Finite Group Databasebeta

← #33093244

Subgroups — #33093244

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×#17988903C211yesmaximal, characteristic
2#22612626C211yesmaximal, characteristic
3#22612287C211yesmaximal, characteristic, complemented
4#22610561C211yesmaximal, characteristic, complemented
5#22610554C211yesmaximal, characteristic, complemented
6#22610519C211yesmaximal, characteristic, complemented
7#22610512C211yesmaximal, characteristic, complemented
8#22610130C211yesmaximal, characteristic, complemented
9#22610018C211yesmaximal, characteristic, complemented
10#22609827C211yesmaximal, characteristic, complemented
11#22609819C211yesmaximal, characteristic, complemented
12#22609758C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic, complemented
14#22609699C211yesmaximal, characteristic, complemented
15C73×#1875C211yesmaximal, characteristic
16C2×D4×Dic73C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19#17988903C418yescomplemented
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988490—21no—
30#17988490C2211yescharacteristic
31#17988486C2211yescharacteristic
32#17988480—21no—
33#17988467—21no—
34#17988462—21no—
35#17988447C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×C4⋊Dic73—21no—
42C4×C4⋊Dic73—21no—
43C22⋊C4×Dic73—21no—
44C22⋊C4×Dic73—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#121×C73C2211yescharacteristic
54C2×#117×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64D4×Dic73—22no—
65D4×Dic73—22no—
66#14556550—22no—
67#14556550—22no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—22no—
71C2×C4⋊Dic73C2×C414yescomplemented
72C2×C4⋊Dic73D411yescharacteristic
73C2×C4⋊Dic73D411yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73D411yescharacteristic
83C2×C22⋊Dic73D411yescharacteristic
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556524—22no—
92#14556522—22no—
93#14556522C2×C414yescomplemented
94#14556522C2×C414yescomplemented
95#14556512C2×C414yes—
96#14556512C2×C414yes—
97C22⋊#10596831C2×C414yescomplemented
98#14556501—22no—
99C8×Dic73—22no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107C22×D4×C73—21no—
108C22×D4×C73C2311yescharacteristic
109C23×C4×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112#121×C73—22no—
113#121×C73C2×C414yes—
114#117×C73—22no—
115#117×C73—22no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73—21no—
122#102×C73—22no—
123#102×C73—22no—
124C22⋊C8×C73—22no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—24no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—42no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C22×Dic73—81no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C22×C412yes—
144C2×#10596831C22×C412yes—
145C4⋊Dic73—24no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—22no—
149C4⋊Dic73—22no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—81no—
155C4⋊Dic73C22×C412yes—
156C4⋊Dic73C22×C412yes—
157C22⋊Dic73—22no—
158C22⋊Dic73—22no—
159C22⋊Dic73—42no—
160C22⋊Dic73—81no—
161C22⋊Dic73—81no—
162C22⋊Dic73—81no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—81no—
166C4×Dic73—81no—
167C4×Dic73—81no—
168C4×Dic73—81no—
169C4×Dic73—81no—
170C4×Dic73#4112yes—
171C4×Dic73#4112yes—
172C24×C73—21no—
173C24×C73C2×D411yescharacteristic
174C2×D4×C73—24no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—81no—
181C2×D4×C73—81no—
182C2×D4×C73C22×C412yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73C22×C412yes—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73#4112yes—
193C22×C4×C73C2411yescharacteristic
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C412yes—
204C4⋊C4×C73#4112yes—
205C4⋊C4×C73#4112yes—
206C22⋊C4×C73—24no—
207C22⋊C4×C73—24no—
208C42×C73—24no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—22no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73C4×D412yes—
223C23×C73C22×D411yescharacteristic
224D4×C73—22no—
225D4×C73—22no—
226D4×C73—41no—
227D4×C73—41no—
228D4×C73—84no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73C4×D412yes—
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—84no—
274C2×Dic73—84no—
275C2×Dic73#12312yes—
276C2×Dic73#12312yes—
277C8×C73—41no—
278C8×C73—41no—
279#10596831—22no—
280#10596831—22no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—81no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73—81no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73C2×#12311yescharacteristic
297C22×C73#52511yescharacteristic
298C22×C73#57611yescharacteristic
299C22×C73#57711yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57711yescharacteristic
302C4×C73—22no—
303C4×C73—42no—
304C4×C73—42no—
305C4×C73—42no—
306C4×C73C4×#4111yescharacteristic
307C4×C73C4×#4111yescharacteristic
308C4×C73#52911yescharacteristic
309C4×C73#52911yescharacteristic
310Dic73—22no—
311Dic73—22no—
312Dic73—41no—
313Dic73—41no—
314Dic73—42no—
315Dic73—42no—
316Dic73—81no—
317Dic73—81no—
318#21584—731nomaximal, Sylow
319C2×C73—42no—
320C2×C73—82no—
321C2×C73C4×#12311yescharacteristic
322C2×C73#294611yescharacteristic
323C2×C73#294911yescharacteristic
324C2×C73#309611yescharacteristic
325C2×C73#309911yescharacteristic
