Finite Group Databasebeta

← #33015260

Subgroups — #33015260

Order 18688.

1550 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988480C212yesmaximal
2#22608188C212yesmaximal, complemented
3#22608184C212yesmaximal, complemented
4#22608176C211yesmaximal, characteristic
5#22608176C211yesmaximal, characteristic
6#22608165C212yesmaximal, complemented
7#22608160C212yesmaximal, complemented
8#22608154C212yesmaximal
9C2×#385×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C4⋊Dic73C2211yescharacteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C22⋊Dic73C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C4×Dic73C2212yes—
16#17988480—28no—
17C2×#14556498C2212yes—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2211yescharacteristic
27C24×C4×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2212yes—
29C22×C22⋊C4×C73C2211yescharacteristic
30C22×C22⋊C4×C73C2211yescharacteristic
31#385×C73C418yescomplemented
32C2×#95×C73C2212yes—
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C2×C4⋊Dic73—24no—
38C2×C4⋊Dic73—24no—
39C2×C4⋊Dic73—24no—
40C2×C4⋊Dic73—42no—
41C2×C4⋊Dic73—41no—
42C2×C4⋊Dic73—41no—
43C2×C4⋊Dic73—42no—
44C2×C22⋊Dic73—24no—
45C2×C22⋊Dic73—24no—
46C2×C22⋊Dic73—24no—
47C2×C22⋊Dic73—24no—
48C2×C22⋊Dic73—42no—
49C2×C22⋊Dic73—42no—
50C2×C4×Dic73—24no—
51C2×C4×Dic73—24no—
52C2×C4×Dic73—42no—
53#14556498—24no—
54#14556498—24no—
55#14556498—24no—
56#14556498—41no—
57#14556498—41no—
58#14556498—42no—
59#14556498—42no—
60#14556498—42no—
61#14556498—42no—
62#14556498—42no—
63C25×C73C2311yescharacteristic
64C23×C4×C73—42no—
65C23×C4×C73—42no—
66C23×C4×C73—41no—
67C23×C4×C73—41no—
68C23×C4×C73C2×C414yes—
69C23×C4×C73C2312yes—
70C23×C4×C73C2311yescharacteristic
71C23×C4×C73C2311yescharacteristic
72C23×C4×C73C2312yes—
73C2×C4⋊C4×C73—42no—
74C2×C4⋊C4×C73—42no—
75C2×C4⋊C4×C73C2×C418yes—
76C2×C22⋊C4×C73—41no—
77C2×C22⋊C4×C73—41no—
78C2×C22⋊C4×C73C2×C414yes—
79C2×C22⋊C4×C73C2×C414yes—
80C2×C22⋊C4×C73D414yes—
81C2×C22⋊C4×C73Q814yes—
82#95×C73C2×C418yes—
83C22×Dic73—22no—
84C22×Dic73—22no—
85C22×Dic73—22no—
86C22×Dic73—22no—
87C22×Dic73—22no—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73#4114yes—
110C22×Dic73#4114yes—
111C4⋊Dic73—44no—
112C4⋊Dic73—48no—
113C4⋊Dic73—44no—
114C22⋊Dic73—48no—
115C22⋊Dic73—48no—
116C4×Dic73—48no—
117C24×C73—22no—
118C24×C73—22no—
119C24×C73—21no—
120C24×C73—21no—
121C24×C73—22no—
122C24×C73—22no—
123C24×C73—21no—
124C24×C73—21no—
125C24×C73C22×C412yes—
126C24×C73C2×D412yes—
127C24×C73C2×Q812yes—
128C24×C73C2411yescharacteristic
129C22×C4×C73—22no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—22no—
134C22×C4×C73—22no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—21no—
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—48no—
148C22×C4×C73—44no—
149C22×C4×C73—48no—
150C22×C4×C73—44no—
151C22×C4×C73—44no—
152C22×C4×C73C22×C412yes—
153C22×C4×C73C22×C414yes—
154C22×C4×C73C22×C412yes—
155C22×C4×C73C22×C414yes—
156C22×C4×C73C2×D414yes—
157C22×C4×C73C2×Q814yes—
158C4⋊C4×C73—48no—
159C22⋊C4×C73—44no—
160C22⋊C4×C73C4⋊C4116yescomplemented
161C23×C73—24no—
162C23×C73—22no—
163C23×C73—24no—
164C23×C73—22no—
165C23×C73—22no—
166C23×C73—22no—
167C23×C73—22no—
168C23×C73—22no—
169C23×C73—24no—
170C23×C73—24no—
171C23×C73—24no—
172C23×C73—24no—
173C23×C73—42no—
174C23×C73—42no—
175C23×C73—42no—
176C23×C73—42no—
177C23×C73—41no—
178C23×C73—42no—
179C23×C73—41no—
180C23×C73—42no—
181C23×C73—41no—
182C23×C73—41no—
183C23×C73C2×C4⋊C414yes—
184C23×C73C23×C411yescharacteristic
