Finite Group Databasebeta

← #33149397

Subgroups — #33149397

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988687C211yesmaximal, characteristic
3#22612851C211yesmaximal, characteristic, complemented
4#22611452C211yesmaximal, characteristic
5#22611412C211yesmaximal, characteristic, complemented
6#22611305C211yesmaximal, characteristic
7#22611266C211yesmaximal, characteristic, complemented
8#22610671C211yesmaximal, characteristic, complemented
9#22610497C211yesmaximal, characteristic, complemented
10#22610289C211yesmaximal, characteristic
11#22610289C211yesmaximal, characteristic
12#22610198C211yesmaximal, characteristic, complemented
13#22610198C211yesmaximal, characteristic, complemented
14#22610192C211yesmaximal, characteristic
15C73×#2454C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988891—24no—
21#17988687—24no—
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988482—21no—
38#17988482—21no—
39#17988453C2211yescharacteristic
40#17988451—21no—
41#17988438—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#399×C73—21no—
53#396×C73—21no—
54#390×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#385×C73—21no—
59#380×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016Q814yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C4×Dic146—22no—
70C4×Dic146—22no—
71C4×Dic146—22no—
72C4×Dic146—22no—
73C4×Dic146—41no—
74C4×Dic146—41no—
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×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86#14556524—22no—
87#14556524—22no—
88#14556522—22no—
89#14556522—22no—
90#14556522—41no—
91C4⋊#10596831—22no—
92#14556514—22no—
93#14556514—22no—
94#14556514D414yes—
95#14556512—22no—
96#14556512—22no—
97#14556512—41no—
98#14556502—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73C2311yescharacteristic
104C23×C4×C73—21no—
105#125×C73—22no—
106#125×C73—41no—
107C4×D4×C73—22no—
108C4×D4×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126#95×C73—21no—
127#95×C73—21no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146C2×D412yes—
134C2×Dic146C2×Q812yes—
135C2×Dic146#4112yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140#11924016—28no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C2×D412yes—
152C2×#10596831C2×Q812yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73#4112yes—
159#11924007—24no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—42no—
164C4×Dic73—24no—
165C4×Dic73—42no—
166C4×Dic73—42no—
167C24×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73C2×Q812yes—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73—81no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—41no—
193C4⋊C4×C73—41no—
194C4⋊C4×C73—42no—
195C4⋊C4×C73—42no—
196C4⋊C4×C73—81no—
197C4⋊C4×C73—81no—
198C4⋊C4×C73C2×D412yes—
199C22⋊C4×C73—42no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—81no—
202C22⋊C4×C73—81no—
203C22⋊C4×C73—81no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C23×C73—22no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73C22×D411yescharacteristic
218D4×C73—22no—
219D4×C73—22no—
220D4×C73—41no—
221D4×C73—41no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—84no—
250C2×C4×C73C22×D411yescharacteristic
251C2×C4×C73C22×Q811yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73H(5,𝔽2)11yescharacteristic
255C2×C4×C732−1+411yescharacteristic
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268Dic146—22no—
269Dic146—22no—
270Dic146—22no—
271Dic146C22⋊Q814yes—
272#10596831—22no—
273#10596831—22no—
274#10596831—41no—
275#10596831—41no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—81no—
283C22×C73#52511yescharacteristic
284C22×C73D4211yescharacteristic
285C22×C73D4×Q811yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57511yescharacteristic
288C22×C73#57711yescharacteristic
289C22×C73#57711yescharacteristic
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—42no—
297C4×C73—81no—
298C4×C73—81no—
299C4×C73C2×C22⋊Q811yescharacteristic
300C4×C73#53911yescharacteristic
301C4×C73#54711yescharacteristic
302C4×C73#55411yescharacteristic
303Dic73—24no—
304Dic73—42no—
305Dic73—81no—
306Dic73—81no—
307#26399—731nomaximal, Sylow
308C2×C73—82no—
309C2×C73#244911yescharacteristic
310C2×C73#311111yescharacteristic
311C2×C73#312611yescharacteristic
312C2×C73#327111yescharacteristic
313C2×C73#327511yescharacteristic
314C2×C73#338811yescharacteristic
315C2×C73#338811yescharacteristic
316C2×#479—731no—
317C2×#440—731no—
318#2454D7311yescharacteristic
319#2351—731no—
320#2092—731no—
321#2085—731no—
322#2065—731no—
