Finite Group Databasebeta

← C59×#11554 #28423663

Subgroups — C59×#11554 #28423663

Order 15104.

408 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2500C211yesmaximal, characteristic
2C59×#1464C211yesmaximal, characteristic, complemented
3C59×#1458C211yesmaximal, characteristic, complemented
4C59×#1325C212yesmaximal, complemented
5C59×#1318C212yesmaximal, complemented
6C2×#121×C59C411yescharacteristic
7C2×#117×C59C2211yescharacteristic
8C59×#432—21no—
9C59×#431—21no—
10C59×#394—21no—
11C59×#394—21no—
12C59×#390C411yescharacteristic
13C59×#390C2211yescharacteristic
14C59×#388—21no—
15C59×#388—21no—
16C59×#384C2211yescharacteristic
17C59×#382C411yescharacteristic
18C4×C22⋊C4×C59C411yescharacteristic
19C59×#324C2212yes—
20C59×#324C2212yes—
21C59×#323—21no—
22C59×#323—21no—
23C22×D4×C59—21no—
24C22×D4×C59C2×C411yescharacteristic
25C23×C4×C59C2×C411yescharacteristic
26#121×C59—41no—
27#121×C59—41no—
28#117×C59—41no—
29#117×C59—41no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59C2×C411yescharacteristic
33C2×C4⋊C4×C59C2×C411yescharacteristic
34C2×C4⋊C4×C59C2311yescharacteristic
35C2×C22⋊C4×C59—21no—
36C2×C22⋊C4×C59—21no—
37C2×C22⋊C4×C59—21no—
38C2×C22⋊C4×C59—21no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47C4⋊C8×C59—41no—
48C4⋊C8×C59—41no—
49C22⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51#97×C59—41no—
52C4×C8×C59—41no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56C24×C59—21no—
57C24×C59C22⋊C411yescharacteristic
58C2×D4×C59—22no—
59C2×D4×C59—41no—
60C2×D4×C59—41no—
61C2×D4×C59—42no—
62C2×D4×C59—81no—
63C2×D4×C59—81no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—22no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—41no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C8×C59—42no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—81no—
83C4⋊C4×C59—81no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—81no—
92C22⋊C4×C59—81no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59—22no—
98C42×C59—41no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59C2×C22⋊C411yescharacteristic
112D4×C59—82no—
113D4×C59—84no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—22no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59#9911yescharacteristic
138C2×C4×C59#9911yescharacteristic
139C2×C4×C59#13112yes—
140C8×C59—82no—
141C8×C59—82no—
142#11554C5911yesmaximal, characteristic, direct product factor, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59—81no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59#40811yescharacteristic
158C22×C59C2×#9911yescharacteristic
159C22×C59#41311yescharacteristic
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—22no—
163C4×C59—42no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—81no—
169C4×C59—81no—
170#2500C2×C5911yescharacteristic
171#1464C2×C5911yescharacteristic, complemented
172#1458C2×C5911yescharacteristic, complemented
173#1325C2×C5912yescomplemented
174#1318C2×C5912yescomplemented
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—81no—
178C2×C59—161no—
179C2×C59#146811yescharacteristic
180C2×C59#212811yescharacteristic
181C2×C59#213211yescharacteristic
182C2×#121C4×C5911yescharacteristic
183C2×#117C22×C5911yescharacteristic
184#432—21no—
185#431—21no—
186#394—21no—
187#394—21no—
188#390C4×C5911yescharacteristic
189#390C22×C5911yescharacteristic
190#388—21no—
191#388—21no—
192#384C22×C5911yescharacteristic
193#382C4×C5911yescharacteristic
194C4×C22⋊C4C4×C5911yescharacteristic
195#324C22×C5912yes—
196#324C22×C5912yes—
197#323—21no—
198#323—21no—
199C59#1155411yescharacteristic, direct product factor, Sylow
200C22×D4—21no—
201C22×D4C2×C4×C5911yescharacteristic
202C23×C4C2×C4×C5911yescharacteristic
203#121—41no—
204#121—41no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×C4×C5911yescharacteristic
210C2×C4⋊C4C2×C4×C5911yescharacteristic
211C2×C4⋊C4C23×C5911yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4D4×C5911yescharacteristic
219C2×C22⋊C4D4×C5911yescharacteristic
220C2×C42C2×C4×C5911yescharacteristic
221C2×C42C2×C4×C5911yescharacteristic
222C2×C42D4×C5911yescharacteristic
223C2×C42D4×C5911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#97—41no—
229C4×C8—41no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C24C22⋊C4×C5911yescharacteristic
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—81no—
240C2×D4—81no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4C22⋊C4×C5911yescharacteristic
249C22×C4C22⋊C4×C5911yescharacteristic
250C22×C4C22⋊C4×C5911yescharacteristic
251C22×C4C22×C4×C5911yescharacteristic
252C22×C4C2×D4×C5911yescharacteristic
253C22×C4C2×D4×C5911yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊C4×C5911yescharacteristic
289D4—82no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#99×C5911yescharacteristic
315C2×C4#99×C5911yescharacteristic
316C2×C4#131×C5912yes—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C59×#40811yescharacteristic
334C22C2×#99×C5911yescharacteristic
335C22C59×#41311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C59×#146811yescharacteristic
351C2C59×#212811yescharacteristic
352C2C59×#213211yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC2×#121×C59C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C2×#117×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1464C2semidirect
3. C59×#1458C2semidirect
4. C59×#1325C2semidirect
5. C59×#1318C2semidirect
142. #11554C59direct (N × H)
171. #1464C2×C59semidirect
172. #1458C2×C59semidirect
173. #1325C2×C59semidirect
174. #1318C2×C59semidirect
199. C59#11554direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#25002
C59×#14642
C59×#14582
C59×#13252
C59×#13182
#1155459