Finite Group Databasebeta

← C59×#10549 #28422658

Subgroups — C59×#10549 #28422658

Order 15104.

502 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3151C211yesmaximal, characteristic
2C59×#1950C211yesmaximal, characteristic, complemented
3C59×#1547C211yesmaximal, characteristic, complemented
4C59×#1545C211yesmaximal, characteristic, complemented
5C2×C59×#331C211yesmaximal, characteristic, complemented
6C59×#1480C211yesmaximal, characteristic, complemented
7C59×#1283C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C59—21no—
9C2×AGL(1,ℤ/8)×C59—21no—
10C59×#532C2211yescharacteristic
11C2×#127×C59C2211yescharacteristic
12C59×#496—22no—
13C59×#493—21no—
14C59×#492—21no—
15C59×#431C2211yescharacteristic
16C2×#108×C59—21no—
17C59×#423C2211yescharacteristic
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×C4⋊C8×C59C2211yescharacteristic
20C4×M16×C59C2211yescharacteristic
21C59×#341—21no—
22C59×#332—21no—
23C59×#331—22no—
24C59×#331—21no—
25C59×#331—22no—
26C59×D4⋊C8—22no—
27C2×#41×C59—21no—
28C22×D4×C59—21no—
29C22×D4×C59—21no—
30C22×D4×C59—21no—
31AGL(1,ℤ/8)×C59—42no—
32AGL(1,ℤ/8)×C59—42no—
33AGL(1,ℤ/8)×C59—42no—
34AGL(1,ℤ/8)×C59—42no—
35C2×SD16×C59—22no—
36C2×SD16×C59—22no—
37C2×D8×C59—22no—
38C2×D8×C59—22no—
39C2×M16×C59—21no—
40C2×M16×C59D411yescharacteristic
41C2×M16×C59D411yescharacteristic
42C22×C8×C59—21no—
43C22×C8×C59—21no—
44C4⋊Q8×C59D411yescharacteristic
45#127×C59—21no—
46#127×C59—21no—
47#127×C59—21no—
48#127×C59D411yescharacteristic
49#127×C59D412yes—
50#124×C59—21no—
51#121×C59—21no—
52#121×C59—21no—
53C4×D4×C59—21no—
54C4×D4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56#108×C59—42no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D412yes—
64C22⋊C8×C59—21no—
65#97×C59—21no—
66C4×C8×C59—21no—
67C4×C8×C59—21no—
68C24×C59—41no—
69#41×C59—41no—
70#41×C59—41no—
71#41×C59—41no—
72#41×C59—41no—
73C2×Q8×C59—21no—
74C2×D4×C59—21no—
75C2×D4×C59—22no—
76C2×D4×C59—21no—
77C2×D4×C59—22no—
78C2×D4×C59—21no—
79C2×D4×C59—22no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—81no—
89C2×D4×C59—81no—
90C2×D4×C59—82no—
91C22×C4×C59—41no—
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×C4×C59#4111yescharacteristic
95SD16×C59—44no—
96SD16×C59—44no—
97D8×C59—44no—
98D8×C59—44no—
99M16×C59—21no—
100M16×C59—21no—
101M16×C59—41no—
102M16×C59—41no—
103M16×C59D812yes—
104M16×C59SD1612yes—
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C2×C8×C59—22no—
108C2×C8×C59—41no—
109C2×C8×C59—41no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C2×C8×C59—81no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—21no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59#4111yescharacteristic
122C23×C59—41no—
123C23×C59—42no—
124C23×C59—82no—
125C23×C59—82no—
126C23×C59—81no—
127C23×C59—81no—
128C23×C59#12411yescharacteristic
129Q8×C59—41no—
130Q8×C59—41no—
131D4×C59—41no—
132D4×C59—42no—
133D4×C59—41no—
134D4×C59—42no—
135D4×C59—41no—
136D4×C59—41no—
137D4×C59—81no—
138D4×C59—82no—
139D4×C59—82no—
140D4×C59—82no—
141D4×C59—81no—
142D4×C59—81no—
143D4×C59—81no—
144D4×C59—81no—
145D4×C59—81no—
146D4×C59—81no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59#12011yescharacteristic
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12311yescharacteristic
158C2×C4×C59#12311yescharacteristic
159C2×C4×C59C2×D811yescharacteristic
160C2×C4×C59C2×SD1611yescharacteristic
161C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C8×C59—22no—
164C8×C59—22no—
165C8×C59—42no—
166C8×C59—81no—
167C8×C59—81no—
168C8×C59—81no—
169#10549C5911yesmaximal, characteristic, direct product factor, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—82no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—82no—
176C22×C59—81no—
177C22×C59—82no—
178C22×C59—162no—
179C22×C59—162no—
180C22×C59#39411yescharacteristic
181C22×C59#48611yescharacteristic
182C22×C59#49011yescharacteristic
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—81no—
187C4×C59#44711yescharacteristic
188C4×C59#45011yescharacteristic
189C4×C59#45911yescharacteristic
190C4×C59#46011yescharacteristic
191C4×C59#47111yescharacteristic
192C4×C59#47211yescharacteristic
193#3151C2×C5911yescharacteristic
