Finite Group Databasebeta

← #33158699

Subgroups — #33158699

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614353C211yesmaximal, characteristic
2#22612934C211yesmaximal, characteristic
3#22611341C211yesmaximal, characteristic, complemented
4#22611311C211yesmaximal, characteristic, complemented
5#22611128C211yesmaximal, characteristic
6#22611128C211yesmaximal, characteristic
7#22610695C211yesmaximal, characteristic, complemented
8#22610659C211yesmaximal, characteristic, complemented
9#22610517C211yesmaximal, characteristic, complemented
10#22610510C211yesmaximal, characteristic, complemented
11#22610202C211yesmaximal, characteristic, complemented
12#22610126C211yesmaximal, characteristic
13#22610022C211yesmaximal, characteristic
14C4×#14556516C211yesmaximal, characteristic, complemented
15C73×#1923C211yesmaximal, characteristic
16C2×#14556584—21no—
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28#17988527—21no—
29#17988512—21no—
30#17988493C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988484—21no—
35#17988474—21no—
36#17988472—21no—
37#17988459—21no—
38#17988457C2211yescharacteristic
39#17988447C2211yescharacteristic
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C22⋊C4×Dic73—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×C4×C8×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C23×Dic73—21no—
61#14556584—44no—
62#14556570—22no—
63#14556570—22no—
64C4×Dic146—22no—
65C4×Dic146—41no—
66C2×C4⋊Dic73—22no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—41no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556523—22no—
90#14556523—22no—
91#14556523—22no—
92#14556523—41no—
93#14556516—22no—
94#14556516—22no—
95#14556516—22no—
96#14556516—41no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106C23×C4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C4⋊C8×C73—22no—
116C22⋊C8×C73—22no—
117C22⋊C8×C73—41no—
118C4×C8×C73—41no—
119#95×C73D411yescharacteristic
120#95×C73D411yescharacteristic
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—21no—
126C2×Dic146#4112yes—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—42no—
136C22×Dic73—42no—
137C22×Dic73—81no—
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—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—42no—
151C4⋊Dic73—42no—
152C4⋊Dic73—42no—
153C4⋊Dic73#4112yes—
154C4⋊Dic73#4112yes—
155#11924007—24no—
156#11924007—42no—
157#11924007—42no—
158#11924007—42no—
159#11924007—81no—
160C22⋊Dic73—24no—
161C22⋊Dic73—42no—
162C22⋊Dic73—42no—
163C4×Dic73—42no—
164C4×Dic73—42no—
165C4×Dic73—81no—
166C24×C73C2×D411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
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—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73#4112yes—
191C22⋊C4×C73—41no—
192C22⋊C4×C73SD1614yes—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73C2×SD1612yes—
204C23×C73C22×D411yescharacteristic
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—21no—
217C2×C4×C73—21no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C2×SD1612yes—
223C2×C4×C73C2×SD1612yes—
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73H(5,𝔽2)11yescharacteristic
229C2×C4×C732−1+411yescharacteristic
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
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—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—84no—
255Dic146—22no—
256Dic146—22no—
257Dic146—41no—
258Dic146—41no—
259C8×C73—22no—
260C8×C73—41no—
261C8×C73—41no—
262C8×C73—41no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73#52511yescharacteristic
271C22×C73#54611yescharacteristic
272C22×C73#54711yescharacteristic
273C22×C73C22×SD1611yescharacteristic
274C22×C73#57511yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57811yescharacteristic
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73#52911yescharacteristic
283C4×C73#53911yescharacteristic
284C4×C73#55311yescharacteristic
285C4×C73#55611yescharacteristic
286Dic73—42no—
287Dic73—42no—
288Dic73—42no—
289Dic73—81no—
290Dic73—81no—
291Dic73—81no—
292Dic73—81no—
293Dic73—81no—
294#27034—731nomaximal, Sylow
295C2×C73—42no—
296C2×C73#246611yescharacteristic
297C2×C73#310311yescharacteristic
298C2×C73#312011yescharacteristic
299C2×C73#329811yescharacteristic
300C2×C73#331111yescharacteristic
301C2×C73#333511yescharacteristic
302C2×C73#335211yescharacteristic
303#2496—731no—
304#2360—731no—
305#2072—731no—
306#2066—731no—
307#2035—731no—
308#2035—731no—
309#1948—731no—
310#1941—731no—
311#1923D7311yescharacteristic
312#1920—731no—
313#1919—731no—
314#1865—731no—
315#1853—731no—
316#1824—731no—
317C4×#103—731no—
318C73#2703411yescharacteristic, complemented, Sylow
319C2×#126—1461no—
320C2×C22⋊Q8—731no—
321C2×C4×Q8—731no—
322C22×C4⋊C4—731no—
323C2×#106—731no—
