Finite Group Databasebeta

← #33084502

Subgroups — #33084502

Order 18688.

1130 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2C2×#17988686C211yesmaximal, characteristic
3C2×#17988674C211yesmaximal, characteristic
4#22612295C211yesmaximal, characteristic, complemented
5#22612291C211yesmaximal, characteristic
6#22609831C211yesmaximal, characteristic, complemented
7#107×Dic73C211yesmaximal, characteristic, complemented
8#22609760C211yesmaximal, characteristic, complemented
9#22609755C211yesmaximal, characteristic
10C4×#14556514C211yesmaximal, characteristic, complemented
11#22609750C211yesmaximal, characteristic, complemented
12C4×#14556511C211yesmaximal, characteristic
13#22609745C211yesmaximal, characteristic
14#22609619C211yesmaximal, characteristic, complemented
15C73×#1776C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556556C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21C2×#14556553C2211yescharacteristic
22C2×#14556553C2211yescharacteristic
23#17988689C418yescomplemented
24#17988686C418yescomplemented
25#17988674C418yescomplemented
26C2×#14556525C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556513C2211yescharacteristic
31C2×#14556511C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556501C2211yescharacteristic
34C2×#14556501C2211yescharacteristic
35C2×C4×#10596831C2211yescharacteristic
36C2×C8×Dic73C2211yescharacteristic
37C2×C8×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007—21no—
43C4×#11924007—21no—
44#17988438—21no—
45C4×#11924005—21no—
46C4×#11924005—21no—
47#17988434—21no—
48#17988433C2211yescharacteristic
49#17988432—21no—
50#95×D73—21no—
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×#97×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×D292C2311yescharacteristic
64C22×C4×D73—21no—
65C22×C4×D73—21no—
66C22×C4×D73—21no—
67C2×#11924017C2×C414yes—
68C2×#11924017C2×C414yes—
69C2×#11924017C2×C414yes—
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C4×Dic146—22no—
73C4×Dic146C2×C414yescomplemented
74#14556556—22no—
75#14556556—22no—
76#14556556C2×C414yescomplemented
77#14556556C2×C414yescomplemented
78C4×D292—22no—
79C4×D292C2×C414yes—
80#14556553—22no—
81#14556553—22no—
82#14556553C2×C414yes—
83#14556553C2×C414yes—
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924005—21no—
90C2×#11924005—21no—
91C2×#11924005—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101#14556525C2×C414yescomplemented
102#14556516C2×C414yescomplemented
103#14556514C2×C414yescomplemented
104#14556514C2×C414yescomplemented
105#14556513C2×C414yes—
106#14556511C2×C414yes—
107#14556511C2×C414yes—
108#14556501—22no—
109#14556501C2×C414yescomplemented
110C4×#10596831—22no—
111C8×Dic73—22no—
112C8×Dic73C2×C414yescomplemented
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498D411yescharacteristic
118#14556498D411yescharacteristic
119#14556498D411yescharacteristic
120#14556498D411yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C42×C73—21no—
128C2×C42×C73—21no—
129C2×C42×C73C2311yescharacteristic
130#107×C73C2×C414yescomplemented
131#106×C73C2×C414yescomplemented
132#97×C73C2×C414yescomplemented
133#95×C73—21no—
134#95×C73—21no—
135C23×D73—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146C22×C412yes—
