Finite Group Databasebeta

← #33093248

Subgroups — #33093248

Order 18688.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988904C211yesmaximal, characteristic
2#22612627C211yesmaximal, characteristic
3#22612285C211yesmaximal, characteristic, complemented
4#22610555C211yesmaximal, characteristic, complemented
5#22610517C211yesmaximal, characteristic, complemented
6#22610509C211yesmaximal, characteristic, complemented
7#22610250C211yesmaximal, characteristic
8#22610127C211yesmaximal, characteristic, complemented
9#22610016C211yesmaximal, characteristic, complemented
10#22609830C211yesmaximal, characteristic, complemented
11#22609819C211yesmaximal, characteristic, complemented
12#22609758C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic, complemented
14#22609696C211yesmaximal, characteristic, complemented
15C73×#1925C211yesmaximal, characteristic, complemented
16C2×#14556562C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×#14556548C2211yescharacteristic
20C2×C4⋊C4×D73C2211yescharacteristic
21#17988904C418yescomplemented
22C2×#14556523C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988487—21no—
32#17988485C2211yescharacteristic
33#17988475—21no—
34#17988464—21no—
35#17988459—21no—
36#17988444C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924007—21no—
41C4×C4⋊Dic73C2211yescharacteristic
42#17988438—21no—
43C4×#11924005—21no—
44#17988434—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C22⋊D73—21no—
61C23×Dic73—21no—
62C22×C4×D73—21no—
63C22×C4×D73C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556562—22no—
67#14556562C2×C414yes—
68D4×Dic73—22no—
69D4×Dic73—22no—
70#14556549—22no—
71#14556549—22no—
72#14556548—22no—
73#14556548—22no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
82C4⋊C4×D73—22no—
83C4⋊C4×D73C2×C414yescomplemented
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×#11924005D411yescharacteristic
89C2×#11924005D411yescharacteristic
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73C2311yescharacteristic
96#14556523C2×C414yescomplemented
97#14556522—22no—
98#14556522C2×C414yescomplemented
99#14556512—22no—
100#14556512C2×C414yes—
101#14556503—22no—
102#14556503C2×C414yescomplemented
103#14556501—22no—
104C8×Dic73—22no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111C22×D4×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#107×C73—22no—
122#102×C73—22no—
123#102×C73C2×C414yes—
124#95×C73—21no—
125C23×D73C2×D411yescharacteristic
126C2×C22⋊D73—24no—
127C2×C22⋊D73—81no—
128C2×C22⋊D73—81no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—42no—
134C22×Dic73—81no—
135C22×Dic73—81no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—24no—
144C2×C4×D73—81no—
145C2×C4×D73—81no—
146C2×C4×D73C22×C412yes—
147C2×C4×D73#4112yes—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831C22×C412yes—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73C22×C412yes—
161C4⋊Dic73#4112yes—
162#11924007—24no—
163#11924007—22no—
164#11924007—22no—
165#11924007—81no—
166C22⋊Dic73—24no—
167C22⋊Dic73—42no—
168#11924005—24no—
169#11924005—22no—
170#11924005—22no—
171#11924005—81no—
172C4×Dic73—24no—
173C4×Dic73—81no—
174C4×Dic73—81no—
175C4×Dic73—81no—
176C4×Dic73—81no—
177C4×Dic73#4112yes—
178C4×Dic73#4112yes—
179C24×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—21no—
185C2×D4×C73C22×C412yes—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C412yes—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73C22×C412yes—
203C4⋊C4×C73#4112yes—
204C22⋊C4×C73—81no—
205C22⋊C4×C73—81no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—81no—
209C23×C73—22no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73C22×D411yescharacteristic
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73C4×D412yes—
