Finite Group Databasebeta

← #137×ΓL(1,𝔽16) #14416845

Subgroups — #137×ΓL(1,𝔽16) #14416845

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×ΓL(1,𝔽16)C211yesmaximal, characteristic
2C2×Q8×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×D5×#137C211yesmaximal, characteristic
4#11752452C211yesmaximal, characteristic
5#11752441C211yesmaximal, characteristic, complemented
6#11752398C212yesmaximal, complemented
7Q16×ΓL(1,𝔽16)C212yesmaximal, complemented
8#11752394C212yesmaximal, complemented
9SD16×ΓL(1,𝔽16)C212yesmaximal, complemented
10#11752386C211yesmaximal, characteristic, complemented
11M16×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
12AGL(1,𝔽5)×#137—31nomaximal, Hall
13C3×D5×#41C2211yescharacteristic
14C2×C3×Q8×D5C2211yescharacteristic
15Q8×ΓL(1,𝔽16)—21no—
16Q8×ΓL(1,𝔽16)—21no—
17Q8×ΓL(1,𝔽16)C2212yes—
18Q8×ΓL(1,𝔽16)C2212yes—
19Q8×ΓL(1,𝔽16)C2211yescharacteristic
20Q8×ΓL(1,𝔽16)C2211yescharacteristic
21D4×ΓL(1,𝔽16)—21no—
22D4×ΓL(1,𝔽16)—21no—
23D4×ΓL(1,𝔽16)C2211yescharacteristic
24D4×ΓL(1,𝔽16)C2211yescharacteristic
25#91989—21no—
26#91989C2211yescharacteristic
27C2×C4⋊ΓL(1,𝔽16)—21no—
28C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
29C2×C4×ΓL(1,𝔽16)—21no—
30C2×C4×ΓL(1,𝔽16)—21no—
31C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
32D5⋊#208C2211yescharacteristic
33C3×C5×#137C411yescharacteristic, complemented
34C3×#4097C411yescharacteristic, complemented
35C3×#4088C411yescharacteristic, complemented
36C3×#4064C412yescomplemented
37C3×D5×Q16C2212yes—
38C3×#4061C412yescomplemented
39C3×D5×SD16C2212yes—
40C3×#4055C411yescharacteristic, complemented
41C3×D5×M16C2211yescharacteristic
42#91238C2212yes—
43#91238C2212yes—
44#91238C2212yes—
45#91236C2212yes—
46#91223C2212yes—
47#91222C2212yes—
48#91221C2212yes—
49C8×ΓL(1,𝔽16)C2212yes—
50Dic3×#137—51nomaximal, Hall
51#41×AGL(1,𝔽5)—31no—
52C2×Q8×AGL(1,𝔽5)—31no—
53D5×#137D311yescharacteristic
54#65793—31no—
55#65782—31no—
56#65739—32no—
57Q16×AGL(1,𝔽5)—32no—
58#65735—32no—
59SD16×AGL(1,𝔽5)—32no—
60#65727—31no—
61M16×AGL(1,𝔽5)—31no—
62C22×ΓL(1,𝔽16)—21no—
63C22×ΓL(1,𝔽16)—21no—
64C22×ΓL(1,𝔽16)—21no—
65C22×ΓL(1,𝔽16)D411yescharacteristic
66C22×ΓL(1,𝔽16)D411yescharacteristic
67C3×C5×#41C2×C411yescharacteristic
68C2×C3×C5×Q8C2×C411yescharacteristic, complemented
69C3×#713C2×C411yescharacteristic
70C3×Q8×D5—21no—
71C3×Q8×D5—21no—
72C3×Q8×D5C2×C412yescomplemented
73C3×Q8×D5C2312yes—
