Finite Group Databasebeta

← C59×#18997 #28431106

Subgroups — C59×#18997 #28431106

Order 15104.

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 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×C59×H(3,𝔽4)C211yesmaximal, characteristic
2C59×#3215C212yesmaximal, complemented
3C59×#3140C211yesmaximal, characteristic, complemented
4C59×#1659C218yesmaximal, complemented
5C59×#1533C211yesmaximal, characteristic, complemented
6C59×#1527C212yesmaximal, complemented
7C59×H(3,𝔽4)—24no—
8C59×H(3,𝔽4)—22no—
9C59×H(3,𝔽4)C2214yes—
10C59×#533C2211yescharacteristic, complemented
11C2×#124×C59—21no—
12C2×#124×C59—22no—
13C2×#124×C59C2212yes—
14C2×#124×C59C2211yescharacteristic
15C2×#120×C59—21no—
16C2×#120×C59—22no—
17C59×#497C2214yescomplemented
18C59×#489C2214yescomplemented
19C59×#489C2214yescomplemented
20C59×#488—22no—
21C59×#464—24no—
22C59×#451—22no—
23C59×#450—22no—
24C59×#432C2212yescomplemented
25C59×#419—21no—
26C59×#418—21no—
27C4×M16×C59C2211yescharacteristic, complemented
28C59×#329C2218yescomplemented
29C59×#329C2214yescomplemented
30C25×C59—21no—
31C22×Q8×C59D411yescharacteristic
32C22×Q8×C59D412yes—
33C22×D4×C59—22no—
34C22×D4×C59—21no—
35C22×D4×C59D411yescharacteristic
36C22×D4×C59D412yes—
37C2×Q16×C59—24no—
38C2×SD16×C59—24no—
39C2×M16×C59—21no—
40C4⋊Q8×C59C2312yescomplemented
41#125×C59—21no—
42#124×C59—24no—
43#124×C59—28no—
44#124×C59—24no—
45#124×C59—28no—
46#124×C59—22no—
47#124×C59—22no—
48#124×C59—22no—
49#124×C59D414yes—
50#124×C59D412yes—
51#124×C59C2312yes—
52#124×C59C2314yes—
53C22⋊Q8×C59—22no—
54#120×C59—24no—
55#120×C59—28no—
56#120×C59—24no—
57#120×C59—28no—
58C4×Q8×C59—22no—
59#117×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—22no—
62C2×C22⋊C4×C59—22no—
63C2×C22⋊C4×C59—22no—
64C2×C22⋊C4×C59—22no—
65C2×C42×C59—21no—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—22no—
68#103×C59—24no—
69#103×C59—24no—
70#102×C59—24no—
71#102×C59—24no—
72C22⋊C8×C59—22no—
73#97×C59C2312yescomplemented
74#97×C59C2314yescomplemented
75C4×C8×C59—21no—
76C24×C59—24no—
77C24×C59—21no—
78C24×C59—22no—
79C24×C59—84no—
80C24×C59—82no—
81C2×Q8×C59—21no—
82C2×Q8×C59—22no—
83C2×Q8×C59—22no—
84C2×Q8×C59—42no—
85C2×Q8×C59—82no—
86C2×Q8×C59C2×D412yes—
87C2×Q8×C59C2×D414yes—
88C2×D4×C59—24no—
89C2×D4×C59—28no—
90C2×D4×C59—21no—
91C2×D4×C59—22no—
92C2×D4×C59—42no—
93C2×D4×C59—82no—
94C2×D4×C59—84no—
95C2×D4×C59—82no—
96C2×D4×C59C2×D412yes—
97C2×D4×C59C2×D414yes—
98C22×C4×C59—22no—
99C22×C4×C59—21no—
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D412yes—
102Q16×C59—84no—
103SD16×C59—84no—
104M16×C59—42no—
105C2×C8×C59—22no—
106C2×C8×C59—24no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59—42no—
110C22⋊C4×C59—28no—
111C22⋊C4×C59—24no—
112C22⋊C4×C59—24no—
113C22⋊C4×C59—24no—
114C22⋊C4×C59—24no—
115C22⋊C4×C59—24no—
116C22⋊C4×C59—24no—
