Finite Group Databasebeta

← C59×#13411 #28425520

Subgroups — C59×#13411 #28425520

Order 15104.

462 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3155C211yesmaximal, characteristic
2C2×C59×#361C211yesmaximal, characteristic, complemented
3C2×C59×#359C211yesmaximal, characteristic, complemented
4C59×#2139C211yesmaximal, characteristic, complemented
5C2×C59×#357C211yesmaximal, characteristic, complemented
6C59×#1952C211yesmaximal, characteristic, complemented
7C59×#1379C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C59—21no—
9C2×#135×C59C2211yescharacteristic
10C22×D8×C59C2211yescharacteristic
11C59×#535—21no—
12C2×M32×C59C2211yescharacteristic
13C22×C16×C59C2211yescharacteristic
14C59×#496—22no—
15C59×#495—21no—
16C59×#493—21no—
17C2×#108×C59C2211yescharacteristic
18C59×#428—21no—
19C2×#107×C59C2211yescharacteristic
20C59×#405C2211yescharacteristic
21C59×#361—22no—
22C59×#359—22no—
23C59×#358—21no—
24C59×#357—22no—
25C59×#357—21no—
26C59×#343—21no—
27C2×#41×C59—21no—
28C22×D4×C59—21no—
29AGL(1,ℤ/8)×C59—42no—
30AGL(1,ℤ/8)×C59—42no—
31#135×C59—21no—
32#135×C59—21no—
33#135×C59—21no—
34#135×C59—21no—
35C2×Q16×C59D411yescharacteristic
36C2×SD16×C59—22no—
37C2×SD16×C59—21no—
38C2×D8×C59—22no—
39C2×D8×C59—21no—
40C2×D8×C59—21no—
41C2×D8×C59—21no—
42C2×D8×C59—21no—
43C2×D8×C59—21no—
44C2×D8×C59D411yescharacteristic
45C2×D8×C59D412yes—
46C2×M16×C59—21no—
47C2×M16×C59D411yescharacteristic
48C22×C8×C59C2311yescharacteristic
49#127×C59—22no—
50#124×C59—22no—
51#121×C59—41no—
52#121×C59—41no—
53#120×C59—42no—
54#117×C59D411yescharacteristic
55C2×C4⋊C4×C59—21no—
56M32×C59—21no—
57M32×C59—21no—
58C2×C16×C59—21no—
59C2×C16×C59—21no—
60C2×C16×C59—21no—
61C2×C16×C59—21no—
62#108×C59—21no—
63#108×C59D412yescomplemented
64#107×C59—21no—
65#107×C59—21no—
66#107×C59—41no—
67#106×C59—41no—
68#97×C59—21no—
69C4×C8×C59—21no—
70C24×C59—41no—
71#41×C59—41no—
72#41×C59—41no—
73#41×C59—41no—
74#41×C59—41no—
75C2×Q8×C59—21no—
76C2×D4×C59—22no—
77C2×D4×C59—21no—
78C2×D4×C59—42no—
79C2×D4×C59—42no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89Q16×C59—21no—
90Q16×C59—21no—
91SD16×C59—44no—
92SD16×C59—41no—
93SD16×C59—41no—
94D8×C59—22no—
95D8×C59—21no—
96D8×C59—22no—
97D8×C59—21no—
98D8×C59—44no—
99D8×C59—41no—
100D8×C59—41no—
101M16×C59—21no—
102M16×C59—22no—
103M16×C59—41no—
104M16×C59SD1612yes—
105C2×C8×C59—21no—
106C2×C8×C59—41no—
107C2×C8×C59C2×D411yescharacteristic
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59#4111yescharacteristic
111C2×C8×C59#4111yescharacteristic
112C2×C8×C59#4111yescharacteristic
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59D811yescharacteristic
116C4⋊C4×C59D811yescharacteristic
117C22⋊C4×C59—21no—
118C22⋊C4×C59—42no—
119C22⋊C4×C59—42no—
120C42×C59—21no—
121C16×C59—41no—
122C16×C59—41no—
123C23×C59—42no—
124C23×C59—41no—
125C23×C59—81no—
126C23×C59—81no—
127C23×C59—161no—
128C23×C59—161no—
129C23×C59AGL(1,ℤ/8)11yescharacteristic
130Q8×C59—41no—
131Q8×C59—41no—
132D4×C59—42no—
133D4×C59—41no—
134D4×C59—41no—
135D4×C59—42no—
136D4×C59—81no—
137D4×C59—81no—
138D4×C59—81no—
139D4×C59—81no—
140D4×C59—162no—
141D4×C59—162no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—41no—
145C2×C4×C59—81no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59—161no—
149C2×C4×C59#12011yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12411yescharacteristic
152C2×C4×C59C2×D811yescharacteristic
153C2×C4×C59C2×SD1611yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C8×C59—22no—
156C8×C59—42no—
157C8×C59#12111yescharacteristic
158C8×C59#12111yescharacteristic
159C8×C59#12311yescharacteristic
160C8×C59#12311yescharacteristic
161#13411C5911yesmaximal, characteristic, direct product factor, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—81no—
165C22×C59—82no—
166C22×C59—81no—
167C22×C59—82no—
168C22×C59—81no—
169C22×C59—82no—
170C22×C59—162no—
171C22×C59—162no—
172C22×C59#44711yescharacteristic
173C22×C59#46011yescharacteristic
174C22×C59#49011yescharacteristic
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—81no—
178C4×C59—161no—
179C4×C59#39411yescharacteristic
180C4×C59#45011yescharacteristic
181C4×C59#45911yescharacteristic
182C4×C59#48611yescharacteristic
183#3155C2×C5911yescharacteristic
184C2×#361C2×C5911yescharacteristic, complemented
185C2×#359C2×C5911yescharacteristic, complemented
186#2139C2×C5911yescharacteristic, complemented
