Finite Group Databasebeta

← #33104493

Subgroups — #33104493

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988707C211yesmaximal, characteristic
2#22618812C211yesmaximal, characteristic
3#22618804C211yesmaximal, characteristic
4#22618523C211yesmaximal, characteristic, complemented
5#22610713C211yesmaximal, characteristic, complemented
6#22610708C211yesmaximal, characteristic, complemented
7#22610164C211yesmaximal, characteristic, complemented
8#22610161C211yesmaximal, characteristic, complemented
9#22610157C211yesmaximal, characteristic, complemented
10#22610154C211yesmaximal, characteristic, complemented
11#22610098C211yesmaximal, characteristic, complemented
12#22609996C211yesmaximal, characteristic, complemented
13#22609990C211yesmaximal, characteristic, complemented
14#22609817C211yesmaximal, characteristic, complemented
15C73×#2913C211yesmaximal, characteristic, complemented
16C2×#14556611C2211yescharacteristic
17C2×#14556597C2211yescharacteristic
18#17989067C2211yescharacteristic
19#17989067C2211yescharacteristic
20#17988707C418yescomplemented
21#17988697—21no—
22#17988692—21no—
23#17988687—21no—
24#17988685—21no—
25C4×#11924018—21no—
26C4×#11924015—21no—
27C4×#11924016—21no—
28C4×#11924013—21no—
29C2×#14556527C2211yescharacteristic
30C2×#14556526C2211yescharacteristic
31#17988630C2211yescharacteristic
32#17988630C2211yescharacteristic
33#17988628—21no—
34#17988628—21no—
35#17988622—21no—
36C2×#14556524C2211yescharacteristic
37C2×#14556523C2211yescharacteristic
38C2×#14556522C2211yescharacteristic
39#106×D73—21no—
40C2×#14556518C2211yescharacteristic
41C2×#14556517C2211yescharacteristic
42#17988588C2211yescharacteristic
43#17988588C2211yescharacteristic
44#17988586C2211yescharacteristic
45#17988586C2211yescharacteristic
46#17988543C2211yescharacteristic
47#17988543C2211yescharacteristic
48C2×#14556501C2211yescharacteristic
49#17988456C2211yescharacteristic
50#17988452C2211yescharacteristic
51#17988356C2211yescharacteristic
52#17988356C2211yescharacteristic
53#17988355C2211yescharacteristic
54#17988355C2211yescharacteristic
55#17988346C2211yescharacteristic
56#17988346C2211yescharacteristic
57C2×#131×C73C2211yescharacteristic
58#520×C73C2211yescharacteristic
59#520×C73C2211yescharacteristic
60#428×C73—21no—
61#428×C73—21no—
62#428×C73C2211yescharacteristic
63#428×C73C2211yescharacteristic
64#427×C73—21no—
65#427×C73C2211yescharacteristic
66C2×#107×C73—21no—
67C2×#107×C73C2211yescharacteristic
68C2×#11924028C2311yescharacteristic
69#14556611C2×C414yes—
70#14556611Q814yes—
71C2×#11924018D411yescharacteristic
72C2×#11924016D411yescharacteristic
73C2×#11924015D411yescharacteristic
74C2×#11924013D411yescharacteristic
75#14556597—22no—
76#14556597C2×C414yescomplemented
77C2×#11924012C2311yescharacteristic
78C2×#11924012C2311yescharacteristic
79C2×#11924012C2311yescharacteristic
80C2×#11924011—21no—
81C22×#10596831C2311yescharacteristic
82C2×C8×D73—21no—
83C4×Dic146—21no—
84C4×Dic146—21no—
85C4×D292—21no—
86C4×D292—21no—
87#14556551—41no—
