Finite Group Databasebeta

← C59×#13408 #28425517

Subgroups — C59×#13408 #28425517

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#495C211yesmaximal, characteristic
2C2×C59×#358C211yesmaximal, characteristic, complemented
3C2×C59×#358C211yesmaximal, characteristic, complemented
4C2×C59×#357C211yesmaximal, characteristic
5C2×C59×#357C211yesmaximal, characteristic
6C59×#1942C211yesmaximal, characteristic, complemented
7C59×#1374C211yesmaximal, characteristic, complemented
8C22×Q16×C59C2211yescharacteristic
9C22×SD16×C59—21no—
10C22×D8×C59C2211yescharacteristic
11C2×#124×C59—21no—
12C22×C16×C59C2211yescharacteristic
13C22×C16×C59C2211yescharacteristic
14C59×#495—22no—
15C59×#495—22no—
16C2×#107×C59C2211yescharacteristic
17C2×#107×C59C2211yescharacteristic
18C2×C4×C8×C59C2211yescharacteristic
19C59×#389—21no—
20C59×#358—22no—
21C59×#358—22no—
22C59×#357—22no—
23C59×#357—22no—
24C59×#340—21no—
25C22×Q8×C59—21no—
26C22×D4×C59—21no—
27C2×Q16×C59—21no—
28C2×Q16×C59—21no—
29C2×Q16×C59—21no—
30C2×Q16×C59—21no—
31C2×Q16×C59—21no—
32C2×Q16×C59D412yes—
33C2×SD16×C59—24no—
34C2×SD16×C59—42no—
35C2×SD16×C59—42no—
36C2×D8×C59—21no—
37C2×D8×C59—21no—
38C2×D8×C59—21no—
39C2×D8×C59—21no—
40C2×D8×C59—21no—
41C2×D8×C59D412yes—
42C22×C8×C59D411yescharacteristic
43C22×C8×C59C2311yescharacteristic
44#124×C59—24no—
45#124×C59—42no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—41no—
49C2×C22⋊C4×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C42×C59D411yescharacteristic
52C2×C16×C59—21no—
53C2×C16×C59—21no—
54C2×C16×C59—21no—
55C2×C16×C59—21no—
56C2×C16×C59—21no—
57C2×C16×C59—21no—
58#107×C59—21no—
59#107×C59—21no—
60#107×C59—21no—
61#107×C59D412yes—
62C4×C8×C59—21no—
63C4×C8×C59—21no—
64#95×C59—41no—
65C24×C59—41no—
66C2×Q8×C59—22no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C2×D4×C59—22no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59D811yescharacteristic
82C22×C4×C59D811yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84Q16×C59—22no—
85Q16×C59—22no—
86Q16×C59—41no—
87Q16×C59—41no—
88SD16×C59—48no—
89D8×C59—22no—
90D8×C59—22no—
91D8×C59—41no—
92D8×C59—41no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59SD1612yes—
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C4⋊C4×C59—22no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—82no—
107C22⋊C4×C59—44no—
108C22⋊C4×C59—44no—
109C42×C59—21no—
110C42×C59—21no—
111C16×C59—41no—
112C16×C59—41no—
113C23×C59—42no—
114C23×C59—81no—
115C23×C59—81no—
116C23×C59—81no—
117C23×C59—81no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59C2×D811yescharacteristic
121Q8×C59—42no—
122Q8×C59—42no—
123Q8×C59—81no—
124Q8×C59—81no—
125D4×C59—42no—
126D4×C59—42no—
127D4×C59—81no—
128D4×C59—81no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59#12011yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12411yescharacteristic
147C2×C4×C59C2×SD1611yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C8×C59—22no—
151C8×C59#12111yescharacteristic
152C8×C59#12111yescharacteristic
153C8×C59#12311yescharacteristic
154C8×C59#12311yescharacteristic
155#13408C5911yesmaximal, characteristic, direct product factor, Sylow
156C22×C59—82no—
157C22×C59—82no—
158C22×C59—82no—
159C22×C59—161no—
160C22×C59—161no—
161C22×C59—161no—
162C22×C59—161no—
163C22×C59#44711yescharacteristic
164C22×C59#45911yescharacteristic
165C22×C59#48611yescharacteristic
166C22×C59#50811yescharacteristic
167C22×C59#50811yescharacteristic
168C22×C59#50911yescharacteristic
169C22×C59#51011yescharacteristic
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—82no—
173C4×C59—82no—
174C4×C59—161no—
175C4×C59#39411yescharacteristic
176C4×C59#45011yescharacteristic
177C4×C59#46011yescharacteristic
178C4×C59#49011yescharacteristic
179C2×#495C2×C5911yescharacteristic
180C2×#358C2×C5911yescharacteristic, complemented
181C2×#358C2×C5911yescharacteristic, complemented
182C2×#357C2×C5911yescharacteristic
183C2×#357C2×C5911yescharacteristic
