Finite Group Databasebeta

← #33089205

Subgroups — #33089205

Order 18688.

800 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic73C211yesmaximal, characteristic
2#22618571C211yesmaximal, characteristic, complemented
3#131×Dic73C211yesmaximal, characteristic, complemented
4#22610582C211yesmaximal, characteristic, complemented
5#22610576C211yesmaximal, characteristic, complemented
6#22610531C211yesmaximal, characteristic, complemented
7#22610529C211yesmaximal, characteristic, complemented
8#22610260C211yesmaximal, characteristic
9#22610206C211yesmaximal, characteristic, complemented
10#22610204C211yesmaximal, characteristic, complemented
11#22609832C211yesmaximal, characteristic, complemented
12#104×Dic73C211yesmaximal, characteristic, complemented
13#104×Dic73C211yesmaximal, characteristic, complemented
14#22609675C211yesmaximal, characteristic, complemented
15C73×#1876C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#41×Dic73C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19Q16×Dic73C411yescharacteristic, complemented
20Q16×Dic73C411yescharacteristic, complemented
21SD16×Dic73C411yescharacteristic, complemented
22SD16×Dic73C411yescharacteristic, complemented
23SD16×Dic73C411yescharacteristic, complemented
24SD16×Dic73C411yescharacteristic, complemented
25D8×Dic73C411yescharacteristic, complemented
26D8×Dic73C411yescharacteristic, complemented
27#17988645—21no—
28#17988645—21no—
29#17988637—21no—
30#17988637—21no—
31C2×#14556526C2211yescharacteristic
32C2×#14556526C2211yescharacteristic
33#17988625C2211yescharacteristic
34#17988615C2211yescharacteristic
35#17988613C2211yescharacteristic
36C2×#14556518C2211yescharacteristic
37C2×#14556518C2211yescharacteristic
38#17988583C2211yescharacteristic
39#17988583C2211yescharacteristic
40#17988554C2211yescharacteristic
41#17988545—21no—
42#17988545C2211yescharacteristic
43#17988542—21no—
44#17988542C2211yescharacteristic
45M16×Dic73—21no—
46M16×Dic73C2211yescharacteristic
47M16×Dic73C2211yescharacteristic
48C2×C8×Dic73—21no—
49C2×C8×Dic73C2211yescharacteristic
50C2×C8×Dic73C2211yescharacteristic
51C42×Dic73C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988342C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#135×C73C2211yescharacteristic
61C2×#131×C73C2211yescharacteristic
62C2×#104×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64#408×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67#14556598—22no—
68#14556598—22no—
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C22×#10596831—21no—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75Q8×Dic73C2×C411yescharacteristic
76Q8×Dic73C2×C411yescharacteristic
77Q8×Dic73C2×C411yescharacteristic
78Q8×Dic73C2×C411yescharacteristic
79D4×Dic73—21no—
80D4×Dic73—21no—
81D4×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73C2×C411yescharacteristic
84D4×Dic73C2×C411yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87#14556550—21no—
88#14556550—21no—
89#14556550C2311yescharacteristic
90#14556550C2311yescharacteristic
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97#14556526—22no—