88#14556551—41no—
89#14556549—21no—
90#14556549—21no—
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×#11924007—41no—
94C4⋊C4×D73—21no—
95C4⋊C4×D73—21no—
96C2×C4×Dic73C2311yescharacteristic
97#14556527—22no—
98#14556526C2×C414yescomplemented
99#14556524—21no—
100#14556524—21no—
101#14556524—21no—
102#14556524—21no—
103#14556523—21no—
104#14556523—21no—
105#14556522—21no—
106#14556522—21no—
107#14556522—21no—
108#14556522—21no—
109#14556518C2×C414yescomplemented
110#14556517C2×C414yescomplemented
111#14556515—21no—
112#14556515—21no—
113#14556514—21no—
114#14556514—21no—
115#14556512—21no—
116#14556512—21no—
117#14556511—21no—
118#14556511—21no—
119#14556502—21no—
120#14556502—21no—
121#14556501C2×C414yescomplemented
122C4×#10596831—21no—
123C4×#10596831—21no—
124#14556498—41no—
125C2×#41×C73C2311yescharacteristic
126#131×C73—22no—
127#131×C73C2×C414yescomplemented
128C2×M16×C73—21no—
129C2×M16×C73C2311yescharacteristic
130C22×C8×C73—21no—
131C22×C8×C73C2311yescharacteristic
132C4×Q8×C73—21no—
133C4×Q8×C73—21no—
134C4×D4×C73—21no—
135C4×D4×C73—21no—
136C4×D4×C73—41no—
137C4×D4×C73—41no—
138#117×C73—21no—
139#117×C73—21no—
140#117×C73C2311yescharacteristic
141#117×C73C2311yescharacteristic
142C2×C4⋊C4×C73—21no—
143C2×C4⋊C4×C73—21no—
144C2×C4⋊C4×C73C2311yescharacteristic
145C2×C4⋊C4×C73C2311yescharacteristic
146#107×C73—21no—
147#107×C73—21no—
148#107×C73—21no—
149#107×C73—21no—
150#107×C73D411yescharacteristic
151#107×C73D411yescharacteristic
152#106×C73—21no—
153#106×C73—21no—
154#106×C73D411yescharacteristic
155#106×C73D411yescharacteristic
156#11924028—21no—
157#11924028—21no—
158#11924028C22×C412yes—
159#11924028C2×Q812yes—
160C2×Dic146C2×D411yescharacteristic
161C2×D292C2×D411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—21no—
166C2×C4×D73—21no—
167C2×C4×D73—41no—
168C2×C4×D73—41no—
169#11924018C4⋊C414yescomplemented
170#11924016C4⋊C414yescomplemented
171#11924015C4⋊C414yescomplemented
172#11924013C4⋊C414yescomplemented
173#11924012—21no—
174#11924012—21no—
175#11924012—21no—
176#11924012—21no—
177#11924012—21no—
178#11924012C22×C412yes—
179#11924011—24no—
180C2×#10596831—21no—
181C2×#10596831—21no—
182C2×#10596831—21no—
183C2×#10596831C22×C412yes—
184C2×#10596831C2×D411yescharacteristic
185C2×#10596831C2×D411yescharacteristic
186C2×#10596831C2×Q812yes—
187C8×D73—24no—
188C4⋊Dic73—21no—
189C4⋊Dic73—21no—
190C4⋊Dic73—21no—
191C4⋊Dic73—21no—
192#11924007—41no—
193#11924007—41no—
194#11924007—82no—
195#11924005—41no—
196#11924005—41no—
197C4×Dic73—21no—
198C4×Dic73—41no—
199C4×Dic73—41no—
200C4×Dic73C22×C412yes—
201#41×C73—21no—
202#41×C73—21no—
203#41×C73C22×C412yes—
204#41×C73C2×Q812yes—
205C2×Q8×C73C2×D411yescharacteristic
206C2×D4×C73—21no—
207C2×D4×C73C2×D411yescharacteristic
208C22×C4×C73—21no—
209C22×C4×C73—21no—
210C22×C4×C73—21no—
211C22×C4×C73—41no—
212C22×C4×C73—41no—
213C22×C4×C73C2411yescharacteristic
214M16×C73—21no—
215M16×C73—24no—
216M16×C73C22×C412yes—
217C2×C8×C73—21no—
218C2×C8×C73—24no—
