Finite Group Databasebeta

← #33086562

Subgroups — #33086562

Order 18688.

810 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988544C211yesmaximal, characteristic
2#22612541C211yesmaximal, characteristic
3#22612539C211yesmaximal, characteristic
4#22610099C211yesmaximal, characteristic, complemented
5#22610061C211yesmaximal, characteristic, complemented
6#22610058C211yesmaximal, characteristic, complemented
7#22609965C211yesmaximal, characteristic
8#22609961C211yesmaximal, characteristic
9#22609957C211yesmaximal, characteristic
10#22609953C211yesmaximal, characteristic
11#22609831C211yesmaximal, characteristic, complemented
12#22609830C211yesmaximal, characteristic, complemented
13#107×Dic73C211yesmaximal, characteristic, complemented
14#106×Dic73C211yesmaximal, characteristic, complemented
15C73×#1847C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556548C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23#17988544C418yescomplemented
24C2×#14556501C2211yescharacteristic
25C2×C4×#10596831C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988479—21no—
28#17988479—21no—
29#17988477—21no—
30#17988477—21no—
31#17988474—21no—
32#17988474—21no—
33#17988468—21no—
34#17988468—21no—
35#17988457C2211yescharacteristic
36#17988453C2211yescharacteristic
37#17988444C2211yescharacteristic
38#17988444C2211yescharacteristic
39#17988444C2211yescharacteristic
40#17988444C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#106×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×#97×C73C2211yescharacteristic
58#384×C73C2211yescharacteristic
59#382×C73C2211yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C2×#11924011—22no—
62C2×#11924011—22no—
63C2×#11924011C2×C414yes—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C2×C8×D73—22no—
67C2×C8×D73—22no—
68C2×C8×D73C2×C414yes—
69#14556548C2×C414yes—
70#14556548Q814yes—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007D411yescharacteristic
78C2×#11924007D411yescharacteristic
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005D411yescharacteristic
84C2×#11924005D411yescharacteristic
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90#14556525—22no—
91#14556525—22no—
92#14556523—22no—
93#14556523—22no—
94#14556501C2×C414yes—
95C4×#10596831C2×C414yes—
96C8×Dic73C2×C414yes—
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—
106#14556498—21no—
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116#107×C73—22no—
117#107×C73—22no—
118#106×C73—22no—
119#106×C73—22no—
120#97×C73C2×C414yes—
121#95×C73—21no—
122#95×C73—21no—
123C23×D73C2×D411yescharacteristic
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73C22×C412yes—
136C2×C4×D73C2×Q812yes—
137#11924011—28no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831C22×C412yes—
143C2×#10596831C22×C412yes—
144C8×D73—28no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—81no—
154C4⋊Dic73—81no—
155#11924007C4⋊C414yes—
156#11924007C4⋊C414yes—
157#11924005—81no—
158#11924005—81no—
159#11924005—81no—
160#11924005—81no—
161#11924005C4⋊C414yes—
162#11924005C4⋊C414yes—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73—81no—
166C4×Dic73—81no—
167C4×Dic73C22×C412yes—
168C4×Dic73C2×Q812yes—
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×C73C22×C412yes—
177C2×C8×C73C22×C412yes—
178C2×C8×C73#4112yes—
179C2×C8×C73#4112yes—
180C2×C8×C73#4112yes—
181C2×C8×C73#4112yes—
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—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—81no—
191C4⋊C4×C73—81no—
192C42×C73C22×C412yes—
193C42×C73C2×Q812yes—
194C23×C73C22×D411yescharacteristic
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73C2×C4⋊C412yes—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
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×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73C2×C4⋊C412yes—
219C2×C4×C73C2×C4⋊C412yes—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C22×Q811yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
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×Dic73C2×C4⋊C412yes—
245C2×Dic73C2×C4⋊C412yes—
246C2×Dic73C2×C4⋊C412yes—
247C4×D73—22no—
248C4×D73—22no—
249C4×D73—22no—
250C4×D73—22no—
251C8×C73#11714yes—
252C8×C73#11714yes—
253#10596831—22no—
254#10596831—22no—
255C22×C73C22×C4⋊C411yescharacteristic
256C22×C73#53211yescharacteristic
257C22×C73#53211yescharacteristic
258C22×C73#57611yescharacteristic
259C22×C73#57711yescharacteristic
260C22×C73#57711yescharacteristic
261C22×C73#57811yescharacteristic
262C2×D73—22no—
263C2×D73—22no—
264C2×D73—22no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C4×C73—22no—
270C4×C73—22no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73C2×#11711yescharacteristic
276C4×C73C2×#11711yescharacteristic
277C4×C73#53311yescharacteristic
278C4×C73#53311yescharacteristic
279Dic73—22no—
280Dic73—22no—
281Dic73—22no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285Dic73—81no—
286#20822—731nomaximal, Sylow
287C2×C73#228511yescharacteristic
288C2×C73#291211yescharacteristic
289C2×C73#291311yescharacteristic
290C2×C73#317811yescharacteristic
291C2×C73#317811yescharacteristic
292C2×C73#317911yescharacteristic