98#14556526—22no—
99#14556525C2×C411yescharacteristic
100#14556525C2×C411yescharacteristic
101#14556523C2×C411yescharacteristic
102#14556523C2×C411yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556518—22no—
108#14556518—22no—
109#14556515C2×C411yescharacteristic
110#14556515C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556515C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556512C2×C411yescharacteristic
116#14556512C2×C411yescharacteristic
117C22⋊#10596831—21no—
118C22⋊#10596831—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501—21no—
122#14556501—21no—
123C8×Dic73—21no—
124C8×Dic73—21no—
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73C2×C411yescharacteristic
128C8×Dic73C2×C411yescharacteristic
129C8×Dic73C2×C411yescharacteristic
130C8×Dic73C2×C411yescharacteristic
131C8×Dic73D411yescharacteristic
132C8×Dic73D411yescharacteristic
133C8×Dic73D411yescharacteristic
134C8×Dic73D411yescharacteristic
135C2×#41×C73C2311yescharacteristic
136C2×#41×C73C2311yescharacteristic
137#135×C73—22no—
138#135×C73—22no—
139C2×Q16×C73C2×C411yescharacteristic
140C2×SD16×C73C2×C411yescharacteristic
141C2×SD16×C73C2×C411yescharacteristic
142C2×D8×C73C2×C411yescharacteristic
143#131×C73—22no—
144#131×C73—22no—
145C2×M16×C73—21no—
146C2×M16×C73C2311yescharacteristic
147C2×M16×C73C2311yescharacteristic
148C22×C8×C73—21no—
149C22×C8×C73C2311yescharacteristic
150C22×C8×C73C2311yescharacteristic
151C2×C42×C73C2311yescharacteristic
152#104×C73—22no—
153#104×C73—22no—
154C22⋊C8×C73—21no—
155C22⋊C8×C73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73#4111yescharacteristic
162C22×Dic73#4111yescharacteristic
163#11924012—24no—
164#11924012—21no—
165#11924012—21no—
166#11924012—21no—
167#11924012#4112yes—
168C2×#10596831—24no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831—21no—
174C2×#10596831—21no—
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C22×C411yescharacteristic
177C2×#10596831C2×D411yescharacteristic
178C2×#10596831C2×D411yescharacteristic
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—21no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C4×Dic73—24no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—42no—
203C4×Dic73C22×C411yescharacteristic
204C4×Dic73C22×C411yescharacteristic
205C4×Dic73C2×D411yescharacteristic
206C4×Dic73C2×D411yescharacteristic
207#41×C73—21no—
208#41×C73—21no—
209#41×C73—21no—
210#41×C73—21no—
211#41×C73—21no—
212#41×C73—21no—
213#41×C73#4112yes—
214#41×C73#4112yes—
215C2×Q8×C73C22×C411yescharacteristic
216C2×Q8×C73C22×C411yescharacteristic
217C2×D4×C73—21no—
218C2×D4×C73—21no—
219C2×D4×C73C22×C411yescharacteristic
220C2×D4×C73C22×C411yescharacteristic
221C22×C4×C73—21no—
222C22×C4×C73—21no—
223C22×C4×C73—21no—
224C22×C4×C73C2411yescharacteristic
225Q16×C73—22no—
226SD16×C73—22no—
227SD16×C73—22no—
228D8×C73—22no—
229M16×C73—24no—
230M16×C73—21no—
231M16×C73—21no—
232M16×C73—21no—
233M16×C73#4112yes—
234C2×C8×C73—24no—
235C2×C8×C73—21no—
236C2×C8×C73—21no—
237C2×C8×C73—21no—
238C2×C8×C73—21no—
239C2×C8×C73—21no—
240C2×C8×C73—21no—