224D4×C73—22no—
225D4×C73—22no—
226D4×C73—41no—
227D4×C73—41no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C22⋊D73—84no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—84no—
269C2×Dic73C4×D412yes—
270C2×Dic73#12312yes—
271C2×Dic73#12312yes—
272C4×D73—22no—
273C4×D73—22no—
274C4×D73—22no—
275C4×D73—22no—
276C4×D73—84no—
277C8×C73—22no—
278C8×C73—41no—
279#10596831—22no—
280#10596831—41no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73C2×C4×D411yescharacteristic
289C22×C73C2×#12311yescharacteristic
290C22×C73#52511yescharacteristic
291C22×C73#57611yescharacteristic
292C22×C73#57711yescharacteristic
293C22×C73#57711yescharacteristic
294C22×C73#57711yescharacteristic
295C2×D73—22no—
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—41no—
299C2×D73—41no—
300C2×D73—41no—
301C2×D73—41no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73—42no—
305C4×C73—42no—
306C4×C73—81no—
307C4×C73C4×#4111yescharacteristic
308C4×C73C4×#4111yescharacteristic
309C4×C73#52911yescharacteristic
310C4×C73#52911yescharacteristic
311Dic73—22no—
312Dic73—22no—
313Dic73—22no—
314Dic73—42no—
315Dic73—42no—
316Dic73—42no—
317Dic73—81no—
318#21584—731nomaximal, Sylow
319C2×C73—82no—
320C2×C73C4×#12311yescharacteristic
321C2×C73#294611yescharacteristic
322C2×C73#294911yescharacteristic
323C2×C73#309611yescharacteristic
324C2×C73#309911yescharacteristic
325C2×C73#311911yescharacteristic
326C2×C73#312511yescharacteristic
327D73—42no—
328C2×#481—731no—
329#2324—731no—
330#2274—731no—
331#1926—731no—
332#1925D7311yescharacteristic, complemented
333#1920—731no—
334#1919—731no—
335#1875—731no—
336#1854—731no—
337#1823—731no—
338#1776—731no—
339C4×#106—731no—
340#1762—731no—
341C4×#102—731no—
342#1750—731no—
343C73#2158411yescharacteristic, complemented, Sylow
344C2×#121—731no—
345C2×C4×D4—731no—
346C2×#117—731no—
347C2×#117—731no—
348C22×C4⋊C4—731no—
349#481—738no—
350C2×#107C2×D7311yescharacteristic
351C2×#106—731no—
352C2×#106—731no—
353C2×#106—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102C2×D7311yescharacteristic
357C2×#102C2×D7311yescharacteristic
358C2×C22⋊C8—731no—
359C2×C22⋊C8—731no—
360C2×#97—731no—
361C2×C4×C8—731no—
362#390—1461no—
363#390C2×D7311yescharacteristic
364#389—731no—
365#388—1461no—
366#386—1461no—
367#385—1461no—
368#380—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4—731no—
372C4×C4⋊C4—731no—
373C4×C4⋊C4—1461no—
374C4×C4⋊C4—1461no—
375C4×C4⋊C4C2×D7311yescharacteristic
376C4×C22⋊C4—1461no—
377C4×C22⋊C4—1461no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340C2×D7311yescharacteristic
386#336—731no—
387#336C2×D7311yescharacteristic
388C22×D4—1461no—
389C22×D4C22×D7311yescharacteristic
390C23×C4—731no—
391C23×C4—1461no—
392C23×C4—1461no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8C22×D7311yescharacteristic
396C22×C8C22×D7311yescharacteristic
397#121—734no—
398#121—1462no—
399C4×D4—1462no—
400C4×D4—1462no—
401#117—1462no—
402#117—1462no—
403#117—1462no—
404#117—1462no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—734no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—1462no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—731no—
421C2×C22⋊C4—731no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4—1461no—
426C2×C42—21no—
427C2×C42—731no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42C22×D7311yescharacteristic
434#107—22no—
435#106—734no—
436#106—734no—
437#106—1462no—
438#102—22no—
439#102—734no—
440#102—1462no—
441#102C4×D7314yes—
442C22⋊C8—734no—
443C22⋊C8—1462no—
444#97—1462no—
445C4×C8—1462no—
446#95—21no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453C24—21no—
454C24—731no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—5841no—
461C2×D4—5841no—
462C2×D4—1464no—
