Finite Group Databasebeta

← C59×#7304 #28419413

Subgroups — C59×#7304 #28419413

Order 15104.

764 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C59×#1783C211yesmaximal, characteristic
2C59×#1781C211yesmaximal, characteristic
3C59×#1438C211yesmaximal, characteristic
4C23×D4×C59C2211yescharacteristic
5C22×C22⋊C4×C59—21no—
6C22×C22⋊C4×C59C412yescomplemented
7C2×#100×C59—21no—
8C2×#100×C59C412yescomplemented
9C2×#99×C59—21no—
10C2×#99×C59C412yescomplemented
11C59×#407—21no—
12C59×#379—21no—
13C2×#95×C59—21no—
14C25×C59C2×C411yescharacteristic
15C25×C59D411yescharacteristic
16C22×D4×C59—42no—
17C22×D4×C59—42no—
18C22×D4×C59—42no—
19C22×D4×C59C2×C411yescharacteristic
20C22×D4×C59C2×C411yescharacteristic
21C22×D4×C59D411yescharacteristic
22C22×D4×C59Q811yescharacteristic
23C23×C4×C59—22no—
24C23×C4×C59D411yescharacteristic
25C2×M16×C59—22no—
26C2×C22⋊C4×C59—22no—
27C2×C22⋊C4×C59—44no—
28C2×C22⋊C4×C59—42no—
29C2×C22⋊C4×C59—42no—
30C2×C22⋊C4×C59—44no—
31C2×C22⋊C4×C59—42no—
32#100×C59—42no—
33#100×C59—42no—
34#99×C59—42no—
35#99×C59—42no—
36C22⋊C8×C59—42no—
37#95×C59—42no—
38C24×C59—22no—
39C24×C59—22no—
40C24×C59—22no—
41C24×C59—41no—
42C24×C59—41no—
43C24×C59—41no—
44C24×C59—42no—
45C24×C59—42no—
46C24×C59—42no—
47C24×C59—42no—
48C24×C59C4211yescharacteristic
49C24×C59C22⋊C411yescharacteristic
50C24×C59C22⋊C411yescharacteristic
51C24×C59C4⋊C411yescharacteristic
52C24×C59C4⋊C411yescharacteristic
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—22no—
56C2×D4×C59—22no—
57C2×D4×C59—41no—
58C2×D4×C59—41no—
59C2×D4×C59—41no—
60C2×D4×C59—41no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C2×D4×C59—416no—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—24no—
69C22×C4×C59—41no—
70C22×C4×C59—42no—
71C22×C4×C59—42no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59—42no—
76C22×C4×C59—42no—
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C4⋊C411yescharacteristic
79M16×C59—42no—
80M16×C59—42no—
81C2×C8×C59—42no—
82C22⋊C4×C59—48no—
83C22⋊C4×C59—44no—
84C22⋊C4×C59—44no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—82no—
88C22⋊C4×C59—82no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—44no—
105C23×C59—42no—
106C23×C59—42no—
107C23×C59—42no—
108C23×C59—44no—
109C23×C59—44no—
110C23×C59—42no—
111C23×C59—44no—
112C23×C59—42no—
113C23×C59—42no—
114C23×C59—42no—
115C23×C59—42no—
116C23×C59—42no—
117C23×C59—42no—
118C23×C59—82no—
119C23×C59—81no—
120C23×C59—82no—
121C23×C59—82no—
122C23×C59—81no—
123C23×C59—81no—
124C23×C59—82no—
125C23×C59—81no—
126C23×C59—81no—
127C23×C59—81no—
128C23×C59#9511yescharacteristic
129C23×C59#9912yes—
130C23×C59#9912yes—
131D4×C59—44no—
132D4×C59—44no—
133D4×C59—44no—
134D4×C59—44no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—44no—
141C2×C4×C59—44no—
142C2×C4×C59—44no—
143C2×C4×C59—44no—
144C2×C4×C59—82no—
145C2×C4×C59—82no—
146C2×C4×C59—82no—
147C2×C4×C59—82no—
148C2×C4×C59—82no—
149C2×C4×C59—84no—
150C2×C4×C59#9912yes—
151C8×C59—82no—
152#7304C5911yesmaximal, characteristic, direct product factor, Sylow
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—21no—
156C22×C59—22no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—22no—
160C22×C59—22no—
161C22×C59—22no—
162C22×C59—22no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—42no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59—81no—
