Finite Group Databasebeta

← #33008687

Subgroups — #33008687

Order 18688.

2078 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556548C211yesmaximal, characteristic
2C2×#17988449C211yesmaximal, characteristic
3C2×#95×D73C211yesmaximal, characteristic
4#22608197C212yesmaximal, complemented
5#22608197C212yesmaximal, complemented
6#22608158C211yesmaximal, characteristic, complemented
7#22608155C211yesmaximal, characteristic, complemented
8#22608155C211yesmaximal, characteristic, complemented
9#22608152C212yesmaximal
10#22608152C212yesmaximal
11C73×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D73C2212yes—
13C23×C4×D73C2211yescharacteristic
14C2×#14556548—24no—
15C2×#14556548—24no—
16C2×#14556548C418yescomplemented
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×#11924005C2212yes—
20C22×#11924005C2211yescharacteristic
21C22×#11924005C2211yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23#17988449—24no—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34#95×D73—24no—
35C22×C42×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2211yescharacteristic
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C24×D73C2311yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2312yes—
45C23×Dic73C2311yescharacteristic
46C22×C4×D73—24no—
47C22×C4×D73—24no—
48C22×C4×D73—24no—
49C22×C4×D73—24no—
50C22×C4×D73—24no—
51C22×C4×D73—24no—
52C22×C4×D73—24no—
53C22×C4×D73—42no—
54C22×C4×D73—42no—
55C22×C4×D73C2×C414yes—
56C22×C4×D73D414yes—
57C22×C4×D73D414yes—
58#14556548—232no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007—22no—
62C2×#11924007—22no—
63C2×#11924007C2×C414yescomplemented
64C2×#11924007C2×C414yescomplemented
65C2×#11924005—22no—
66C2×#11924005—22no—
67C2×#11924005—22no—
68C2×#11924005—22no—
69C2×#11924005—24no—
70C2×#11924005—24no—
71C2×#11924005—24no—
72C2×#11924005C2×C414yes—
73C2×#11924005C2×C414yes—
74C2×C4×Dic73—22no—
75C2×C4×Dic73—22no—
76C2×C4×Dic73C2×C414yescomplemented
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—22no—
83#14556498—24no—
84#14556498—42no—
85#14556498—42no—
86#14556498—42no—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2312yes—
91C23×C4×C73C2311yescharacteristic
92C2×C42×C73—22no—
93C2×C42×C73—22no—
94C2×C42×C73C2×C414yescomplemented
95#95×C73—22no—
96#95×C73—22no—
97#95×C73—24no—
98#95×C73—42no—
99C23×D73—22no—
100C23×D73—22no—
101C23×D73—21no—
102C23×D73—21no—
103C23×D73—21no—
104C23×D73—21no—
105C23×D73C22×C412yes—
106C23×D73C2×D412yes—
107C23×D73C2×D412yes—
108C23×D73#4114yes—
109C23×D73#4114yes—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73—22no—
135C22×Dic73C22×C412yes—
136C22×Dic73C22×C412yes—
137C22×Dic73C22×C412yes—
138C22×Dic73C2×D412yes—
139C22×Dic73C2×D412yes—
140C22×Dic73#4114yes—
141C2×C4×D73—28no—
142C2×C4×D73—28no—
143C2×C4×D73—216no—
144C2×C4×D73—216no—
145C2×C4×D73—216no—
146C2×C4×D73—48no—
147C2×C4×D73—48no—
148C2×C4×D73—48no—
149C2×C4×D73—48no—
150C2×C4×D73—48no—
151C2×C4×D73—48no—
152C2×C4×D73C22⋊C4116yes—
153#11924007—28no—
154#11924007—28no—
155#11924005—28no—
156#11924005—28no—
157#11924005—48no—
158C4×Dic73—28no—