140C2×Dic146C22×C412yes—
141C2×Dic146C22×C412yes—
142C2×D292—21no—
143C2×D292—21no—
144C2×D292—21no—
145C2×D292C22×C412yes—
146C2×D292C22×C412yes—
147C2×D292C22×C412yes—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C2×C4×D73—24no—
156C2×C4×D73—24no—
157C2×C4×D73—24no—
158C2×C4×D73—81no—
159C2×C4×D73—81no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×C4×D73—81no—
163C2×C4×D73—81no—
164#11924017C42116yescomplemented
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831C22×C412yes—
170C2×#10596831C22×C412yes—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73C22×C412yes—
174C4⋊Dic73C22×C412yes—
175#11924007—24no—
176#11924007—24no—
177#11924007—24no—
178#11924005—24no—
179#11924005—24no—
180#11924005—24no—
181C4×Dic73—24no—
182C4×Dic73—24no—
183C4×Dic73—24no—
184C4×Dic73—81no—
185C4×Dic73—81no—
186C4×Dic73—81no—
187C4×Dic73—81no—
188C4×Dic73C22×C412yes—
189C4×Dic73C22×C412yes—
190C4×Dic73#4112yes—
191C4×Dic73#4112yes—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73C22×C412yes—
202C2×C8×C73C22×C412yes—
203C2×C8×C73C22×C412yes—
204C2×C8×C73C22×C412yes—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C412yes—
210C4⋊C4×C73C22×C412yes—
211C4⋊C4×C73C22×C412yes—
212C4⋊C4×C73C22×C412yes—
213C42×C73—81no—
214C42×C73—81no—
215C42×C73C22×C412yes—
216C42×C73#4112yes—
217C23×C73C22×D411yescharacteristic
218C22×D73—22no—
219C22×D73—22no—
220C22×D73—22no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—41no—
224C22×D73—41no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73C4×D412yes—
242C2×C4×C73C4×D412yes—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73C4×D412yes—
267C2×Dic73C4×D412yes—
268C2×Dic73C4×D412yes—
269C2×Dic73C4×D412yes—
270D292—22no—
271D292—22no—
272D292—22no—
273D292C2×C4214yes—
274C4×D73—84no—
275C4×D73—84no—
276C4×D73—84no—
277Dic146—22no—
278Dic146—22no—
279Dic146—22no—
280Dic146C2×C4214yes—
281C8×C73—22no—
282C8×C73C2×C4214yes—
283#10596831—22no—
284#10596831—22no—
285C22×C73C2×C4×D411yescharacteristic
286C22×C73C2×C4×D411yescharacteristic
287C22×C73C2×C4×D411yescharacteristic
288C22×C73#57611yescharacteristic
289C22×C73#57711yescharacteristic
290C22×C73#57711yescharacteristic
291C22×C73#57811yescharacteristic
292C2×D73—24no—
293C2×D73—42no—
294C2×D73—42no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—42no—
299C4×C73—22no—
300C4×C73—22no—
301C4×C73—42no—
302C4×C73—42no—
303C4×C73—42no—
304C4×C73—42no—
305C4×C73C22×C4211yescharacteristic
306C4×C73C4×#4111yescharacteristic
307C4×C73C4×#4111yescharacteristic
308C4×C73C4×#4111yescharacteristic
309Dic73—22no—
310Dic73—22no—
311Dic73—22no—
312Dic73—22no—
313Dic73—24no—
314Dic73—42no—
315Dic73—42no—
316#20504—731nomaximal, Sylow
317C2×C73C42×D411yescharacteristic
318C2×C73#294611yescharacteristic
319C2×C73#294611yescharacteristic
320C2×C73#294711yescharacteristic
321C2×C73#294711yescharacteristic
322C2×C73#294911yescharacteristic
323C2×C73#295111yescharacteristic
324D73—44no—
325C2×#440—731no—
326C2×#440—731no—
327C2×C4×SD16—731no—
328#2276—731no—
329#2275—731no—
330#1776—731no—
331#1776D7311yescharacteristic, complemented
332C4×#107—731no—
333#1763—731no—
334#1762—731no—
335C4×#103—731no—
336C4×#103—731no—
337C4×#102—731no—
338C4×#102—731no—
339#1726—731no—
