Finite Group Databasebeta

← #32978737

Subgroups — #32978737

Order 18688.

742 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic, complemented
2C2×#17988321C212yesmaximal
3#22608161C211yesmaximal, characteristic, complemented
4#22607809C212yesmaximal, complemented
5C4×#121×C73C211yesmaximal, characteristic
6C22×C4⋊Dic73—21no—
7C2×C4⋊#10596831—21no—
8C2×C22⋊#10596831C2212yes—
9C2×#14556498—21no—
10C2×#14556498C2212yes—
11#17988342—21no—
12#17988321—24no—
13C2×#121×C73C2211yescharacteristic
14C2×C4×D4×C73—21no—
15C2×C4×D4×C73C411yescharacteristic, complemented
16C22×C42×C73C2211yescharacteristic
17#390×C73C411yescharacteristic, complemented
18#389×C73C411yescharacteristic, complemented
19#388×C73—21no—
20#386×C73—21no—
21#380×C73C2211yescharacteristic
22C4×C4⋊C4×C73C411yescharacteristic, complemented
23C4×C22⋊C4×C73—21no—
24C23×Dic73—22no—
25C22×#10596831—22no—
26C2×C4⋊Dic73—24no—
27C2×C4⋊Dic73—42no—
28C2×C4⋊Dic73—42no—
29C4⋊#10596831—42no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32#14556498—22no—
33#14556498—22no—
34#14556498—42no—
35C22×D4×C73—21no—
36C22×D4×C73C2×C411yescharacteristic
37C23×C4×C73—21no—
38C23×C4×C73D411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41#121×C73—21no—
42#121×C73—21no—
43#121×C73D412yes—
44#121×C73Q812yes—
45C4×D4×C73—42no—
46C4×D4×C73—42no—
47C4×D4×C73—42no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—21no—
57C2×C42×C73—21no—
58C2×C42×C73—42no—
59C2×C42×C73C2×C411yescharacteristic
60C2×C42×C73C2×C411yescharacteristic
61C2×C42×C73D411yescharacteristic
62C2×C42×C73D411yescharacteristic
63#95×C73—21no—
64#95×C73—21no—
65#95×C73—21no—
66C22×Dic73—24no—
67C22×Dic73—42no—
68C22×Dic73—42no—
69C22×Dic73—42no—
70C22×Dic73—42no—
71C22×Dic73—42no—
72C22×Dic73—42no—
73C2×#10596831—42no—
74C2×#10596831—42no—
75C2×#10596831—42no—
76C4⋊Dic73—48no—
77C24×C73—21no—
78C24×C73C2×D411yescharacteristic
79C2×D4×C73—22no—
80C2×D4×C73—22no—
81C2×D4×C73—21no—
82C2×D4×C73—41no—
83C2×D4×C73—41no—
84C2×D4×C73—41no—
85C2×D4×C73—41no—
86C2×D4×C73—41no—
87C2×D4×C73—41no—
88C2×D4×C73C4⋊C412yescomplemented
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—22no—
92C22×C4×C73—22no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73—42no—
101C22×C4×C73—42no—
102C22×C4×C73—42no—
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C22⋊C411yescharacteristic
105C22×C4×C73C22⋊C411yescharacteristic
106C22×C4×C73C22⋊C411yescharacteristic
107C22×C4×C73D812yes—
108C22×C4×C73SD1612yes—
109C22×C4×C73C22×C411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×Q811yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C22×C4×C73#4111yescharacteristic
115C4⋊C4×C73—21no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—42no—
119C4⋊C4×C73C4⋊C412yescomplemented
120C22⋊C4×C73—22no—
121C22⋊C4×C73—22no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C22⋊C4×C73—42no—
125C22⋊C4×C73—42no—
126C22⋊C4×C73—42no—
127C42×C73—22no—
128C42×C73—22no—
129C42×C73—41no—
130C42×C73—41no—
131C42×C73—42no—
132C23×C73—21no—
133C23×C73—21no—
134C23×C73—21no—
135C23×C73—21no—
136C23×C73—22no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73—41no—
140C23×C73—41no—
141C23×C73—41no—
142C23×C73—41no—
143C23×C73C2×C22⋊C411yescharacteristic
144C23×C73#12111yescharacteristic
145C23×C73C22⋊Q811yescharacteristic
146C23×C73C2×D811yescharacteristic