326C2×C73#311911yescharacteristic
327C2×C73#312511yescharacteristic
328C2×#481—731no—
329#2324—731no—
330#2274—731no—
331#1926—731no—
332#1925—731no—
333#1920—731no—
334#1919—731no—
335#1875D7311yescharacteristic
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×#121C2×D7311yescharacteristic
345C2×C4×D4—731no—
346C2×#117—731no—
347C2×#117C2×D7311yescharacteristic
348C22×C4⋊C4—731no—
349#481—738no—
350C2×#107—731no—
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⋊C8C2×D7311yescharacteristic
360C2×#97—731no—
361C2×C4×C8—731no—
362#390—731no—
363#390—1461no—
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—731no—
374C4×C4⋊C4—1461no—
375C4×C4⋊C4—1461no—
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#340C2×D7311yescharacteristic
385#340C2×D7311yescharacteristic
386#336—731no—
387#336—731no—
388C22×D4—21no—
389C22×D4C22×D7311yescharacteristic
390C23×C4—1461no—
391C23×C4—1461no—
392C23×C4C22×D7311yescharacteristic
393C22×C8—731no—
394C22×C8—731no—
395C22×C8C22×D7311yescharacteristic
396C22×C8C22×D7311yescharacteristic
397#121—22no—
398#121C4×D7314yes—
399C4×D4—1462no—
400C4×D4—1462no—
401#117—22no—
402#117—22no—
403#117—1462no—
404#117—1462no—
405C2×C4⋊C4—734no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1462no—
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—731no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42—1461no—
434#107—1462no—
435#106—734no—
436#106—734no—
437#106—1462no—
438#102—22no—
439#102—22no—
440#102—734no—
441#102—734no—
442C22⋊C8—22no—
443C22⋊C8—734no—
444#97—1462no—
445C4×C8—1462no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453C24—21no—
454C24D4×D7311yescharacteristic
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×C4×D7312yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—2922no—
472C22×C4—5841no—
473C22×C4—5841no—
474C22×C4—5841no—
475C22×C4—5841no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—731no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1464no—
485C22×C4C2×C4×D7312yes—
486C22×C4D4×D7311yescharacteristic
487C22×C4#1192402712yes—
488C22×C4C23×D7311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—732no—
496C2×C8—732no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—5841no—
505C4⋊C4—732no—
506C4⋊C4—732no—
507C4⋊C4—1462no—
508C4⋊C4—1462no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1464no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4C2×C4×D7312yes—
517C4⋊C4#1192402712yes—
518C4⋊C4#1192402712yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—2922no—
522C22⋊C4—5841no—
523C22⋊C4—5841no—
524C22⋊C4—5841no—
525C22⋊C4—1462no—
526C22⋊C4—1462no—
527C42—24no—
528C42—5841no—
529C42—5841no—
530C42—5841no—
531C42—5841no—
532C42—5841no—
533C42—732no—
534C42—732no—
535C42—1461no—
536C42—1461no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23#1455655112yes—
551C23C2×D4×D7311yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
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—2921no—
582C2×C4—5844no—
583C2×C4—5844no—
584C2×C4—732no—
585C2×C4—732no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4#1455655112yes—
599C2×C4C22×C4×D7311yescharacteristic
600C2×C4#41×D7311yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4C2×#1192402711yescharacteristic
604C2×C4C2×#1192402711yescharacteristic
605C8—41no—
606C8—41no—
607C8—1462no—
608C8—1462no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×#1455655111yescharacteristic
624C22#123×D7311yescharacteristic
625C22#1798909311yescharacteristic
626C22#1798938911yescharacteristic
627C22#1798939411yescharacteristic
628C22#1798939711yescharacteristic
629C22#1798939711yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—2921no—
635C4—2922no—
636C4—2922no—
637C4—2921no—
638C4—5841no—
639C4—5841no—
640C4—1462no—
641C4—1462no—
642C4C4×#1192402711yescharacteristic
643C4C4×#1192402711yescharacteristic
644C4#1798911811yescharacteristic
645C4#1798911811yescharacteristic
646C2—42no—
647C2—82no—
648C2#2261245611yescharacteristic
649C2#2261876611yescharacteristic
650C2#2261879311yescharacteristic
651C2#2261988611yescharacteristic
652C2#2261992311yescharacteristic
653C2#2262012311yescharacteristic
654C2#2262020911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1875C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
cosocleC22×C4×C73C24
hypercenter#1875D73
hypocenterC73#21584
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
3. #22612287C2semidirect
4. #22610561C2semidirect
5. #22610554C2semidirect
6. #22610519C2semidirect
7. #22610512C2semidirect
8. #22610130C2semidirect
9. #22610018C2semidirect
10. #22609827C2semidirect
11. #22609819C2semidirect
12. #22609758C2semidirect
13. #102×Dic73C2semidirect
14. #22609699C2semidirect
19. #17988903C4semidirect
71. C2×C4⋊Dic73C2×C4semidirect
93. #14556522C2×C4semidirect
94. #14556522C2×C4semidirect
97. C22⋊#10596831C2×C4semidirect
343. C73#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#179889032
#226126262
#226122872
#226105612
#226105542
#226105192
#226105122
#226101302
#226100182
#226098272
#226098192
#226097582
#102×Dic732
#226096992
C73×#18752
#2158473