219C2×C8×C73C22×C412yes—
220C2×C8×C73C2×D411yescharacteristic
221C2×C8×C73C2×D411yescharacteristic
222C2×C8×C73#4112yes—
223C4⋊C4×C73—21no—
224C4⋊C4×C73—21no—
225C4⋊C4×C73—21no—
226C4⋊C4×C73—21no—
227C4⋊C4×C73—41no—
228C4⋊C4×C73—41no—
229C4⋊C4×C73C2×D411yescharacteristic
230C4⋊C4×C73C2×D411yescharacteristic
231C4⋊C4×C73C2×D411yescharacteristic
232C4⋊C4×C73C2×D411yescharacteristic
233C22⋊C4×C73—21no—
234C22⋊C4×C73—21no—
235C22⋊C4×C73—41no—
236C22⋊C4×C73—41no—
237C42×C73—21no—
238C42×C73—21no—
239C42×C73—41no—
240C42×C73—41no—
241C23×C73—21no—
242C23×C73H(5,𝔽2)11yescharacteristic
243C22×D73—21no—
244Q8×C73—21no—
245Q8×C73C2×C4⋊C412yes—
246D4×C73—22no—
247D4×C73—22no—
248D4×C73—21no—
249D4×C73C2×C4⋊C412yes—
250C2×C4×C73—21no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—81no—
260C2×C4×C73—81no—
261C2×C4×C73—161no—
262C2×C4×C73—161no—
263C2×C4×C73C23×C411yescharacteristic
264C2×C4×C73C22×D411yescharacteristic
265C2×C4×C73C22×D411yescharacteristic
266C2×C4×C73C22×Q811yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C732−1+411yescharacteristic
269C22⋊D73—22no—
270C22⋊D73—22no—
271C2×Dic73—21no—
272C2×Dic73—22no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—81no—
278C2×Dic73—81no—
279D292—21no—
280D292C2×C4⋊C412yes—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—161no—
284C4×D73—161no—
285Dic146—21no—
286Dic146C2×C4⋊C412yes—
287C8×C73—22no—
288C8×C73C4×D412yes—
289C8×C73C4×D412yes—
290#10596831—22no—
291#10596831—41no—
292#10596831—41no—
293#10596831C2×C4⋊C412yes—
294#10596831C2×C4⋊C412yes—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—22no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73#52011yescharacteristic
301C22×C73D4211yescharacteristic
302C22×C73#55411yescharacteristic
303C2×D73—22no—
304C2×D73—41no—
305C2×D73—41no—
306C4×C73—22no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—81no—
312C4×C73—81no—
313C4×C73C22×C4⋊C411yescharacteristic
314C4×C73C2×C4×D411yescharacteristic
315C4×C73#54711yescharacteristic
316C4×C73D4×Q811yescharacteristic
317Dic73—22no—
318Dic73—42no—
319Dic73—81no—
320Dic73—81no—
321#22693—731nomaximal, Sylow
322C2×C73—21no—
323C2×C73—21no—
324C2×C73—42no—
325C2×C73D4×C4⋊C411yescharacteristic
326C2×C73#328811yescharacteristic
327C2×C73#328911yescharacteristic
328D73—42no—
329C2×#444—731no—
330#2952—731no—
331#2951—731no—
332#2913D7311yescharacteristic, complemented
333#2912—731no—
334#1952—731no—
335#1951—731no—
336#1860—731no—
337#1860—731no—
338#1859—731no—
339#1859—731no—
340#1847—731no—
341#1816—731no—
342#1815—731no—
343#1773—731no—
344C73#2269311yescharacteristic, complemented, Sylow
345C2×#135—731no—
346C2×#131—731no—
347C2×#131C2×D7311yescharacteristic
348#520—731no—
349#520—731no—
350#520C2×D7311yescharacteristic
351#520C2×D7311yescharacteristic
352#444—738no—
353#442—1461no—
354#441—1461no—
355#440—1461no—
356#440—1461no—
357C4×Q16—1461no—
358C4×SD16—1461no—
359C4×SD16—1461no—
360C4×D8—1461no—
361C2×#108—731no—
362C2×#108—731no—
363#428—21no—
364#428—21no—
365#428—731no—
366#428—731no—