117C22⋊C4×C59—24no—
118C22⋊C4×C59—41no—
119C42×C59—24no—
120C42×C59—41no—
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D412yes—
123C42×C59C2411yescharacteristic
124C23×C59—22no—
125C23×C59—24no—
126C23×C59—44no—
127C23×C59—44no—
128C23×C59—44no—
129C23×C59—42no—
130C23×C59—44no—
131C23×C59—84no—
132C23×C59—88no—
133C23×C59—816no—
134C23×C59#12011yescharacteristic
135Q8×C59—44no—
136Q8×C59—42no—
137Q8×C59—44no—
138Q8×C59—81no—
139Q8×C59—82no—
140D4×C59—44no—
141D4×C59—42no—
142D4×C59—81no—
143D4×C59—82no—
144D4×C59—84no—
145D4×C59—88no—
146C2×C4×C59—22no—
147C2×C4×C59—22no—
148C2×C4×C59—24no—
149C2×C4×C59—42no—
150C2×C4×C59—44no—
151C2×C4×C59—42no—
152C2×C4×C59—42no—
153C2×C4×C59#12011yescharacteristic
154C2×C4×C59#12012yes—
155C2×C4×C59C22×D411yescharacteristic
156C2×C4×C59C22×D412yes—
157C8×C59—42no—
158C8×C59—44no—
159#18997C5911yesmaximal, characteristic, direct product factor, Sylow
160C22×C59—22no—
161C22×C59—42no—
162C22×C59—44no—
163C22×C59—44no—
164C22×C59—44no—
165C22×C59—44no—
166C22×C59—84no—
167C22×C59—88no—
168C22×C59—816no—
169C22×C59#45512yes—
170C22×C59C2×#12011yescharacteristic
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—22no—
174C4×C59—22no—
175C4×C59—42no—
176C4×C59—44no—
177C4×C59—82no—
178C2×H(3,𝔽4)C2×C5911yescharacteristic
179#3215C2×C5912yescomplemented
180#3140C2×C5911yescharacteristic, complemented
181#1659C2×C5918yescomplemented
182#1533C2×C5911yescharacteristic, complemented
183#1527C2×C5912yescomplemented
184C2×C59—22no—
185C2×C59—82no—
186C2×C59—84no—
187C2×C59C2×#45511yescharacteristic
188C2×C59#301912yes—
189H(3,𝔽4)—24no—
190H(3,𝔽4)—22no—
191H(3,𝔽4)C22×C5914yes—
192#533C22×C5911yescharacteristic, complemented
193C2×#124—21no—
194C2×#124—22no—
195C2×#124C22×C5911yescharacteristic
196C2×#124C22×C5912yes—
197C2×#120—21no—
198C2×#120—22no—
199#497C22×C5914yescomplemented
200#489C22×C5914yescomplemented
201#489C22×C5914yescomplemented
202#488—22no—
203#464—24no—
204#451—22no—
205#450—22no—
206#432C22×C5912yescomplemented
207#419—21no—
208#418—21no—
209C4×M16C22×C5911yescharacteristic, complemented
210#329C22×C5918yescomplemented
211#329C22×C5914yescomplemented
212C59#1899711yescharacteristic, direct product factor, Sylow
213C25—21no—
214C22×Q8D4×C5911yescharacteristic
215C22×Q8D4×C5912yes—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4D4×C5911yescharacteristic
219C22×D4D4×C5912yes—
220C2×Q16—24no—
221C2×SD16—24no—
222C2×M16—21no—
223C4⋊Q8C23×C5912yescomplemented
224#125—21no—
225#124—24no—
226#124—28no—
227#124—24no—
228#124—28no—
229#124—22no—
230#124—22no—
231#124—22no—
232#124D4×C5914yes—
233#124D4×C5912yes—
234#124C23×C5912yes—
235#124C23×C5914yes—
236C22⋊Q8—22no—
237#120—24no—
238#120—28no—
239#120—24no—
240#120—28no—
241C4×Q8—22no—
242#117—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42—21no—
