Finite Group Databasebeta

← #33014171

Subgroups — #33014171

Order 18688.

1506 conjugacy classes of subgroups.

The subgroup lattice has 602 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#22608180C212yesmaximal
5#22608163C212yesmaximal, complemented
6#22608156C212yesmaximal, complemented
7#22608154C211yesmaximal, characteristic
8#22608154C211yesmaximal, characteristic
9C2×#380×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C22⋊Dic73C2212yes—
12C22×C22⋊Dic73C2211yescharacteristic
13C22×C22⋊Dic73C2211yescharacteristic
14C22×C4×Dic73C2212yes—
15#17988480—28no—
16C2×#14556498C2212yes—
17C2×#14556498C2212yes—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2211yescharacteristic
22C2×#14556498C2212yes—
23C2×#14556498C2211yescharacteristic
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C24×C4×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29C22×C4⋊C4×C73C2211yescharacteristic
30C22×C22⋊C4×C73C2212yes—
31#380×C73C418yescomplemented
32C2×#95×C73C2212yes—
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C2×C4⋊Dic73—24no—
38C2×C4⋊Dic73—42no—
39C2×C4⋊Dic73—42no—
40C2×C22⋊Dic73—24no—
41C2×C22⋊Dic73—24no—
42C2×C22⋊Dic73—22no—
43C2×C22⋊Dic73—41no—
44C2×C22⋊Dic73—42no—
45C2×C22⋊Dic73—41no—
46C2×C22⋊Dic73—41no—
47C2×C22⋊Dic73—41no—
48C2×C22⋊Dic73D414yes—
49C2×C4×Dic73—24no—
50C2×C4×Dic73—24no—
51C2×C4×Dic73—42no—
52#14556498—24no—
53#14556498—24no—
54#14556498—24no—
55#14556498—24no—
56#14556498—24no—
57#14556498—42no—
58#14556498—42no—
59#14556498—41no—
60#14556498—42no—
61#14556498—41no—
62#14556498—42no—
63C25×C73C2311yescharacteristic
64C23×C4×C73—24no—
65C23×C4×C73—41no—
66C23×C4×C73—41no—
67C23×C4×C73—42no—
68C23×C4×C73C2×C414yes—
69C23×C4×C73C2311yescharacteristic
70C23×C4×C73C2312yes—
71C23×C4×C73C2312yes—
72C23×C4×C73C2311yescharacteristic
73C2×C4⋊C4×C73—41no—
74C2×C4⋊C4×C73—42no—
75C2×C4⋊C4×C73—41no—
76C2×C4⋊C4×C73C2×C414yes—
77C2×C4⋊C4×C73C2×C414yes—
78C2×C22⋊C4×C73—24no—
79C2×C22⋊C4×C73—24no—
80C2×C22⋊C4×C73C2×C418yes—
81#95×C73C2×C418yes—
82C22×Dic73—22no—
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×Dic73C2×D414yes—
109C22×Dic73#4114yes—
110C4⋊Dic73—48no—
111C22⋊Dic73—44no—
112C22⋊Dic73—48no—
113C22⋊Dic73—44no—
114C4×Dic73—48no—
115C24×C73—22no—
116C24×C73—22no—
117C24×C73—21no—
118C24×C73—21no—
119C24×C73—22no—
120C24×C73—21no—
121C24×C73—21no—
122C24×C73—22no—
123C24×C73C22×C412yes—
124C24×C73C2×D412yes—
125C24×C73#4112yes—
126C24×C73C2411yescharacteristic
127C22×C4×C73—21no—
128C22×C4×C73—21no—
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—22no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—28no—
147C22×C4×C73—44no—
148C22×C4×C73—44no—
149C22×C4×C73—48no—
150C22×C4×C73—48no—
151C22×C4×C73C22×C412yes—
152C22×C4×C73C22×C414yes—
153C22×C4×C73C22×C414yes—
154C22×C4×C73C22×C412yes—
155C22×C4×C73#4114yes—
156C4⋊C4×C73—44no—
157C4⋊C4×C73—44no—
158C22⋊C4×C73—216no—
159C23×C73—22no—
160C23×C73—22no—
161C23×C73—22no—
162C23×C73—22no—
163C23×C73—22no—
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—42no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73C4×D414yes—
182C23×C73C23×C411yescharacteristic
183C23×C73C22×D411yescharacteristic
184C23×C73C2×#4112yes—
185C23×C73C2×#4111yescharacteristic
186C23×C73C2×#4112yes—