185C23×C73C22×D411yescharacteristic
186C23×C73C22×Q811yescharacteristic
187C23×C73C2×#4112yes—
188C23×C73C2×#4112yes—
189C23×C73H(5,𝔽2)11yescharacteristic
190C23×C73H(5,𝔽2)11yescharacteristic
191C23×C73H(5,𝔽2)12yes—
192C23×C73H(5,𝔽2)12yes—
193C23×C732−1+411yescharacteristic
194C23×C732−1+411yescharacteristic
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—416no—
209C2×C4×C73—41no—
210C2×C4×C73—42no—
211C2×C4×C73—41no—
212C2×C4×C73—42no—
213C2×C4×C73—41no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—41no—
219C2×C4×C73C2×C4⋊C418yes—
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—24no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—42no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
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—
248C22×C73—24no—
249C22×C73—28no—
250C22×C73—44no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—44no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—44no—
258C22×C73—42no—
259C22×C73C22×C4⋊C411yescharacteristic
260C22×C73C4×#4112yes—
261C22×C73#51811yescharacteristic
262C22×C73#51812yes—
263C22×C73#51911yescharacteristic
264C22×C73#53711yescharacteristic
265C22×C73#53711yescharacteristic
266C22×C73#53912yes—
267C22×C73#53912yes—
268C22×C73#54212yes—
269C22×C73#54212yes—
270C22×C73#54411yescharacteristic
271C22×C73#54411yescharacteristic
272C22×C73#54612yes—
273C22×C73#54612yes—
274C22×C73#55111yescharacteristic
275C22×C73#55312yes—
276C22×C73#55412yes—
277C22×C73#55412yes—
278C22×C73#55911yescharacteristic
279C22×C73#56012yes—
280C22×C73#56011yescharacteristic
281C22×C73H(3,𝔽4)11yescharacteristic
282C4×C73—28no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73—44no—
286Dic73—44no—
287Dic73—44no—
288Dic73—82no—
289Dic73—82no—
290#14514—731nomaximal, Sylow
291C2×C73—44no—
292C2×C73#231611yescharacteristic
293C2×C73#233212yes—
294C2×C73#234912yes—
295C2×C73#237411yescharacteristic
296C2×C73#237511yescharacteristic
297C2×C73#269112yes—
298C2×C73#269311yescharacteristic
299C2×C73#271211yescharacteristic
300C2×C73#276311yescharacteristic
301C2×C73#279012yes—
302C2×C73#280711yescharacteristic
303C2×#388—732no—
304C2×#385D7311yescharacteristic
305#1442—732no—
306#1441—732no—
307#1439—731no—
308#1439—731no—
309#1436—732no—
310C4⋊#95—732no—
311C22⋊#95—732no—
312C73#1451411yescharacteristic, complemented, Sylow
313C24×C4C2×D7311yescharacteristic
314C22×C4⋊C4—731no—
315C22×C4⋊C4—732no—
316C22×C4⋊C4—731no—
317C22×C4⋊C4C2×D7312yes—
318C22×C22⋊C4—732no—
319C22×C22⋊C4—732no—
320C22×C22⋊C4C2×D7311yescharacteristic
321C22×C22⋊C4C2×D7311yescharacteristic
322C22×C42—732no—
323#388—1468no—
324#385Dic7318yescomplemented
325C2×#95—732no—
326C2×#95—732no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—731no—
330C2×#95—732no—
331C2×#95—731no—
332C2×#95—732no—
333C2×#95—732no—
334C2×#95—732no—
335C2×#95C2×D7312yes—
336C25C22×D7311yescharacteristic
337C23×C4—41no—
338C23×C4—41no—
339C23×C4—42no—
340C23×C4—42no—
341C23×C4—732no—
342C23×C4—732no—
343C23×C4—732no—
344C23×C4—732no—
345C23×C4C2×Dic7314yes—
346C23×C4C22×D7311yescharacteristic
347C23×C4C22×D7311yescharacteristic
348C23×C4C22×D7312yes—
349C23×C4C22×D7312yes—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—2922no—
353C2×C4⋊C4—2921no—
354C2×C4⋊C4—2922no—
355C2×C4⋊C4—2921no—
356C2×C4⋊C4—1464no—
357C2×C4⋊C4—1464no—
358C2×C4⋊C4—1464no—
359C2×C4⋊C4C2×Dic7318yes—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—2922no—
363C2×C22⋊C4—2922no—
364C2×C22⋊C4—1464no—
