Finite Group Databasebeta

← #32204617

Subgroups — #32204617

Order 18176.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596627C212yesmaximal
2#22595535C214yesmaximal, complemented
3#22595245C214yesmaximal, complemented
4#22594998C214yesmaximal, complemented
5#22594806C214yesmaximal, complemented
6#22594454C211yesmaximal, characteristic, complemented
7#22594310C211yesmaximal, characteristic, complemented
8#22592580C214yesmaximal
9#22592475C214yesmaximal, complemented
10#22591073C212yesmaximal, complemented
11C2×#380×C71C211yesmaximal, characteristic
12C2×#14556121C2214yes—
13C2×#14556118C2214yescomplemented
14C2×#14556107C2212yes—
15C2×#14556107C2212yes—
16C2×#14556107C2212yes—
17#17985941C2214yescomplemented
18#17985937C2214yescomplemented
19#17985933C2214yescomplemented
20#17985932C2214yescomplemented
21#17985931C2214yescomplemented
22#17985930C2214yescomplemented
23#17985929C2214yescomplemented
24#17985927C2214yes—
25#17985927C2214yes—
26#17985924C2214yes—
27#17985922C2214yes—
28#17985920C2214yescomplemented
29#17985903C2214yescomplemented
30#17985896C2214yescomplemented
31#17985894C2214yescomplemented
32#17985893C2214yes—
33#17985892C2214yes—
34#17985887C2214yes—
35#17985886C2214yescomplemented
36#17985885C2214yescomplemented
37#17985882C2214yescomplemented
38#17985880C2214yescomplemented
39#17985872C2214yes—
40#17985872C2214yes—
41#17985872C2214yes—
42#17985872C2214yes—
43#17985872C2214yes—
44#17985859C2212yes—
45#17985859C2212yes—
46#17985859C2212yes—
47#17985859C2214yes—
48#17985855C2214yes—
49C4×C22⋊Dic71C2214yes—
50C24×C4×C71C2211yescharacteristic
51C22×C4⋊C4×C71C2212yes—
52C22×C22⋊C4×C71C2212yes—
53#380×C71C2214yes—
54#380×C71C2214yes—
55C2×#95×C71C2211yescharacteristic
56C2×#95×C71C2211yescharacteristic
57C22×Dic142C2312yes—
58#14556121—48no—
59#14556118—48no—
60#14556107—28no—
61#14556107—44no—
62#14556107—44no—
63C2×C4⋊Dic71C2312yes—
64C2×C4⋊Dic71C2312yes—
65C2×C4⋊Dic71C2312yes—
66C2×C4⋊Dic71C2314yes—
67C2×C4⋊Dic71C2314yes—
68C2×Dic71⋊C4C2314yes—
69C2×Dic71⋊C4C2314yes—
70C2×Dic71⋊C4C2314yes—
71C2×Dic71⋊C4C2314yes—
72C2×Dic71⋊C4C2314yes—
73C2×Dic71⋊C4C2314yes—
74C2×Dic71⋊C4C2314yes—
75C2×C22⋊Dic71C2314yes—
76C2×C22⋊Dic71C2314yes—
77C2×C22⋊Dic71C2314yes—
78C2×C22⋊Dic71C2314yes—
79C2×C22⋊Dic71C2314yes—
80C2×C22⋊Dic71C2314yes—
81C2×C4×Dic71C2314yes—
82C2×C4×Dic71C2314yes—
83C71⋊#95C2314yes—
84C71⋊#95C2314yes—
85C71⋊#95C2314yes—
86C71⋊#95C2314yes—
87C71⋊#95C2314yes—
88C71⋊#95C2314yes—
89C71⋊#95C2314yes—
90C71⋊#95C2314yes—
91C71⋊#95C2314yes—
92C71⋊#95C2314yes—
93C71⋊#95C2314yes—
94C25×C71C2311yescharacteristic
95C23×C4×C71—24no—
96C23×C4×C71—44no—
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2312yes—
100C23×C4×C71C2312yes—
101C23×C4×C71C2312yes—
