Finite Group Databasebeta

← #33013162

Subgroups — #33013162

Order 18688.

1926 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556547C211yesmaximal, characteristic
2C2×#17988456C211yesmaximal, characteristic, complemented
3C2×#17988451C211yesmaximal, characteristic, complemented
4C2×#17988445C212yesmaximal
5#22608185C211yesmaximal, characteristic, complemented
6#22608182C212yesmaximal, complemented
7#22608181C211yesmaximal, characteristic, complemented
8#22608176C211yesmaximal, characteristic
9#22608174C212yesmaximal
10#22608159C212yesmaximal, complemented
11C73×#1439C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C2×#14556547—24no—
14C2×#14556547—24no—
15C2×#14556547C418yescomplemented
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×#11924007C2211yescharacteristic
20C22×#11924007C2211yescharacteristic
21C22×#11924007C2212yes—
22C22×C22⋊Dic73C2211yescharacteristic
23C22×C22⋊Dic73C2212yes—
24C22×C4×Dic73C2211yescharacteristic
25C22×C4×Dic73C2211yescharacteristic
26#17988456—24no—
27#17988451—24no—
28#17988445—28no—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C2×#14556498C2212yes—
36C22×C4⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2211yescharacteristic
38C22×C22⋊C4×C73C2212yes—
39C2×#95×C73C2211yescharacteristic
40C2×#95×C73C2212yes—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73C2×C414yes—
46C23×Dic73D414yes—
47C23×Dic73Q814yes—
48C23×Dic73C2312yes—
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53C23×Dic73C2311yescharacteristic
54#14556547—232no—
55C2×C4⋊Dic73—22no—
56C2×C4⋊Dic73—42no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—24no—
60C2×#11924007—24no—
61C2×#11924007—24no—
62C2×#11924007—24no—
63C2×#11924007—24no—
64C2×#11924007—24no—
65C2×#11924007—24no—
66C2×#11924007—22no—
67C2×#11924007—22no—
68C2×#11924007—42no—
69C2×#11924007—42no—
70C2×#11924007C2×C414yes—
71C2×#11924007C2×C414yes—
72C2×C22⋊Dic73—22no—
73C2×C22⋊Dic73—22no—
74C2×C22⋊Dic73—24no—
75C2×C22⋊Dic73—42no—
76C2×C22⋊Dic73—42no—
77C2×C22⋊Dic73C2×C414yes—
78C2×C4×Dic73—22no—
79C2×C4×Dic73—22no—
80C2×C4×Dic73C2×C414yes—
81C2×C4×Dic73C2×C414yes—
82C2×C4×Dic73D414yes—
83C2×C4×Dic73D414yes—
84#14556498—24no—
85#14556498—24no—
86#14556498—24no—
87#14556498—22no—
88#14556498—42no—
89#14556498—42no—
90#14556498—42no—
91C25×C73C2311yescharacteristic
92C23×C4×C73C2312yes—
93C23×C4×C73C2312yes—
94C23×C4×C73C2311yescharacteristic
95C23×C4×C73C2311yescharacteristic
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—42no—
98C2×C22⋊C4×C73—22no—
99C2×C22⋊C4×C73—22no—
100C2×C22⋊C4×C73—24no—
101C2×C22⋊C4×C73—42no—
102C2×C22⋊C4×C73—42no—
103C2×C22⋊C4×C73C2×C414yes—
104#95×C73—22no—
105#95×C73—42no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—28no—
113C22×Dic73—28no—
114C22×Dic73—28no—
115C22×Dic73—28no—
116C22×Dic73—28no—
117C22×Dic73—28no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73—22no—
135C22×Dic73—22no—
136C22×Dic73C4⋊C4116yescomplemented
137C22×Dic73C22×C412yes—
138C22×Dic73C22×C412yes—
139C22×Dic73C22×C412yes—
140C22×Dic73C22×C412yes—
141C22×Dic73C2×D412yes—
142C22×Dic73C2×D412yes—
143C22×Dic73C2×D412yes—
144C22×Dic73C2×D412yes—
145C22×Dic73C2×D412yes—
146C22×Dic73C2×D412yes—
147C22×Dic73C2×Q812yes—
148C22×Dic73C2×Q812yes—
149C22×Dic73#4112yes—