241C2×C8×C73C22×C411yescharacteristic
242C2×C8×C73C22×C411yescharacteristic
243C2×C8×C73C2×D411yescharacteristic
244C2×C8×C73C2×D411yescharacteristic
245C42×C73—21no—
246C42×C73—21no—
247C23×C73—21no—
248C23×C73—21no—
249C23×C73C2×#4111yescharacteristic
250Q8×C73—21no—
251Q8×C73—21no—
252Q8×C73—21no—
253Q8×C73#11712yes—
254D4×C73—22no—
255D4×C73—22no—
256D4×C73—21no—
257D4×C73—21no—
258D4×C73—21no—
259D4×C73—41no—
260D4×C73—41no—
261D4×C73#11712yes—
262C2×C4×C73—21no—
263C2×C4×C73—22no—
264C2×C4×C73—22no—
265C2×C4×C73—22no—
266C2×C4×C73—21no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—41no—
270C2×C4×C73—41no—
271C2×C4×C73C23×C411yescharacteristic
272C2×C4×C73C22×D411yescharacteristic
273C2×C4×C73C2×#4111yescharacteristic
274C2×C4×C73C2×#4111yescharacteristic
275C2×C4×C73C2×#4111yescharacteristic
276C2×C4×C73C2×#4111yescharacteristic
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—21no—
281C2×Dic73—21no—
282C2×Dic73—21no—
283C2×Dic73—21no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—42no—
287C2×Dic73—42no—
288C2×Dic73—42no—
289C2×Dic73—42no—
290C2×Dic73C4×D411yescharacteristic
291C2×Dic73C4×D411yescharacteristic
292C2×Dic73C4×D411yescharacteristic
293C2×Dic73C4×D411yescharacteristic
294C8×C73—21no—
295C8×C73—21no—
296C8×C73—21no—
297C8×C73—22no—
298C8×C73—21no—
299C8×C73—41no—
300#10596831—21no—
301#10596831—21no—
302#10596831—21no—
303#10596831—22no—
304#10596831—21no—
305#10596831—41no—
306C22×C73—21no—
307C22×C73—21no—
308C22×C73—22no—
309C22×C73—41no—
310C22×C73—41no—
311C22×C73—41no—
312C22×C73—41no—
313C22×C73—41no—
314C22×C73C2×C4×D411yescharacteristic
315C22×C73#52911yescharacteristic
316C22×C73#52911yescharacteristic
317C4×C73—22no—
318C4×C73—42no—
319C4×C73—41no—
320C4×C73C2×#11711yescharacteristic
321C4×C73C4×#4111yescharacteristic
322C4×C73#52511yescharacteristic
323C4×C73#52511yescharacteristic
324Dic73—21no—
325Dic73—21no—
326Dic73—42no—
327Dic73—41no—
328Dic73—41no—
329Dic73—41no—
330Dic73—41no—
331Dic73#44312yes—
332Dic73#44412yes—
333#21137—731nomaximal, Sylow
334C2×C73—21no—
335C2×C73—21no—
336C2×C73—42no—
337C2×C73—81no—
338C2×C73#229011yescharacteristic
339C2×C73C2×#44311yescharacteristic
340C2×C73C2×#44411yescharacteristic
341C4×#135—731no—
342#2922—731no—
343C4×#131—731no—
344#1929—731no—
345#1929—731no—
346#1922—731no—
347#1922—731no—
348#1876—731no—
349#1876D7311yescharacteristic
350#1866—731no—
351#1866—731no—
352C4×#108—731no—
353C4×#104—731no—
354C4×#104—731no—
355#1744—731no—
356C73#2113711yescharacteristic, complemented, Sylow
357C2×#135C2×D7311yescharacteristic
358C2×#131—731no—
359C2×#131C2×D7311yescharacteristic
360C4×#41—731no—
361C4×#41—731no—
362C4×Q16—731no—
363C4×Q16—731no—
364C4×SD16—731no—
365C4×SD16—731no—
366C4×SD16—731no—
367C4×SD16—731no—
368C4×D8—731no—
369C4×D8—731no—
370#433—1461no—
371#433—1461no—
372#431—1461no—
373#431—1461no—
374C2×#108—731no—
375C2×#108—731no—
376#427—731no—
377#424—731no—
378#424—731no—
379C2×#104—731no—
380C2×#104—731no—
381C2×#104C2×D7311yescharacteristic
382C2×#104C2×D7311yescharacteristic
383#416—731no—
