Finite Group Databasebeta

← #33159089

Subgroups — #33159089

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614364C211yesmaximal, characteristic
2#22612956C211yesmaximal, characteristic
3#22611341C211yesmaximal, characteristic, complemented
4#22611325C211yesmaximal, characteristic, complemented
5#22611128C211yesmaximal, characteristic
6#22611128C211yesmaximal, characteristic
7#22610695C211yesmaximal, characteristic, complemented
8#22610674C211yesmaximal, characteristic, complemented
9#22610517C211yesmaximal, characteristic, complemented
10#22610517C211yesmaximal, characteristic, complemented
11#22610202C211yesmaximal, characteristic, complemented
12#22610132C211yesmaximal, characteristic
13#22610027C211yesmaximal, characteristic
14#22609762C211yesmaximal, characteristic, complemented
15C73×#1924C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27#17988529—21no—
28#17988510—21no—
29#17988493C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988481—21no—
34#17988480—21no—
35#17988475—21no—
36#17988474—21no—
37#17988469—21no—
38#17988457C2211yescharacteristic
39#17988447C2211yescharacteristic
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988438—21no—
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C22⋊C8×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60#14556570—22no—
61#14556570—22no—
62C4×Dic146—22no—
63C4×Dic146—41no—
64#14556550—22no—
65#14556550—41no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556525—41no—
89#14556523—22no—
90#14556523—22no—
91#14556523—22no—
92#14556516—22no—
93#14556516—22no—
94#14556516—22no—
95#14556516—41no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C23×C4×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C2×C22⋊C4×C73D411yescharacteristic
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C4⋊C8×C73—22no—
115C22⋊C8×C73—22no—
116C22⋊C8×C73—41no—
117#97×C73—41no—
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146#4112yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—81no—
147C4⋊Dic73—81no—
148C4⋊Dic73#4112yes—
149C4⋊Dic73#4112yes—
150#11924007—24no—
151#11924007—42no—
152#11924007—42no—
153#11924007—42no—
154#11924007—81no—
155C22⋊Dic73—24no—
156C22⋊Dic73—42no—
157C22⋊Dic73—81no—
158C22⋊Dic73—81no—
159C4×Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2411yescharacteristic
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73#4112yes—
188C22⋊C4×C73—22no—
189C22⋊C4×C73—41no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73C22×D411yescharacteristic
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73#13712yes—
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73H(5,𝔽2)11yescharacteristic
226C2×C4×C732−1+411yescharacteristic
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251Dic146—22no—
252Dic146—22no—
253Dic146—41no—
254Dic146—41no—
255C8×C73—22no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73#52511yescharacteristic
267C22×C73#54611yescharacteristic
268C22×C73#54711yescharacteristic
269C22×C73C2×#13711yescharacteristic
270C22×C73#57511yescharacteristic
271C22×C73#57711yescharacteristic
272C22×C73#57711yescharacteristic
273C4×C73—22no—
274C4×C73—41no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73#52911yescharacteristic
279C4×C73#53911yescharacteristic
280C4×C73#55311yescharacteristic
281C4×C73#55611yescharacteristic
282Dic73—42no—
283Dic73—42no—
284Dic73—42no—
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289Dic73—81no—
290#27060—731nomaximal, Sylow
291C2×C73—81no—
292C2×C73#246611yescharacteristic
293C2×C73#311611yescharacteristic
294C2×C73#312611yescharacteristic
295C2×C73#332111yescharacteristic
296C2×C73#333111yescharacteristic
297C2×C73#334411yescharacteristic
298C2×C73#335011yescharacteristic
299#2497—731no—
300#2362—731no—
301#2072—731no—
302#2068—731no—
303#2035—731no—
304#2035—731no—
305#1948—731no—
306#1944—731no—
307#1924D7311yescharacteristic
308#1920—731no—
309#1920—731no—
310#1865—731no—
311#1854—731no—
312#1825—731no—
313#1763—731no—
314C73#2706011yescharacteristic, complemented, Sylow
315C2×C22⋊Q8—731no—
316C2×C4×Q8—731no—
317C2×#117—731no—
318C2×#107—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×#106—731no—
322C2×C4⋊C8C2×D7311yescharacteristic