367#428—1461no—
368#428—1461no—
369#428C2×D7311yescharacteristic
370#428C2×D7311yescharacteristic
371#427—21no—
372#427—1461no—
373#427C2×D7311yescharacteristic
374C2×#107—21no—
375C2×#107—731no—
376C2×#107C2×D7311yescharacteristic
377C2×#106—731no—
378C2×#106—731no—
379C2×#106—1461no—
380C2×#104—731no—
381C2×#104—731no—
382#417—731no—
383#417—731no—
384#417—731no—
385#417—731no—
386#405—731no—
387#405—731no—
388C2×#97—731no—
389#384—731no—
390#382—731no—
391#343—731no—
392#343—731no—
393#343—731no—
394#343—731no—
395#339—731no—
396#339—731no—
397C2×#41—731no—
398C2×#41C22×D7311yescharacteristic
399#135—734no—
400#135—734no—
401C2×Q16—731no—
402C2×SD16—731no—
403C2×SD16—731no—
404C2×D8—731no—
405#131—22no—
406#131—734no—
407#131—1462no—
408#131C4×D7314yescomplemented
409C2×M16—21no—
410C2×M16—731no—
411C2×M16—731no—
412C2×M16—731no—
413C2×M16—1461no—
414C2×M16C22×D7311yescharacteristic
415C22×C8—21no—
416C22×C8—731no—
417C22×C8—1461no—
418C22×C8C22×D7311yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—1461no—
422C4×Q8—1461no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—2921no—
428C4×D4—2921no—
429C4×D4—1461no—
430C4×D4—1461no—
431#117—21no—
432#117—21no—
433#117—1461no—
434#117—1461no—
435#117C22×D7311yescharacteristic
436#117C22×D7311yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—2921no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C42—731no—
447#108—734no—
448#108—1462no—
449#107—21no—
450#107—21no—
451#107—21no—
452#107—21no—
453#107—1461no—
454#107—1461no—
455#107—1461no—
456#107—1461no—
457#107C22⋊D7311yescharacteristic
458#107C22⋊D7311yescharacteristic
459#106—21no—
460#106—21no—
461#106—1461no—
462#106—1461no—
463#106—1461no—
464#106—1461no—
465#106—1461no—
466#106—1461no—
467#106C22⋊D7311yescharacteristic
468#106C22⋊D7311yescharacteristic
469#104—734no—
470#104—734no—
471#103—1461no—
472#103—1461no—
473#103—1461no—
474#103—1461no—
475#102—1461no—
476#102—1461no—
477#102—1461no—
478#102—1461no—
479#97—734no—
480#97—1461no—
481#97—1461no—
482C4×C8—1461no—
483C4×C8—1461no—
484#95—2921no—
485#41—21no—
486#41—21no—
487#41—732no—
488#41—732no—
489#41—1461no—
490#41—1461no—
491#41C2×C4×D7312yes—
492#41Q8×D7312yes—
493C2×Q8—731no—
494C2×Q8D4×D7311yescharacteristic
495C2×D4—21no—
496C2×D4—731no—
497C2×D4—1461no—
498C2×D4D4×D7311yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—2921no—
505C22×C4—2921no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4C23×D7311yescharacteristic
511Q16—734no—
512SD16—734no—
513SD16—734no—
514D8—734no—
515M16—21no—
516M16—24no—
517M16—732no—
518M16—1461no—
519M16—1461no—
520M16—1461no—
521M16—1461no—
522M16—1461no—
523M16—1464no—
524M16C2×C4×D7312yes—
525C2×C8—21no—
526C2×C8—24no—
527C2×C8—732no—
528C2×C8—731no—
529C2×C8—731no—
530C2×C8—732no—
531C2×C8—1461no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1464no—
535C2×C8C2×C4×D7312yes—
536C2×C8C2×C22⋊D7311yescharacteristic
537C2×C8C2×C22⋊D7311yescharacteristic
538C2×C8#1192402812yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—2921no—
