Finite Group Databasebeta

← #33103607

Subgroups — #33103607

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D73C211yesmaximal, characteristic
2#22612804C211yesmaximal, characteristic
3#22612799C211yesmaximal, characteristic
4#22610658C211yesmaximal, characteristic, complemented
5#22610656C211yesmaximal, characteristic, complemented
6#22610517C211yesmaximal, characteristic, complemented
7#22610515C211yesmaximal, characteristic, complemented
8#22610511C211yesmaximal, characteristic, complemented
9#22610020C211yesmaximal, characteristic
10#22610016C211yesmaximal, characteristic
11#22609955C211yesmaximal, characteristic
12C4×#14556522C211yesmaximal, characteristic, complemented
13#22609819C211yesmaximal, characteristic, complemented
14#22609701C211yesmaximal, characteristic
15C73×#1919C211yesmaximal, characteristic, complemented
16C22×C8×D73C2211yescharacteristic
17C2×#14556549—21no—
18C2×C4⋊C4×D73C2211yescharacteristic
19C2×C4⋊C4×D73C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25#106×D73C418yescomplemented
26C2×#14556503C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988474—21no—
33#17988471—21no—
34#17988469—21no—
35#17988468—21no—
36#17988449—21no—
37#17988446—21no—
38#17988444C2211yescharacteristic
39#17988444C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C4×#11924005—21no—
44C4×#11924005C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#106×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C2×C8×D73—22no—
66C2×C8×D73—22no—
67C2×C8×D73C2×C414yes—
68#14556549—44no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C4⋊C4×D73—22no—
79C4⋊C4×D73—22no—
80C4⋊C4×D73C2×C414yes—
81C4⋊C4×D73C2×C414yes—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005D411yescharacteristic
85C2×#11924005D411yescharacteristic
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90#14556523C2×C414yes—
91#14556522—22no—
92#14556522—22no—
93#14556522C2×C414yes—
94#14556522C2×C414yes—
95#14556503—22no—
96C4×#10596831—22no—
97C8×Dic73—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498D411yescharacteristic
107#14556498D411yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117#106×C73C2×C414yes—
118#95×C73—21no—
119#95×C73—21no—
120C23×D73C2×D411yescharacteristic
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—24no—
133C2×C4×D73—24no—
134C2×C4×D73—42no—
135C2×C4×D73—42no—
136C2×C4×D73—81no—
137C2×C4×D73—81no—
138C2×C4×D73C22×C412yes—
139C2×C4×D73#4112yes—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C22×C412yes—
144C2×#10596831#4112yes—
145C2×#10596831#4112yes—
146C8×D73—28no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73C22×C412yes—
156#11924007—24no—
157#11924007—24no—
158#11924007—81no—
159#11924007—81no—
160#11924005—22no—
161#11924005—42no—
162#11924005—42no—
163#11924005SD1614yes—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73—81no—
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×C73C2411yescharacteristic
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73C22×C412yes—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—42no—
189C4⋊C4×C73C22×C412yes—
190C4⋊C4×C73C22×C412yes—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—81no—
194C23×C73C22×D411yescharacteristic
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73C4×D412yes—
200C22×D73C2×SD1612yes—
201C22×D73#13512yes—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
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×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73C2×SD1612yes—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
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×Dic73C4×D412yes—
246C2×Dic73C2×SD1612yes—
247C4×D73—22no—
248C4×D73—22no—
249C4×D73—22no—
250C4×D73—22no—
251C4×D73—48no—
252C4×D73—84no—
253C8×C73—22no—
254C8×C73—41no—
255#10596831—41no—
256#10596831#11714yescomplemented
257C22×C73C2×C4×D411yescharacteristic
258C22×C73#54711yescharacteristic
259C22×C73#54711yescharacteristic
260C22×C73C22×SD1611yescharacteristic
261C22×C73C2×#13511yescharacteristic
262C22×C73#57711yescharacteristic
263C22×C73#57711yescharacteristic
264C2×D73—22no—
265C2×D73—22no—
266C2×D73—22no—
267C2×D73—22no—
268C2×D73—22no—
269C2×D73—22no—
270C2×D73C4×SD1614yes—
271C4×C73—22no—
272C4×C73—41no—
273C4×C73—42no—
274C4×C73—42no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73C2×#11711yescharacteristic
278C4×C73#51911yescharacteristic
279C4×C73#55211yescharacteristic
280C4×C73#55211yescharacteristic
281Dic73—22no—
282Dic73—22no—
283Dic73—41no—
284Dic73—42no—
285Dic73—42no—
286Dic73—81no—
287Dic73—81no—
288#22609—731nomaximal, Sylow
289C2×C73#234411yescharacteristic
290C2×C73C2×C4×SD1611yescharacteristic
291C2×C73#294911yescharacteristic
292C2×C73#333511yescharacteristic
293C2×C73#333511yescharacteristic
294C2×C73#333711yescharacteristic
295C2×C73#333711yescharacteristic
296D73—24no—
297C22×#106—731no—
298#2345—731no—
299#2344—731no—
300#1941—731no—
301#1941—731no—
302#1920—731no—
303#1920—731no—
304#1919—731no—
305#1919D7311yescharacteristic, complemented
306#1824—731no—
307#1823—731no—