293C2×C73#317911yescharacteristic
294D73—42no—
295C2×#405—731no—
296#2314—731no—
297#2314—731no—
298#1847—731no—
299#1847D7311yescharacteristic, complemented
300#1835—731no—
301#1834—731no—
302#1809—731no—
303#1809—731no—
304#1808—731no—
305#1807—731no—
306#1776—731no—
307#1776—731no—
308C4×#107—731no—
309C4×#106—731no—
310C73#2082211yescharacteristic, complemented, Sylow
311C22×M16—731no—
312C23×C8—731no—
313C2×#117—731no—
314C2×#107—731no—
315C2×#107—731no—
316C2×#107C2×D7311yescharacteristic
317C2×#107C2×D7311yescharacteristic
318C2×#106—731no—
319C2×#106—731no—
320C2×#106C2×D7311yescharacteristic
321C2×#106C2×D7311yescharacteristic
322#405—738no—
323C2×#97—731no—
324C2×#97C2×D7311yescharacteristic
325C2×C4×C8—731no—
326C2×C4×C8—731no—
327#388—1461no—
328#388—1461no—
329#388—1461no—
330#388—1461no—
331#387—1461no—
332#387—1461no—
333#387—1461no—
334#387—1461no—
335#384—731no—
336#384C2×D7311yescharacteristic
337#382—731no—
338#382C2×D7311yescharacteristic
339#380—731no—
340#380—731no—
341#380—731no—
342#380—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—731no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355C23×C4—731no—
356C2×M16—734no—
357C2×M16—1462no—
358C2×M16—1462no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—734no—
362C22×C8—1462no—
363C22×C8—1462no—
364C22×C8C22×D7311yescharacteristic
365C22×C8C22×D7311yescharacteristic
366#117—734no—
367#117—734no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
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—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42C22×D7311yescharacteristic
394#107—22no—
395#107—22no—
396#107—1462no—
397#107—1462no—
398#106—22no—
399#106—22no—
400#106—1462no—
401#106—1462no—
402#97—734no—
403#97C4×D7314yes—
404C4×C8—734no—
405C4×C8—734no—
406#95—21no—
407#95—21no—
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#95—1461no—
418C24—731no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—731no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—732no—
427C22×C4—732no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4D4×D7311yescharacteristic
437C22×C4D4×D7311yescharacteristic
438C22×C4C23×D7311yescharacteristic
439M16—1468no—
440C2×C8—732no—
441C2×C8—732no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1468no—
447C2×C8C2×C4×D7312yes—
448C2×C8C2×C4×D7312yes—
449C2×C8#1192402712yes—
450C2×C8#1192402712yes—
451C2×C8#1192402912yes—
452C2×C8#1192402912yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—5841no—
464C4⋊C4—5841no—
465C4⋊C4—734no—
466C4⋊C4—734no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C22⋊C4—5841no—
476C22⋊C4—5841no—
477C22⋊C4—5841no—
478C22⋊C4—5841no—
479C22⋊C4—734no—
480C22⋊C4—734no—
481C42—5841no—
482C42—5841no—
483C42—5841no—
484C42—5841no—
485C42—732no—
486C42—732no—
487C42C2×C4×D7312yes—
488C42Q8×D7312yes—
489C23—732no—
490C23—1461no—
491C23—1461no—
492C23—1461no—
493C23—1461no—
494C23—1461no—
495C23—1461no—
496C23C2×D4×D7311yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
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—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
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—732no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4C4⋊C4×D7312yes—
539C2×C4C4⋊C4×D7312yes—
540C2×C4C22×C4×D7311yescharacteristic
541C2×C4C2×Q8×D7311yescharacteristic
542C2×C4C2×#1192402711yescharacteristic
543C2×C4C2×#1192402711yescharacteristic
544C2×C4C2×#1192402911yescharacteristic
545C2×C4C2×#1192402911yescharacteristic
546C8—1462no—
547C8—1462no—
548C8#1455654914yes—
549C8#1455654914yes—
550C22—2921no—
551C22—2921no—
552C22—2921no—
553C22—2921no—
554C22—1462no—
555C22—1462no—
556C22—1462no—
557C22C2×C4⋊C4×D7311yescharacteristic
558C22#1798913511yescharacteristic
559C22#1798913811yescharacteristic
560C22#1798939111yescharacteristic
561C22#1798939611yescharacteristic
562C22#1798939711yescharacteristic
563C22#1798940311yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—5841no—
571C4—5841no—
572C4—5841no—
573C4—5841no—
574C4—1462no—
575C4—1462no—
576C4—1462no—
577C4C2×#1455654911yescharacteristic
578C4C2×#1455654911yescharacteristic
579C4#1798914211yescharacteristic
580C4#1798914211yescharacteristic
581C2—2922no—
582C2#2261233611yescharacteristic
583C2#2261852511yescharacteristic
584C2#2261853011yescharacteristic
585C2#2262059611yescharacteristic
586C2#2262060411yescharacteristic
587C2#2262061111yescharacteristic
588C2#2262061911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1847C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1847D73
hypocenterC73#20822
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #22610099C2semidirect
5. #22610061C2semidirect
6. #22610058C2semidirect
11. #22609831C2semidirect
12. #22609830C2semidirect
13. #107×Dic73C2semidirect
14. #106×Dic73C2semidirect
15. C73×#1847C2semidirect
23. #17988544C4semidirect
299. #1847D73semidirect
310. C73#20822semidirect

Maximal subgroups

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

subgroupindex
C2×#179885442
#226125412
#226125392
#226100992
#226100612
#226100582
#226099652
#226099612
#226099572
#226099532
#226098312
#226098302
#107×Dic732
#106×Dic732
C73×#18472
#2082273