74C3×Q8×D5C2311yescharacteristic
75C3×#711—21no—
76C3×#711C2×C411yescharacteristic
77C3×D4×D5—21no—
78C3×D4×D5C2311yescharacteristic
79C3×#709—21no—
80C3×#709C2×C411yescharacteristic
81C2×C3×C4×D5—21no—
82C2×C3×C4×D5—21no—
83C2×C3×C4×D5C2311yescharacteristic
84C2×C3×Dic10—21no—
85C2×C3×Dic10C2×C411yescharacteristic, complemented
86#6763—21no—
87#6763—21no—
88#6763—21no—
89#6762—21no—
90C4⋊ΓL(1,𝔽16)—22no—
91C4⋊ΓL(1,𝔽16)—22no—
92C4⋊ΓL(1,𝔽16)—21no—
93C4⋊ΓL(1,𝔽16)—21no—
94C4⋊ΓL(1,𝔽16)—21no—
95C4⋊ΓL(1,𝔽16)C2312yes—
96C4⋊ΓL(1,𝔽16)C2311yescharacteristic
97C4⋊ΓL(1,𝔽16)C2311yescharacteristic
98C4×ΓL(1,𝔽16)—21no—
99C4×ΓL(1,𝔽16)—21no—
100C4×ΓL(1,𝔽16)—22no—
101C4×ΓL(1,𝔽16)—22no—
102C4×ΓL(1,𝔽16)—21no—
103C4×ΓL(1,𝔽16)D411yescharacteristic
104C4×ΓL(1,𝔽16)D411yescharacteristic
105C4×ΓL(1,𝔽16)C2311yescharacteristic
106C4×ΓL(1,𝔽16)C2311yescharacteristic
107#6758—21no—
108D5⋊(C3⋊C8)C2312yes—
109C3×C5×Q16C2×C412yescomplemented
110C3×C5×SD16C2×C412yescomplemented
111C3×C5×M16C2×C411yescharacteristic, complemented
112C3×#689C2×C412yescomplemented
113C3×#689C2×C412yescomplemented
114C3×#688C2×C412yescomplemented
115C3×#687C2×C412yescomplemented
116C3×#681C2×C411yescharacteristic, complemented
117C3×Dic20C2×C412yescomplemented
118C3×#677C2×C412yescomplemented
119C3×#676C2×C412yescomplemented
120C3×C8×D5C2312yes—
121C2×C3×#137—51no—
122Dic3×#41—51no—
123C2×Q8×Dic3—51no—
124#5232—51no—
125#5225—51no—
126#5184—52no—
127Dic3×Q16—52no—
128#5164—52no—
129Dic3×SD16—52no—
130#5121—51no—
131Dic3×M16—51no—
132D5×#41C2×D311yescharacteristic
133C2×Q8×D5C2×D311yescharacteristic
134Q8×AGL(1,𝔽5)—32no—
135Q8×AGL(1,𝔽5)—32no—
136Q8×AGL(1,𝔽5)—31no—
137Q8×AGL(1,𝔽5)—31no—
138Q8×AGL(1,𝔽5)—61no—
139Q8×AGL(1,𝔽5)—61no—
140D4×AGL(1,𝔽5)—31no—
141D4×AGL(1,𝔽5)—31no—
142D4×AGL(1,𝔽5)—61no—
143D4×AGL(1,𝔽5)—61no—
144#4130—31no—
145#4130—61no—
146C2×C4⋊AGL(1,𝔽5)—31no—
147C2×C4⋊AGL(1,𝔽5)—61no—
148C2×C4×AGL(1,𝔽5)—31no—
149C2×C4×AGL(1,𝔽5)—61no—
150C2×C4×AGL(1,𝔽5)—61no—
151D5⋊M16—31no—
152C5×#137Dic311yescharacteristic, complemented
153#4097Dic311yescharacteristic, complemented
154#4088Dic311yescharacteristic, complemented
155#4064Dic312yescomplemented
156D5×Q16C2×D312yes—
157#4061Dic312yescomplemented
158D5×SD16C2×D312yes—
159#4055Dic311yescharacteristic, complemented
160D5×M16C2×D311yescharacteristic
161#4009—32no—
162#4009—32no—
163#4009—32no—
164#4007—32no—
165#3994—32no—
166#3993—32no—
167#3992—32no—
168C8×AGL(1,𝔽5)—32no—