365C2×C22⋊C4—1464no—
366C2×C22⋊C4—1464no—
367C2×C22⋊C4—1464no—
368C2×C22⋊C4Dic14614yes—
369C2×C22⋊C4D29214yes—
370C2×C22⋊C4C2×Dic7314yes—
371C2×C22⋊C4C2×Dic7314yes—
372C2×C42—2922no—
373C2×C42—1464no—
374C2×C42—1464no—
375#95—2921no—
376#95—2921no—
377#95—2922no—
378#95—2922no—
379#95—2922no—
380#95—2922no—
381#95—2922no—
382#95—1464no—
383#95—1464no—
384#95—1464no—
385#95C2×Dic7318yes—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—21no—
393C24—21no—
394C24C22×Dic7312yes—
395C24C2×D29212yes—
396C24C2×Dic14612yes—
397C24C23×D7311yescharacteristic
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—734no—
422C22×C4—734no—
423C22×C4—1462no—
424C22×C4—1462no—
425C22×C4—1462no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1462no—
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×C4C22×Dic7312yes—
450C22×C4C22×Dic7314yes—
451C22×C4C22×Dic7312yes—
452C22×C4C22×Dic7314yes—
453C22×C4C2×D29214yes—
454C22×C4C2×Dic14614yes—
455C4⋊C4—48no—
456C4⋊C4—2928no—
457C4⋊C4—2924no—
458C4⋊C4—2924no—
459C22⋊C4—44no—
460C22⋊C4—2928no—
461C22⋊C4—2928no—
462C22⋊C4C4⋊Dic73116yescomplemented
463C42—2928no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23C2×C4⋊Dic7314yes—
487C23C23×Dic7311yescharacteristic
488C23C22×D29211yescharacteristic
489C23C22×Dic14611yescharacteristic
490C23#41×D7312yes—
491C23#41×D7312yes—
492C23#1455663611yescharacteristic
493C23#1455663611yescharacteristic
494C23#1455663612yes—
495C23#1455663612yes—
496C23#1455663911yescharacteristic
497C23#1455663911yescharacteristic
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
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—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4—1464no—
547C2×C4—1464no—
548C2×C4—1464no—
549C2×C4—1464no—
550C2×C4C2×C4⋊Dic7318yes—
551C22—24no—
552C22—28no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22C22×C4⋊Dic7311yescharacteristic
563C22#41×Dic7312yes—
564C22#1798906211yescharacteristic
565C22#1798906212yes—
566C22#1798906511yescharacteristic
567C22#1798916811yescharacteristic
568C22#1798916811yescharacteristic
569C22#1798918312yes—
570C22#1798918312yes—
571C22#1798920712yes—
572C22#1798920712yes—
573C22#1798921811yescharacteristic
574C22#1798921811yescharacteristic
575C22#1798923212yes—
576C22#1798923212yes—
577C22#1798926511yescharacteristic
578C22#1798928212yes—
579C22#1798929612yes—
580C22#1798929612yes—
581C22#1798932111yescharacteristic
582C22#1798932611yescharacteristic
583C22#1798932612yes—
584C22#1798933011yescharacteristic
585C4—28no—
586C4—42no—
587C4—44no—
588C4—42no—
589C4—2924no—
590C4—2924no—
591C4—5842no—
592C4—5842no—
593C2—44no—
594C2#2261254911yescharacteristic
595C2#2261268912yes—
596C2#2261284112yes—
597C2#2261306711yescharacteristic
598C2#2261307211yescharacteristic
599C2#2261641012yes—
600C2#2261643811yescharacteristic
601C2#2261669911yescharacteristic
602C2#2261734511yescharacteristic
603C2#2261770412yes—
604C2#2261790311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#385×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#385D73
hypocenterC73#14514
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
2. #22608188C2semidirect
3. #22608184C2semidirect
6. #22608165C2semidirect
7. #22608160C2semidirect
31. #385×C73C4semidirect
160. C22⋊C4×C73C4⋊C4semidirect
312. C73#14514semidirect
324. #385Dic73semidirect
462. C22⋊C4C4⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
#226081882
#226081842
#226081762
#226081652
#226081602
#226081542
C2×#385×C732
#1451473