150C22×Dic73#4112yes—
151C4⋊Dic73—44no—
152#11924007—28no—
153#11924007—28no—
154#11924007—48no—
155#11924007—48no—
156#11924007—48no—
157C22⋊Dic73—28no—
158C22⋊Dic73—48no—
159C4×Dic73—28no—
160C4×Dic73C22⋊C4116yescomplemented
161C24×C73—21no—
162C24×C73—21no—
163C24×C73—21no—
164C24×C73—21no—
165C24×C73—22no—
166C24×C73—22no—
167C24×C73—22no—
168C24×C73—22no—
169C24×C73C22×C412yes—
170C24×C73C2×D412yes—
171C24×C73C2×Q812yes—
172C24×C73C2411yescharacteristic
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—22no—
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×C73—22no—
186C22×C4×C73—22no—
187C22×C4×C73—22no—
188C22×C4×C73—22no—
189C22×C4×C73—22no—
190C22×C4×C73—22no—
191C22×C4×C73—22no—
192C22×C4×C73—22no—
193C22×C4×C73C22×C412yes—
194C22×C4×C73C22×C412yes—
195C22×C4×C73C2×D412yes—
196C22×C4×C73C2×D412yes—
197C4⋊C4×C73—44no—
198C22⋊C4×C73—28no—
199C22⋊C4×C73—48no—
200C23×C73—22no—
201C23×C73—22no—
202C23×C73—24no—
203C23×C73—24no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—24no—
209C23×C73—24no—
210C23×C73—24no—
211C23×C73—24no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—42no—
217C23×C73—42no—
218C23×C73—42no—
219C23×C73—42no—
220C23×C73—42no—
221C23×C73—42no—
222C23×C73C2×C4⋊C414yes—
223C23×C73C23×C411yescharacteristic
224C23×C73C22×D411yescharacteristic
225C23×C73C22×D411yescharacteristic
226C23×C73C22×D411yescharacteristic
227C23×C73C22×Q811yescharacteristic
228C23×C73C2×#4111yescharacteristic
229C23×C73C2×#4111yescharacteristic
230C23×C73H(5,𝔽2)12yes—
231C23×C73H(5,𝔽2)12yes—
232C23×C732−1+412yes—
233C23×C732−1+412yes—
234C2×C4×C73—24no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—24no—
238C2×C4×C73—22no—
239C2×C4×C73—24no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
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×C73—42no—
257C2×C4×C73C2×C22⋊C414yes—
258C2×Dic73—22no—
259C2×Dic73—22no—
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—232no—
269C2×Dic73—22no—
270C2×Dic73—24no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
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×Dic73—42no—
288C2×Dic73C2×C22⋊C414yes—
289C2×Dic73C2×C22⋊C414yes—
290C2×Dic73C2×C4⋊C414yes—
291C2×Dic73C2×C4⋊C414yes—
292C2×Dic73#11714yes—
293C2×Dic73#12114yes—
294C2×Dic73#12114yes—
295C2×Dic73C22⋊Q814yes—
296C2×Dic73C22⋊Q814yes—
297C22×C73—28no—
298C22×C73—24no—
299C22×C73—42no—
300C22×C73—42no—
301C22×C73—44no—
302C22×C73—42no—
303C22×C73—44no—
304C22×C73—42no—
305C22×C73—42no—
306C22×C73—42no—
307C22×C73—44no—
308C22×C73C22×C22⋊C411yescharacteristic
309C22×C73C22×C4⋊C411yescharacteristic
310C22×C73C2×#11711yescharacteristic
311C22×C73#52012yes—
312C22×C73#52012yes—
313C22×C73C2×#12111yescharacteristic
314C22×C73C2×#12111yescharacteristic
315C22×C73C2×C22⋊Q811yescharacteristic
316C22×C73C2×C22⋊Q811yescharacteristic
317C22×C73#53511yescharacteristic
318C22×C73#53511yescharacteristic
319C22×C73#53512yes—
320C22×C73#53611yescharacteristic
321C22×C73#53611yescharacteristic
322C22×C73#53612yes—
323C22×C73#53712yes—
324C22×C73#53712yes—
325C22×C73#54012yes—
326C22×C73#54112yes—
327C22×C73#54212yes—
328C22×C73#54212yes—
329C22×C73#54412yes—
330C22×C73#54412yes—
331C4×C73—24no—
332C4×C73—42no—
333C4×C73—82no—
334C4×C73—82no—
335Dic73—24no—
336Dic73—24no—
337Dic73—42no—