169C4×#137—151noSylow
170C22×C3×D5—21no—
171C22×C3×D5C2×D411yescharacteristic
172C2×ΓL(1,𝔽16)—22no—
173C2×ΓL(1,𝔽16)—22no—
174C2×ΓL(1,𝔽16)—21no—
175C2×ΓL(1,𝔽16)—21no—
176C2×ΓL(1,𝔽16)—41no—
177C2×ΓL(1,𝔽16)—42no—
178C2×ΓL(1,𝔽16)—42no—
179C2×ΓL(1,𝔽16)—42no—
180C2×ΓL(1,𝔽16)C2×D411yescharacteristic
181C2×ΓL(1,𝔽16)C2×D411yescharacteristic
182C2×ΓL(1,𝔽16)C2×D411yescharacteristic
183C2×ΓL(1,𝔽16)C2×D411yescharacteristic
184C3×C5×Q8—21no—
185C3×C5×Q8C22×C412yes—
186C3×C5×Q8C22×C411yescharacteristic
187C3×C5×D4—21no—
188C3×C5×D4C22×C411yescharacteristic
189C2×C3×C4×C5—21no—
190C2×C3×C4×C5—21no—
191C2×C3×C4×C5C22×C411yescharacteristic
192C3×C22⋊D5—21no—
193C3×C22⋊D5—21no—
194C3×C22⋊D5—21no—
195C2×C3×Dic5—21no—
196C2×C3×Dic5—21no—
197C2×C3×Dic5#4111yescharacteristic
198C3×D20—21no—
199C3×D20C22×C411yescharacteristic
200C3×C4×D5—22no—
201C3×C4×D5—21no—
202C3×C4×D5—22no—
203C3×C4×D5—21no—
204C3×C4×D5—21no—
205C3×C4×D5C22×C411yescharacteristic
206C3×C4×D5C2×D411yescharacteristic
207C3×C4×D5#4111yescharacteristic
208C3×C4×D5C2411yescharacteristic
209C3×Dic10—21no—
210C3×Dic10—22no—
211C3×Dic10—22no—
212C3×Dic10—21no—
213C3×Dic10C22×C412yes—
214C3×Dic10C22×C411yescharacteristic
215C3⋊#170—22no—
216C3×C5×C8C22×C412yes—
217C3×#168C22×C412yes—
218C2×C3×#41—51no—
219C22×C3×Q8—51no—
220C3×#137—51no—
221C3×#137—51no—
222C3×#137—52no—
223C3×#137—52no—
224C3×#137—51no—
225C3×#137AGL(1,𝔽5)11yescharacteristic, complemented
226C2×C3×Q16—52no—
227C2×C3×SD16—52no—
228C2×C3×M16—51no—
229Q8×Dic3—52no—
230Q8×Dic3—52no—
231Q8×Dic3—51no—
232Q8×Dic3—51no—
233Q8×Dic3—101no—
234Q8×Dic3—101no—
235D4×Dic3—51no—
236D4×Dic3—51no—
237D4×Dic3—101no—
238D4×Dic3—101no—
239#926—51no—
240#926—101no—
241C2×C4⋊Dic3—51no—
242C2×C4⋊Dic3—101no—
243C2×C4×Dic3—51no—
244C2×C4×Dic3—101no—
245C2×C4×Dic3—101no—
246C2×#208—51no—
247#835—52no—
248#835—52no—
249#835—52no—
250#832—52no—
251#818—52no—
252#817—52no—
253#815—52no—
254C8×Dic3—52no—
255C22×AGL(1,𝔽5)—31no—
256C22×AGL(1,𝔽5)—31no—
257C22×AGL(1,𝔽5)—61no—
258C22×AGL(1,𝔽5)—61no—
259C22×AGL(1,𝔽5)—61no—
260C5×#41C2×Dic311yescharacteristic
261C2×C5×Q8C2×Dic311yescharacteristic, complemented
262#713C2×Dic311yescharacteristic
263Q8×D5—21no—
264Q8×D5—21no—
265Q8×D5C2×Dic312yescomplemented
266Q8×D5C22×D312yes—
267Q8×D5C22×D311yescharacteristic
268#711—21no—
269#711C2×Dic311yescharacteristic
270D4×D5—21no—
271D4×D5C22×D311yescharacteristic
272#709—21no—
