Finite Group Databasebeta

← C2×C59×#2196 #28445652

Subgroups — C2×C59×#2196 #28445652

Order 15104.

1956 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42×C59C211yesmaximal, characteristic, complemented
2C2×C59×B(4,𝔽2)C212yesmaximal, complemented
3C2×C59×#453C211yesmaximal, characteristic, complemented
4C59×#2196C218yesmaximal, direct product factor
5C2×C59×#353C211yesmaximal, characteristic, complemented
6C2×C59×AΣL(1,𝔽16)C212yesmaximal, complemented
7C2×H(5,𝔽2)×C59C2211yescharacteristic, complemented
8C23×D4×C59—22no—
9C2×AGL(1,ℤ/8)×C59—21no—
10D42×C59—22no—
11D42×C59—24no—
12D42×C59C2214yescomplemented
13C2×#127×C59C2211yescharacteristic
14C2×#121×C59—22no—
15C2×#120×C59—21no—
16C2×#120×C59—22no—
17C2×#120×C59C2212yes—
18C2×C4×D4×C59—21no—
19C59×B(4,𝔽2)—24no—
20C59×B(4,𝔽2)C2218yescomplemented
21C59×#453—22no—
22C59×#453C2214yescomplemented
23C2×#104×C59—21no—
24C2×#100×C59C2211yescharacteristic, complemented
25C2×#99×C59—22no—
26C2×#99×C59C2212yescomplemented
27C59×#353C2214yescomplemented
28C59×AΣL(1,𝔽16)C2218yescomplemented
29C25×C59—21no—
30C25×C59D412yescomplemented
31H(5,𝔽2)×C59—22no—
32H(5,𝔽2)×C59—21no—
33H(5,𝔽2)×C59—42no—
34H(5,𝔽2)×C59C2312yescomplemented
35C2×#41×C59—21no—
36C2×#41×C59—41no—
37C22×D4×C59—21no—
38C22×D4×C59—22no—
39C22×D4×C59—28no—
40C22×D4×C59—22no—
41C22×D4×C59—22no—
42C22×D4×C59—22no—
43C22×D4×C59—28no—
44C22×D4×C59—41no—
45C22×D4×C59—416no—
46C22×D4×C59—42no—
47C22×D4×C59D411yescharacteristic, complemented
48C22×D4×C59C2311yescharacteristic
49C23×C4×C59—22no—
50AGL(1,ℤ/8)×C59—24no—
51AGL(1,ℤ/8)×C59—42no—
52C2×SD16×C59—41no—
53C2×D8×C59—41no—
54C2×M16×C59—21no—
55#127×C59—22no—
56#127×C59D412yescomplemented
57#127×C59C2312yes—
58#121×C59—28no—
59#121×C59—28no—
60#120×C59—24no—
61#120×C59—22no—
62#120×C59—22no—
63#120×C59—24no—
64#120×C59—28no—
65#120×C59—44no—
66#120×C59D414yescomplemented
67#120×C59C2314yes—
68C4×D4×C59—24no—
69C4×D4×C59—24no—
70C2×C4⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72C2×C22⋊C4×C59—22no—
73C2×C22⋊C4×C59—22no—
74C2×C22⋊C4×C59—42no—
75C2×C22⋊C4×C59D412yescomplemented
76C2×C42×C59D411yescharacteristic, complemented
77#104×C59—24no—
78#100×C59—21no—
79#100×C59C2312yescomplemented
80#99×C59—22no—
81#99×C59—28no—
82#99×C59C2314yescomplemented
83C24×C59—22no—
84C24×C59—24no—
85C24×C59—22no—
86C24×C59—21no—
87C24×C59—22no—
88C24×C59—42no—
89C24×C59—48no—
90C24×C59—48no—
91C24×C59—42no—
92C24×C59—81no—
93C24×C59—81no—
94C24×C59C2×D412yes—
95C24×C59C2×D414yescomplemented
96#41×C59—22no—
97#41×C59—22no—
98#41×C59—42no—
99#41×C59—41no—
100#41×C59—44no—
101#41×C59—42no—
102#41×C59—41no—
103C2×Q8×C59—21no—
104C2×D4×C59—24no—
105C2×D4×C59—28no—
106C2×D4×C59—21no—
107C2×D4×C59—28no—
108C2×D4×C59—24no—
109C2×D4×C59—24no—
110C2×D4×C59—24no—
111C2×D4×C59—24no—
112C2×D4×C59—24no—
113C2×D4×C59—24no—
114C2×D4×C59—21no—
115C2×D4×C59—41no—
116C2×D4×C59—41no—
117C2×D4×C59—48no—
118C2×D4×C59—41no—
119C2×D4×C59—41no—
120C2×D4×C59—42no—
121C2×D4×C59—432no—
122C2×D4×C59—432no—
123C2×D4×C59—42no—
124C2×D4×C59—48no—
125C2×D4×C59—48no—
126C2×D4×C59—42no—
