Finite Group Databasebeta

← C59×#8304 #28420413

Subgroups — C59×#8304 #28420413

Order 15104.

1238 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C59C211yesmaximal, characteristic
2C59×#1779C212yesmaximal, complemented
3C2×C59×#324C212yesmaximal, complemented
4C2×C59×#323C212yesmaximal, complemented
5C23×D4×C59C411yescharacteristic
6C24×C4×C59C2211yescharacteristic
7C2×#123×C59—24no—
8C2×#123×C59—24no—
9C2×#123×C59—24no—
10C22×C4⋊C4×C59C411yescharacteristic
11C22×C4⋊C4×C59C2211yescharacteristic
12C22×C22⋊C4×C59C411yescharacteristic
13C22×C22⋊C4×C59C411yescharacteristic
14C22×C22⋊C4×C59C2211yescharacteristic
15C2×C22⋊C8×C59—24no—
16C2×C22⋊C8×C59C2214yes—
17C59×#324—24no—
18C59×#323—24no—
19C25×C59C2×C411yescharacteristic
20C25×C59C2×C411yescharacteristic
21C22×D4×C59—22no—
22C22×D4×C59—22no—
23C22×D4×C59—22no—
24C22×D4×C59—44no—
25C23×C4×C59—22no—
26C23×C4×C59—22no—
27C23×C4×C59—44no—
28C23×C4×C59C2×C411yescharacteristic
29C23×C4×C59C2×C411yescharacteristic
30C23×C4×C59C2×C411yescharacteristic
31C23×C4×C59C2×C411yescharacteristic
32C23×C4×C59D414yes—
33C23×C4×C59C2311yescharacteristic
34C22×C8×C59—24no—
35#123×C59—232no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59—22no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59D414yes—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59D414yes—
51C22⋊C8×C59—24no—
52C22⋊C8×C59—24no—
53C22⋊C8×C59—48no—
54C24×C59—24no—
55C24×C59—21no—
56C24×C59—21no—
57C24×C59—21no—
58C24×C59—21no—
59C24×C59—24no—
60C24×C59—24no—
61C24×C59—42no—
62C24×C59—42no—
63C24×C59—42no—
64C24×C59C22⋊C412yes—
65C24×C59C22⋊C412yes—
66C24×C59C22×C411yescharacteristic
67C2×D4×C59—28no—
68C2×D4×C59—48no—
69C2×D4×C59—44no—
70C2×D4×C59—48no—
71C2×D4×C59—44no—
72C22×C4×C59—28no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—24no—
82C22×C4×C59—28no—
83C22×C4×C59—42no—
84C22×C4×C59—44no—
85C22×C4×C59—42no—
86C22×C4×C59—48no—
87C22×C4×C59—48no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59C22⋊C412yes—
91C22×C4×C59C22⋊C412yes—
92C22×C4×C59C22⋊C412yes—
93C22×C4×C59C22⋊C412yes—
94C22×C4×C59C2×D412yes—
95C22×C4×C59C2×D412yes—
96C22×C4×C59C2×D412yes—
97C2×C8×C59—48no—
98C2×C8×C59—44no—
99C4⋊C4×C59—28no—
100C4⋊C4×C59—28no—
101C22⋊C4×C59—28no—
102C22⋊C4×C59—28no—
103C22⋊C4×C59—28no—
104C23×C59—22no—
105C23×C59—22no—
106C23×C59—24no—
107C23×C59—24no—
108C23×C59—22no—
109C23×C59—24no—
110C23×C59—22no—
111C23×C59—24no—
112C23×C59—22no—
113C23×C59—24no—
114C23×C59—416no—
115C23×C59—42no—
116C23×C59—42no—
117C23×C59—42no—
118C23×C59—42no—
119C23×C59—42no—
120C23×C59—42no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—42no—
124C23×C59—42no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—88no—
128C23×C59#9914yes—
129C23×C59C2×C22⋊C411yescharacteristic
130C23×C59C2×C22⋊C411yescharacteristic
131C23×C59C2×C22⋊C411yescharacteristic
132C23×C59#13114yes—
133D4×C59—416no—
134C2×C4×C59—24no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—24no—
140C2×C4×C59—24no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—416no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—42no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59#12014yes—
155C8×C59—84no—
156#8304C5911yesmaximal, characteristic, direct product factor, Sylow
157C22×C59—28no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—28no—
163C22×C59—24no—
164C22×C59—48no—
165C22×C59—48no—
166C22×C59—44no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59—84no—
173C22×C59—84no—
174C22×C59—84no—
175C22×C59#37911yescharacteristic
176C22×C59#40812yes—
177C22×C59#40812yes—
178C22×C59#40812yes—
179C22×C59C2×#9912yes—