308#1807—731no—
309C4×#106—731no—
310C4×#106—731no—
311C8×C22⋊C4—731no—
312C73#2260911yescharacteristic, complemented, Sylow
313C23×C8—731no—
314C2×#117—1461no—
315C22×C4⋊C4—731no—
316C22×C4⋊C4—731no—
317C2×#106—731no—
318C2×#106—731no—
319C2×#106—731no—
320C2×#106—738no—
321C2×#106—731no—
322C2×#106—731no—
323C2×#106C2×D7311yescharacteristic
324C2×C22⋊C8—731no—
325C2×C4×C8—731no—
326C2×C4×C8—731no—
327#389—731no—
328#389—731no—
329#389—731no—
330#389C2×D7311yescharacteristic
331#387—1461no—
332#387—1461no—
333#387—1461no—
334#387—1461no—
335#381—1461no—
336#380—731no—
337#380—731no—
338#380—1461no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4C2×D7311yescharacteristic
343C4×C22⋊C4—731no—
344C4×C22⋊C4—1461no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340C2×D7311yescharacteristic
353#340C2×D7311yescharacteristic
354#340C2×D7311yescharacteristic
355#336—731no—
356#336C2×D7311yescharacteristic
357C23×C4—731no—
358C23×C4—1461no—
359C23×C4—1461no—
360C22×C8—731no—
361C22×C8—734no—
362C22×C8—731no—
363C22×C8—1462no—
364C22×C8—1462no—
365C22×C8C22×D7311yescharacteristic
366C22×C8C22×D7311yescharacteristic
367#117—2924no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—734no—
372C2×C4⋊C4—734no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1462no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C42—21no—
391C2×C42—731no—
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42—1461no—
395C2×C42C22×D7311yescharacteristic
396#106—734no—
397#106—734no—
398#106—734no—
399#106—1462no—
400#106—1462no—
401#106C4×D7314yes—
402C22⋊C8—1462no—
403C4×C8—1462no—
404C4×C8—1462no—
405#95—21no—
406#95—21no—
407#95—731no—
408#95—731no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417C24—731no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—2922no—
423C22×C4—2922no—
424C22×C4—5841no—
425C22×C4—5841no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—731no—
431C22×C4—1464no—
432C22×C4—1464no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4D4×D7311yescharacteristic
442C22×C4D4×D7311yescharacteristic
443C22×C4C23×D7311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—732no—
450C2×C8—732no—
451C2×C8—732no—
452C2×C8—1468no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8C2×C4×D7312yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—2922no—
465C4⋊C4—5841no—
466C4⋊C4—5841no—
467C4⋊C4—732no—
468C4⋊C4—732no—
469C4⋊C4—1464no—
470C4⋊C4—1464no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4C2×C4×D7312yes—
478C4⋊C4C2×C4×D7312yes—
479C22⋊C4—2922no—
480C22⋊C4—2922no—
481C22⋊C4—734no—
482C22⋊C4—1462no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—2922no—
487C42—2922no—
488C42—5841no—
489C42—1461no—
490C42—1461no—
491C23—732no—
492C23—732no—
493C23—732no—
494C23—1461no—
495C23—1461no—
496C23—1461no—
497C23—1461no—
498C23C2×D4×D7311yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
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—41no—
515C2×C4—41no—
516C2×C4—2928no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—5844no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1462no—
540C2×C4—1462no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4SD16×D7312yes—
544C2×C4C22×C4×D7311yescharacteristic
545C2×C4#41×D7311yescharacteristic
546C2×C4#41×D7311yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4#1455663811yescharacteristic
549C2×C4#1455663911yescharacteristic
550C8—22no—
551C8—41no—
552C8—2921no—
553C8—734no—
554C22—734no—
555C22—1462no—
556C22—1462no—
557C22—1462no—
558C22—1462no—
559C22—1462no—
560C22—1462no—
561C22C4×D4×D7311yescharacteristic
562C22#1798923511yescharacteristic
563C22#1798923911yescharacteristic
564C22C2×SD16×D7311yescharacteristic
565C22#135×D7311yescharacteristic
566C22#1798939611yescharacteristic
567C22#1798939711yescharacteristic
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—2922no—
575C4—2921no—
576C4—2922no—
577C4—5841no—
578C4—5841no—
579C4—1462no—
580C4—1462no—
581C4#117×D7311yescharacteristic
582C4#1798906311yescharacteristic
583C4#1798926911yescharacteristic
584C4#1798927311yescharacteristic
585C2—1464no—
586C2#2261279711yescharacteristic
587C2C4×SD16×D7311yescharacteristic
588C2#2261878711yescharacteristic
589C2#2262228611yescharacteristic
590C2#2262229111yescharacteristic
591C2#2262231311yescharacteristic
592C2#2262231711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1919C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1919D73
hypocenterC73#22609
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #22610658C2semidirect
5. #22610656C2semidirect
6. #22610517C2semidirect
7. #22610515C2semidirect
8. #22610511C2semidirect
12. C4×#14556522C2semidirect
13. #22609819C2semidirect
15. C73×#1919C2semidirect
25. #106×D73C4semidirect
256. #10596831#117semidirect
305. #1919D73semidirect
312. C73#22609semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D732
#226128042
#226127992
#226106582
#226106562
#226105172
#226105152
#226105112
#226100202
#226100162
#226099552
C4×#145565222
#226098192
#226097012
C73×#19192
#2260973