338Dic73—82no—
339Dic73—82no—
340Dic73#38018yes—
341#14340—731nomaximal, Sylow
342C2×C73—44no—
343C2×C73C2×#38011yescharacteristic
344C2×C73#231111yescharacteristic
345C2×C73#231111yescharacteristic
346C2×C73#231412yes—
347C2×C73#231712yes—
348C2×C73#261911yescharacteristic
349C2×C73#262012yes—
350C2×C73#262111yescharacteristic
351C2×C73#262811yescharacteristic
352C2×C73#262811yescharacteristic
353C2×C73#262912yes—
354C22×#117—731no—
355C2×#384—731no—
356C2×#382—731no—
357C2×#380—732no—
358#1441—732no—
359#1441—731no—
360#1441—731no—
361#1439—732no—
362#1439—731no—
363#1439D7311yescharacteristic
364C4⋊#95—732no—
365C73#1434011yescharacteristic, complemented, Sylow
366C24×C4—731no—
367C2×#117—738no—
368C2×#117—1464no—
369C2×#117—1464no—
370C22×C4⋊C4—732no—
371C22×C4⋊C4—731no—
372C22×C4⋊C4—731no—
373C22×C4⋊C4—731no—
374C22×C4⋊C4—732no—
375C22×C4⋊C4—732no—
376C22×C4⋊C4C2×D7311yescharacteristic
377C22×C22⋊C4—732no—
378C22×C22⋊C4—731no—
379C22×C22⋊C4C2×D7311yescharacteristic
380C22×C22⋊C4C2×D7312yes—
381C22×C42—731no—
382C22×C42—731no—
383#384—1464no—
384#382—1464no—
385#380—1468no—
386C2×#95—732no—
387C2×#95—732no—
388C2×#95—732no—
389C2×#95—732no—
390C2×#95—731no—
391C2×#95—732no—
392C2×#95—732no—
393C2×#95C2×D7311yescharacteristic
394C2×#95C2×D7312yes—
395C25C22×D7311yescharacteristic
396C23×C4—731no—
397C23×C4—734no—
398C23×C4—734no—
399C23×C4—734no—
400C23×C4—731no—
401C23×C4—731no—
402C23×C4—731no—
403C23×C4—732no—
404C23×C4—732no—
405C23×C4—1462no—
406C23×C4—1462no—
407C23×C4—1462no—
408C23×C4—1462no—
409C23×C4C22×D7312yes—
410C23×C4C22×D7312yes—
411C23×C4C22×D7311yescharacteristic
412C23×C4C22×D7311yescharacteristic
413#117—14632no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—42no—
416C2×C4⋊C4—2922no—
417C2×C4⋊C4—2922no—
418C2×C4⋊C4—2922no—
419C2×C4⋊C4—734no—
420C2×C4⋊C4—734no—
421C2×C4⋊C4—1464no—
422C2×C4⋊C4—1464no—
423C2×C4⋊C4—1464no—
424C2×C4⋊C4—1464no—
425C2×C4⋊C4—1464no—
426C2×C4⋊C4—1464no—
427C2×C4⋊C4—1464no—
428C2×C4⋊C4—1462no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4—1462no—
432C2×C4⋊C4—1462no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—42no—
437C2×C22⋊C4—42no—
438C2×C22⋊C4—2922no—
439C2×C22⋊C4—2922no—
440C2×C22⋊C4—734no—
441C2×C22⋊C4—1464no—
442C2×C22⋊C4—1462no—
443C2×C22⋊C4—1462no—
444C2×C22⋊C4C4×D7314yes—
445C2×C42—734no—
446C2×C42—734no—
447C2×C42—734no—
448C2×C42—734no—
449C2×C42—1462no—
450C2×C42—1462no—
451#95—22no—
452#95—42no—
453#95—2922no—
454#95—2922no—
455#95—2922no—
456#95—1464no—
457#95—1464no—
458#95—1464no—
459#95—1462no—
460C24—21no—
461C24—21no—
462C24—21no—
463C24—21no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24C2×C4×D7312yes—
469C24D4×D7312yes—
470C24Q8×D7312yes—
471C24C23×D7311yescharacteristic
472C22×C4—22no—
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—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—732no—
493C22×C4—732no—
494C22×C4—732no—
495C22×C4—732no—
496C22×C4—732no—
497C22×C4—732no—
498C22×C4—732no—
499C22×C4—7316no—
500C22×C4—732no—
501C22×C4—732no—
502C22×C4—732no—
503C22×C4—732no—
504C22×C4—732no—
505C22×C4—732no—
506C22×C4—732no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1461no—
522C22×C4—1461no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1468no—
526C22×C4—1468no—
