Finite Group Databasebeta

← #33008710

Subgroups — #33008710

Order 18688.

1622 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic146C211yesmaximal, characteristic
2C2×#17988454C211yesmaximal, characteristic
3C2×#17988432C211yesmaximal, characteristic
4#22608201C212yesmaximal
5#22608197C212yesmaximal
6#22608165C212yesmaximal
7#22608162C212yesmaximal
8#22608158C211yesmaximal, characteristic
9#22608155C211yesmaximal, characteristic
10#22608155C211yesmaximal, characteristic
11C73×#1434C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×C4×Dic146—24no—
14C2×C4×Dic146—24no—
15C2×C4×Dic146C418yescomplemented
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988454—24no—
23#17988432—24no—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C42×C73C2211yescharacteristic
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2212yes—
40C22×Dic146—22no—
41C22×Dic146—22no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146—22no—
45C22×Dic146—22no—
46C22×Dic146C2×C414yes—
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53C4×Dic146—232no—
54C2×C4⋊Dic73—22no—
55C2×C4⋊Dic73—22no—
56C2×C4⋊Dic73C2×C414yes—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007C2×C414yes—
62C2×#11924007C2×C414yes—
63C2×C4×Dic73—24no—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—22no—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73C2×C414yes—
69C2×C4×Dic73C2×C414yes—
70C2×C4×Dic73D414yes—
71C2×C4×Dic73D414yes—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—24no—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—22no—
80#14556498—22no—
81#14556498—24no—
82#14556498—24no—
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C2×C42×C73—22no—
89C2×C42×C73—22no—
90C2×C42×C73C2×C414yes—
91#95×C73—22no—
92#95×C73—22no—
93#95×C73—24no—
94#95×C73—24no—
95C2×Dic146—28no—
96C2×Dic146—28no—
97C2×Dic146—28no—
98C2×Dic146—28no—
99C2×Dic146—28no—
100C2×Dic146—28no—
101C2×Dic146—28no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
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×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C412yes—
132C22×Dic73C22×C412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C22×Dic73C2×D412yes—
137C22×Dic73#4114yes—
138C22×Dic73#4112yes—
139C22×Dic73#4112yes—
140C4⋊Dic73—28no—
141#11924007—28no—
142#11924007—28no—
143C4×Dic73—216no—
144C4×Dic73—28no—
145C4×Dic73C22⋊C4116yes—
146C24×C73C2411yescharacteristic
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C22×C412yes—
173C22×C4×C73C2×D412yes—
174C22×C4×C73C2×D412yes—
175C22×C4×C73#4112yes—
176C22×C4×C73#4114yes—
177C22×C4×C73#4112yes—
178C42×C73—28no—
179C23×C73C23×C411yescharacteristic
180C23×C73C22×D411yescharacteristic
181C23×C73C22×D411yescharacteristic
182C23×C73C2×#4111yescharacteristic
183C23×C73C2×#4111yescharacteristic
184C23×C73C2×#4112yes—
185C23×C73C2×#4111yescharacteristic
186C23×C73C2×#4112yes—
187C23×C73C2×#4111yescharacteristic
188C23×C732−1+412yes—
189C23×C732−1+412yes—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73C2×C22⋊C414yes—
220C2×C4×C73#11714yes—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—24no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—24no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73C2×C22⋊C414yes—
257C2×Dic73C2×C22⋊C414yes—
258C2×Dic73#11714yes—
259Dic146—232no—
260C22×C73C22×C22⋊C411yescharacteristic
261C22×C73C2×#11711yescharacteristic
262C22×C73C2×#11711yescharacteristic
263C22×C73C4×#4111yescharacteristic
264C22×C73C4×#4111yescharacteristic
265C22×C73#51911yescharacteristic
266C22×C73#51911yescharacteristic
267C22×C73#52512yes—
268C22×C73#52511yescharacteristic
269C22×C73#52511yescharacteristic
270C22×C73#52511yescharacteristic
271C22×C73#52511yescharacteristic
272C22×C73#53212yes—
273C22×C73#53612yes—
274C22×C73#53612yes—
275C22×C73#55212yes—
276C22×C73#55212yes—
277C22×C73#55212yes—
278C22×C73#55212yes—
279C22×C73#55612yes—
280C22×C73#55612yes—
281C22×C73#55612yes—
282C22×C73#55612yes—
283C4×C73—24no—
284C4×C73—24no—
285C4×C73—44no—
286C4×C73—42no—
287C4×C73—82no—
288Dic73—24no—
289Dic73—24no—
290Dic73—24no—
291Dic73—44no—
292Dic73—42no—
293Dic73—82no—
294#13961—731nomaximal, Sylow
295C2×C73#228011yescharacteristic
296C2×C73#229711yescharacteristic
297C2×C73#231011yescharacteristic
298C2×C73#235611yescharacteristic
299C2×C73#235611yescharacteristic
300C2×C73#235611yescharacteristic
301C2×C73#235611yescharacteristic
302C2×C73#259912yes—
303C2×C73#259912yes—
304C2×C73#260412yes—
305C2×C73#260412yes—
306C22×C4×Q8—731no—
307C2×#383—731no—
308C2×#376—731no—
309#1446—732no—
310#1445—732no—
311#1436—732no—
312#1436—732no—
313#1434—731no—
314#1434—731no—
315#1434—731no—
316#1434D7311yescharacteristic