184#1942C2×C5911yescharacteristic, complemented
185#1374C2×C5911yescharacteristic, complemented
186C2×C59—162no—
187C2×C59#204111yescharacteristic
188C2×C59#218611yescharacteristic
189C2×C59#218911yescharacteristic
190C2×C59#221011yescharacteristic
191C2×C59#221111yescharacteristic
192C2×C59#224211yescharacteristic
193C2×C59#224311yescharacteristic
194C22×Q16C22×C5911yescharacteristic
195C22×SD16—21no—
196C22×D8C22×C5911yescharacteristic
197C2×#124—21no—
198C22×C16C22×C5911yescharacteristic
199C22×C16C22×C5911yescharacteristic
200#495—22no—
201#495—22no—
202C2×#107C22×C5911yescharacteristic
203C2×#107C22×C5911yescharacteristic
204C2×C4×C8C22×C5911yescharacteristic
205#389—21no—
206#358—22no—
207#358—22no—
208#357—22no—
209#357—22no—
210#340—21no—
211C59#1340811yescharacteristic, direct product factor, Sylow
212C22×Q8—21no—
213C22×D4—21no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16D4×C5912yes—
220C2×SD16—24no—
221C2×SD16—42no—
222C2×SD16—42no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8D4×C5912yes—
229C22×C8D4×C5911yescharacteristic
230C22×C8C23×C5911yescharacteristic
231#124—24no—
232#124—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42D4×C5911yescharacteristic
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107D4×C5912yes—
249C4×C8—21no—
250C4×C8—21no—
251#95—41no—
252C24—41no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4D8×C5911yescharacteristic
269C22×C4D8×C5911yescharacteristic
270C22×C4C2×D4×C5911yescharacteristic
271Q16—22no—
272Q16—22no—
273Q16—41no—
274Q16—41no—
275SD16—48no—
276D8—22no—
277D8—22no—
278D8—41no—
279D8—41no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8SD16×C5912yes—
283C2×C8C2×D4×C5911yescharacteristic
284C2×C8C2×D4×C5911yescharacteristic
285C2×C8C2×D4×C5911yescharacteristic
286C2×C8#41×C5911yescharacteristic
287C2×C8#41×C5911yescharacteristic
288C2×C8#41×C5911yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—82no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C42—21no—
297C42—21no—
298C16—41no—
299C16—41no—
300C23—42no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23C2×D8×C5911yescharacteristic
308Q8—42no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#120×C5911yescharacteristic
332C2×C4#121×C5911yescharacteristic
333C2×C4#124×C5911yescharacteristic
334C2×C4C2×SD16×C5911yescharacteristic
335C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
336C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
337C8—22no—
338C8#121×C5911yescharacteristic
339C8#121×C5911yescharacteristic
340C8#123×C5911yescharacteristic
341C8#123×C5911yescharacteristic
342C22—82no—
343C22—82no—
344C22—82no—
345C22—161no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22C59×#44711yescharacteristic
350C22C59×#45911yescharacteristic
351C22C59×#48611yescharacteristic
352C22C59×#50811yescharacteristic
353C22C59×#50811yescharacteristic
354C22C59×#50911yescharacteristic
355C22C59×#51011yescharacteristic
356C4—41no—
357C4—41no—
358C4—82no—
359C4—82no—
360C4—161no—
361C4C59×#39411yescharacteristic
362C4C59×#45011yescharacteristic
363C4C59×#46011yescharacteristic
364C4C59×#49011yescharacteristic
365C2—162no—
366C2C59×#204111yescharacteristic
367C2C59×#218611yescharacteristic
368C2C59×#218911yescharacteristic
369C2C59×#221011yescharacteristic
370C2C59×#221111yescharacteristic
371C2C59×#224211yescharacteristic
372C2C59×#224311yescharacteristic
subgroupG/H
center, socleC23×C59C2×D8
commutator, Frattini, cosocleC22×C8C23×C59
prime radicalC22×D8×C59C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C42×C59 ◁ C2×C4×C8×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#358C2semidirect
3. C2×C59×#358C2semidirect
6. C59×#1942C2semidirect
7. C59×#1374C2semidirect
155. #13408C59direct (N × H)
180. C2×#358C2×C59semidirect
181. C2×#358C2×C59semidirect
184. #1942C2×C59semidirect
185. #1374C2×C59semidirect
211. C59#13408direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#4952
C2×C59×#3582
C2×C59×#3572
C59×#19422
C59×#13742
#1340859