187C23×C73H(5,𝔽2)11yescharacteristic
188C23×C73H(5,𝔽2)11yescharacteristic
189C23×C73H(5,𝔽2)12yes—
190C23×C73H(5,𝔽2)11yescharacteristic
191C23×C732−1+411yescharacteristic
192C23×C732−1+412yes—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—24no—
206C2×C4×C73—416no—
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#11714yes—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—24no—
223C2×Dic73—24no—
224C2×Dic73—42no—
225C2×Dic73—42no—
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—
246C22×C73—24no—
247C22×C73—24no—
248C22×C73—24no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—44no—
254C22×C73—44no—
255C22×C73—44no—
256C22×C73—44no—
257C22×C73C2×#11711yescharacteristic
258C22×C73C2×C4×D411yescharacteristic
259C22×C73C4×#4111yescharacteristic
260C22×C73#51811yescharacteristic
261C22×C73#51811yescharacteristic
262C22×C73#51911yescharacteristic
263C22×C73#52011yescharacteristic
264C22×C73#53411yescharacteristic
265C22×C73#53711yescharacteristic
266C22×C73#53811yescharacteristic
267C22×C73#53812yes—
268C22×C73#53812yes—
269C22×C73#53911yescharacteristic
270C22×C73#53912yes—
271C22×C73#54212yes—
272C22×C73#54612yes—
273C22×C73#54712yes—
274C22×C73#55212yes—
275C22×C73#55212yes—
276C22×C73#55312yes—
277C22×C73#55512yes—
278C22×C73#56212yes—
279C22×C73#56312yes—
280C4×C73—24no—
281C4×C73—42no—
282C4×C73—44no—
283C4×C73—44no—
284Dic73—44no—
285Dic73—44no—
286Dic73—82no—
287Dic73—82no—
288#14422—731nomaximal, Sylow
289C2×C73—44no—
290C2×C73#231211yescharacteristic
291C2×C73#232511yescharacteristic
292C2×C73#233211yescharacteristic
293C2×C73#234411yescharacteristic
294C2×C73#234611yescharacteristic
295C2×C73#236411yescharacteristic
296C2×C73#237811yescharacteristic
297C2×C73#268612yes—
298C2×C73#269512yes—
299C2×C73#277012yes—
300C2×C73#278112yes—
301C2×#388—732no—
302C2×#380D7311yescharacteristic
303#1442—732no—
304#1441—732no—
305#1440—732no—
306#1436—732no—
307#1434—732no—
308C22⋊#95—731no—
309C22⋊#95—731no—
310C73#1442211yescharacteristic, complemented, Sylow
311C24×C4C2×D7311yescharacteristic
312C22×C4⋊C4—732no—
313C22×C4⋊C4C2×D7311yescharacteristic
314C22×C4⋊C4C2×D7311yescharacteristic
315C22×C22⋊C4—731no—
316C22×C22⋊C4—731no—
317C22×C22⋊C4—732no—
318C22×C22⋊C4C2×D7312yes—
319C22×C42—732no—
320#388—1468no—
321#380Dic7318yescomplemented
322C2×#95—731no—
323C2×#95—732no—
324C2×#95—731no—
325C2×#95—732no—
326C2×#95—732no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—732no—
330C2×#95—732no—
331C2×#95—732no—
332C2×#95—732no—
333C2×#95C2×D7312yes—
334C25C22×D7311yescharacteristic
335C23×C4—24no—
336C23×C4—41no—
337C23×C4—41no—
338C23×C4—42no—
339C23×C4—732no—
340C23×C4—732no—
341C23×C4—732no—
342C23×C4—732no—
343C23×C4C2×Dic7314yes—
344C23×C4C22×D7312yes—
345C23×C4C22×D7312yes—
346C23×C4C22×D7311yescharacteristic
347C23×C4C22×D7311yescharacteristic
348C2×C4⋊C4—41no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—41no—
351C2×C4⋊C4—2922no—
352C2×C4⋊C4—2922no—
353C2×C4⋊C4—1464no—
354C2×C4⋊C4C2×Dic7314yes—
355C2×C4⋊C4C2×Dic7314yes—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—2921no—
359C2×C22⋊C4—2921no—
360C2×C22⋊C4—2922no—
361C2×C22⋊C4—2921no—
362C2×C22⋊C4—2921no—
363C2×C22⋊C4—734no—
364C2×C22⋊C4—1464no—
365C2×C22⋊C4—1464no—