194#1950C2×C5911yescharacteristic, complemented
195#1547C2×C5911yescharacteristic, complemented
196#1545C2×C5911yescharacteristic, complemented
197C2×#331C2×C5911yescharacteristic, complemented
198#1480C2×C5911yescharacteristic, complemented
199#1283C2×C5911yescharacteristic, complemented
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—161no—
203C2×C59—162no—
204C2×C59#204111yescharacteristic
205C2×C59#204811yescharacteristic
206C2×C59#205111yescharacteristic
207C2×AGL(1,ℤ/8)—21no—
208C2×AGL(1,ℤ/8)—21no—
209#532C22×C5911yescharacteristic
210C2×#127C22×C5911yescharacteristic
211#496—22no—
212#493—21no—
213#492—21no—
214#431C22×C5911yescharacteristic
215C2×#108—21no—
216#423C22×C5911yescharacteristic
217C2×C4⋊C8C22×C5911yescharacteristic
218C2×C4⋊C8C22×C5911yescharacteristic
219C4×M16C22×C5911yescharacteristic
220#341—21no—
221#332—21no—
222#331—22no—
223#331—21no—
224#331—22no—
225D4⋊C8—22no—
226C59#1054911yescharacteristic, direct product factor, Sylow
227C2×#41—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231AGL(1,ℤ/8)—42no—
232AGL(1,ℤ/8)—42no—
233AGL(1,ℤ/8)—42no—
234AGL(1,ℤ/8)—42no—
235C2×SD16—22no—
236C2×SD16—22no—
237C2×D8—22no—
238C2×D8—22no—
239C2×M16—21no—
240C2×M16D4×C5911yescharacteristic
241C2×M16D4×C5911yescharacteristic
242C22×C8—21no—
243C22×C8—21no—
244C4⋊Q8D4×C5911yescharacteristic
245#127—21no—
246#127—21no—
247#127—21no—
248#127D4×C5912yes—
249#127D4×C5911yescharacteristic
250#124—21no—
251#121—21no—
252#121—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C42C23×C5911yescharacteristic
256#108—42no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C5912yes—
264C22⋊C8—21no—
265#97—21no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4—82no—
291C22×C4—41no—
292C22×C4C2×D4×C5911yescharacteristic
293C22×C4#41×C5911yescharacteristic
294C22×C4#41×C5911yescharacteristic
295SD16—44no—
296SD16—44no—
297D8—44no—
298D8—44no—
299M16—21no—
300M16—21no—
301M16—41no—
302M16—41no—
303M16D8×C5912yes—
304M16SD16×C5912yes—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—81no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42C2×D4×C5911yescharacteristic
319C42C2×D4×C5911yescharacteristic
320C42C2×D4×C5911yescharacteristic
321C42#41×C5911yescharacteristic
322C23—41no—
323C23—42no—
324C23—82no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23#124×C5911yescharacteristic
329Q8—41no—
330Q8—41no—
331D4—41no—
332D4—42no—
333D4—41no—
334D4—42no—
335D4—41no—
336D4—41no—
337D4—81no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#120×C5911yescharacteristic
354C2×C4#121×C5911yescharacteristic
355C2×C4#121×C5911yescharacteristic
356C2×C4#121×C5911yescharacteristic
357C2×C4#123×C5911yescharacteristic
358C2×C4#123×C5911yescharacteristic
359C2×C4C2×D8×C5911yescharacteristic
360C2×C4C2×SD16×C5911yescharacteristic
361C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
362C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
363C8—22no—
364C8—22no—
365C8—42no—
366C8—81no—
367C8—81no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—82no—
377C22—162no—
378C22—162no—
379C22C59×#39411yescharacteristic
380C22C59×#48611yescharacteristic
381C22C59×#49011yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4C59×#44711yescharacteristic
387C4C59×#45011yescharacteristic
388C4C59×#45911yescharacteristic
389C4C59×#46011yescharacteristic
390C4C59×#47111yescharacteristic
391C4C59×#47211yescharacteristic
392C2—21no—
393C2—21no—
394C2—162no—
395C2—161no—
396C2C59×#204111yescharacteristic
397C2C59×#204811yescharacteristic
398C2C59×#205111yescharacteristic
subgroupG/H
center, socleC22×C59#394
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3151C2
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
2. C59×#1950C2semidirect
3. C59×#1547C2semidirect
4. C59×#1545C2semidirect
5. C2×C59×#331C2semidirect
6. C59×#1480C2semidirect
7. C59×#1283C2semidirect
169. #10549C59direct (N × H)
194. #1950C2×C59semidirect
195. #1547C2×C59semidirect
196. #1545C2×C59semidirect
197. C2×#331C2×C59semidirect
198. #1480C2×C59semidirect
199. #1283C2×C59semidirect
226. C59#10549direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31512
C59×#19502
C59×#15472
C59×#15452
C2×C59×#3312
C59×#14802
C59×#12832
#1054959