340C73#2050411yescharacteristic, complemented, Sylow
341C22×SD16—731no—
342C2×C4×Q8—731no—
343C2×C4×Q8—731no—
344C2×C4×Q8—731no—
345C2×C4×D4—731no—
346C2×C4×D4—731no—
347C2×C4×D4—731no—
348#440—738no—
349#440—738no—
350C4×SD16—738no—
351C2×#107—731no—
352C2×#107C2×D7311yescharacteristic
353C2×#106C2×D7311yescharacteristic
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×#102—731no—
360C2×#97—731no—
361C2×#97—731no—
362C2×#97C2×D7311yescharacteristic
363C2×C4×C8—731no—
364C2×C4×C8—731no—
365C2×C4×C8—731no—
366C4×C4⋊C4—731no—
367C4×C4⋊C4—731no—
368C4×C4⋊C4—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—1461no—
371C4×C4⋊C4—1461no—
372C4×C4⋊C4—1461no—
373C4×C4⋊C4C2×D7311yescharacteristic
374C4×C4⋊C4C2×D7311yescharacteristic
375C4×C22⋊C4—1461no—
376C4×C22⋊C4—1461no—
377C4×C22⋊C4—1461no—
378#376—731no—
379#376—1461no—
380C2×#95—1461no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340C2×D7311yescharacteristic
386#340C2×D7311yescharacteristic
387C22×Q8—731no—
388C22×D4—731no—
389C23×C4—1461no—
390C23×C4—1461no—
391C23×C4—1461no—
392C2×SD16—734no—
393C2×SD16—734no—
394C2×SD16—734no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C4×Q8—734no—
400C4×Q8—734no—
401C4×Q8—734no—
402C4×Q8—1462no—
403C4×Q8—1462no—
404C4×Q8—1462no—
405C4×D4—734no—
406C4×D4—734no—
407C4×D4—734no—
408C4×D4—1462no—
409C4×D4—1462no—
410C4×D4—1462no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C4⋊C4C22×D7311yescharacteristic
420C2×C22⋊C4—1461no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—731no—
426C2×C42—731no—
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42C22×D7311yescharacteristic
435#107—734no—
436#107C4×D7314yescomplemented
437#106C4×D7314yescomplemented
438#103—734no—
439#103—734no—
440#103—734no—
441#102—734no—
442#102—734no—
443#102—734no—
444#97—734no—
445#97—1462no—
446#97C4×D7314yescomplemented
447C4×C8—734no—
448C4×C8—1462no—
449C4×C8—1462no—
450#95—21no—
451#95—21no—
452#95—731no—
453#95—731no—
454#95—731no—
455#95—731no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460C24—1461no—
461C2×Q8—732no—
462C2×Q8—732no—
463C2×Q8—732no—
464C2×Q8—1461no—
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×D4—732no—
468C2×D4—732no—
469C2×D4—732no—
470C2×D4—1461no—
471C2×D4—1461no—
472C2×D4—1461no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—5841no—
480C22×C4—5841no—
481C22×C4—5841no—
482C22×C4—5841no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—1464no—
488C22×C4—1464no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1464no—
492C22×C4—1461no—
493C22×C4D4×D7311yescharacteristic
494C22×C4D4×D7311yescharacteristic
495C22×C4C23×D7311yescharacteristic
496SD16—7316no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—732no—
500C2×C8—732no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8C2×C4×D7312yes—
506C2×C8C2×C4×D7312yes—
507C2×C8C2×C4×D7312yes—
508C2×C8C2×C4×D7312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—732no—
514C4⋊C4—732no—
515C4⋊C4—1464no—
516C4⋊C4—1464no—
517C4⋊C4—1464no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4C2×C4×D7312yes—
521C4⋊C4C2×C4×D7312yes—
522C4⋊C4C2×C4×D7312yes—
523C4⋊C4C2×C4×D7312yes—
524C22⋊C4—1464no—
525C22⋊C4—1464no—
526C22⋊C4—1464no—
527C42—81no—
528C42—81no—
529C42—5841no—
530C42—5841no—