102C23×C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71—44no—
104C2×C4⋊C4×C71C2314yes—
105C2×C4⋊C4×C71C2314yes—
106C2×C22⋊C4×C71—44no—
107C2×C22⋊C4×C71C2314yes—
108C2×C22⋊C4×C71C2314yes—
109#95×C71C2314yes—
110#95×C71C2314yes—
111C2×Dic142—24no—
112C2×Dic142—42no—
113C2×Dic142—42no—
114C22×Dic71C2414yes—
115C22×Dic71C2414yes—
116C22×Dic71C2414yes—
117C22×Dic71C2414yes—
118C4⋊Dic71—24no—
119C4⋊Dic71—42no—
120C4⋊Dic71—42no—
121C4⋊Dic71—44no—
122C4⋊Dic71—44no—
123Dic71⋊C4—28no—
124Dic71⋊C4—28no—
125Dic71⋊C4—28no—
126Dic71⋊C4—44no—
127Dic71⋊C4—44no—
128Dic71⋊C4—44no—
129Dic71⋊C4—44no—
130C22⋊Dic71—28no—
131C22⋊Dic71—28no—
132C22⋊Dic71—28no—
133C22⋊Dic71—28no—
134C22⋊Dic71—44no—
135C22⋊Dic71—44no—
136C4×Dic71—44no—
137C4×Dic71—44no—
138C24×C71—22no—
139C24×C71—22no—
140C24×C71—22no—
141C24×C71—24no—
142C24×C71—42no—
143C24×C71C2411yescharacteristic
144C24×C71C2411yescharacteristic
145C24×C71C2411yescharacteristic
146C22×C4×C71—22no—
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71—24no—
150C22×C4×C71—24no—
151C22×C4×C71—24no—
152C22×C4×C71—24no—
153C22×C4×C71—24no—
154C22×C4×C71—42no—
155C22×C4×C71—42no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71—416no—
162C22×C4×C71#4114yes—
163C22×C4×C71C2414yes—
164C22×C4×C71C2414yes—
165C22×C4×C71C2412yes—
166C22×C4×C71C2412yes—
167C4⋊C4×C71—28no—
168C4⋊C4×C71—44no—
169C22⋊C4×C71—28no—
170C22⋊C4×C71—44no—
171C23×C71—22no—
172C23×C71—21no—
173C23×C71—21no—
174C23×C71—28no—
175C23×C71—21no—
176C23×C71—21no—
177C23×C71—21no—
178C23×C71—22no—
179C23×C71—22no—
180C23×C71—22no—
181C23×C71—22no—
182C23×C71—21no—
183C23×C71—48no—
184C23×C71—48no—
185C23×C71—44no—
186C23×C71—44no—
187C23×C71C2×#4112yes—
188C23×C71H(5,𝔽2)12yes—
189C23×C712−1+414yes—
190C23×C712−1+412yes—
191C23×C71C2511yescharacteristic
192C2×C4×C71—28no—
193C2×C4×C71—22no—
194C2×C4×C71—22no—
195C2×C4×C71—22no—
196C2×C4×C71—22no—
197C2×C4×C71—24no—
198C2×C4×C71—24no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
202C2×C4×C71—24no—
203C2×C4×C71—44no—
204C2×C4×C71—48no—
205C2×C4×C71—48no—
206C2×C4×C71—84no—
207C2×C4×C71—84no—
208C2×C4×C71C2×#4114yes—
209C2×C4×C71H(5,𝔽2)14yes—
210C2×Dic71—24no—
211C2×Dic71—24no—
212C2×Dic71—24no—
213C2×Dic71—24no—
214C2×Dic71—24no—
215C2×Dic71—24no—
216C2×Dic71—24no—
217C2×Dic71—24no—
218C2×Dic71—24no—
219C2×Dic71—24no—
220C2×Dic71—24no—
221C2×Dic71—24no—
222Dic142—84no—
223C22×C71—22no—
224C22×C71—22no—
225C22×C71—24no—
226C22×C71—24no—
227C22×C71—22no—
228C22×C71—22no—
229C22×C71—24no—
230C22×C71—24no—
231C22×C71—416no—
232C22×C71—41no—
233C22×C71—41no—
234C22×C71—41no—
235C22×C71—41no—
236C22×C71—42no—
237C22×C71—42no—
238C22×C71#54114yes—
239C22×C71C22×#4111yescharacteristic
240C22×C71C2×H(5,𝔽2)12yes—
241C22×C71C2×H(5,𝔽2)11yescharacteristic