187C2×#357C2×C5911yescharacteristic, complemented
188#1952C2×C5911yescharacteristic, complemented
189#1379C2×C5911yescharacteristic, complemented
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—161no—
193C2×C59—162no—
194C2×C59#204111yescharacteristic
195C2×C59#219211yescharacteristic
196C2×C59#219411yescharacteristic
197C2×AGL(1,ℤ/8)—21no—
198C2×#135C22×C5911yescharacteristic
199C22×D8C22×C5911yescharacteristic
200#535—21no—
201C2×M32C22×C5911yescharacteristic
202C22×C16C22×C5911yescharacteristic
203#496—22no—
204#495—21no—
205#493—21no—
206C2×#108C22×C5911yescharacteristic
207#428—21no—
208C2×#107C22×C5911yescharacteristic
209#405C22×C5911yescharacteristic
210#361—22no—
211#359—22no—
212#358—21no—
213#357—22no—
214#357—21no—
215#343—21no—
216C59#1341111yescharacteristic, direct product factor, Sylow
217C2×#41—21no—
218C22×D4—21no—
219AGL(1,ℤ/8)—42no—
220AGL(1,ℤ/8)—42no—
221#135—21no—
222#135—21no—
223#135—21no—
224#135—21no—
225C2×Q16D4×C5911yescharacteristic
226C2×SD16—22no—
227C2×SD16—21no—
228C2×D8—22no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8D4×C5912yes—
235C2×D8D4×C5911yescharacteristic
236C2×M16—21no—
237C2×M16D4×C5911yescharacteristic
238C22×C8C23×C5911yescharacteristic
239#127—22no—
240#124—22no—
241#121—41no—
242#121—41no—
243#120—42no—
244#117D4×C5911yescharacteristic
245C2×C4⋊C4—21no—
246M32—21no—
247M32—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252#108—21no—
253#108D4×C5912yescomplemented
254#107—21no—
255#107—21no—
256#107—41no—
257#106—41no—
258#97—21no—
259C4×C8—21no—
260C24—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×D4—22no—
267C2×D4—21no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×D4×C5911yescharacteristic
279Q16—21no—
280Q16—21no—
281SD16—44no—
282SD16—41no—
283SD16—41no—
284D8—22no—
285D8—21no—
286D8—22no—
287D8—21no—
288D8—44no—
289D8—41no—
290D8—41no—
291M16—21no—
292M16—22no—
293M16—41no—
294M16SD16×C5912yes—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8C2×D4×C5911yescharacteristic
298C2×C8C2×D4×C5911yescharacteristic
299C2×C8C2×D4×C5911yescharacteristic
300C2×C8#41×C5911yescharacteristic
301C2×C8#41×C5911yescharacteristic
302C2×C8#41×C5911yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4D8×C5911yescharacteristic
306C4⋊C4D8×C5911yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C42—21no—
311C16—41no—
312C16—41no—
313C23—42no—
314C23—41no—
315C23—81no—
316C23—81no—
317C23—161no—
318C23—161no—
319C23AGL(1,ℤ/8)×C5911yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—42no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—162no—
331D4—162no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—161no—
339C2×C4#120×C5911yescharacteristic
340C2×C4#121×C5911yescharacteristic
341C2×C4#124×C5911yescharacteristic
342C2×C4C2×D8×C5911yescharacteristic
343C2×C4C2×SD16×C5911yescharacteristic
344C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
345C8—22no—
346C8—42no—
347C8#121×C5911yescharacteristic
348C8#121×C5911yescharacteristic
349C8#123×C5911yescharacteristic
350C8#123×C5911yescharacteristic
351C22—21no—
352C22—21no—
353C22—81no—
354C22—82no—
355C22—81no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22—162no—
360C22—162no—
361C22C59×#44711yescharacteristic
362C22C59×#46011yescharacteristic
363C22C59×#49011yescharacteristic
364C4—41no—
365C4—41no—
366C4—81no—
367C4—161no—
368C4C59×#39411yescharacteristic
369C4C59×#45011yescharacteristic
370C4C59×#45911yescharacteristic
371C4C59×#48611yescharacteristic
372C2—21no—
373C2—21no—
374C2—162no—
375C2—161no—
376C2C59×#204111yescharacteristic
377C2C59×#219211yescharacteristic
378C2C59×#219411yescharacteristic
subgroupG/H
center, socleC22×C59#447
commutator, Frattini, cosocleC22×C8C23×C59
prime radicalC59×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C59×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#361C2semidirect
3. C2×C59×#359C2semidirect
4. C59×#2139C2semidirect
5. C2×C59×#357C2semidirect
6. C59×#1952C2semidirect
7. C59×#1379C2semidirect
63. #108×C59D4semidirect
161. #13411C59direct (N × H)
184. C2×#361C2×C59semidirect
185. C2×#359C2×C59semidirect
186. #2139C2×C59semidirect
187. C2×#357C2×C59semidirect
188. #1952C2×C59semidirect
189. #1379C2×C59semidirect
216. C59#13411direct (N × H)
253. #108D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#31552
C2×C59×#3612
C2×C59×#3592
C59×#21392
C2×C59×#3572
C59×#19522
C59×#13792
#1341159