Finite Group Databasebeta

← #33108000

Subgroups — #33108000

Order 18688.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988886C211yesmaximal, characteristic
2#22613023C211yesmaximal, characteristic, complemented
3#22612832C211yesmaximal, characteristic
4#22610898C211yesmaximal, characteristic, complemented
5#22610693C211yesmaximal, characteristic, complemented
6#22610666C211yesmaximal, characteristic, complemented
7#22610660C211yesmaximal, characteristic, complemented
8#22610505C211yesmaximal, characteristic, complemented
9#22610195C211yesmaximal, characteristic
10#22610193C211yesmaximal, characteristic
11#22609804C211yesmaximal, characteristic, complemented
12#22609759C211yesmaximal, characteristic
13#22609755C211yesmaximal, characteristic
14#22609745C211yesmaximal, characteristic
15C73×#1920C211yesmaximal, characteristic, complemented
16C2×#14556582C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556553C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×C4⋊C4×D73—21no—
21#17988886C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556511C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988486C2211yescharacteristic
34#17988486C2211yescharacteristic
35#17988485—21no—
36#17988485C2211yescharacteristic
37#17988474—21no—
38#17988470—21no—
39#17988461—21no—
40#17988458—21no—
41#17988456C2211yescharacteristic
42#17988446—21no—
43C4⋊C4×Dic73C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45C4×#11924007—21no—
46C4×#11924007C2211yescharacteristic
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×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556582—22no—
67#14556582C2×C414yescomplemented
68C4×D292—22no—
69C4×D292—41no—
70#14556553—22no—
71#14556553—22no—
72#14556553—41no—
73#14556553C2×C414yes—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007D411yescharacteristic
85C2×#11924007D411yescharacteristic
86C4⋊C4×D73—44no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556524C2×C414yescomplemented
95#14556523—41no—
96#14556522—41no—
97#14556520C2×C414yescomplemented
98#14556513—22no—
99#14556513C2×C414yes—
100#14556511C2×C414yes—
101#14556511Q814yes—
102#14556502—41no—
103#14556501—22no—
104C8×Dic73—41no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498D411yescharacteristic
111#14556498D411yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#106×C73C2×C414yescomplemented
122#95×C73—21no—
123#95×C73—21no—
124C23×D73—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292C22×C412yes—
129C2×D292C2×Q812yes—
130C2×D292#4112yes—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C2×C4×D73—24no—
139C2×C4×D73—42no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145C2×C4×D73—81no—
146C2×C4×D73—81no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C22×C412yes—
152C2×#10596831C2×Q812yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—24no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—42no—
160C4⋊Dic73—81no—
161C4⋊Dic73C22×C412yes—
162#11924007—24no—
163#11924007—22no—
164#11924007—41no—
165#11924007—42no—
166#11924007—42no—
167#11924007—81no—
168#11924005—24no—
169#11924005—42no—
170#11924005—42no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—42no—
174C4×Dic73—81no—
175C4×Dic73C22×C412yes—
176C4×Dic73#4112yes—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73C22×C412yes—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—42no—
196C4⋊C4×C73C22×C412yes—
197C4⋊C4×C73C22×C412yes—
198C4⋊C4×C73C2×Q812yes—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—81no—
202C23×C73C22×D411yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—41no—
207C22×D73—41no—
208C22×D73—41no—
209C22×D73—41no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73C23×C411yescharacteristic
229C2×C4×C73C22×Q811yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—22no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
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×Dic73C4×D412yes—
253C2×Dic73C4×D412yes—
254C2×Dic73AGL(1,ℤ/8)12yes—
255D292—22no—
256D292—22no—
257D292—22no—
258D292C4×Q814yes—
259C4×D73—48no—
260C4×D73—84no—
261C4×D73—84no—
262C8×C73—22no—
263C8×C73—41no—
264#10596831—22no—
265#10596831—22no—
266C22×C73C2×C4×D411yescharacteristic
267C22×C73#54611yescharacteristic
268C22×C73D4×Q811yescharacteristic
269C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C73#57511yescharacteristic
271C22×C73#57711yescharacteristic
272C22×C73#57711yescharacteristic
273C2×D73—24no—
274C2×D73—42no—
275C2×D73—42no—
276C2×D73—42no—
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—42no—
280C4×C73—42no—
281C4×C73—42no—
282C4×C73—42no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73—81no—
286C4×C73C2×C4×Q811yescharacteristic
287C4×C73#52011yescharacteristic
288C4×C73#55211yescharacteristic
289C4×C73#55411yescharacteristic
290Dic73—22no—
291Dic73—22no—
292Dic73—22no—
293Dic73—42no—
294Dic73—41no—
295Dic73—81no—
296Dic73—81no—
297#22978—731nomaximal, Sylow
298C2×C73#236011yescharacteristic
299C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
300C2×C73#295211yescharacteristic
301C2×C73#330711yescharacteristic
302C2×C73#331011yescharacteristic
303C2×C73#338411yescharacteristic
304C2×C73#338811yescharacteristic
305D73—44no—
306C2×#478—731no—
307#2370—731no—
308D4×C4⋊C4—731no—
309#1992—731no—
310#1948—731no—
311#1943—731no—
312#1941—731no—
313#1920D7311yescharacteristic, complemented