147C23×C73C2×SD1611yescharacteristic
148C23×C73AGL(1,ℤ/8)11yescharacteristic
149C23×C73#13711yescharacteristic
150D4×C73—42no—
151D4×C73—42no—
152D4×C73—42no—
153D4×C73—42no—
154D4×C73—42no—
155D4×C73—42no—
156C2×C4×C73—24no—
157C2×C4×C73—24no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—21no—
169C2×C4×C73—21no—
170C2×C4×C73—22no—
171C2×C4×C73—42no—
172C2×C4×C73—42no—
173C2×C4×C73—41no—
174C2×C4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—84no—
186C2×C4×C73#9912yes—
187C2×C4×C73#10212yes—
188C2×C4×C73#10212yes—
189C2×C4×C73#10412yes—
190C2×C4×C73C2×C4⋊C411yescharacteristic
191C2×C4×C73#11711yescharacteristic
192C2×C4×C73#12111yescharacteristic
193C2×C4×C73C22⋊Q811yescharacteristic
194C2×Dic73—44no—
195C2×Dic73—44no—
196C2×Dic73—44no—
197C2×Dic73—82no—
198C2×Dic73—82no—
199C2×Dic73—82no—
200C2×Dic73—82no—
201#10596831—82no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73—42no—
215C22×C73—42no—
216C22×C73—42no—
217C22×C73—81no—
218C22×C73—81no—
219C22×C73—81no—
220C22×C73—81no—
221C22×C73#38011yescharacteristic
222C22×C73C2×#9911yescharacteristic
223C22×C73#41011yescharacteristic
224C22×C73C2×#10211yescharacteristic
225C22×C73#41711yescharacteristic
226C22×C73C2×#10411yescharacteristic
227C22×C73#42111yescharacteristic
228C22×C73#45911yescharacteristic
229C22×C73#46111yescharacteristic
230C22×C73#47411yescharacteristic
231C22×C73#47611yescharacteristic
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—22no—
237C4×C73—22no—
238C4×C73—22no—
239C4×C73—22no—
240C4×C73—42no—
241C4×C73—81no—
242C4×C73—81no—
243Dic73—84no—
244#8577—731nomaximal, Sylow
245C2×C73—21no—
246C2×C73—21no—
247C2×C73—21no—
248C2×C73—21no—
249C2×C73—82no—
250C2×C73#150211yescharacteristic
251C2×C73#150411yescharacteristic
252C2×C73#151211yescharacteristic
253C2×C73#151911yescharacteristic
254C2×C73#179911yescharacteristic
255C2×C73#180211yescharacteristic
256C2×C73#180611yescharacteristic
257C4×#121D7311yescharacteristic
258#1797—731no—
259C2×#327—732no—
260C4⋊#95—731no—
261#1294—732no—
262C73#857711yescharacteristic, complemented, Sylow
263C2×#121C2×D7311yescharacteristic
264C2×C4×D4—21no—
265C2×C4×D4Dic7311yescharacteristic, complemented
266C22×C4⋊C4—1461no—
267C22×C42C2×D7311yescharacteristic
268C2×C4⋊C8—1461no—
269C2×C22⋊C8—732no—
270#390Dic7311yescharacteristic, complemented
271#389Dic7311yescharacteristic, complemented
272#388—21no—
273#386—21no—
274#380C2×D7311yescharacteristic
275C4×C4⋊C4Dic7311yescharacteristic, complemented
276C4×C22⋊C4—21no—
277C2×#95—732no—
278C2×#95—1461no—
279#336—1461no—
280#327—1464no—
281C22×D4—21no—
282C22×D4C2×Dic7311yescharacteristic
283C23×C4—21no—
284C23×C4—1462no—
285C23×C4C22⋊D7311yescharacteristic
286C23×C4C22⋊D7311yescharacteristic
287C23×C4C22×D7311yescharacteristic
288C22×C8—1462no—
289#121—21no—
290#121—21no—
291#121Dic14612yes—
292#121D29212yes—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—42no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—2922no—
298C2×C4⋊C4—2922no—
299C2×C4⋊C4—1464no—
300C2×C4⋊C4C2×Dic7311yescharacteristic
301C2×C4⋊C4C2×Dic7311yescharacteristic
302C2×C4⋊C4C2×Dic7311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42C2×Dic7311yescharacteristic
311C2×C42C2×Dic7311yescharacteristic
312C2×C42C22⋊D7311yescharacteristic
313C2×C42C22⋊D7311yescharacteristic
314C4⋊C8—2922no—
315C22⋊C8—1462no—