127C2×D4×C59—42no—
128C2×D4×C59—44no—
129C2×D4×C59—42no—
130C2×D4×C59—41no—
131C2×D4×C59—48no—
132C2×D4×C59—88no—
133C2×D4×C59C2×D412yes—
134C2×D4×C59C2×D411yescharacteristic
135C2×D4×C59C2×D412yescomplemented
136C2×D4×C59C2411yescharacteristic
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—28no—
140C22×C4×C59—22no—
141C22×C4×C59—48no—
142C22×C4×C59—41no—
143C22×C4×C59—42no—
144C22×C4×C59C2×D411yescharacteristic
145SD16×C59—44no—
146D8×C59—44no—
147M16×C59—22no—
148M16×C59—41no—
149C2×C8×C59—41no—
150C4⋊C4×C59—24no—
151C22⋊C4×C59—24no—
152C22⋊C4×C59—22no—
153C22⋊C4×C59—24no—
154C22⋊C4×C59—24no—
155C22⋊C4×C59—24no—
156C22⋊C4×C59—48no—
157C22⋊C4×C59—42no—
158C22⋊C4×C59C2×D414yescomplemented
159C42×C59—21no—
160C42×C59C2×D412yescomplemented
161C23×C59—28no—
162C23×C59—24no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—24no—
166C23×C59—22no—
167C23×C59—44no—
168C23×C59—44no—
169C23×C59—44no—
170C23×C59—48no—
171C23×C59—44no—
172C23×C59—48no—
173C23×C59—416no—
174C23×C59—42no—
175C23×C59—42no—
176C23×C59—42no—
177C23×C59—48no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59—82no—
182C23×C59—84no—
183C23×C59—88no—
184C23×C59—84no—
185C23×C59—88no—
186C23×C59—88no—
187C23×C59—84no—
188C23×C59#12012yes—
189C23×C59#12011yescharacteristic
190C23×C59C22×D412yes—
191Q8×C59—22no—
192Q8×C59—41no—
193D4×C59—216no—
194D4×C59—22no—
195D4×C59—41no—
196D4×C59—42no—
197D4×C59—41no—
198D4×C59—42no—
199D4×C59—48no—
200D4×C59—48no—
201D4×C59—41no—
202D4×C59—48no—
203D4×C59—48no—
204D4×C59—416no—
205D4×C59—44no—
206D4×C59—44no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—81no—
210D4×C59—816no—
211C2×C4×C59—24no—
212C2×C4×C59—22no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—24no—
216C2×C4×C59—24no—
217C2×C4×C59—416no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—44no—
221C2×C4×C59—42no—
222C2×C4×C59—41no—
223C2×C4×C59—44no—
224C2×C4×C59—84no—
225C2×C4×C59#12011yescharacteristic
226C2×C4×C59C22×D411yescharacteristic
227C8×C59—42no—
228C2×#2196C5911yesmaximal, characteristic, direct product factor, Sylow
229C22×C59—22no—
230C22×C59—28no—
231C22×C59—22no—
232C22×C59—22no—
233C22×C59—42no—
234C22×C59—42no—
235C22×C59—44no—
236C22×C59—48no—
237C22×C59—44no—
238C22×C59—48no—
239C22×C59—44no—
240C22×C59—48no—
241C22×C59—44no—
242C22×C59—42no—
243C22×C59—42no—
244C22×C59—82no—
245C22×C59—82no—
246C22×C59—84no—
247C22×C59—88no—
248C22×C59—816no—
249C22×C59—84no—
250C22×C59—82no—
251C22×C59—82no—
252C22×C59—162no—
253C22×C59B(4,𝔽2)11yescharacteristic
254C22×C59B(4,𝔽2)11yescharacteristic
255C22×C59C2×#12011yescharacteristic
256C4×C59—24no—
257C4×C59—21no—
258C4×C59—21no—
259C4×C59—44no—
260C4×C59—42no—
261C4×C59—84no—
262C2×D42C2×C5911yescharacteristic, complemented
263C2×B(4,𝔽2)C2×C5912yescomplemented
264C2×#453C2×C5911yescharacteristic, complemented
265#2196C2×C5918yesdirect product factor
266C2×#353C2×C5911yescharacteristic, complemented
267C2×AΣL(1,𝔽16)C2×C5912yescomplemented
268C2×C59—21no—
269C2×C59—21no—
270C2×C59—42no—
271C2×C59—48no—
272C2×C59—42no—
273C2×C59—82no—