159C24×C73C2411yescharacteristic
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—21no—
173C22×C4×C73—22no—
174C22×C4×C73—21no—
175C22×C4×C73—22no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73C22×C412yes—
186C22×C4×C73C22×C412yes—
187C22×C4×C73C22×C412yes—
188C22×C4×C73C2×D412yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73#4114yes—
191C42×C73—28no—
192C23×C73C23×C411yescharacteristic
193C23×C73C22×D411yescharacteristic
194C23×C73C22×D411yescharacteristic
195C23×C73C2×#4112yes—
196C23×C73C2×#4112yes—
197C23×C73C2×#4112yes—
198C23×C73C2×#4112yes—
199C23×C73H(5,𝔽2)11yescharacteristic
200C23×C73H(5,𝔽2)11yescharacteristic
201C23×C732−1+411yescharacteristic
202C23×C732−1+411yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—22no—
207C22×D73—24no—
208C22×D73—24no—
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—22no—
213C22×D73—24no—
214C22×D73—24no—
215C22×D73—24no—
216C22×D73—24no—
217C22×D73—24no—
218C22×D73—24no—
219C22×D73—24no—
220C22×D73—24no—
221C22×D73C2×C22⋊C414yes—
222C22×D73#11714yes—
223C22×D73#11714yes—
224C22×D73#12318yes—
225C22×D73#12318yes—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—24no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73—42no—
255C2×C4×C73—42no—
256C2×C4×C73C2×C22⋊C414yes—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—22no—
267C2×Dic73—24no—
268C2×Dic73—24no—
269C2×Dic73—24no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—42no—
279C2×Dic73—42no—
280C2×Dic73—42no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C2×Dic73—42no—
284C2×Dic73—42no—
285C2×Dic73—42no—
286C2×Dic73—42no—
287C2×Dic73C2×C22⋊C414yes—
288C4×D73—232no—
289C4×D73—432no—
290C22×C73C22×C22⋊C411yescharacteristic
291C22×C73C2×#11711yescharacteristic
292C22×C73C2×#11711yescharacteristic
293C22×C73C4×#4111yescharacteristic
294C22×C73C4×#4111yescharacteristic
295C22×C73#52011yescharacteristic
296C22×C73#52011yescharacteristic
297C22×C73C2×#12312yes—
298C22×C73C2×#12312yes—
299C22×C73#52512yes—
300C22×C73#53212yes—
301C22×C73#53511yescharacteristic
302C22×C73#53611yescharacteristic
303C22×C73#53711yescharacteristic
304C22×C73#53711yescharacteristic
305C22×C73#55112yes—
306C22×C73#55112yes—
307C22×C73#55112yes—
308C22×C73#55112yes—
309C22×C73#55212yes—
310C22×C73#55212yes—
311C22×C73#55212yes—
312C22×C73#55212yes—
313C2×D73—24no—
314C2×D73—24no—
315C2×D73—24no—
316C2×D73—24no—
317C2×D73—24no—
318C2×D73—28no—
319C2×D73—28no—
320C2×D73—28no—
321C2×D73—28no—
322C2×D73—24no—
323C2×D73#38118yes—
324C4×C73—24no—
325C4×C73—42no—
326C4×C73—44no—
327C4×C73—42no—
328C4×C73—82no—
329Dic73—24no—
330Dic73—42no—
331Dic73—44no—
332Dic73—42no—
333Dic73—82no—
334#13958—731nomaximal, Sylow
335C2×C73C2×#38111yescharacteristic
336C2×C73#229711yescharacteristic
337C2×C73#231111yescharacteristic
338C2×C73#234311yescharacteristic
339C2×C73#234311yescharacteristic
340C2×C73#234311yescharacteristic
341C2×C73#234311yescharacteristic
342C2×C73#248212yes—
343C2×C73#248412yes—
344C2×C73#249812yes—
345C2×C73#249812yes—
346D73—28no—
347C22×#117—731no—
348C2×#381—731no—
349C22×#95—731no—
350#1445—732no—
351#1445—732no—
352#1434—731no—
