942—731no—
301#1936—731no—
302#1934—731no—
303#1926—731no—
304#1925—731no—
305#1921—731no—
306#1918—731no—
307#1804—731no—
308#1768—731no—
309C73#2212711yescharacteristic, complemented, Sylow
310C2×C4⋊Q8—731no—
311C2×C4×D4—731no—
312#474—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107C2×D7311yescharacteristic
316C2×#106C2×D7311yescharacteristic
317C2×C4⋊C8—731no—
318C2×C4⋊C8C2×D7311yescharacteristic
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#102—731no—
323C2×#97—731no—
324C2×C4×C8—731no—
325#391—1461no—
326#391—1461no—
327#390—731no—
328#390—731no—
329#389—731no—
330#389—731no—
331#389—731no—
332#389—731no—
333#388—1461no—
334#388—1461no—
335#387—1461no—
336#387—1461no—
337#382C2×D7311yescharacteristic
338#381—1461no—
339#380—1461no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4C2×D7311yescharacteristic
343#376—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340C2×D7311yescharacteristic
351#340C2×D7311yescharacteristic
352#336—731no—
353#336—731no—
354C22×Q8—1461no—
355C22×D4—731no—
356C23×C4—1461no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C4⋊Q8—734no—
362C4⋊Q8—1462no—
363C4×D4—734no—
364C4×D4—1462no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4C22×D7311yescharacteristic
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C42—21no—
383C2×C42—731no—
384C2×C42—731no—
385C2×C42—1461no—
386C2×C42—1461no—
387C2×C42C22×D7311yescharacteristic
388#107—41no—
389#107—734no—
390#107—734no—
391#106—41no—
392C4⋊C8—1462no—
393C4⋊C8C4×D7314yes—
394#102—2921no—
395#102—2921no—
396#102—734no—
397#102—734no—
398#97—2921no—
399C4×C8—2921no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—731no—
404#95—731no—
405#95—731no—
406#95—731no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413C24—1461no—
414C2×Q8—5841no—
415C2×Q8—5841no—
416C2×Q8—1464no—
417C2×D4—732no—
418C2×D4—1461no—
419C2×D4—1461no—
420C2×D4—1461no—
421C2×D4—1461no—
422C2×D4—1461no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—5841no—
428C22×C4—5841no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1464no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8C2×C4×D7312yes—
451C2×C8C2×C4×D7312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—5841no—
461C4⋊C4—5841no—
462C4⋊C4—5841no—
463C4⋊C4—5841no—
464C4⋊C4—732no—
465C4⋊C4—732no—
466C4⋊C4—732no—
467C4⋊C4—1464no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C22⋊C4—5841no—
472C22⋊C4—5841no—
473C22⋊C4—5841no—
474C22⋊C4—5841no—
475C22⋊C4—1464no—
476C42—81no—
477C42—5841no—
478C42—5841no—
479C42—1461no—
480C42—1461no—
481C42—1461no—
482C42—1461no—
483C42C2×C4×D7312yes—
484C42#1192402912yes—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—2921no—
491C23—1462no—
492C23C2×D4×D7311yescharacteristic
493Q8—5844no—
494D4—2921no—
495D4—2921no—
496D4—1462no—
497D4—1462no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—5844no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4#1455655312yes—
537C2×C4#1455655312yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4C2×#1192402911yescharacteristic
540C2×C4#1455663611yescharacteristic
541C2×C4#1455663811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663911yescharacteristic
544C8—22no—
545C8—22no—
546C8—2921no—
547C8—2921no—
548C22—2922no—
549C22—2922no—
550C22—2922no—
551C22—5841no—
552C22—5841no—
553C22—5841no—
554C22—5841no—
555C22C2×#1455655311yescharacteristic
556C22#1798916011yescharacteristic
557C22#1798916611yescharacteristic
558C22#1798938911yescharacteristic
559C22#1798938911yescharacteristic
560C22#1798939711yescharacteristic
561C22#1798940311yescharacteristic
562C4—22no—
563C4—22no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—2921no—
569C4—2921no—
570C4—2922no—
571C4—2922no—
572C4—2922no—
573C4—2921no—
574C4—2921no—
575C4#1798906311yescharacteristic
576C4#1798906611yescharacteristic
577C4#1798920011yescharacteristic
578C4#1798921111yescharacteristic
579C2—5842no—
580C2#2261261111yescharacteristic
581C2#2261877211yescharacteristic
582C2#2261878211yescharacteristic
583C2#2262118111yescharacteristic
584C2#2262122111yescharacteristic
585C2#2262133911yescharacteristic
586C2#2262139011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1949C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1949D73
hypocenterC73#22127
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #22612677C2semidirect
4. #22610879C2semidirect
5. #22610667C2semidirect
6. #22610662C2semidirect
11. #22610522C2semidirect
12. #22610501C2semidirect
14. #22609790C2semidirect
15. C73×#1949C2semidirect
18. #17988868C4semidirect
298. #1949D73semidirect
309. C73#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#179888682
#226126772
#226125692
#226108792
#226106672
#226106622
#226106262
#226106122
#226105572
#226105502
#226105222
#226105012
#226099382
#226097902
C73×#19492
#2212773