366C2×C22⋊C4—1462no—
367C2×C22⋊C4C2×Dic7318yes—
368C2×C42—2922no—
369C2×C42—1464no—
370C2×C42—1464no—
371#95—2922no—
372#95—2922no—
373#95—2922no—
374#95—2921no—
375#95—2921no—
376#95—2922no—
377#95—1464no—
378#95—1464no—
379#95—1464no—
380#95—1464no—
381#95—1464no—
382#95C2×Dic7318yes—
383C24—22no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24C22×Dic7312yes—
392C24D4×D7312yes—
393C24#1192402712yes—
394C24C23×D7311yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—734no—
420C22×C4—734no—
421C22×C4—1462no—
422C22×C4—1462no—
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×C4C22×Dic7312yes—
448C22×C4C22×Dic7314yes—
449C22×C4C22×Dic7314yes—
450C22×C4C22×Dic7312yes—
451C22×C4#1192402814yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—2928no—
455C22⋊C4—216no—
456C22⋊C4—2924no—
457C22⋊C4—2928no—
458C22⋊C4—2924no—
459C42—2928no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23D4×Dic7314yes—
483C23C23×Dic7311yescharacteristic
484C23C2×D4×D7311yescharacteristic
485C23#41×D7312yes—
486C23C2×#1192402711yescharacteristic
487C23C2×#1192402812yes—
488C23#1455663611yescharacteristic
489C23#1455663611yescharacteristic
490C23#1455663612yes—
491C23#1455663711yescharacteristic
492C23#1455663811yescharacteristic
493C23#1455663912yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—416no—
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—2922no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
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—1464no—
541C2×C4—1464no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4#1455655014yes—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—44no—
558C22C2×#1455655011yescharacteristic
559C22C2×D4×Dic7311yescharacteristic
560C22#41×Dic7311yescharacteristic
561C22#1798906211yescharacteristic
562C22#1798906211yescharacteristic
563C22#1798906511yescharacteristic
564C22#1798907011yescharacteristic
565C22#1798914511yescharacteristic
566C22#1798916411yescharacteristic
567C22#1798917411yescharacteristic
568C22#1798917512yes—
569C22#1798917512yes—
570C22#1798918211yescharacteristic
571C22#1798918312yes—
572C22#1798920712yes—
573C22#1798922612yes—
574C22#1798923812yes—
575C22#1798927212yes—
576C22#1798927612yes—
577C22#1798928012yes—
578C22#1798930012yes—
579C22#1798933312yes—
580C22#1798933812yes—
581C4—24no—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—2924no—
586C4—2924no—
587C4—5842no—
588C4—5842no—
589C2—44no—
590C2#2261252911yescharacteristic
591C2#2261263611yescharacteristic
592C2#2261268911yescharacteristic
593C2#2261280011yescharacteristic
594C2#2261281611yescharacteristic
595C2#2261298311yescharacteristic
596C2#2261309711yescharacteristic
597C2#2261633412yes—
598C2#2261646312yes—
599C2#2261743012yes—
600C2#2261759512yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#380×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#380D73
hypocenterC73#14422
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #22608188C2semidirect
3. #22608184C2semidirect
5. #22608163C2semidirect
6. #22608156C2semidirect
31. #380×C73C4semidirect
310. C73#14422semidirect
321. #380Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
#226081882
#226081842
#226081802
#226081632
#226081562
#226081542
C2×#380×C732
#1442273