Finite Group Databasebeta

← C59×#10176 #28422285

Subgroups — C59×#10176 #28422285

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3205C211yesmaximal, characteristic, complemented
2C59×#3036C211yesmaximal, characteristic, complemented
3C2×C59×#459C211yesmaximal, characteristic, complemented
4C59×#1545C211yesmaximal, characteristic, complemented
5C59×#1480C211yesmaximal, characteristic, complemented
6C59×#1460C211yesmaximal, characteristic, complemented
7C59×#1301C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C59—21no—
9C22×D8×C59—21no—
10C59×#532C2211yescharacteristic
11C2×#127×C59C2211yescharacteristic
12C2×C4×D4×C59C2211yescharacteristic
13C59×#493—21no—
14C59×#486—21no—
15C59×#463—22no—
16C59×#461—21no—
17C59×#459—22no—
18C59×#459—21no—
19C59×#459—22no—
20C59×#459—22no—
21C59×#447—22no—
22C59×#431C2211yescharacteristic
23C59×#431C2211yescharacteristic
24C59×#423C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#417—21no—
27C2×#102×C59—21no—
28C59×#331—22no—
29C59×D4⋊C8—22no—
30C59×#323—22no—
31C2×#41×C59—21no—
32C22×D4×C59—21no—
33C22×D4×C59—21no—
34C22×D4×C59—21no—
35C22×D4×C59D411yescharacteristic
36C23×C4×C59—21no—
37AGL(1,ℤ/8)×C59—42no—
38AGL(1,ℤ/8)×C59—42no—
39C2×SD16×C59—22no—
40C2×D8×C59—24no—
41C2×D8×C59—22no—
42C2×D8×C59—24no—
43C2×D8×C59—42no—
44C2×D8×C59—42no—
45C2×M16×C59—21no—
46C22×C8×C59—21no—
47C4⋊Q8×C59D411yescharacteristic
48#127×C59—21no—
49#127×C59—21no—
50#127×C59—21no—
51#127×C59D412yes—
52#127×C59D411yescharacteristic
53#124×C59—21no—
54#121×C59—21no—
55#121×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C22⋊C4×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59D412yes—
70C4⋊C8×C59D412yes—
71#103×C59—22no—
72#102×C59—24no—
73#102×C59—22no—
74#102×C59—24no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59D412yes—
77C4×C8×C59—21no—
78C4×C8×C59—21no—
79C24×C59—21no—
80C24×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83#41×C59—41no—
84#41×C59—41no—
85C2×Q8×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59—21no—
88C2×D4×C59—21no—
89C2×D4×C59—22no—
90C2×D4×C59—22no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—22no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—21no—
97C2×D4×C59—41no—
98C2×D4×C59—41no—
99C2×D4×C59—41no—
100C2×D4×C59—41no—
101C2×D4×C59—41no—
102C2×D4×C59—41no—
103C2×D4×C59—41no—
104C2×D4×C59—41no—
105C2×D4×C59—82no—
106C2×D4×C59—82no—
107C2×D4×C59D812yescomplemented
108C22×C4×C59—21no—
109C22×C4×C59—41no—
110C22×C4×C59—41no—
111C22×C4×C59—41no—
112C22×C4×C59—82no—
113C22×C4×C59C2×D411yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59#4111yescharacteristic
116SD16×C59—44no—
117D8×C59—48no—
118D8×C59—44no—
119D8×C59—84no—
120M16×C59—41no—
121M16×C59—41no—
122C2×C8×C59—22no—
123C2×C8×C59—22no—
124C2×C8×C59—22no—
125C2×C8×C59—41no—
126C2×C8×C59—41no—
127C2×C8×C59—41no—
128C4⋊C4×C59—21no—
129C4⋊C4×C59—21no—
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C22⋊C4×C59—41no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—41no—
135C22⋊C4×C59—41no—
136C42×C59C2×D411yescharacteristic
137C42×C59C2×D411yescharacteristic
138C42×C59C2×D411yescharacteristic
139C42×C59C2×D411yescharacteristic
140C23×C59—22no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59—42no—