180C22×C59#41312yes—
181C4×C59—24no—
182C4×C59—42no—
183C4×C59—42no—
184C4×C59—42no—
185C4×C59—42no—
186C22×#123C2×C5911yescharacteristic
187#1779C2×C5912yescomplemented
188C2×#324C2×C5912yescomplemented
189C2×#323C2×C5912yescomplemented
190C2×C59—22no—
191C2×C59—22no—
192C2×C59—44no—
193C2×C59—82no—
194C2×C59#146814yes—
195C2×C59#178011yescharacteristic
196C2×C59#178111yescharacteristic
197C2×C59#178511yescharacteristic
198C23×D4C4×C5911yescharacteristic
199C24×C4C22×C5911yescharacteristic
200C2×#123—24no—
201C2×#123—24no—
202C2×#123—24no—
203C22×C4⋊C4C4×C5911yescharacteristic
204C22×C4⋊C4C22×C5911yescharacteristic
205C22×C22⋊C4C4×C5911yescharacteristic
206C22×C22⋊C4C4×C5911yescharacteristic
207C22×C22⋊C4C22×C5911yescharacteristic
208C2×C22⋊C8—24no—
209C2×C22⋊C8C22×C5914yes—
210#324—24no—
211#323—24no—
212C59#830411yescharacteristic, direct product factor, Sylow
213C25C2×C4×C5911yescharacteristic
214C25C2×C4×C5911yescharacteristic
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4C2×C4×C5911yescharacteristic
223C23×C4C2×C4×C5911yescharacteristic
224C23×C4C2×C4×C5911yescharacteristic
225C23×C4C2×C4×C5911yescharacteristic
226C23×C4D4×C5914yes—
227C23×C4C23×C5911yescharacteristic
228C22×C8—24no—
229#123—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4D4×C5914yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4D4×C5914yes—
245C22⋊C8—24no—
246C22⋊C8—24no—
247C22⋊C8—48no—
248C24—24no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—24no—
254C24—24no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C22⋊C4×C5912yes—
259C24C22⋊C4×C5912yes—
260C24C22×C4×C5911yescharacteristic
261C2×D4—28no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—48no—
265C2×D4—44no—
266C22×C4—28no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C22⋊C4×C5912yes—
285C22×C4C22⋊C4×C5912yes—
286C22×C4C22⋊C4×C5912yes—
287C22×C4C22⋊C4×C5912yes—
288C22×C4C2×D4×C5912yes—
289C22×C4C2×D4×C5912yes—
290C22×C4C2×D4×C5912yes—
291C2×C8—48no—
292C2×C8—44no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—28no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—24no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—24no—
308C23—416no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—88no—
322C23#99×C5914yes—
323C23C2×C22⋊C4×C5911yescharacteristic
324C23C2×C22⋊C4×C5911yescharacteristic
325C23C2×C22⋊C4×C5911yescharacteristic
326C23#131×C5914yes—
327D4—416no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—416no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4#120×C5914yes—
349C8—84no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—28no—
356C22—24no—
357C22—48no—
358C22—48no—
359C22—44no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22C59×#37911yescharacteristic
369C22C59×#40812yes—
370C22C59×#40812yes—
371C22C59×#40812yes—
372C22C2×#99×C5912yes—
373C22C59×#41312yes—
374C4—24no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C2—22no—
380C2—22no—
381C2—44no—
382C2—82no—
383C2C59×#146814yes—
384C2C59×#178011yescharacteristic
385C2C59×#178111yescharacteristic
386C2C59×#178511yescharacteristic
subgroupG/H
center, socleC23×C59C2×C22⋊C4
commutatorC24C22×C4×C59
Frattini, cosocleC23×C4C23×C59
prime radicalC23×D4×C59C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C24×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1779C2semidirect
3. C2×C59×#324C2semidirect
4. C2×C59×#323C2semidirect
156. #8304C59direct (N × H)
187. #1779C2×C59semidirect
188. C2×#324C2×C59semidirect
189. C2×#323C2×C59semidirect
212. C59#8304direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#123×C592
C59×#17792
C2×C59×#3242
C2×C59×#3232
#830459