Finite Group Databasebeta

← #33013203

Subgroups — #33013203

Order 18688.

1542 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic146C211yesmaximal, characteristic
2C2×#17988454C211yesmaximal, characteristic
3C2×#17988451C211yesmaximal, characteristic
4#22608201C212yesmaximal
5#22608197C212yesmaximal
6#22608183C212yesmaximal
7#22608182C212yesmaximal
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×#11924007C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988454—24no—
23#17988451—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—24no—
43C22×Dic146—24no—
44C22×Dic146C2×C414yes—
45C23×Dic73C2312yes—
46C23×Dic73C2312yes—
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51C4×Dic146—232no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73C2×C414yes—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—24no—
60C2×#11924007—42no—
61C2×#11924007—42no—
62C2×#11924007C2×C414yes—
63C2×#11924007C2×C414yes—
64C2×C4×Dic73—22no—
65C2×C4×Dic73—22no—
66C2×C4×Dic73C2×C414yes—
67C2×C4×Dic73C2×C414yes—
68C2×C4×Dic73D414yes—
69C2×C4×Dic73D414yes—
70#14556498—24no—
71#14556498—24no—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—22no—
76#14556498—22no—
77#14556498—24no—
78#14556498—42no—
79#14556498—42no—
80#14556498—42no—
81C23×C4×C73C2312yes—
82C23×C4×C73C2312yes—
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C2×C42×C73—22no—
87C2×C42×C73—22no—
88C2×C42×C73C2×C414yes—
89#95×C73—22no—
90#95×C73—22no—
91#95×C73—24no—
92#95×C73—42no—
93C2×Dic146—216no—
94C2×Dic146—216no—
95C2×Dic146—28no—
96C2×Dic146—28no—
97C2×Dic146—28no—
98C22×Dic73—21no—
99C22×Dic73—21no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—22no—
114C22×Dic73—22no—
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×Dic73C22×C412yes—
128C22×Dic73C22×C412yes—
129C22×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C2×D412yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73#4114yes—
136C4⋊Dic73—28no—
137#11924007—28no—
138#11924007—28no—
139#11924007—48no—
140C4×Dic73—28no—
141C4×Dic73C22⋊C4116yes—
142C24×C73C2411yescharacteristic
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
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×C73C22×C412yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73#4114yes—
174C42×C73—28no—
175C23×C73C23×C411yescharacteristic
176C23×C73C22×D411yescharacteristic
177C23×C73C22×D411yescharacteristic
178C23×C73C2×#4112yes—
179C23×C73C2×#4112yes—
180C23×C73H(5,𝔽2)11yescharacteristic
181C23×C73H(5,𝔽2)11yescharacteristic
182C23×C732−1+412yes—
183C23×C732−1+412yes—
184C23×C732−1+411yescharacteristic
185C23×C732−1+411yescharacteristic
186C2×C4×C73—22no—
187C2×C4×C73—22no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—24no—
191C2×C4×C73—24no—
192C2×C4×C73—24no—
193C2×C4×C73—24no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
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×C73C2×C22⋊C414yes—
216C2×C4×C73#11714yes—
217C2×Dic73—22no—
218C2×Dic73—22no—
219C2×Dic73—22no—
220C2×Dic73—22no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
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×Dic73C2×C22⋊C414yes—
253C2×Dic73C2×C22⋊C414yes—
254C2×Dic73#11714yes—
255Dic146—232no—
256C22×C73C22×C22⋊C411yescharacteristic
257C22×C73C2×#11711yescharacteristic
258C22×C73C2×#11711yescharacteristic
259C22×C73#51911yescharacteristic
260C22×C73#51911yescharacteristic
261C22×C73#52011yescharacteristic
262C22×C73#52011yescharacteristic
263C22×C73#52512yes—
264C22×C73#52512yes—
265C22×C73#53511yescharacteristic
266C22×C73#53612yes—
267C22×C73#53612yes—
268C22×C73#53611yescharacteristic
269C22×C73#53711yescharacteristic
270C22×C73#53711yescharacteristic
271C22×C73#54112yes—
272C22×C73#54112yes—
273C22×C73#54112yes—
274C22×C73#54112yes—
275C22×C73#54212yes—
276C22×C73#54212yes—
277C22×C73#54212yes—
278C22×C73#54212yes—
279C4×C73—24no—
280C4×C73—24no—
281C4×C73—42no—
282C4×C73—82no—
283C4×C73—82no—
284Dic73—24no—
285Dic73—24no—
286Dic73—24no—
287Dic73—42no—
288Dic73—82no—
289Dic73—82no—
290#14346—731nomaximal, Sylow
291C2×C73#228011yescharacteristic
292C2×C73#231011yescharacteristic
293C2×C73#231111yescharacteristic
294C2×C73#232011yescharacteristic
295C2×C73#232011yescharacteristic
296C2×C73#232011yescharacteristic
297C2×C73#232011yescharacteristic
298C2×C73#263212yes—
299C2×C73#263412yes—
300C2×C73#263712yes—
301C2×C73#263712yes—
302C22×C4×Q8—731no—
303C2×#383—731no—
304C2×#382—731no—