273#709C2×Dic311yescharacteristic
274C2×C4×D5—21no—
275C2×C4×D5—21no—
276C2×C4×D5C22×D311yescharacteristic
277C2×Dic10—21no—
278C2×Dic10C2×Dic311yescharacteristic, complemented
279C22⋊AGL(1,𝔽5)—61no—
280C22⋊AGL(1,𝔽5)—61no—
281C22⋊AGL(1,𝔽5)—61no—
282#704—61no—
283C4⋊AGL(1,𝔽5)—32no—
284C4⋊AGL(1,𝔽5)—31no—
285C4⋊AGL(1,𝔽5)—31no—
286C4⋊AGL(1,𝔽5)—62no—
287C4⋊AGL(1,𝔽5)—62no—
288C4⋊AGL(1,𝔽5)—61no—
289C4⋊AGL(1,𝔽5)—61no—
290C4⋊AGL(1,𝔽5)—61no—
291C4×AGL(1,𝔽5)—31no—
292C4×AGL(1,𝔽5)—31no—
293C4×AGL(1,𝔽5)—31no—
294C4×AGL(1,𝔽5)—31no—
295C4×AGL(1,𝔽5)—61no—
296C4×AGL(1,𝔽5)—61no—
297C4×AGL(1,𝔽5)—62no—
298C4×AGL(1,𝔽5)—62no—
299C4×AGL(1,𝔽5)—61no—
300#700—61no—
301D5⋊C8—32no—
302C5×Q16C2×Dic312yescomplemented
303C5×SD16C2×Dic312yescomplemented
304C5×M16C2×Dic311yescharacteristic, complemented
305#689C2×Dic312yescomplemented
306#689C2×Dic312yescomplemented
307#688C2×Dic312yescomplemented
308#687C2×Dic312yescomplemented
309#681C2×Dic311yescharacteristic, complemented
310Dic20C2×Dic312yescomplemented
311#677C2×Dic312yescomplemented
312#676C2×Dic312yescomplemented
313C8×D5C22×D312yes—
314C2×#137—51no—
315C4×#41—151no—
316C2×C4×Q8—151no—
317#441—152no—
318#440—152no—
319C4×Q16—152no—
320C4×SD16—152no—
321#428—151no—
322#419—151no—
323#417—151no—
324C4×M16—151no—
325C22×C3×C5—21no—
326C22×C3×C5C4×D411yescharacteristic
327C2×C3×D5—21no—
328C2×C3×D5—21no—
329C2×C3×D5—21no—
330C2×C3×D5—41no—
331C2×C3×D5—41no—
332C2×C3×D5C4×D411yescharacteristic
333C2×C3×D5C22×D411yescharacteristic
334ΓL(1,𝔽16)—21no—
335ΓL(1,𝔽16)—21no—
336ΓL(1,𝔽16)—21no—
337ΓL(1,𝔽16)—41no—
338ΓL(1,𝔽16)—42no—
339ΓL(1,𝔽16)—41no—
340ΓL(1,𝔽16)#13712yesdirect product factor
341C3×C4×C5—22no—
342C3×C4×C5—21no—
343C3×C4×C5C4×D411yescharacteristic
344C3×C4×C5C23×C411yescharacteristic
345C3×Dic5—22no—
346C3×Dic5—21no—
347C3×Dic5C4×D411yescharacteristic
348C3×Dic5C2×#4111yescharacteristic
349C3×#41—51no—
350C3×#41—51no—
351C3×#41—51no—
352C3×#41—101no—
353C3×#41—101no—
354C3×#41C2×AGL(1,𝔽5)11yescharacteristic
355C2×C3×Q8—52no—
356C2×C3×Q8—51no—
357C2×C3×Q8—52no—
358C2×C3×Q8—51no—
359C2×C3×Q8—101no—
360C2×C3×Q8—101no—
361C2×C3×Q8—101no—
362C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
363C2×C3×D4—51no—
364C2×C3×D4—101no—
365C22×C3×C4—51no—
366C22×C3×C4—101no—
367C22×C3×C4—101no—
368C22×Dic3—51no—
369C22×Dic3—51no—
370C22×Dic3—101no—
371C22×Dic3—101no—
372C22×Dic3—101no—
373C3×Q16—52no—
374C3×Q16—52no—
375C3×Q16—52no—