274C2×C59—82no—
275C2×C59#219612yesdirect product factor
276C2×C59C2×B(4,𝔽2)11yescharacteristic
277C2×H(5,𝔽2)C22×C5911yescharacteristic, complemented
278C23×D4—22no—
279C2×AGL(1,ℤ/8)—21no—
280D42—22no—
281D42—24no—
282D42C22×C5914yescomplemented
283C2×#127C22×C5911yescharacteristic
284C2×#121—22no—
285C2×#120—21no—
286C2×#120—22no—
287C2×#120C22×C5912yes—
288C2×C4×D4—21no—
289B(4,𝔽2)—24no—
290B(4,𝔽2)C22×C5918yescomplemented
291#453—22no—
292#453C22×C5914yescomplemented
293C2×#104—21no—
294C2×#100C22×C5911yescharacteristic, complemented
295C2×#99—22no—
296C2×#99C22×C5912yescomplemented
297#353C22×C5914yescomplemented
298AΣL(1,𝔽16)C22×C5918yescomplemented
299C59C2×#219611yescharacteristic, direct product factor, Sylow
300C25—21no—
301C25D4×C5912yescomplemented
302H(5,𝔽2)—22no—
303H(5,𝔽2)—21no—
304H(5,𝔽2)—42no—
305H(5,𝔽2)C23×C5912yescomplemented
306C2×#41—21no—
307C2×#41—41no—
308C22×D4—21no—
309C22×D4—22no—
310C22×D4—28no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—28no—
315C22×D4—41no—
316C22×D4—416no—
317C22×D4—42no—
318C22×D4D4×C5911yescharacteristic, complemented
319C22×D4C23×C5911yescharacteristic
320C23×C4—22no—
321AGL(1,ℤ/8)—24no—
322AGL(1,ℤ/8)—42no—
323C2×SD16—41no—
324C2×D8—41no—
325C2×M16—21no—
326#127—22no—
327#127D4×C5912yescomplemented
328#127C23×C5912yes—
329#121—28no—
330#121—28no—
331#120—24no—
332#120—22no—
333#120—22no—
334#120—24no—
335#120—28no—
336#120—44no—
337#120D4×C5914yescomplemented
338#120C23×C5914yes—
339C4×D4—24no—
340C4×D4—24no—
341C2×C4⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4D4×C5912yescomplemented
347C2×C42D4×C5911yescharacteristic, complemented
348#104—24no—
349#100—21no—
350#100C23×C5912yescomplemented
351#99—22no—
352#99—28no—
353#99C23×C5914yescomplemented
354C24—22no—
355C24—24no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—42no—
360C24—48no—
361C24—48no—
362C24—42no—
363C24—81no—
364C24—81no—
365C24C2×D4×C5912yes—
366C24C2×D4×C5914yescomplemented
367#41—22no—
368#41—22no—
369#41—42no—
370#41—41no—
371#41—44no—
372#41—42no—
373#41—41no—
374C2×Q8—21no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—48no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—432no—
393C2×D4—432no—
394C2×D4—42no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—48no—
403C2×D4—88no—
404C2×D4C2×D4×C5912yes—
405C2×D4C2×D4×C5911yescharacteristic
406C2×D4C2×D4×C5912yescomplemented
407C2×D4C24×C5911yescharacteristic
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—48no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4C2×D4×C5911yescharacteristic
416SD16—44no—
417D8—44no—
418M16—22no—
419M16—41no—
420C2×C8—41no—
421C4⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—48no—
428C22⋊C4—42no—
429C22⋊C4C2×D4×C5914yescomplemented
430C42—21no—
431C42C2×D4×C5912yescomplemented
432C23—28no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—48no—
442C23—44no—
443C23—48no—
444C23—416no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—82no—
453C23—84no—
454C23—88no—
455C23—84no—
456C23—88no—
457C23—88no—
458C23—84no—
459C23#120×C5911yescharacteristic