463C2×D4C2×C4×D7312yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—2922no—
468C22×C4—5841no—
469C22×C4—5841no—
470C22×C4—5841no—
471C22×C4—5841no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—732no—
476C22×C4—732no—
477C22×C4—1464no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4D4×D7311yescharacteristic
487C22×C4D4×D7311yescharacteristic
488C22×C4C23×D7311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—732no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8C2×C4×D7312yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—5841no—
506C4⋊C4—732no—
507C4⋊C4—732no—
508C4⋊C4—732no—
509C4⋊C4—1464no—
510C4⋊C4—1462no—
511C4⋊C4—1462no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4C2×C4×D7312yes—
518C4⋊C4#1192402712yes—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—2922no—
522C22⋊C4—5841no—
523C22⋊C4—1462no—
524C22⋊C4—1462no—
525C22⋊C4—1464no—
526C22⋊C4—1464no—
527C42—21no—
528C42—21no—
529C42—81no—
530C42—5841no—
531C42—5841no—
532C42—5841no—
533C42—5841no—
534C42—732no—
535C42—732no—
536C42—1464no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—732no—
545C23—1461no—
546C23—1461no—
547C23—1461no—
548C23—1461no—
549C23—1461no—
550C23—1461no—
551C23C2×D4×D7311yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—5844no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—5844no—
582C2×C4—5844no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—732no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4—1462no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1462no—
596C2×C4—1462no—
597C2×C4—1462no—
598C2×C4—1462no—
599C2×C4C22×C4×D7311yescharacteristic
600C2×C4#41×D7311yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4#41×D7311yescharacteristic
603C2×C4#41×D7311yescharacteristic
604C2×C4C2×#1192402711yescharacteristic
605C8—22no—
606C8—41no—
607C8—2921no—
608C8—1462no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—2921no—
617C22—2921no—
618C22—2921no—
619C22—2921no—
620C22—1462no—
621C22—1462no—
622C22—1462no—
623C22C4×D4×D7311yescharacteristic
624C22#123×D7311yescharacteristic
625C22#1798909711yescharacteristic
626C22#1798938811yescharacteristic
627C22#1798939411yescharacteristic
628C22#1798939611yescharacteristic
629C22#1798939711yescharacteristic
630C4—41no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—2922no—
636C4—2922no—
637C4—2922no—
638C4—5841no—
639C4—1462no—
640C4—1462no—
641C4—1462no—
642C4C4×#1192402711yescharacteristic
643C4#1798905511yescharacteristic
644C4#1798911611yescharacteristic
645C4#1798912011yescharacteristic
646C2—82no—
647C2—2922no—
648C2#2261245811yescharacteristic
649C2#2261875611yescharacteristic
650C2#2261878711yescharacteristic
651C2#2261988211yescharacteristic
652C2#2261992311yescharacteristic
653C2#2262012711yescharacteristic
654C2#2262020711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1925C2
socleC23×C73C22×D4
prime radicalC2×#14556562C22
cosocleC22×C4×C73C24
hypercenter#1925D73
hypocenterC73#21584
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
3. #22612285C2semidirect
4. #22610555C2semidirect
5. #22610517C2semidirect
6. #22610509C2semidirect
8. #22610127C2semidirect
9. #22610016C2semidirect
10. #22609830C2semidirect
11. #22609819C2semidirect
12. #22609758C2semidirect
13. #102×Dic73C2semidirect
14. #22609696C2semidirect
15. C73×#1925C2semidirect
21. #17988904C4semidirect
83. C4⋊C4×D73C2×C4semidirect
96. #14556523C2×C4semidirect
98. #14556522C2×C4semidirect
102. #14556503C2×C4semidirect
332. #1925D73semidirect
343. C73#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#179889042
#226126272
#226122852
#226105552
#226105172
#226105092
#226102502
#226101272
#226100162
#226098302
#226098192
#226097582
#102×Dic732
#226096962
C73×#19252
#2158473