353#1434—731no—
354#1434—731no—
355#1434D7311yescharacteristic, complemented
356C22⋊#95—732no—
357C22⋊#95—732no—
358C73#1395811yescharacteristic, complemented, Sylow
359C24×C4—732no—
360C24×C4—731no—
361C2×#117—738no—
362C2×#117—1464no—
363C2×#117—1464no—
364C22×C4⋊C4—731no—
365C22×C4⋊C4—731no—
366C22×C22⋊C4—731no—
367C22×C22⋊C4—731no—
368C22×C22⋊C4—732no—
369C22×C42—731no—
370C22×C42C2×D7311yescharacteristic
371#381—1464no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95—732no—
375C2×#95—732no—
376C2×#95—732no—
377C2×#95—732no—
378C2×#95—731no—
379C2×#95—731no—
380C2×#95—732no—
381C2×#95—732no—
382C2×#95—1464no—
383C2×#95C2×D7312yes—
384C2×#95C2×D7311yescharacteristic
385C2×#95C2×D7311yescharacteristic
386C2×#95C2×D7312yes—
387C25—731no—
388C23×C4—2922no—
389C23×C4—2922no—
390C23×C4—731no—
391C23×C4—734no—
392C23×C4—734no—
393C23×C4—734no—
394C23×C4—731no—
395C23×C4—731no—
396C23×C4—732no—
397C23×C4—732no—
398C23×C4—1464no—
399C23×C4—1464no—
400C23×C4—1464no—
401C23×C4—1464no—
402C23×C4—1464no—
403C23×C4—1464no—
404C23×C4—1464no—
405C23×C4C22×D7312yes—
406C23×C4C22×D7311yescharacteristic
407C23×C4C22×D7311yescharacteristic
408C23×C4C22×D7312yes—
409C23×C4C22×D7311yescharacteristic
410#117—14632no—
411C2×C4⋊C4—734no—
412C2×C4⋊C4—734no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4—1462no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4—1462no—
417C2×C22⋊C4—734no—
418C2×C22⋊C4—734no—
419C2×C22⋊C4—1464no—
420C2×C22⋊C4—1464no—
421C2×C22⋊C4—1464no—
422C2×C22⋊C4—1462no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4—1462no—
425C2×C22⋊C4—1462no—
426C2×C42—22no—
427C2×C42—22no—
428C2×C42—734no—
429C2×C42—1462no—
430C2×C42—1462no—
431C2×C42C4×D7314yescomplemented
432#95—22no—
433#95—22no—
434#95—24no—
435#95—42no—
436#95—2922no—
437#95—2922no—
438#95—2922no—
439#95—1464no—
440#95—1464no—
441#95—1464no—
442#95—1464no—
443#95—1462no—
444#95—1462no—
445#95—1464no—
446C24—732no—
447C24—732no—
448C24—732no—
449C24—734no—
450C24—734no—
451C24—1462no—
452C24—1462no—
453C24—1461no—
454C24—1461no—
455C24—1461no—
456C24—1461no—
457C24C23×D7311yescharacteristic
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—2928no—
484C22×C4—2928no—
485C22×C4—2928no—
486C22×C4—2928no—
487C22×C4—2928no—
488C22×C4—2928no—
489C22×C4—732no—
490C22×C4—732no—
491C22×C4—732no—
492C22×C4—7316no—
493C22×C4—732no—
494C22×C4—732no—
495C22×C4—734no—
496C22×C4—14616no—
497C22×C4—1468no—
498C22×C4—1468no—
499C22×C4—14616no—
500C22×C4—14616no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1461no—
510C22×C4—1461no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1462no—
518C22×C4—1461no—
519C22×C4—1462no—
520C22×C4—1461no—
521C22×C4—1462no—
522C22×C4—1461no—
523C22×C4—1461no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4C2×C4×D7312yes—
527C22×C4C2×C4×D7312yes—
528C22×C4C2×C4×D7312yes—
529C22×C4D4×D7312yes—
530C22×C4D4×D7312yes—
531C22×C4#1192402814yes—
532C4⋊C4—1468no—
533C4⋊C4—1468no—
534C22⋊C4—2928no—
535C22⋊C4—1468no—
536C22⋊C4—1468no—
537C42—28no—
538C42—1468no—
539C23—734no—
540C23—734no—
541C23—734no—
542C23—738no—
543C23—738no—