376C3×Q16C2×AGL(1,𝔽5)12yescomplemented
377C3×SD16—52no—
378C3×SD16—52no—
379C3×SD16—52no—
380C3×SD16C2×AGL(1,𝔽5)12yescomplemented
381C3×M16—51no—
382C3×M16—52no—
383C3×M16C2×AGL(1,𝔽5)11yescharacteristic, complemented
384C2×C3×C8—52no—
385#217—101no—
386#217—101no—
387#217—101no—
388C4⋊Dic3—51no—
389C4⋊Dic3—52no—
390C4⋊Dic3—51no—
391C4⋊Dic3—102no—
392C4⋊Dic3—102no—
393C4⋊Dic3—101no—
394C4⋊Dic3—101no—
395C4⋊Dic3—101no—
396C4×Dic3—51no—
397C4×Dic3—51no—
398C4×Dic3—51no—
399C4×Dic3—51no—
400C4×Dic3—101no—
401C4×Dic3—101no—
402C4×Dic3—102no—
403C4×Dic3—102no—
404C4×Dic3—101no—
405#208—101no—
406#208—101no—
407C2×C3⋊C8—52no—
408C22×D5—21no—
409C22×D5D3×D411yescharacteristic
410C2×AGL(1,𝔽5)—31no—
411C2×AGL(1,𝔽5)—31no—
412C2×AGL(1,𝔽5)—31no—
413C2×AGL(1,𝔽5)—31no—
414C2×AGL(1,𝔽5)—62no—
415C2×AGL(1,𝔽5)—62no—
416C2×AGL(1,𝔽5)—61no—
417C2×AGL(1,𝔽5)—61no—
418C2×AGL(1,𝔽5)—122no—
419C2×AGL(1,𝔽5)—121no—
420C2×AGL(1,𝔽5)—122no—
421C2×AGL(1,𝔽5)—122no—
422C5×Q8—21no—
423C5×Q8C22×Dic312yes—
424C5×Q8C22×Dic311yescharacteristic
425C5×D4—21no—
426C5×D4C22×Dic311yescharacteristic
427C2×C4×C5—21no—
428C2×C4×C5—21no—
429C2×C4×C5C22×Dic311yescharacteristic
430C22⋊D5—21no—
431C22⋊D5—21no—
432C22⋊D5—21no—
433C2×Dic5—21no—
434C2×Dic5—21no—
435C2×Dic5#23711yescharacteristic
436D20—21no—
437D20C22×Dic311yescharacteristic
438C4×D5—22no—
439C4×D5—21no—
440C4×D5—22no—
441C4×D5—21no—
442C4×D5—21no—
443C4×D5D3×D411yescharacteristic
444C4×D5#23711yescharacteristic
445C4×D5C22×Dic311yescharacteristic
446C4×D5C23×D311yescharacteristic
447Dic10—21no—
448Dic10—22no—
449Dic10—22no—
450Dic10—21no—
451Dic10C22×Dic312yes—
452Dic10C22×Dic311yescharacteristic
453#170—62no—
454C5×C8C22×Dic312yes—
455#168C22×Dic312yes—
456C2×#41—51no—
457C22×Q8—51no—
458#137—51no—
459#137—51no—
460#137—52no—
461#137—52no—
462#137—51no—
463#137ΓL(1,𝔽16)11yescharacteristic, direct product factor
464C2×Q16—52no—
465C2×SD16—52no—
466C2×M16—51no—
467C2×M16—151no—
468C4×Q8—152no—
469C4×Q8—152no—
470C4×Q8—151no—
471C4×Q8—151no—
472C4×Q8—301no—
473C4×Q8—301no—
474C4×D4—151no—
475C4×D4—151no—
476C4×D4—301no—
477C4×D4—301no—
478#117—151no—
479#117—301no—
480C2×C4⋊C4—151no—
481C2×C4⋊C4—301no—
482C2×C42—151no—
483C2×C42—301no—
484C2×C42—301no—
485#107—152no—
486#106—152no—
487#103—152no—
488#103—152no—
489#103—152no—
490#102—152no—
491#97—152no—
492C4×C8—152no—
493C2×C3×C5—21no—
494C2×C3×C5—41no—
495C2×C3×C5C2×C4×D411yescharacteristic