242C22×C71C2×2−1+412yes—
243C22×C71C2×2−1+411yescharacteristic
244C4×C71—24no—
245C4×C71—24no—
246C4×C71—44no—
247C4×C71—44no—
248Dic71—44no—
249Dic71—44no—
250Dic71—44no—
251Dic71—44no—
252#36728—711nomaximal, Sylow
253C2×C71—24no—
254C2×C71—42no—
255C2×C71—42no—
256C2×C71C2×#54111yescharacteristic
257C2×C71#349011yescharacteristic
258C2×C71#349211yescharacteristic
259C2×C71#353612yes—
260C2×C71#353812yes—
261#2848—712no—
262#2747—714no—
263#2722—714no—
264#2704—714no—
265#2690—714no—
266#2659—711no—
267#2646—711no—
268#2480—714no—
269#2473—714no—
270#2329—712no—
271C2×#380D7111yescharacteristic
272C71#3672811yescharacteristic, complemented, Sylow
273C24×C4C2×D7111yescharacteristic
274C2×#126—714no—
275C2×#125—714no—
276C2×C22⋊Q8—712no—
277C2×C22⋊Q8—712no—
278C2×C22⋊Q8—712no—
279C22×C4⋊C4C2×D7112yes—
280C22×C22⋊C4C2×D7112yes—
281#400—714no—
282#400—714no—
283#398—714no—
284#398—714no—
285#398—714no—
286#398—714no—
287#398—714no—
288#397—714no—
289#397—714no—
290#397—714no—
291#396—714no—
292#395—714no—
293#391—714no—
294#389—714no—
295#389—714no—
296#388—714no—
297#388—714no—
298#388—714no—
299#387—714no—
300#387—714no—
301#387—714no—
302#387—714no—
303#385—714no—
304#385—714no—
305#385—714no—
306#385—714no—
307#385—714no—
308#380—712no—
309#380—712no—
310#380—712no—
311#380—714no—
312#380C2×D7114yes—
313#380C2×D7114yes—
314#379—714no—
315C4×C22⋊C4—714no—
316C2×#95C2×D7111yescharacteristic
317C2×#95C2×D7111yescharacteristic
318C25C22×D7111yescharacteristic
319C22×Q8—712no—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C22×D7111yescharacteristic
323C23×C4C22×D7112yes—
324C23×C4C22×D7112yes—
325C23×C4C22×D7112yes—
326C23×C4C22×D7112yes—
327C23×C4C22×D7111yescharacteristic
328#126—2848no—
329#125—2848no—
330C22⋊Q8—2844no—
331C22⋊Q8—2844no—
332C22⋊Q8—1428no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—712no—
335C2×C4⋊C4—712no—
336C2×C4⋊C4—714no—
337C2×C4⋊C4—714no—
338C2×C4⋊C4—714no—
339C2×C4⋊C4—714no—
340C2×C4⋊C4—714no—
341C2×C4⋊C4—714no—
342C2×C4⋊C4—714no—
343C2×C4⋊C4—714no—
344C2×C4⋊C4—712no—
345C2×C4⋊C4—714no—
346C2×C4⋊C4C22×D7114yes—
347C2×C4⋊C4C22×D7114yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—714no—
350C2×C22⋊C4—714no—
351C2×C22⋊C4—714no—
352C2×C22⋊C4—714no—
353C2×C22⋊C4—714no—
354C2×C22⋊C4—714no—
355C2×C22⋊C4C22×D7114yes—
356C2×C22⋊C4C22×D7114yes—
357C2×C42—714no—
358C2×C42—714no—
359#95—714no—
360#95—714no—
361#95—714no—
362#95—714no—
363#95—714no—
364#95—714no—
365#95—714no—
366#95—714no—
367#95—714no—
368#95—714no—
369#95—714no—
370#95C22×D7114yes—
371#95C22×D7114yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D7111yescharacteristic
378C24C23×D7111yescharacteristic
379C24C23×D7111yescharacteristic
380C2×Q8—2842no—