324C2×#106—731no—
325C2×#106—731no—
326C2×#106—731no—
327C2×C4⋊C8C2×D7311yescharacteristic
328C2×#103—731no—
329C2×#103—731no—
330C2×#103—731no—
331C2×#103—731no—
332C2×C22⋊C8C2×D7311yescharacteristic
333C2×C22⋊C8C2×D7311yescharacteristic
334C2×C4×C8C2×D7311yescharacteristic
335#400—1461no—
336#397—1461no—
337#391—731no—
338#389—731no—
339#389—731no—
340#389—731no—
341#389—1461no—
342#387—1461no—
343#387—1461no—
344#385—1461no—
345#384—731no—
346#380—731no—
347#380—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C22⋊C4—1461no—
351C4×C22⋊C4C2×D7311yescharacteristic
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#336C2×D7311yescharacteristic
361#336C2×D7311yescharacteristic
362C22×Q8—731no—
363C23×C4—1461no—
364C23×C4C22×D7311yescharacteristic
365C22×C8C22×D7311yescharacteristic
366C22×C8C22×D7311yescharacteristic
367C22×C8C22×D7311yescharacteristic
368C22×C8C22×D7311yescharacteristic
369#126—2924no—
370C22⋊Q8—1462no—
371C22⋊Q8—1462no—
372C4×Q8—2921no—
373C4×Q8—1462no—
374C2×C4⋊C4—2921no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1462no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4D29211yescharacteristic
395C2×C22⋊C4D29211yescharacteristic
396C2×C42—1461no—
397C2×C42—1461no—
398C2×C42—1461no—
399C2×C42C22×D7311yescharacteristic
400C2×C42C22×D7311yescharacteristic
401#106—2921no—
402#106—1462no—
403#106—1462no—
404#106—1462no—
405C4⋊C8—22no—
406#103—2921no—
407#103—1462no—
408#103—1462no—
409#103—1462no—
410C22⋊C8—22no—
411C22⋊C8—41no—
412C4×C8—41no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95D29211yescharacteristic
423#95D29211yescharacteristic
424C24C2×D29211yescharacteristic
425C2×Q8—732no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8—1461no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—2922no—
438C22×C4—2922no—
439C22×C4—5841no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4C2×D29211yescharacteristic
449C22×C4C2×D29211yescharacteristic
450C22×C4C2×D29211yescharacteristic
451C22×C4C2×D29211yescharacteristic
452C22×C4C2×D29211yescharacteristic
453C22×C4C23×D7311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8#1192402712yes—
466C4⋊C4—2922no—
467C4⋊C4—2922no—
468C4⋊C4—2922no—
469C4⋊C4—2922no—
470C4⋊C4—2922no—
471C4⋊C4—2922no—
472C4⋊C4—5841no—
473C4⋊C4—732no—
474C4⋊C4—732no—
475C4⋊C4—1464no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C22⋊C4—41no—
489C22⋊C4—2922no—
490C22⋊C4—2922no—
491C22⋊C4—1464no—
492C22⋊C4#1192401714yes—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—2922no—
498C42—2922no—
499C42—5841no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23C2×#1192401712yes—
507C23C22×D29211yescharacteristic
508Q8—2921no—
509Q8—2921no—
510Q8—1462no—
511Q8—1462no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
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—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—5844no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4—1462no—
554C2×C4C2×#1192401712yes—
555C2×C4C2×#1192401712yes—
556C2×C4#41×D7311yescharacteristic
557C2×C4#41×D7311yescharacteristic
558C2×C4#41×D7311yescharacteristic
559C2×C4C2×#1192402711yescharacteristic
560C2×C4#1455663611yescharacteristic
561C2×C4#1455663811yescharacteristic
562C8—22no—
563C8—41no—
564C8—41no—
565C8—41no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1798909911yescharacteristic
574C22#1798923211yescharacteristic
575C22#1798924111yescharacteristic
576C22C22×#1192401711yescharacteristic
577C22#1798938711yescharacteristic
578C22#1798939911yescharacteristic
579C22#1798940411yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—2922no—
586C4—2922no—
587C4—2922no—
588C4—5841no—
589C4—5841no—
590C4—5841no—
591C4—5841no—
592C4—5841no—
593C4#1798911611yescharacteristic
594C4#1798918011yescharacteristic
595C4#1798928011yescharacteristic
596C4#1798930411yescharacteristic
597C2—42no—
598C2#2261396511yescharacteristic
599C2#2261996311yescharacteristic
600C2#2262014711yescharacteristic
601C2#2262179711yescharacteristic
602C2#2262196911yescharacteristic
603C2#2262228911yescharacteristic
604C2#2262250511yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1923C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1923D73
hypocenterC73#27034
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #22611341C2semidirect
4. #22611311C2semidirect
7. #22610695C2semidirect
8. #22610659C2semidirect
9. #22610517C2semidirect
10. #22610510C2semidirect
11. #22610202C2semidirect
14. C4×#14556516C2semidirect
318. C73#27034semidirect

Maximal subgroups

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

subgroupindex
#226143532
#226129342
#226113412
#226113112
#226111282
#226106952
#226106592
#226105172
#226105102
#226102022
#226101262
#226100222
C4×#145565162
C73×#19232
#2703473