317C73#1396111yescharacteristic, complemented, Sylow
318C23×Q8—731no—
319C2×C4×Q8—738no—
320C2×C4×Q8—1464no—
321C2×C4×Q8—1464no—
322C22×C4⋊C4—731no—
323C22×C4⋊C4—731no—
324C22×C4⋊C4—731no—
325C22×C42—731no—
326C22×C42—731no—
327C22×C42—732no—
328C22×C42C2×D7311yescharacteristic
329#383—1464no—
330#376—1464no—
331C2×#95—731no—
332C2×#95—731no—
333C2×#95—732no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95—732no—
337C2×#95—732no—
338C2×#95—732no—
339C2×#95—732no—
340C2×#95—732no—
341C2×#95—732no—
342C2×#95C2×D7311yescharacteristic
343C2×#95C2×D7311yescharacteristic
344C2×#95C2×D7312yes—
345C2×#95C2×D7312yes—
346C22×Q8—734no—
347C22×Q8—1462no—
348C22×Q8—1462no—
349C22×Q8—1462no—
350C22×Q8—1462no—
351C22×Q8—1462no—
352C22×Q8—1462no—
353C23×C4—731no—
354C23×C4—731no—
355C23×C4—731no—
356C23×C4—731no—
357C23×C4—732no—
358C23×C4—732no—
359C23×C4C22×D7312yes—
360C23×C4C22×D7311yescharacteristic
361C23×C4C22×D7312yes—
362C23×C4C22×D7311yescharacteristic
363C23×C4C22×D7311yescharacteristic
364C4×Q8—14632no—
365C2×C4⋊C4—734no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—734no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1462no—
373C2×C4⋊C4—1462no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—734no—
377C2×C42—734no—
378C2×C42—734no—
379C2×C42—734no—
380C2×C42—1464no—
381C2×C42—1464no—
382C2×C42—1462no—
383C2×C42—1462no—
384C2×C42—1464no—
385C2×C42C4×D7314yes—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—1464no—
391#95—1464no—
392#95—1464no—
393#95—1464no—
394#95—1464no—
395#95—1464no—
396#95—1464no—
397#95—1462no—
398#95—1462no—
399#95—1464no—
400#95—1464no—
401C24C23×D7311yescharacteristic
402C2×Q8—1468no—
403C2×Q8—1468no—
404C2×Q8—1468no—
405C2×Q8—1468no—
406C2×Q8—1468no—
407C2×Q8—1468no—
408C2×Q8—1468no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—732no—
433C22×C4—732no—
434C22×C4—732no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—732no—
438C22×C4—732no—
439C22×C4—732no—
440C22×C4—732no—
441C22×C4—732no—
442C22×C4—734no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1462no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1462no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4C2×C4×D7312yes—
471C22×C4C2×C4×D7312yes—
472C22×C4C2×C4×D7312yes—
473C22×C4D4×D7312yes—
474C22×C4D4×D7312yes—
475C22×C4#1192402712yes—
476C22×C4#1192402814yes—
477C22×C4#1192402912yes—
478C4⋊C4—1468no—
479C4⋊C4—1468no—
480C4⋊C4—1468no—
481C42—28no—
482C42—7316no—
483C42—14616no—
484C42—1468no—
485C23C22×C4×D7311yescharacteristic
486C23C2×D4×D7311yescharacteristic
487C23C2×D4×D7311yescharacteristic
488C23#41×D7312yes—
489C23#41×D7311yescharacteristic
490C23#41×D7311yescharacteristic
491C23C2×#1192402711yescharacteristic
492C23C2×#1192402812yes—
493C23C2×#1192402911yescharacteristic
494C23#1455663812yes—
495C23#1455663812yes—
496Q8—14632no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—734no—
540C2×C4—734no—
541C2×C4—734no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1464no—
548C2×C4—1464no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4—1462no—
554C2×C4—1462no—
555C2×C4—1464no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4—1462no—
563C2×C4—1462no—
564C2×C4C22⋊C4×D7314yes—
565C2×C4#1455654914yes—
566C22C2×C22⋊C4×D7311yescharacteristic
567C22#117×D7311yescharacteristic
568C22C2×#1455654911yescharacteristic
569C22#1798905511yescharacteristic
570C22C4×#1192402811yescharacteristic
571C22#1798906411yescharacteristic
572C22#1798906411yescharacteristic
573C22#1798909612yes—
574C22#1798909711yescharacteristic
575C22#1798909711yescharacteristic
576C22#1798910011yescharacteristic
577C22#1798910011yescharacteristic
578C22#1798913412yes—
579C22#1798915912yes—
580C22#1798915912yes—
581C22#1798927012yes—
582C22#1798927312yes—
583C22#1798927512yes—
584C22#1798927712yes—
585C22#1798930412yes—
586C22#1798930512yes—
587C22#1798930612yes—
588C22#1798931012yes—
589C4—24no—
590C4—24no—
591C4—42no—
592C4—44no—
593C4—82no—
594C4—2922no—
595C4—2924no—
596C4—5842no—
597C4—1464no—
598C4—1464no—
599C4—1464no—
600C2#2261231311yescharacteristic
601C2#2261241711yescharacteristic
602C2#2261251611yescharacteristic
603C2#2261289111yescharacteristic
604C2#2261289411yescharacteristic
605C2#2261289711yescharacteristic
606C2#2261289811yescharacteristic
607C2#2261550512yes—
608C2#2261550712yes—
609C2#2261556612yes—
610C2#2261556812yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1434D73
hypocenterC73#13961
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic146C4semidirect
317. C73#13961semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1462
C2×#179884542
C2×#179884322
#226082012
#226081972
#226081652
#226081622
#226081582
#226081552
C73×#14342
#1396173