527C22×C4—1468no—
528C22×C4—1468no—
529C22×C4—1468no—
530C22×C4—1468no—
531C22×C4—1461no—
532C22×C4—1461no—
533C22×C4—1461no—
534C22×C4—1461no—
535C22×C4—1462no—
536C22×C4—1462no—
537C22×C4C2×C4×D7312yes—
538C22×C4C2×C4×D7312yes—
539C22×C4D4×D7312yes—
540C22×C4D4×D7312yes—
541C4⋊C4—44no—
542C4⋊C4—2924no—
543C4⋊C4—2928no—
544C4⋊C4—2928no—
545C4⋊C4—2928no—
546C4⋊C4—1468no—
547C4⋊C4—1468no—
548C22⋊C4—28no—
549C22⋊C4—48no—
550C22⋊C4—2928no—
551C22⋊C4—1468no—
552C42—7316no—
553C42—1468no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C4⋊C4×D7314yes—
577C23C22×C4×D7311yescharacteristic
578C23C2×D4×D7311yescharacteristic
579C23C2×D4×D7311yescharacteristic
580C23C2×D4×D7311yescharacteristic
581C23C2×Q8×D7311yescharacteristic
582C23#41×D7311yescharacteristic
583C23#41×D7311yescharacteristic
584C23#1455663612yes—
585C23#1455663612yes—
586C23#1455663812yes—
587C23#1455663812yes—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—2921no—
612C2×C4—2921no—
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—2922no—
622C2×C4—2922no—
623C2×C4—2922no—
624C2×C4—2922no—
625C2×C4—2922no—
626C2×C4—2922no—
627C2×C4—2922no—
628C2×C4—734no—
629C2×C4—734no—
630C2×C4—734no—
631C2×C4—734no—
632C2×C4—734no—
633C2×C4—734no—
634C2×C4—734no—
635C2×C4—734no—
636C2×C4—734no—
637C2×C4—1462no—
638C2×C4—1464no—
639C2×C4—14632no—
640C2×C4—1462no—
641C2×C4—1462no—
642C2×C4—1464no—
643C2×C4—1462no—
644C2×C4—1462no—
645C2×C4—1464no—
646C2×C4—1462no—
647C2×C4—1462no—
648C2×C4—1462no—
649C2×C4—1462no—
650C2×C4C22⋊C4×D7314yes—
651C22—28no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22C2×C22⋊C4×D7311yescharacteristic
663C22C2×C4⋊C4×D7311yescharacteristic
664C22#117×D7311yescharacteristic
665C22#1798906612yes—
666C22#1798906612yes—
667C22#121×D7311yescharacteristic
668C22#121×D7311yescharacteristic
669C22C22⋊Q8×D7311yescharacteristic
670C22C22⋊Q8×D7311yescharacteristic
671C22#1798915011yescharacteristic
672C22#1798915211yescharacteristic
673C22#1798915212yes—
674C22#1798915711yescharacteristic
675C22#1798915911yescharacteristic
676C22#1798915912yes—
677C22#1798916412yes—
678C22#1798916612yes—
679C22#1798918712yes—
680C22#1798919412yes—
681C22#1798920112yes—
682C22#1798920512yes—
683C22#1798921612yes—
684C22#1798921612yes—
685C4—24no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—2922no—
690C4—5842no—
691C4—5842no—
692C4—738no—
693C4—1464no—
694C4—1464no—
695C2—44no—
696C2#380×D7311yescharacteristic
697C2#2261251811yescharacteristic
698C2#2261252111yescharacteristic
699C2#2261253512yes—
700C2#2261255012yes—
701C2#2261568011yescharacteristic
702C2#2261568712yes—
703C2#2261569611yescharacteristic
704C2#2261575111yescharacteristic
705C2#2261575711yescharacteristic
706C2#2261576012yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14340
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#17988456C2semidirect
3. C2×#17988451C2semidirect
5. #22608185C2semidirect
6. #22608182C2semidirect
7. #22608181C2semidirect
10. #22608159C2semidirect
15. C2×#14556547C4semidirect
136. C22×Dic73C4⋊C4semidirect
160. C4×Dic73C22⋊C4semidirect
365. C73#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#145565472
C2×#179884562
C2×#179884512
C2×#179884452
#226081852
#226081822
#226081812
#226081762
#226081742
#226081592
C73×#14392
#1434073