249C2×C42C23×C5911yescharacteristic
250C4⋊C8—22no—
251#103—24no—
252#103—24no—
253#102—24no—
254#102—24no—
255C22⋊C8—22no—
256#97C23×C5912yescomplemented
257#97C23×C5914yescomplemented
258C4×C8—21no—
259C24—24no—
260C24—21no—
261C24—22no—
262C24—84no—
263C24—82no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×Q8C2×D4×C5912yes—
270C2×Q8C2×D4×C5914yes—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—82no—
277C2×D4—84no—
278C2×D4—82no—
279C2×D4C2×D4×C5912yes—
280C2×D4C2×D4×C5914yes—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4C2×D4×C5911yescharacteristic
284C22×C4C2×D4×C5912yes—
285Q16—84no—
286SD16—84no—
287M16—42no—
288C2×C8—22no—
289C2×C8—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C42—24no—
303C42—41no—
304C42C2×D4×C5911yescharacteristic
305C42C2×D4×C5912yes—
306C42C24×C5911yescharacteristic
307C23—22no—
308C23—24no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—44no—
314C23—84no—
315C23—88no—
316C23—816no—
317C23#120×C5911yescharacteristic
318Q8—44no—
319Q8—42no—
320Q8—44no—
321Q8—81no—
322Q8—82no—
323D4—44no—
324D4—42no—
325D4—81no—
326D4—82no—
327D4—84no—
328D4—88no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#120×C5911yescharacteristic
337C2×C4#120×C5912yes—
338C2×C4C22×D4×C5911yescharacteristic
339C2×C4C22×D4×C5912yes—
340C8—42no—
341C8—44no—
342C22—22no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22C59×#45512yes—
352C22C2×#120×C5911yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—44no—
359C4—82no—
360C2—22no—
361C2—82no—
362C2—84no—
363C2C2×C59×#45511yescharacteristic
364C2C59×#301912yes—
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC2×C59×H(3,𝔽4)C2
LCSG ▷ 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×#3215C2semidirect
3. C59×#3140C2semidirect
4. C59×#1659C2semidirect
5. C59×#1533C2semidirect
6. C59×#1527C2semidirect
10. C59×#533C22semidirect
17. C59×#497C22semidirect
18. C59×#489C22semidirect
19. C59×#489C22semidirect
24. C59×#432C22semidirect
27. C4×M16×C59C22semidirect
28. C59×#329C22semidirect
29. C59×#329C22semidirect
40. C4⋊Q8×C59C23semidirect
73. #97×C59C23semidirect
74. #97×C59C23semidirect
159. #18997C59direct (N × H)
179. #3215C2×C59semidirect
180. #3140C2×C59semidirect
181. #1659C2×C59semidirect
182. #1533C2×C59semidirect
183. #1527C2×C59semidirect
192. #533C22×C59semidirect
199. #497C22×C59semidirect
200. #489C22×C59semidirect
201. #489C22×C59semidirect
206. #432C22×C59semidirect
209. C4×M16C22×C59semidirect
210. #329C22×C59semidirect
211. #329C22×C59semidirect
212. C59#18997direct (N × H)
223. C4⋊Q8C23×C59semidirect
256. #97C23×C59semidirect
257. #97C23×C59semidirect

Maximal subgroups

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

subgroupindex
C2×C59×H(3,𝔽4)2
C59×#32152
C59×#31402
C59×#16592
C59×#15332
C59×#15272
#1899759