544C23—1464no—
545C23—1462no—
546C23—1462no—
547C23—1462no—
548C23—1462no—
549C23—1464no—
550C23—1464no—
551C23—1464no—
552C23—1464no—
553C23—1464no—
554C23—1464no—
555C23—1464no—
556C23—1464no—
557C23—1462no—
558C23—1462no—
559C23—1462no—
560C23—1462no—
561C23—1464no—
562C23C22×C4×D7311yescharacteristic
563C23C2×D4×D7311yescharacteristic
564C23C2×D4×D7311yescharacteristic
565C23#41×D7312yes—
566C23#41×D7312yes—
567C23#41×D7312yes—
568C23C2×#1192402812yes—
569C23#1455663611yescharacteristic
570C23#1455663711yescharacteristic
571C23#1455663811yescharacteristic
572C23#1455663911yescharacteristic
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—29232no—
604C2×C4—2922no—
605C2×C4—2922no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2922no—
611C2×C4—2922no—
612C2×C4—2922no—
613C2×C4—2922no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2922no—
618C2×C4—2922no—
619C2×C4—2922no—
620C2×C4—2922no—
621C2×C4—734no—
622C2×C4—14632no—
623C2×C4—1464no—
624C2×C4—1464no—
625C2×C4—1462no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4—1464no—
632C2×C4—1462no—
633C2×C4—1462no—
634C2×C4—1464no—
635C2×C4—1464no—
636C2×C4C22⋊C4×D7314yes—
637C22—738no—
638C22—1464no—
639C22—1464no—
640C22—1464no—
641C22—1464no—
642C22—1464no—
643C22—1468no—
644C22—1468no—
645C22—1468no—
646C22—1464no—
647C22—1468no—
648C22C2×C22⋊C4×D7311yescharacteristic
649C22#117×D7311yescharacteristic
650C22#117×D7311yescharacteristic
651C22#1798905511yescharacteristic
652C22C4×#1192402811yescharacteristic
653C22#1798906711yescharacteristic
654C22#1798906811yescharacteristic
655C22#123×D7312yes—
656C22#123×D7312yes—
657C22#1798909612yes—
658C22#1798913412yes—
659C22#1798915311yescharacteristic
660C22#1798916011yescharacteristic
661C22#1798916611yescharacteristic
662C22#1798916711yescharacteristic
663C22#1798926112yes—
664C22#1798926212yes—
665C22#1798926612yes—
666C22#1798926712yes—
667C22#1798926912yes—
668C22#1798927312yes—
669C22#1798927612yes—
670C22#1798927812yes—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—82no—
676C4—2922no—
677C4—2924no—
678C4—2922no—
679C4—5842no—
680C4—1464no—
681C2—1468no—
682C2#381×D7311yescharacteristic
683C2#2261241711yescharacteristic
684C2#2261252011yescharacteristic
685C2#2261278411yescharacteristic
686C2#2261278811yescharacteristic
687C2#2261279211yescharacteristic
688C2#2261279411yescharacteristic
689C2#2261420312yes—
690C2#2261421612yes—
691C2#2261437212yes—
692C2#2261437412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1434D73
hypocenterC73#13958
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #22608197C2semidirect
5. #22608197C2semidirect
6. #22608158C2semidirect
7. #22608155C2semidirect
8. #22608155C2semidirect
11. C73×#1434C2semidirect
16. C2×#14556548C4semidirect
63. C2×#11924007C2×C4semidirect
64. C2×#11924007C2×C4semidirect
76. C2×C4×Dic73C2×C4semidirect
94. C2×C42×C73C2×C4semidirect
355. #1434D73semidirect
358. C73#13958semidirect
431. C2×C42C4×D73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565482
C2×#179884492
C2×#95×D732
#226081972
#226081582
#226081552
#226081522
C73×#14342
#1395873