496C3×D5—21no—
497C3×D5—41no—
498C3×D5C2×#13711yescharacteristic
499C3×D5#57511yescharacteristic
500C23×C3—51no—
501C23×C3—101no—
502C3×Q8—21no—
503C3×Q8—52no—
504C3×Q8—51no—
505C3×Q8—102no—
506C3×Q8—102no—
507C3×Q8—101no—
508C3×Q8—101no—
509C3×Q8C22×AGL(1,𝔽5)12yes—
510C3×Q8C22×AGL(1,𝔽5)11yescharacteristic
511C3×D4—21no—
512C3×D4—51no—
513C3×D4—101no—
514C3×D4—101no—
515C3×D4—101no—
516C3×D4—101no—
517C3×D4C22×AGL(1,𝔽5)11yescharacteristic
518C2×C3×C4—21no—
519C2×C3×C4—21no—
520C2×C3×C4—51no—
521C2×C3×C4—51no—
522C2×C3×C4—51no—
523C2×C3×C4—51no—
524C2×C3×C4—51no—
525C2×C3×C4—101no—
526C2×C3×C4—101no—
527C2×C3×C4—102no—
528C2×C3×C4—102no—
529C2×C3×C4—101no—
530C2×C3×C4—101no—
531C2×C3×C4—101no—
532C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
533C2×Dic3—51no—
534C2×Dic3—51no—
535C2×Dic3—51no—
536C2×Dic3—51no—
537C2×Dic3—102no—
538C2×Dic3—102no—
539C2×Dic3—101no—
540C2×Dic3—101no—
541C2×Dic3—202no—
542C2×Dic3—201no—
543C2×Dic3—202no—
544C2×Dic3—202no—
545C3×C8—52no—
546C3×C8C22×AGL(1,𝔽5)12yes—
547C3⋊C8—102no—
548C22×C5—21no—
549C22×C5D4×Dic311yescharacteristic
550C2×D5—21no—
551C2×D5—21no—
552C2×D5—21no—
553C2×D5—41no—
554C2×D5—41no—
555C2×D5D4×Dic311yescharacteristic
556C2×D5C2×D3×D411yescharacteristic
557AGL(1,𝔽5)—32no—
558AGL(1,𝔽5)—61no—
559AGL(1,𝔽5)—61no—
560AGL(1,𝔽5)—61no—
561AGL(1,𝔽5)—121no—
562AGL(1,𝔽5)—122no—
563AGL(1,𝔽5)—121no—
564C4×C5—22no—
565C4×C5—21no—
566C4×C5D4×Dic311yescharacteristic
567C4×C5C23×Dic311yescharacteristic
568Dic5—22no—
569Dic5—21no—
570Dic5D4×Dic311yescharacteristic
571Dic5C2×#23711yescharacteristic
572#41—51no—
573#41—51no—
574#41—51no—
575#41—101no—
576#41—101no—
577#41C2×ΓL(1,𝔽16)11yescharacteristic
578C2×Q8—52no—
579C2×Q8—51no—
580C2×Q8—52no—
581C2×Q8—51no—
582C2×Q8—101no—
583C2×Q8—101no—
584C2×Q8—101no—
585C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic, complemented
586C2×D4—51no—
587C2×D4—101no—
588C22×C4—51no—
589C22×C4—101no—
590C22×C4—101no—
591C22×C4—151no—
592C22×C4—151no—
593C22×C4—301no—
594C22×C4—301no—
595C22×C4—301no—
596Q16—52no—
597Q16—52no—
598Q16—52no—
599Q16C2×ΓL(1,𝔽16)12yescomplemented
600SD16—52no—
601SD16—52no—
602SD16—52no—
603SD16C2×ΓL(1,𝔽16)12yescomplemented
604M16—51no—
605M16—52no—
606M16—301no—
607M16—301no—
608M16C2×ΓL(1,𝔽16)11yescharacteristic, complemented
609C2×C8—52no—
610C2×C8—152no—
611C4⋊C4—152no—
612C4⋊C4—151no—
613C4⋊C4—151no—
614C4⋊C4—301no—
615C4⋊C4—302no—
616C4⋊C4—302no—