381C2×Q8—2842no—
382C2×Q8—1424no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—714no—
400C22×C4—714no—
401C22×C4—714no—
402C22×C4—714no—
403C22×C4#1176586514yes—
404C22×C4C23×D7114yes—
405C22×C4C23×D7114yes—
406C22×C4C23×D7112yes—
407C22×C4C23×D7112yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—2844no—
411C4⋊C4—2844no—
412C4⋊C4—2844no—
413C4⋊C4—2844no—
414C4⋊C4—2842no—
415C4⋊C4—2842no—
416C4⋊C4—2844no—
417C4⋊C4—2844no—
418C4⋊C4—1428no—
419C4⋊C4—1424no—
420C4⋊C4—1428no—
421C4⋊C4—1428no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—2844no—
425C22⋊C4—2844no—
426C22⋊C4—1428no—
427C22⋊C4—1428no—
428C22⋊C4—1428no—
429C22⋊C4—1428no—
430C42—2844no—
431C42—2844no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#1176586512yes—
449C23#1455617312yes—
450C23#1455617514yes—
451C23#1455617612yes—
452C23C24×D7111yescharacteristic
453Q8—5684no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1424no—
471C2×C4—1424no—
472C2×C4—1424no—
473C2×C4—1424no—
474C2×C4—1424no—
475C2×C4—1424no—
476C2×C4—1424no—
477C2×C4—1424no—
478C2×C4—1424no—
479C2×C4—1424no—
480C2×C4—1424no—
481C2×C4—1424no—
482C2×C4C2×#1176586514yes—
483C2×C4#1455617314yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22#1798661114yes—
500C22C22×#1176586511yescharacteristic
501C22C2×#1455617312yes—
502C22C2×#1455617311yescharacteristic
503C22C2×#1455617512yes—
504C22C2×#1455617611yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—2844no—
510C4—2844no—
511C4—2844no—
512C4—2844no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1798661111yescharacteristic
517C2#2260202911yescharacteristic
518C2#2260204911yescharacteristic
519C2#2260256212yes—
520C2#2260258512yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC2×#380×C71C2
prime radicalC25×C71C23
hypercenterC2×#380D71
hypocenterC71#36728
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #22595535C2semidirect
3. #22595245C2semidirect
4. #22594998C2semidirect
5. #22594806C2semidirect
6. #22594454C2semidirect
7. #22594310C2semidirect
9. #22592475C2semidirect
10. #22591073C2semidirect
13. C2×#14556118C22semidirect
17. #17985941C22semidirect
18. #17985937C22semidirect
19. #17985933C22semidirect
20. #17985932C22semidirect
21. #17985931C22semidirect
22. #17985930C22semidirect
23. #17985929C22semidirect
28. #17985920C22semidirect
29. #17985903C22semidirect
30. #17985896C22semidirect
31. #17985894C22semidirect
35. #17985886C22semidirect
36. #17985885C22semidirect
37. #17985882C22semidirect
38. #17985880C22semidirect
272. C71#36728semidirect

Maximal subgroups

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

subgroupindex
#225966272
#225955352
#225952452
#225949982
#225948062
#225944542
#225943102
#225925802
#225924752
#225910732
C2×#380×C712
#3672871