546C4⋊C4—2921no—
547C4⋊C4—5842no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4C2×C22⋊D7311yescharacteristic
553C4⋊C4C2×C22⋊D7311yescharacteristic
554C4⋊C4C2×C22⋊D7311yescharacteristic
555C4⋊C4C2×C22⋊D7311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—2921no—
561C22⋊C4—2921no—
562C42—21no—
563C42—21no—
564C42—41no—
565C42—41no—
566C42—2921no—
567C42—2921no—
568C42—732no—
569C42—1461no—
570C23—21no—
571C23—1461no—
572C23#1455663611yescharacteristic
573Q8—21no—
574Q8—732no—
575Q8—1461no—
576Q8C4⋊C4×D7312yes—
577D4—22no—
578D4—22no—
579D4—21no—
580D4—732no—
581D4—1462no—
582D4—1462no—
583D4—1461no—
584D4C4⋊C4×D7312yes—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—161no—
597C2×C4—161no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—5841no—
603C2×C4—5841no—
604C2×C4—11681no—
605C2×C4—11681no—
606C2×C4—1462no—
607C2×C4—1461no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4C22×C4×D7311yescharacteristic
611C2×C4C2×D4×D7311yescharacteristic
612C2×C4C22×C22⋊D7311yescharacteristic
613C2×C4C2×Q8×D7311yescharacteristic
614C2×C4C2×#1192402811yescharacteristic
615C2×C4#1455663911yescharacteristic
616C8—22no—
617C8—2921no—
618C8—2921no—
619C8—732no—
620C8—732no—
621C8—1462no—
622C8C4×C22⋊D7312yes—
623C8C4×C22⋊D7312yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—2921no—
630C22—2921no—
631C22—1462no—
632C22#1798906811yescharacteristic
633C22D4×C22⋊D7311yescharacteristic
634C22#1798929211yescharacteristic
635C4—22no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—2922no—
643C4—5841no—
644C4—5841no—
645C4—1462no—
646C4C2×C4⋊C4×D7311yescharacteristic
647C4C2×C4×C22⋊D7311yescharacteristic
648C4#1798923811yescharacteristic
649C4Q8×C22⋊D7311yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—2922no—
654C2C4⋊C4×C22⋊D7311yescharacteristic
655C2#2262169511yescharacteristic
656C2#2262170211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2913C2
socleC22×C73D42
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#2913D73
hypocenterC73#22693
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
4. #22618523C2semidirect
5. #22610713C2semidirect
6. #22610708C2semidirect
7. #22610164C2semidirect
8. #22610161C2semidirect
9. #22610157C2semidirect
10. #22610154C2semidirect
11. #22610098C2semidirect
12. #22609996C2semidirect
13. #22609990C2semidirect
14. #22609817C2semidirect
15. C73×#2913C2semidirect
20. #17988707C4semidirect
76. #14556597C2×C4semidirect
98. #14556526C2×C4semidirect
109. #14556518C2×C4semidirect
110. #14556517C2×C4semidirect
121. #14556501C2×C4semidirect
127. #131×C73C2×C4semidirect
169. #11924018C4⋊C4semidirect
170. #11924016C4⋊C4semidirect
171. #11924015C4⋊C4semidirect
172. #11924013C4⋊C4semidirect
332. #2913D73semidirect
344. C73#22693semidirect
408. #131C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887072
#226188122
#226188042
#226185232
#226107132
#226107082
#226101642
#226101612
#226101572
#226101542
#226100982
#226099962
#226099902
#226098172
C73×#29132
#2269373