145C23×C59—81no—
146C23×C59—81no—
147C23×C59—82no—
148C23×C59—82no—
149C23×C59—81no—
150C23×C59—81no—
151C23×C59#12111yescharacteristic
152Q8×C59—41no—
153Q8×C59—41no—
154D4×C59—22no—
155D4×C59—22no—
156D4×C59—42no—
157D4×C59—44no—
158D4×C59—42no—
159D4×C59—41no—
160D4×C59—41no—
161D4×C59—41no—
162D4×C59—42no—
163D4×C59—41no—
164D4×C59—42no—
165D4×C59—82no—
166D4×C59—82no—
167D4×C59—81no—
168D4×C59—81no—
169D4×C59—81no—
170D4×C59—81no—
171D4×C59—82no—
172C2×C4×C59—22no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—82no—
177C2×C4×C59—81no—
178C2×C4×C59—81no—
179C2×C4×C59—82no—
180C2×C4×C59—81no—
181C2×C4×C59#12011yescharacteristic
182C2×C4×C59#12011yescharacteristic
183C2×C4×C59#12011yescharacteristic
184C2×C4×C59#12111yescharacteristic
185C2×C4×C59#12111yescharacteristic
186C2×C4×C59#12711yescharacteristic
187C2×C4×C59C2×D811yescharacteristic
188C2×C4×C59#13511yescharacteristic
189C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
191C8×C59—42no—
192C8×C59—42no—
193C8×C59—42no—
194C8×C59—81no—
195#10176C5911yesmaximal, characteristic, direct product factor, Sylow
196C22×C59—21no—
197C22×C59—21no—
198C22×C59—42no—
199C22×C59—82no—
200C22×C59—82no—
201C22×C59—81no—
202C22×C59—81no—
203C22×C59—81no—
204C22×C59—82no—
205C22×C59—82no—
206C22×C59—81no—
207C22×C59—81no—
208C22×C59—82no—
209C22×C59—162no—
210C22×C59—162no—
211C22×C59#39211yescharacteristic
212C22×C59#46611yescharacteristic
213C22×C59#46911yescharacteristic
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—81no—
218C4×C59—81no—
219C4×C59#44711yescharacteristic
220C4×C59#44711yescharacteristic
221C4×C59#44911yescharacteristic
222C4×C59#44911yescharacteristic
223C4×C59#47212yes—
224#3205C2×C5911yescharacteristic, complemented
225#3036C2×C5911yescharacteristic, complemented
226C2×#459C2×C5911yescharacteristic, complemented
227#1545C2×C5911yescharacteristic, complemented
228#1480C2×C5911yescharacteristic, complemented
229#1460C2×C5911yescharacteristic, complemented
230#1301C2×C5911yescharacteristic, complemented
231C2×C59—21no—
232C2×C59—21no—
233C2×C59—82no—
234C2×C59—161no—
235C2×C59—162no—
236C2×C59#199911yescharacteristic
237C2×C59#200811yescharacteristic
238C2×C59#201811yescharacteristic
239C2×AGL(1,ℤ/8)—21no—
240C22×D8—21no—
241#532C22×C5911yescharacteristic
242C2×#127C22×C5911yescharacteristic
243C2×C4×D4C22×C5911yescharacteristic
244#493—21no—
245#486—21no—
246#463—22no—
247#461—21no—
248#459—22no—
249#459—22no—
250#459—21no—
251#459—22no—
252#447—22no—
253#431C22×C5911yescharacteristic
254#431C22×C5911yescharacteristic
255#423C22×C5911yescharacteristic
256C2×C4⋊C8C22×C5911yescharacteristic
257#417—21no—
258C2×#102—21no—
259#331—22no—
260D4⋊C8—22no—
261#323—22no—
262C59#1017611yescharacteristic, direct product factor, Sylow
263C2×#41—21no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4D4×C5911yescharacteristic
268C23×C4—21no—
269AGL(1,ℤ/8)—42no—
270AGL(1,ℤ/8)—42no—
271C2×SD16—22no—
272C2×D8—24no—
273C2×D8—22no—
274C2×D8—24no—
275C2×D8—42no—
276C2×D8—42no—
277C2×M16—21no—
278C22×C8—21no—
279C4⋊Q8D4×C5911yescharacteristic
280#127—21no—
281#127—21no—
282#127—21no—
283#127D4×C5912yes—
284#127D4×C5911yescharacteristic