323#2059—731no—
324#1944—731no—
325#1917—731no—
326#1880—731no—
327#1880—731no—
328#1865—731no—
329#1865—731no—
330#1864—731no—
331C73#2639911yescharacteristic, complemented, Sylow
332C22×SD16—731no—
333C2×#125C2×D7311yescharacteristic
334C2×C4×Q8—731no—
335C2×C4×Q8—731no—
336C2×C4×Q8—731no—
337C2×C4×D4C2×D7311yescharacteristic
338#479—1464no—
339#440—1464no—
340C2×#107—731no—
341C2×#107—731no—
342C2×#106—731no—
343C2×#106—731no—
344C2×#106—731no—
345C2×C4⋊C8—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×#102—731no—
350C2×#102—731no—
351C2×#102—731no—
352C2×#97—731no—
353#399—21no—
354#396—21no—
355#390C2×D7311yescharacteristic
356#390C2×D7311yescharacteristic
357#389—731no—
358#389—731no—
359#389—1461no—
360#389—1461no—
361#389C2×D7311yescharacteristic
362#389C2×D7311yescharacteristic
363#385—21no—
364#382—731no—
365#382—1461no—
366#380—21no—
367C4×C4⋊C4—1461no—
368C4×C4⋊C4C2×D7311yescharacteristic
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377C22×Q8—731no—
378C22×D4C22×D7311yescharacteristic
379C23×C4—21no—
380C2×SD16—734no—
381C2×SD16—1462no—
382C2×SD16—1462no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387#125—22no—
388#125—41no—
389C4×Q8—2921no—
390C4×Q8—2921no—
391C4×Q8—1462no—
392C4×Q8—1462no—
393C4×Q8—1462no—
394C4×Q8—1462no—
395C4×D4—22no—
396C4×D4—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4C22⋊D7311yescharacteristic
410C2×C4⋊C4C22⋊D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C4⋊C4C22×D7311yescharacteristic
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42C22×D7311yescharacteristic
423C2×C42C22×D7311yescharacteristic
424C2×C42C22×D7311yescharacteristic
425#107—1462no—
426#107—1462no—
427#106—2921no—
428#106—1462no—
429#106—1462no—
430C4⋊C8—1462no—
431#103—734no—
432#103—1462no—
433#103—1462no—
434#102—2921no—
435#102—1462no—
436#102—1462no—
437#97—1462no—
438#95—21no—
439#95—21no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95—1461no—
444#95C22⋊D7311yescharacteristic
445#95C22⋊D7311yescharacteristic
446C24—21no—
447C2×Q8—732no—
448C2×Q8—732no—
449C2×Q8—732no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4Q8×D7312yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4C2×C22⋊D7311yescharacteristic
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C23×D7311yescharacteristic
476SD16—1468no—
477C2×C8—732no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—2922no—
501C4⋊C4—2922no—
502C4⋊C4—2922no—
503C4⋊C4—2922no—
504C4⋊C4—732no—
505C4⋊C4—1464no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4D4×D7312yes—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—2922no—
524C42—2922no—
525C42—1464no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×C22⋊D7311yescharacteristic
534Q8—734no—
535Q8—1462no—
536Q8—1462no—
537Q8—1462no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—1462no—
579C2×C4—1462no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4C2×D4×D7311yescharacteristic
583C2×C4C2×Q8×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4#41×D7311yescharacteristic
586C2×C4#1455663611yescharacteristic
587C2×C4#1455663911yescharacteristic
588C8—2921no—
589C8—2921no—
590C8—1462no—
591C8—1462no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1798909411yescharacteristic
600C22D4×C22⋊D7311yescharacteristic
601C22Q8×C22⋊D7311yescharacteristic
602C22#1798938511yescharacteristic
603C22#1798938511yescharacteristic
604C22#1798939811yescharacteristic
605C22#1798939811yescharacteristic
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—2922no—
616C4—5841no—
617C4—5841no—
618C4—1464no—
619C4C22⋊Q8×D7311yescharacteristic
620C4#1798918311yescharacteristic
621C4#1798923911yescharacteristic
622C4#1798929111yescharacteristic
623C2—82no—
624C2#2261373711yescharacteristic
625C2#2262003411yescharacteristic
626C2#2262021711yescharacteristic
627C2#2262151211yescharacteristic
628C2#2262155811yescharacteristic
629C2#2262287611yescharacteristic
630C2#2262287611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2454C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2454D73
hypocenterC73#26399
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
3. #22612851C2semidirect
5. #22611412C2semidirect
7. #22611266C2semidirect
8. #22610671C2semidirect
9. #22610497C2semidirect
12. #22610198C2semidirect
13. #22610198C2semidirect
331. C73#26399semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×#179886872
#226128512
#226114522
#226114122
#226113052
#226112662
#226106712
#226104972
#226102892
#226101982
#226101922
C73×#24542
#2639973