531C42—5841no—
532C42—5841no—
533C42—732no—
534C42—732no—
535C42—732no—
536C42—732no—
537C42—1464no—
538C42—1464no—
539C42—1464no—
540C42C2×C4×D7312yes—
541C42#1192402912yes—
542C23—2921no—
543C23—2921no—
544C23—2921no—
545C23—2921no—
546C23—1462no—
547C23—1462no—
548C23—1462no—
549C23C2×D4×D7311yescharacteristic
550Q8—734no—
551Q8—1462no—
552Q8—1462no—
553Q8—1462no—
554D4—734no—
555D4—1462no—
556D4—1462no—
557D4—1462no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
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—41no—
573C2×C4—41no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—5844no—
579C2×C4—5844no—
580C2×C4—5844no—
581C2×C4—732no—
582C2×C4—732no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4—1461no—
588C2×C4—1461no—
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#1455655312yes—
599C2×C4#1455655312yes—
600C2×C4C22×C4×D7311yescharacteristic
601C2×C4C22×C4×D7311yescharacteristic
602C2×C4C22×C4×D7311yescharacteristic
603C2×C4#41×D7311yescharacteristic
604C2×C4#41×D7311yescharacteristic
605C2×C4C2×#1192402911yescharacteristic
606C8—22no—
607C8—1462no—
608C8—1462no—
609C8C42×D7314yes—
610C22—2922no—
611C22—2922no—
612C22—2922no—
613C22—2922no—
614C22—2922no—
615C22—2922no—
616C22—1464no—
617C22C4×D4×D7311yescharacteristic
618C22C4×D4×D7311yescharacteristic
619C22C2×#1455655311yescharacteristic
620C22#1798939111yescharacteristic
621C22#1798939611yescharacteristic
622C22#1798939711yescharacteristic
623C22#1798940311yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—2922no—
631C4—2922no—
632C4—1464no—
633C4—1462no—
634C4—1462no—
635C4—1462no—
636C4—1462no—
637C4C2×C42×D7311yescharacteristic
638C4#1798905511yescharacteristic
639C4C4×#1192402911yescharacteristic
640C4C4×#1192402911yescharacteristic
641C2—2924no—
642C2C4×#1455655311yescharacteristic
643C2#2261876011yescharacteristic
644C2#2261876111yescharacteristic
645C2#2261877211yescharacteristic
646C2#2261877311yescharacteristic
647C2#2261878711yescharacteristic
648C2#2261880311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC4×C73C22×C42
FrattiniC22×C4C23×D73
FittingC73×#1776C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1776D73
hypocenterC73#20504
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #22612295C2semidirect
6. #22609831C2semidirect
7. #107×Dic73C2semidirect
8. #22609760C2semidirect
10. C4×#14556514C2semidirect
11. #22609750C2semidirect
14. #22609619C2semidirect
15. C73×#1776C2semidirect
23. #17988689C4semidirect
24. #17988686C4semidirect
25. #17988674C4semidirect
73. C4×Dic146C2×C4semidirect
76. #14556556C2×C4semidirect
77. #14556556C2×C4semidirect
101. #14556525C2×C4semidirect
102. #14556516C2×C4semidirect
103. #14556514C2×C4semidirect
104. #14556514C2×C4semidirect
109. #14556501C2×C4semidirect
112. C8×Dic73C2×C4semidirect
130. #107×C73C2×C4semidirect
131. #106×C73C2×C4semidirect
132. #97×C73C2×C4semidirect
164. #11924017C42semidirect
331. #1776D73semidirect
340. C73#20504semidirect
436. #107C4×D73semidirect
437. #106C4×D73semidirect
446. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
C2×#179886862
C2×#179886742
#226122952
#226122912
#226098312
#107×Dic732
#226097602
#226097552
C4×#145565142
#226097502
C4×#145565112
#226097452
#226096192
C73×#17762
#2050473