285#124—21no—
286#121—21no—
287#121—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C2×C4⋊C4D4×C5911yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C23×C5911yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8D4×C5912yes—
302C4⋊C8D4×C5912yes—
303#103—22no—
304#102—24no—
305#102—22no—
306#102—24no—
307C22⋊C8—21no—
308C22⋊C8D4×C5912yes—
309C4×C8—21no—
310C4×C8—21no—
311C24—21no—
312C24—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317C2×Q8—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4D8×C5912yescomplemented
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—82no—
345C22×C4C2×D4×C5911yescharacteristic
346C22×C4C2×D4×C5911yescharacteristic
347C22×C4#41×C5911yescharacteristic
348SD16—44no—
349D8—48no—
350D8—44no—
351D8—84no—
352M16—41no—
353M16—41no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C42C2×D4×C5911yescharacteristic
369C42C2×D4×C5911yescharacteristic
370C42C2×D4×C5911yescharacteristic
371C42C2×D4×C5911yescharacteristic
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—81no—
378C23—81no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—81no—
383C23#121×C5911yescharacteristic
384Q8—41no—
385Q8—41no—
386D4—22no—
387D4—22no—
388D4—42no—
389D4—44no—
390D4—42no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—42no—
395D4—41no—
396D4—42no—
397D4—82no—
398D4—82no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—82no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4#120×C5911yescharacteristic
414C2×C4#120×C5911yescharacteristic
415C2×C4#120×C5911yescharacteristic
416C2×C4#121×C5911yescharacteristic
417C2×C4#121×C5911yescharacteristic
418C2×C4#127×C5911yescharacteristic
419C2×C4C2×D8×C5911yescharacteristic
420C2×C4#135×C5911yescharacteristic
421C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
422C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
423C8—42no—
424C8—42no—
425C8—42no—
426C8—81no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—82no—
431C22—82no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—82no—
436C22—82no—
437C22—81no—
438C22—81no—
439C22—82no—
440C22—162no—
441C22—162no—
442C22C59×#39211yescharacteristic
443C22C59×#46611yescharacteristic
444C22C59×#46911yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4C59×#44711yescharacteristic
451C4C59×#44711yescharacteristic
452C4C59×#44911yescharacteristic
453C4C59×#44911yescharacteristic
454C4C59×#47212yes—
455C2—21no—
456C2—21no—
457C2—82no—
458C2—162no—
459C2—161no—
460C2C59×#199911yescharacteristic
461C2C59×#200811yescharacteristic
462C2C59×#201811yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3205C2semidirect
2. C59×#3036C2semidirect
3. C2×C59×#459C2semidirect
4. C59×#1545C2semidirect
5. C59×#1480C2semidirect
6. C59×#1460C2semidirect
7. C59×#1301C2semidirect
107. C2×D4×C59D8semidirect
195. #10176C59direct (N × H)
224. #3205C2×C59semidirect
225. #3036C2×C59semidirect
226. C2×#459C2×C59semidirect
227. #1545C2×C59semidirect
228. #1480C2×C59semidirect
229. #1460C2×C59semidirect
230. #1301C2×C59semidirect
262. C59#10176direct (N × H)
339. C2×D4D8×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32052
C59×#30362
C2×C59×#4592
C59×#15452
C59×#14802
C59×#14602
C59×#13012
#1017659