316C22⋊C8—1462no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—2922no—
321#95—1462no—
322#95—1462no—
323C24—21no—
324C24C2×C22⋊D7311yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C4⋊Dic7312yescomplemented
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—2922no—
350C22×C4—2922no—
351C22×C4—2922no—
352C22×C4—2922no—
353C22×C4—2922no—
354C22×C4—2922no—
355C22×C4—1464no—
356C22×C4C22⋊Dic7311yescharacteristic
357C22×C4C22⋊Dic7311yescharacteristic
358C22×C4C22⋊Dic7311yescharacteristic
359C22×C4C22⋊Dic7311yescharacteristic
360C22×C4#1192401312yes—
361C22×C4#1192401612yes—
362C22×C4C22×Dic7311yescharacteristic
363C22×C4C2×C22⋊D7311yescharacteristic
364C22×C4C2×D29211yescharacteristic
365C22×C4C2×Dic14611yescharacteristic
366C22×C4#1192402811yescharacteristic
367C22×C4#1192402811yescharacteristic
368C2×C8—2922no—
369C2×C8—2922no—
370C2×C8—2922no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—2928no—
376C4⋊C4C4⋊Dic7312yescomplemented
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—42no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C2×C22⋊Dic7311yescharacteristic
401C23#1455656511yescharacteristic
402C23#1455657011yescharacteristic
403C23C2×#1192401311yescharacteristic
404C23C2×#1192401611yescharacteristic
405C23#1455661611yescharacteristic
406C23#1455662111yescharacteristic
407D4—42no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—2924no—
444C2×C4—2924no—
445C2×C4—2924no—
446C2×C4—5842no—
447C2×C4—5842no—
448C2×C4—5842no—
449C2×C4—5842no—
450C2×C4#1455650612yes—
451C2×C4#1455651212yes—
452C2×C4#1455651212yes—
453C2×C4#1455651812yes—
454C2×C4C2×C4⋊Dic7311yescharacteristic
455C2×C4#1455655011yescharacteristic
456C2×C4#1455656511yescharacteristic
457C2×C4#1455657011yescharacteristic
458C8—5842no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#1798844711yescharacteristic
479C22C2×#1455650611yescharacteristic
480C22#1798856111yescharacteristic
481C22C2×#1455651211yescharacteristic
482C22#1798858811yescharacteristic
483C22C2×#1455651811yescharacteristic
484C22#1798860411yescharacteristic
485C22#1798878211yescharacteristic
486C22#1798879211yescharacteristic
487C22#1798886711yescharacteristic
488C22#1798887711yescharacteristic
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—5844no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—82no—
506C2#2260842611yescharacteristic
507C2#2260843511yescharacteristic
508C2#2260846911yescharacteristic
509C2#2260849911yescharacteristic
510C2#2260991911yescharacteristic
511C2#2260993111yescharacteristic
512C2#2260995111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC4×#121×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#121×C73C22
cosocleC23×C4×C73C23
hypercenterC4×#121D73
hypocenterC73#8577
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #22609911C2semidirect
3. #22608161C2semidirect
4. #22607809C2semidirect
15. C2×C4×D4×C73C4semidirect
17. #390×C73C4semidirect
18. #389×C73C4semidirect
22. C4×C4⋊C4×C73C4semidirect
88. C2×D4×C73C4⋊C4semidirect
119. C4⋊C4×C73C4⋊C4semidirect
262. C73#8577semidirect
265. C2×C4×D4Dic73semidirect
270. #390Dic73semidirect
271. #389Dic73semidirect
275. C4×C4⋊C4Dic73semidirect
334. C2×D4C4⋊Dic73semidirect
376. C4⋊C4C4⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099112
C2×#179883212
#226081612
#226078092
C4×#121×C732
#857773