305#1446—732no—
306#1445—732no—
307#1441—732no—
308#1441—732no—
309#1434—731no—
310#1434—731no—
311#1434—731no—
312#1434D7311yescharacteristic
313C73#1434611yescharacteristic, complemented, Sylow
314C23×Q8—731no—
315C2×C4×Q8—738no—
316C2×C4×Q8—1464no—
317C2×C4×Q8—1464no—
318C22×C4⋊C4—732no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—731no—
322C22×C42—731no—
323C22×C42—731no—
324C22×C42C2×D7311yescharacteristic
325#383—1464no—
326#382—1464no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—732no—
330C2×#95—731no—
331C2×#95—731no—
332C2×#95—732no—
333C2×#95—732no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95—732no—
337C2×#95—732no—
338C2×#95C2×D7311yescharacteristic
339C2×#95C2×D7311yescharacteristic
340C2×#95C2×D7312yes—
341C2×#95C2×D7312yes—
342C22×Q8—734no—
343C22×Q8—1462no—
344C22×Q8—1462no—
345C22×Q8—1464no—
346C22×Q8—1464no—
347C23×C4—731no—
348C23×C4—731no—
349C23×C4—731no—
350C23×C4—731no—
351C23×C4—732no—
352C23×C4—732no—
353C23×C4C22×D7312yes—
354C23×C4C22×D7312yes—
355C23×C4C22×D7311yescharacteristic
356C23×C4C22×D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C4×Q8—14632no—
359C2×C4⋊C4—2922no—
360C2×C4⋊C4—2922no—
361C2×C4⋊C4—734no—
362C2×C4⋊C4—734no—
363C2×C4⋊C4—734no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—734no—
374C2×C42—734no—
375C2×C42—734no—
376C2×C42—734no—
377C2×C42—1462no—
378C2×C42—1462no—
379C2×C42C4×D7314yes—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—42no—
384#95—2922no—
385#95—2922no—
386#95—2922no—
387#95—1464no—
388#95—1464no—
389#95—1464no—
390#95—1464no—
391#95—1464no—
392#95—1462no—
393#95—1462no—
394#95—1464no—
395C24C23×D7311yescharacteristic
396C2×Q8—1468no—
397C2×Q8—1468no—
398C2×Q8—14616no—
399C2×Q8—14616no—
400C2×Q8—1468no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
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—732no—
427C22×C4—732no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—732no—
432C22×C4—732no—
433C22×C4—734no—
434C22×C4—732no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4C2×C4×D7312yes—
465C22×C4C2×C4×D7312yes—
466C22×C4C2×C4×D7312yes—
467C22×C4D4×D7312yes—
468C22×C4D4×D7312yes—
469C22×C4#1192402814yes—
470C4⋊C4—2928no—
471C4⋊C4—1468no—
472C4⋊C4—1468no—
473C4⋊C4—1468no—
474C42—28no—
475C42—7316no—
476C42—1468no—
477C23C22×C4×D7311yescharacteristic
478C23C2×D4×D7311yescharacteristic
479C23C2×D4×D7311yescharacteristic
480C23#41×D7312yes—
481C23C2×#1192402812yes—
482C23#1455663611yescharacteristic
483C23#1455663711yescharacteristic
484C23#1455663812yes—
485C23#1455663812yes—
486C23#1455663811yescharacteristic
487C23#1455663911yescharacteristic
488Q8—14632no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2922no—
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—734no—
537C2×C4—734no—
538C2×C4—734no—
539C2×C4—1462no—
540C2×C4—1462no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1464no—
553C2×C4—1464no—
554C2×C4—1464no—
555C2×C4—1464no—
556C2×C4C22⋊C4×D7314yes—
557C2×C4#1455654814yes—
558C22C2×C22⋊C4×D7311yescharacteristic
559C22#117×D7311yescharacteristic
560C22C2×#1455654811yescharacteristic
561C22#1798906411yescharacteristic
562C22#1798906411yescharacteristic
563C22#1798906711yescharacteristic
564C22#1798906811yescharacteristic
565C22#1798909612yes—
566C22#1798909612yes—
567C22#1798915311yescharacteristic
568C22#1798915912yes—
569C22#1798915912yes—
570C22#1798916011yescharacteristic
571C22#1798916611yescharacteristic
572C22#1798916711yescharacteristic
573C22#1798919412yes—
574C22#1798919512yes—
575C22#1798919612yes—
576C22#1798919912yes—
577C22#1798920112yes—
578C22#1798920212yes—
579C22#1798920312yes—
580C22#1798920812yes—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—2922no—
587C4—5842no—
588C4—5842no—
589C4—1464no—
590C4—1464no—
591C4—1464no—
592C2#2261231211yescharacteristic
593C2#2261251611yescharacteristic
594C2#2261252011yescharacteristic
595C2#2261257711yescharacteristic
596C2#2261257911yescharacteristic
597C2#2261258111yescharacteristic
598C2#2261258311yescharacteristic
599C2#2261579512yes—
600C2#2261581112yes—
601C2#2261584312yes—
602C2#2261584512yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1434D73
hypocenterC73#14346
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
313. C73#14346semidirect

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×#179884512
#226082012
#226081972
#226081832
#226081822
#226081582
#226081552
C73×#14342
#1434673