173C22×C59—84no—
174C22×C59—84no—
175C22×C59—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59—84no—
181C22×C59—84no—
182C22×C59#34211yescharacteristic
183C22×C59#34211yescharacteristic
184C22×C59#34211yescharacteristic
185C22×C59AΣL(1,𝔽16)12yes—
186C22×C59#35212yes—
187C4×C59—22no—
188C4×C59—41no—
189C4×C59—84no—
190C4×C59—162no—
191#1783C2×C5911yescharacteristic
192#1781C2×C5911yescharacteristic
193#1438C2×C5911yescharacteristic
194C2×C59—22no—
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—22no—
198C2×C59—42no—
199C2×C59—81no—
200C2×C59—81no—
201C2×C59—81no—
202C2×C59#130411yescharacteristic
203C2×C59#139012yes—
204C23×D4C22×C5911yescharacteristic
205C22×C22⋊C4—21no—
206C22×C22⋊C4C4×C5912yescomplemented
207C2×#100—21no—
208C2×#100C4×C5912yescomplemented
209C2×#99—21no—
210C2×#99C4×C5912yescomplemented
211#407—21no—
212#379—21no—
213C2×#95—21no—
214C59#730411yescharacteristic, direct product factor, Sylow
215C25C2×C4×C5911yescharacteristic
216C25D4×C5911yescharacteristic
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4C2×C4×C5911yescharacteristic
221C22×D4C2×C4×C5911yescharacteristic
222C22×D4D4×C5911yescharacteristic
223C22×D4Q8×C5911yescharacteristic
224C23×C4—22no—
225C23×C4D4×C5911yescharacteristic
226C2×M16—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233#100—42no—
234#100—42no—
235#99—42no—
236#99—42no—
237C22⋊C8—42no—
238#95—42no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C42×C5911yescharacteristic
250C24C22⋊C4×C5911yescharacteristic
251C24C22⋊C4×C5911yescharacteristic
252C24C4⋊C4×C5911yescharacteristic
253C24C4⋊C4×C5911yescharacteristic
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—416no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C22⋊C4×C5911yescharacteristic
279C22×C4C4⋊C4×C5911yescharacteristic
280M16—42no—
281M16—42no—
282C2×C8—42no—
283C22⋊C4—48no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—44no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—82no—
320C23—81no—
321C23—82no—
322C23—82no—
323C23—81no—
324C23—81no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23#95×C5911yescharacteristic
330C23#99×C5912yes—
331C23#99×C5912yes—
332D4—44no—
333D4—44no—
334D4—44no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4#99×C5912yes—
352C8—82no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—84no—
374C22—84no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C59×#34211yescharacteristic
383C22C59×#34211yescharacteristic
384C22C59×#34211yescharacteristic
385C22C59×AΣL(1,𝔽16)12yes—
386C22C59×#35212yes—
387C4—22no—
388C4—41no—
389C4—84no—
390C4—162no—
391C2—22no—
392C2—21no—
393C2—21no—
394C2—22no—
395C2—42no—
396C2—81no—
397C2—81no—
398C2—81no—
399C2C59×#130411yescharacteristic
400C2C59×#139012yes—
subgroupG/H
center, socleC22×C59#342
commutatorC24C42×C59
Frattini, cosocleC23×D4C22×C59
prime radicalC23×D4×C59C22
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C24×C59 ◁ C23×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
6. C22×C22⋊C4×C59C4semidirect
8. C2×#100×C59C4semidirect
10. C2×#99×C59C4semidirect
152. #7304C59direct (N × H)
206. C22×C22⋊C4C4×C59semidirect
208. C2×#100C4×C59semidirect
210. C2×#99C4×C59semidirect
214. C59#7304direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#17832
C59×#17812
C59×#14382
#730459