323C2×#103—731no—
324C2×#103—731no—
325C2×#103—731no—
326C2×#103—731no—
327C2×C22⋊C8C2×D7311yescharacteristic
328C2×C22⋊C8C2×D7311yescharacteristic
329C2×#97C2×D7311yescharacteristic
330#400—1461no—
331#396—1461no—
332#391—731no—
333#389—731no—
334#389—731no—
335#389—731no—
336#388—1461no—
337#388—1461no—
338#388—1461no—
339#387—1461no—
340#387—1461no—
341#384—731no—
342#380—731no—
343#380—731no—
344C4×C4⋊C4—731no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4—1461no—
347C4×C22⋊C4C2×D7311yescharacteristic
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#336C2×D7311yescharacteristic
357#336C2×D7311yescharacteristic
358C22×Q8—731no—
359C23×C4C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363C22×C8C22×D7311yescharacteristic
364C22⋊Q8—1462no—
365C22⋊Q8—1462no—
366C4×Q8—2921no—
367C4×Q8—1462no—
368#117—2921no—
369#117—1462no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4D29211yescharacteristic
389C2×C22⋊C4D29211yescharacteristic
390C2×C42—1461no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42C22×D7311yescharacteristic
395C2×C42C22×D7311yescharacteristic
396#107—2921no—
397#106—1462no—
398#106—1462no—
399#106—1462no—
400C4⋊C8—22no—
401#103—2921no—
402#103—1462no—
403#103—1462no—
404#103—1462no—
405C22⋊C8—22no—
406C22⋊C8—41no—
407#97—41no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95D29211yescharacteristic
418#95D29211yescharacteristic
419C24C2×D29211yescharacteristic
420C2×Q8—732no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×Q8—1461no—
424C2×Q8—1461no—
425C2×Q8—1461no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C2×D29211yescharacteristic
441C22×C4C2×D29211yescharacteristic
442C22×C4C2×D29211yescharacteristic
443C22×C4C2×D29211yescharacteristic
444C22×C4C2×D29211yescharacteristic
445C22×C4C23×D7311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8#1192402712yes—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—2922no—
461C4⋊C4—5841no—
462C4⋊C4—5841no—
463C4⋊C4—5841no—
464C4⋊C4—732no—
465C4⋊C4—732no—
466C4⋊C4—1464no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—2922no—
482C22⋊C4—5841no—
483C22⋊C4—5841no—
484C22⋊C4—1464no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—2922no—
490C42—2922no—
491C42—2922no—
492C42—5841no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×D29211yescharacteristic
501Q8—2921no—
502Q8—2921no—
503Q8—1462no—
504Q8—1462no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4#1455662412yes—
548C2×C4#41×D7311yescharacteristic
549C2×C4#41×D7311yescharacteristic
550C2×C4#41×D7311yescharacteristic
551C2×C4C2×#1192402711yescharacteristic
552C2×C4#1455663611yescharacteristic
553C2×C4#1455663811yescharacteristic
554C8—22no—
555C8—41no—
556C8—41no—
557C8—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1798909911yescharacteristic
566C22#1798923211yescharacteristic
567C22#1798924111yescharacteristic
568C22C2×#1455662411yescharacteristic
569C22#1798938711yescharacteristic
570C22#1798939911yescharacteristic
571C22#1798939911yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—2922no—
578C4—2922no—
579C4—2922no—
580C4—5841no—
581C4—5841no—
582C4—5841no—
583C4—5841no—
584C4—5841no—
585C4#1798911611yescharacteristic
586C4#1798918011yescharacteristic
587C4#1798928011yescharacteristic
588C4#1798930411yescharacteristic
589C2—81no—
590C2#2261396511yescharacteristic
591C2#2262009411yescharacteristic
592C2#2262022311yescharacteristic
593C2#2262211911yescharacteristic
594C2#2262224811yescharacteristic
595C2#2262238811yescharacteristic
596C2#2262247811yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1924C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1924D73
hypocenterC73#27060
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
3. #22611341C2semidirect
4. #22611325C2semidirect
7. #22610695C2semidirect
8. #22610674C2semidirect
9. #22610517C2semidirect
10. #22610517C2semidirect
11. #22610202C2semidirect
14. #22609762C2semidirect
314. C73#27060semidirect

Maximal subgroups

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

subgroupindex
#226143642
#226129562
#226113412
#226113252
#226111282
#226106952
#226106742
#226105172
#226102022
#226101322
#226100272
#226097622
C73×#19242
#2706073