384#416—731no—
385#408—731no—
386#408C2×D7311yescharacteristic
387#405—731no—
388#405—731no—
389#405—1461no—
390#405—1461no—
391C4×M16—731no—
392C4×M16—731no—
393C4×M16—1461no—
394C2×C4×C8—731no—
395C2×C4×C8—731no—
396C2×C4×C8—1461no—
397C43—731no—
398#341—731no—
399#341C2×D7311yescharacteristic
400#339—731no—
401#339—731no—
402#339—731no—
403#339—731no—
404#336—731no—
405#336—731no—
406#336—731no—
407#336C2×D7311yescharacteristic
408C2×#41C22×D7311yescharacteristic
409C2×#41C22×D7311yescharacteristic
410#135—22no—
411#135—22no—
412C2×Q16C4×D7311yescharacteristic
413C2×SD16C4×D7311yescharacteristic
414C2×SD16C4×D7311yescharacteristic
415C2×D8C4×D7311yescharacteristic
416#131—22no—
417#131—22no—
418#131—1462no—
419#131—1462no—
420C2×M16—21no—
421C2×M16—731no—
422C2×M16—731no—
423C2×M16—1461no—
424C2×M16C22×D7311yescharacteristic
425C2×M16C22×D7311yescharacteristic
426C22×C8—21no—
427C22×C8—731no—
428C22×C8—731no—
429C22×C8—1461no—
430C22×C8C22×D7311yescharacteristic
431C22×C8C22×D7311yescharacteristic
432C4×Q8—731no—
433C4×Q8—731no—
434C4×Q8—731no—
435C4×Q8—731no—
436C4×D4—731no—
437C4×D4—731no—
438C4×D4—731no—
439C4×D4—731no—
440C4×D4—1461no—
441C4×D4—1461no—
442C4×D4—1461no—
443C4×D4—1461no—
444#117—731no—
445#117—731no—
446#117—1461no—
447#117—1461no—
448C2×C42—731no—
449C2×C42—731no—
450C2×C42—1461no—
451C2×C42—1461no—
452C2×C42—1461no—
453C2×C42—1461no—
454C2×C42C22×D7311yescharacteristic
455#108—1462no—
456#108—1462no—
457#107—731no—
458#107—731no—
459#106—731no—
460#106—731no—
461C4⋊C8—1461no—
462C4⋊C8—1461no—
463C4⋊C8—1461no—
464C4⋊C8—1461no—
465#104—22no—
466#104—22no—
467#104—1462no—
468#104—1462no—
469#103—731no—
470#103—731no—
471#103—731no—
472#103—731no—
473#102—731no—
474#102—731no—
475#102—731no—
476#102—731no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479C22⋊C8—1461no—
480C22⋊C8—1461no—
481#97—1461no—
482#97—1461no—
483#97—1461no—
484#97—1461no—
485C4×C8—731no—
486C4×C8—731no—
487C4×C8—731no—
488C4×C8—731no—
489C4×C8—731no—
490C4×C8—731no—
491C4×C8—731no—
492C4×C8—731no—
493C4×C8—1461no—
494C4×C8—1461no—
495C4×C8—1461no—
496C4×C8—1461no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41—21no—
502#41—21no—
503#41#1192402712yes—
504#41#1192402712yes—
505C2×Q8C2×C4×D7311yescharacteristic
506C2×Q8C2×C4×D7311yescharacteristic
507C2×D4—21no—
508C2×D4—21no—
509C2×D4C2×C4×D7311yescharacteristic
510C2×D4C2×C4×D7311yescharacteristic
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—731no—
515C22×C4—731no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4—1461no—
520C22×C4—1461no—
521C22×C4C23×D7311yescharacteristic
522Q16—22no—
523SD16—22no—
524SD16—22no—
525D8—22no—
526M16—24no—
527M16—21no—
528M16—21no—
529M16—21no—
530M16—732no—
531M16—1461no—
532M16—1461no—
533M16—1464no—
534M16—1461no—
535M16#1192402712yes—
536C2×C8—24no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—731no—
544C2×C8—731no—
545C2×C8—731no—
546C2×C8—731no—
547C2×C8—1461no—
548C2×C8—1464no—
549C2×C8—1461no—
550C2×C8—1461no—
551C2×C8—1461no—
552C2×C8—1461no—
553C2×C8—1461no—
554C2×C8C2×C4×D7311yescharacteristic