460C23#120×C5912yes—
461C23C22×D4×C5912yes—
462Q8—22no—
463Q8—41no—
464D4—216no—
465D4—22no—
466D4—41no—
467D4—42no—
468D4—41no—
469D4—42no—
470D4—48no—
471D4—48no—
472D4—41no—
473D4—48no—
474D4—48no—
475D4—416no—
476D4—44no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4—81no—
481D4—816no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—416no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—84no—
496C2×C4#120×C5911yescharacteristic
497C2×C4C22×D4×C5911yescharacteristic
498C8—42no—
499C22—22no—
500C22—28no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—48no—
507C22—44no—
508C22—48no—
509C22—44no—
510C22—44no—
511C22—48no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22—88no—
518C22—816no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—162no—
523C22C59×B(4,𝔽2)11yescharacteristic
524C22C59×B(4,𝔽2)11yescharacteristic
525C22C2×#120×C5911yescharacteristic
526C4—24no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—84no—
532C2—21no—
533C2—21no—
534C2—42no—
535C2—48no—
536C2—42no—
537C2—82no—
538C2—82no—
539C2C59×#219612yesdirect product factor
540C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C59
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C22×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×D42×C59C2semidirect
2. C2×C59×B(4,𝔽2)C2semidirect
3. C2×C59×#453C2semidirect
4. C59×#2196C2direct (N × H)
5. C2×C59×#353C2semidirect
6. C2×C59×AΣL(1,𝔽16)C2semidirect
7. C2×H(5,𝔽2)×C59C22semidirect
12. D42×C59C22semidirect
20. C59×B(4,𝔽2)C22semidirect
22. C59×#453C22semidirect
24. C2×#100×C59C22semidirect
26. C2×#99×C59C22semidirect
27. C59×#353C22semidirect
28. C59×AΣL(1,𝔽16)C22semidirect
30. C25×C59D4semidirect
34. H(5,𝔽2)×C59C23semidirect
47. C22×D4×C59D4semidirect
56. #127×C59D4semidirect
66. #120×C59D4semidirect
75. C2×C22⋊C4×C59D4semidirect
76. C2×C42×C59D4semidirect
79. #100×C59C23semidirect
82. #99×C59C23semidirect
95. C24×C59C2×D4semidirect
135. C2×D4×C59C2×D4semidirect
158. C22⋊C4×C59C2×D4semidirect
160. C42×C59C2×D4semidirect
228. C2×#2196C59direct (N × H)
262. C2×D42C2×C59semidirect
263. C2×B(4,𝔽2)C2×C59semidirect
264. C2×#453C2×C59semidirect
265. #2196C2×C59direct (N × H)
266. C2×#353C2×C59semidirect
267. C2×AΣL(1,𝔽16)C2×C59semidirect
275. C2×C59#2196direct (N × H)
277. C2×H(5,𝔽2)C22×C59semidirect
282. D42C22×C59semidirect
290. B(4,𝔽2)C22×C59semidirect
292. #453C22×C59semidirect
294. C2×#100C22×C59semidirect
296. C2×#99C22×C59semidirect
297. #353C22×C59semidirect
298. AΣL(1,𝔽16)C22×C59semidirect
299. C59C2×#2196direct (N × H)
301. C25D4×C59semidirect
305. H(5,𝔽2)C23×C59semidirect
318. C22×D4D4×C59semidirect
327. #127D4×C59semidirect
337. #120D4×C59semidirect
346. C2×C22⋊C4D4×C59semidirect
347. C2×C42D4×C59semidirect
350. #100C23×C59semidirect
353. #99C23×C59semidirect
366. C24C2×D4×C59semidirect
406. C2×D4C2×D4×C59semidirect
429. C22⋊C4C2×D4×C59semidirect
431. C42C2×D4×C59semidirect
539. C2C59×#2196direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D42×C592
C2×C59×B(4,𝔽2)2
C2×C59×#4532
C59×#21962
C2×C59×#3532
C2×C59×AΣL(1,𝔽16)2
C2×#219659