314#1918—731no—
315#1864—731no—
316#1864—731no—
317#1771—731no—
318#1762—731no—
319#1762—731no—
320C4×#102—731no—
321C73#2297811yescharacteristic, complemented, Sylow
322C2×#125—731no—
323C2×C4×D4—731no—
324C2×C4×D4—731no—
325C2×C4×D4—731no—
326C22×C4⋊C4—1461no—
327#478—738no—
328C2×#107—731no—
329C2×#106—731no—
330C2×#106—731no—
331C2×#106C2×D7311yescharacteristic
332C2×C4⋊C8—731no—
333C2×#102—731no—
334C2×#102—731no—
335C2×#102—731no—
336C2×#102—731no—
337C2×#97—731no—
338C2×#97—731no—
339C2×C4×C8—731no—
340#389—731no—
341#389—731no—
342#389—731no—
343#389—1461no—
344#389C2×D7311yescharacteristic
345#387—1461no—
346#387—1461no—
347#385—1461no—
348#385—1461no—
349#384—731no—
350#380—1461no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—1461no—
355C4×C4⋊C4C2×D7311yescharacteristic
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340C2×D7311yescharacteristic
362#340C2×D7311yescharacteristic
363#340C2×D7311yescharacteristic
364#336—731no—
365#336C2×D7311yescharacteristic
366C22×D4—731no—
367C23×C4—1461no—
368C23×C4—1461no—
369C23×C4—1461no—
370C22×C8—731no—
371C22×C8—731no—
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374#125—734no—
375#125—1462no—
376C4×D4—2921no—
377C4×D4—2921no—
378C4×D4—734no—
379C4×D4—1462no—
380C4×D4—1462no—
381C4×D4—1462no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—2924no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C42—21no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42C22×D7311yescharacteristic
409#107—734no—
410#106—2921no—
411#106—2921no—
412#106C4×D7314yescomplemented
413C4⋊C8—734no—
414#102—734no—
415#102—734no—
416#102—734no—
417#102—1462no—
418#97—2921no—
419#97—1462no—
420C4×C8—2921no—
421#95—21no—
422#95—21no—
423#95—731no—
424#95—731no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430C24—1461no—
431C2×D4—732no—
432C2×D4—732no—
433C2×D4—732no—
434C2×D4—1461no—
435C2×D4—1461no—
436C2×D4—1461no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2922no—
442C22×C4—2922no—
443C22×C4—2922no—
444C22×C4—2922no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—5841no—
448C22×C4—5841no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—1464no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4D4×D7311yescharacteristic
458C22×C4D4×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—732no—
466C2×C8—732no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8C2×C4×D7312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—2922no—
479C4⋊C4—2921no—
480C4⋊C4—2922no—
481C4⋊C4—2922no—
482C4⋊C4—5841no—
483C4⋊C4—5841no—
484C4⋊C4—732no—
485C4⋊C4—1462no—
486C4⋊C4—1464no—
487C4⋊C4—1464no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C2×C4×D7312yes—
494C4⋊C4C2×C4×D7312yes—
495C4⋊C4Q8×D7312yes—
496C22⋊C4—2922no—
497C22⋊C4—2922no—
498C22⋊C4—1464no—
499C42—21no—
500C42—21no—
501C42—81no—
502C42—2922no—
503C42—5841no—
504C42—732no—
505C42—732no—
506C42—1461no—
507C42—1461no—
508C23—2921no—
509C23—2921no—
510C23—2921no—
511C23—2921no—
512C23—1462no—
513C23—1462no—
514C23—1462no—
515C23C2×D4×D7311yescharacteristic
516D4—734no—
517D4—1462no—
518D4—1462no—
519D4—1462no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—2928no—
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—5844no—
548C2×C4—5844no—
549C2×C4—732no—
550C2×C4—732no—
551C2×C4—732no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1462no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C2×Q8×D7311yescharacteristic
564C2×C4#41×D7311yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4#1455663711yescharacteristic
567C2×C4#1455663811yescharacteristic
568C8—22no—
569C8—41no—
570C8—1462no—
571C8—1462no—
572C22—2922no—
573C22—2922no—
574C22—2922no—
575C22—2922no—
576C22—2922no—
577C22—2922no—
578C22—1464no—
579C22C4×D4×D7311yescharacteristic
580C22#1798922911yescharacteristic
581C22#1798925011yescharacteristic
582C22AGL(1,ℤ/8)×D7311yescharacteristic
583C22#1798938611yescharacteristic
584C22#1798939311yescharacteristic
585C22#1798939411yescharacteristic
586C4—41no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—81no—
592C4—2922no—
593C4—2921no—
594C4—5841no—
595C4—5841no—
596C4—1462no—
597C4—1462no—
598C4—1462no—
599C4C4×Q8×D7311yescharacteristic
600C4#1798906711yescharacteristic
601C4#1798927311yescharacteristic
602C4#1798929311yescharacteristic
603C2—2924no—
604C2#2261293911yescharacteristic
605C2#2261873611yescharacteristic
606C2#2261880911yescharacteristic
607C2#2262191011yescharacteristic
608C2#2262195511yescharacteristic
609C2#2262282511yescharacteristic
610C2#2262287911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1920C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1920D73
hypocenterC73#22978
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
2. #22613023C2semidirect
4. #22610898C2semidirect
5. #22610693C2semidirect
6. #22610666C2semidirect
7. #22610660C2semidirect
8. #22610505C2semidirect
11. #22609804C2semidirect
15. C73×#1920C2semidirect
21. #17988886C4semidirect
67. #14556582C2×C4semidirect
94. #14556524C2×C4semidirect
97. #14556520C2×C4semidirect
121. #106×C73C2×C4semidirect
313. #1920D73semidirect
321. C73#22978semidirect
412. #106C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888862
#226130232
#226128322
#226108982
#226106932
#226106662
#226106602
#226105052
#226101952
#226101932
#226098042
#226097592
#226097552
#226097452
C73×#19202
#2297873