Finite Group Databasebeta

← C59×#11563 #28423672

Subgroups — C59×#11563 #28423672

Order 15104.

616 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2541C211yesmaximal, characteristic
2C59×#1465C211yesmaximal, characteristic, complemented
3C59×#1459C211yesmaximal, characteristic, complemented
4C59×#1318C214yesmaximal, complemented
5C2×#127×C59C411yescharacteristic
6C2×#127×C59C412yes—
7C2×#121×C59—21no—
8C2×#121×C59—21no—
9C2×C4×D4×C59C2211yescharacteristic
10C59×#407—21no—
11C2×C22⋊C8×C59—21no—
12C59×#392—21no—
13C59×#392—21no—
14C59×#390C2212yes—
15C4×C4⋊C4×C59C411yescharacteristic
16C59×#324C2212yes—
17C59×#324C2212yes—
18C59×#323—21no—
19C59×#323—21no—
20C22×D4×C59—22no—
21C22×D4×C59—22no—
22C22×D4×C59—21no—
23C22×D4×C59—21no—
24C22×D4×C59C2×C411yescharacteristic
25C22×D4×C59C2×C412yes—
26C23×C4×C59D411yescharacteristic
27C23×C4×C59D411yescharacteristic
28C2×M16×C59—41no—
29C22×C8×C59—41no—
30#127×C59—41no—
31#127×C59—41no—
32#127×C59—44no—
33#121×C59—44no—
34#121×C59—44no—
35C4×D4×C59—41no—
36C4×D4×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59C2×C412yes—
45C22⋊C8×C59—22no—
46C22⋊C8×C59—22no—
47C22⋊C8×C59—41no—
48C22⋊C8×C59—41no—
49#95×C59—21no—
50C24×C59—22no—
51C24×C59C22⋊C411yescharacteristic
52C24×C59C22⋊C411yescharacteristic
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—24no—
56C2×D4×C59—41no—
57C2×D4×C59—42no—
58C2×D4×C59—42no—
59C2×D4×C59—44no—
60C2×D4×C59—44no—
61C2×D4×C59—42no—
62C2×D4×C59—42no—
63C2×D4×C59—42no—
64C2×D4×C59—44no—
65C2×D4×C59—44no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—82no—
69C2×D4×C59—82no—
70C2×D4×C59—82no—
71C22×C4×C59—22no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—42no—
77C22×C4×C59—42no—
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83M16×C59—82no—
84C2×C8×C59—42no—
85C2×C8×C59—42no—
86C2×C8×C59—82no—
87C4⋊C4×C59—24no—
88C4⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—44no—
92C22⋊C4×C59—44no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—42no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—42no—
106C23×C59—42no—
107C23×C59—42no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—42no—
112C23×C59#10012yes—
113C23×C59C2×C22⋊C411yescharacteristic
114D4×C59—44no—
115D4×C59—84no—
116D4×C59—82no—
117D4×C59—88no—
118D4×C59—88no—
119D4×C59—84no—
120D4×C59—82no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—22no—
124C2×C4×C59—24no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—42no—
129C2×C4×C59—42no—
130C2×C4×C59—42no—
131C2×C4×C59—42no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59#9911yescharacteristic
138C2×C4×C59#9911yescharacteristic
139C2×C4×C59#13112yes—
140C8×C59—82no—
141C8×C59—82no—
142#11563C5911yesmaximal, characteristic, direct product factor, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—82no—
156C22×C59—82no—
157C22×C59—82no—
158C22×C59—82no—
159C22×C59—82no—
160C22×C59—82no—
161C22×C59—82no—
162C22×C59#40811yescharacteristic
163C22×C59C2×#9911yescharacteristic
164C22×C59C2×#10011yescharacteristic
165C4×C59—21no—
166C4×C59—22no—
167C4×C59—44no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59#35312yes—
172#2541C2×C5911yescharacteristic
173#1465C2×C5911yescharacteristic, complemented
174#1459C2×C5911yescharacteristic, complemented
175#1318C2×C5914yescomplemented
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—42no—
179C2×C59—81no—
180C2×C59—162no—
181C2×C59#147111yescharacteristic
182C2×C59C2×#35311yescharacteristic
183C2×C59#213411yescharacteristic
184C2×#127C4×C5911yescharacteristic
185C2×#127C4×C5912yes—
186C2×#121—21no—
187C2×#121—21no—
188C2×C4×D4C22×C5911yescharacteristic
189#407—21no—
190C2×C22⋊C8—21no—
191#392—21no—
192#392—21no—
193#390C22×C5912yes—
194C4×C4⋊C4C4×C5911yescharacteristic
195#324C22×C5912yes—
196#324C22×C5912yes—
197#323—21no—
198#323—21no—
199C59#1156311yescharacteristic, direct product factor, Sylow
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×C4×C5911yescharacteristic
205C22×D4C2×C4×C5912yes—
206C23×C4D4×C5911yescharacteristic
207C23×C4D4×C5911yescharacteristic
208C2×M16—41no—
209C22×C8—41no—
210#127—41no—
211#127—41no—
212#127—44no—
213#121—44no—
214#121—44no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C23×C5911yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4D4×C5911yescharacteristic
222C2×C22⋊C4D4×C5911yescharacteristic
223C2×C42C2×C4×C5911yescharacteristic
224C2×C42C2×C4×C5912yes—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229#95—21no—
230C24—22no—
231C24C22⋊C4×C5911yescharacteristic
232C24C22⋊C4×C5911yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C22⋊C4×C5911yescharacteristic
259C22×C4C22⋊C4×C5911yescharacteristic
260C22×C4C22×C4×C5911yescharacteristic
261C22×C4C2×D4×C5911yescharacteristic
262C22×C4C2×D4×C5911yescharacteristic
263M16—82no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—82no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—21no—
274C42—21no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23#100×C5912yes—
293C23C2×C22⋊C4×C5911yescharacteristic
294D4—44no—
295D4—84no—
296D4—82no—
297D4—88no—
298D4—88no—
299D4—84no—
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#99×C5911yescharacteristic
318C2×C4#99×C5911yescharacteristic
319C2×C4#131×C5912yes—
320C8—82no—
321C8—82no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22C59×#40811yescharacteristic
342C22C2×#99×C5911yescharacteristic
343C22C2×#100×C5911yescharacteristic
344C4—22no—
345C4—21no—
346C4—44no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4C59×#35312yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—162no—
356C2C59×#147111yescharacteristic
357C2C2×C59×#35311yescharacteristic
358C2C59×#213411yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC59×#2541C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1465C2semidirect
3. C59×#1459C2semidirect
4. C59×#1318C2semidirect
142. #11563C59direct (N × H)
173. #1465C2×C59semidirect
174. #1459C2×C59semidirect
175. #1318C2×C59semidirect
199. C59#11563direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#25412
C59×#14652
C59×#14592
C59×#13182
#1156359