555C2×C8C2×C4×D7311yescharacteristic
556C2×C8D4×D7311yescharacteristic
557C2×C8D4×D7311yescharacteristic
558C4⋊C4—731no—
559C4⋊C4—731no—
560C4⋊C4—731no—
561C4⋊C4—731no—
562C4⋊C4—1461no—
563C4⋊C4—1461no—
564C4⋊C4—1461no—
565C4⋊C4—1461no—
566C22⋊C4—1461no—
567C22⋊C4—1461no—
568C22⋊C4—1461no—
569C22⋊C4—1461no—
570C22⋊C4—1461no—
571C22⋊C4—1461no—
572C42—21no—
573C42—21no—
574C42—2922no—
575C42—731no—
576C42—731no—
577C42—731no—
578C42—731no—
579C42—1461no—
580C42—1461no—
581C42—1461no—
582C42—1461no—
583C42—1464no—
584C42—1461no—
585C42—1461no—
586C42—1461no—
587C42—1461no—
588C23—21no—
589C23—21no—
590C23#41×D7311yescharacteristic
591Q8—21no—
592Q8—21no—
593Q8—21no—
594Q8#1455654712yes—
595D4—22no—
596D4—22no—
597D4—21no—
598D4—21no—
599D4—21no—
600D4—41no—
601D4—41no—
602D4#1455654712yes—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2922no—
618C2×C4—731no—
619C2×C4—731no—
620C2×C4—731no—
621C2×C4—731no—
622C2×C4—1462no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4C22×C4×D7311yescharacteristic
630C2×C4C2×D4×D7311yescharacteristic
631C2×C4#41×D7311yescharacteristic
632C2×C4C2×#1192402711yescharacteristic
633C2×C4C2×#1192402711yescharacteristic
634C2×C4C2×#1192402711yescharacteristic
635C8—21no—
636C8—21no—
637C8—21no—
638C8—22no—
639C8—21no—
640C8—41no—
641C8—2921no—
642C8—1461no—
643C8—1461no—
644C8—1461no—
645C8—1461no—
646C8—1462no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22C4×D4×D7311yescharacteristic
656C22#1798911411yescharacteristic
657C22#1798911811yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—2922no—
662C4—2921no—
663C4—2921no—
664C4—2921no—
665C4—2921no—
666C4—732no—
667C4—732no—
668C4—1461no—
669C4—1461no—
670C4C2×#1455654711yescharacteristic
671C4C4×#1192402711yescharacteristic
672C4#1798909311yescharacteristic
673C4#1798909711yescharacteristic
674C2—21no—
675C2—21no—
676C2—42no—
677C2—81no—
678C2#2261237111yescharacteristic
679C2#443×D7311yescharacteristic
680C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1876C2
socleC22×C73C2×C4×D4
prime radicalC2×#135×C73C22
cosocleC22×C4×C73C24
hypercenter#1876D73
hypocenterC73#21137
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1876

Complements

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

NH ≅ G/Ntype
2. #22618571C2semidirect
3. #131×Dic73C2semidirect
4. #22610582C2semidirect
5. #22610576C2semidirect
6. #22610531C2semidirect
7. #22610529C2semidirect
9. #22610206C2semidirect
10. #22610204C2semidirect
11. #22609832C2semidirect
12. #104×Dic73C2semidirect
13. #104×Dic73C2semidirect
14. #22609675C2semidirect
19. Q16×Dic73C4semidirect
20. Q16×Dic73C4semidirect
21. SD16×Dic73C4semidirect
22. SD16×Dic73C4semidirect
23. SD16×Dic73C4semidirect
24. SD16×Dic73C4semidirect
25. D8×Dic73C4semidirect
26. D8×Dic73C4semidirect
356. C73#21137semidirect

Maximal subgroups

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

subgroupindex
#135×Dic732
#226185712
#131×Dic732
#226105822
#226105762
#226105312
#226105292
#226102602
#226102062
#226102042
#226098322
#104×Dic732
#226096752
C73×#18762
#2113773