617C4⋊C4—301no—
618C4⋊C4—301no—
619C22⋊C4—301no—
620C22⋊C4—301no—
621C22⋊C4—301no—
622C42—151no—
623C42—151no—
624C42—151no—
625C42—151no—
626C42—301no—
627C42—301no—
628C42—301no—
629C42—302no—
630C42—302no—
631C3×C5C4×#13711yescharacteristic, complemented, Hall
632C22×C3—21no—
633C22×C3—51no—
634C22×C3—51no—
635C22×C3—101no—
636C22×C3—101no—
637C22×C3—101no—
638C22×C3—201no—
639C22×C3—201no—
640C22×C3D4×AGL(1,𝔽5)11yescharacteristic
641C3×C4—21no—
642C3×C4—22no—
643C3×C4—51no—
644C3×C4—51no—
645C3×C4—101no—
646C3×C4—102no—
647C3×C4D4×AGL(1,𝔽5)11yescharacteristic
648C3×C4C23×AGL(1,𝔽5)11yescharacteristic
649Dic3—52no—
650Dic3—101no—
651Dic3—101no—
652Dic3—101no—
653Dic3—201no—
654Dic3—202no—
655Dic3—201no—
656C2×C5—21no—
657C2×C5—41no—
658C2×C5C2×D4×Dic311yescharacteristic
659D5—21no—
660D5—41no—
661D5D3×#13711yescharacteristic
662D5#577711yescharacteristic
663C23—51no—
664C23—101no—
665Q8—21no—
666Q8—52no—
667Q8—51no—
668Q8—102no—
669Q8—102no—
670Q8—101no—
671Q8—101no—
672Q8C22×ΓL(1,𝔽16)12yes—
673Q8C22×ΓL(1,𝔽16)11yescharacteristic
674D4—21no—
675D4—51no—
676D4—101no—
677D4—101no—
678D4—101no—
679D4—101no—
680D4C22×ΓL(1,𝔽16)11yescharacteristic
681C2×C4—21no—
682C2×C4—21no—
683C2×C4—51no—
684C2×C4—51no—
685C2×C4—51no—
686C2×C4—51no—
687C2×C4—51no—
688C2×C4—101no—
689C2×C4—101no—
690C2×C4—102no—
691C2×C4—102no—
692C2×C4—101no—
693C2×C4—101no—
694C2×C4—101no—
695C2×C4—151no—
696C2×C4—151no—
697C2×C4—151no—
698C2×C4—151no—
699C2×C4—302no—
700C2×C4—301no—
701C2×C4—301no—
702C2×C4—302no—
703C2×C4—602no—
704C2×C4—601no—
705C2×C4—602no—
706C2×C4—602no—
707C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
708C8—52no—
709C8—302no—
710C8C22×ΓL(1,𝔽16)12yes—
711C2×C3—21no—
712C2×C3—41no—
713C2×C3—51no—
714C2×C3—51no—
715C2×C3—101no—
716C2×C3—201no—
717C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
718C5Dic3×#13711yescharacteristic, complemented, Sylow
719C22—21no—
720C22—51no—
721C22—51no—
722C22—101no—
723C22—101no—
724C22—101no—
725C22—201no—
726C22—201no—
727C22D4×ΓL(1,𝔽16)11yescharacteristic
728C4—21no—
729C4—22no—
730C4—51no—
731C4—51no—
732C4—101no—
733C4—102no—
734C4—152no—
735C4—301no—
736C4—301no—
737C4—301no—
738C4—601no—
739C4—601no—
740C4—602no—
741C4D4×ΓL(1,𝔽16)11yescharacteristic
742C4C23×ΓL(1,𝔽16)11yescharacteristic
743C3AGL(1,𝔽5)×#13711yescharacteristic, complemented, Sylow
744C2—21no—
745C2—41no—
746C2—51no—
747C2—51no—
748C2—101no—
749C2—201no—
750C2C2×D4×ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC2C2×D4×ΓL(1,𝔽16)
commutatorC3×C4×C5C23×C4
FrattiniC4C23×ΓL(1,𝔽16)
FittingC3×C5×#137C4
socleC2×C3×C5C2×C4×D4
prime radicalC3×D5×#41C22
cosocleC3×C4×D5C24
hypercenter#137ΓL(1,𝔽16)
hypocenterC3×C5C4×#137
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C2×Q8×ΓL(1,𝔽16)C2semidirect
5. #11752441C2semidirect
6. #11752398C2semidirect
7. Q16×ΓL(1,𝔽16)C2semidirect
8. #11752394C2semidirect
9. SD16×ΓL(1,𝔽16)C2semidirect
10. #11752386C2semidirect
11. M16×ΓL(1,𝔽16)C2semidirect
33. C3×C5×#137C4semidirect
34. C3×#4097C4semidirect
35. C3×#4088C4semidirect
36. C3×#4064C4semidirect
38. C3×#4061C4semidirect
40. C3×#4055C4semidirect
68. C2×C3×C5×Q8C2×C4semidirect
72. C3×Q8×D5C2×C4semidirect
85. C2×C3×Dic10C2×C4semidirect
109. C3×C5×Q16C2×C4semidirect
110. C3×C5×SD16C2×C4semidirect
111. C3×C5×M16C2×C4semidirect
112. C3×#689C2×C4semidirect
113. C3×#689C2×C4semidirect
114. C3×#688C2×C4semidirect
115. C3×#687C2×C4semidirect
116. C3×#681C2×C4semidirect
117. C3×Dic20C2×C4semidirect
118. C3×#677C2×C4semidirect
119. C3×#676C2×C4semidirect
152. C5×#137Dic3semidirect
153. #4097Dic3semidirect
154. #4088Dic3semidirect
155. #4064Dic3semidirect
157. #4061Dic3semidirect
159. #4055Dic3semidirect
225. C3×#137AGL(1,𝔽5)semidirect
261. C2×C5×Q8C2×Dic3semidirect
265. Q8×D5C2×Dic3semidirect
278. C2×Dic10C2×Dic3semidirect
302. C5×Q16C2×Dic3semidirect
303. C5×SD16C2×Dic3semidirect
304. C5×M16C2×Dic3semidirect
305. #689C2×Dic3semidirect
306. #689C2×Dic3semidirect
307. #688C2×Dic3semidirect
308. #687C2×Dic3semidirect
309. #681C2×Dic3semidirect
310. Dic20C2×Dic3semidirect
311. #677C2×Dic3semidirect
312. #676C2×Dic3semidirect
340. ΓL(1,𝔽16)#137direct (N × H)
362. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
376. C3×Q16C2×AGL(1,𝔽5)semidirect
380. C3×SD16C2×AGL(1,𝔽5)semidirect
383. C3×M16C2×AGL(1,𝔽5)semidirect
463. #137ΓL(1,𝔽16)direct (N × H)
585. C2×Q8C2×ΓL(1,𝔽16)semidirect
599. Q16C2×ΓL(1,𝔽16)semidirect
603. SD16C2×ΓL(1,𝔽16)semidirect
608. M16C2×ΓL(1,𝔽16)semidirect
631. C3×C5C4×#137semidirect
718. C5Dic3×#137semidirect
743. C3AGL(1,𝔽5)×#137semidirect

Maximal subgroups

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

subgroupindex
#41×ΓL(1,𝔽16)2
C2×Q8×ΓL(1,𝔽16)2
C3×D5×#1372
#117524522
#117524412
#117523982
Q16×ΓL(1,𝔽16)2
#117523942
SD16×ΓL(1,𝔽16)2
#117523862
M16×ΓL(1,𝔽16)2
AGL(1,𝔽5)×#1373
Dic3×#1375