Finite Group Databasebeta

← C59×#11822 #28423931

Subgroups — C59×#11822 #28423931

Order 15104.

434 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2395C211yesmaximal, characteristic
2C59×#1799C211yesmaximal, characteristic
3C59×#1466C211yesmaximal, characteristic
4C59×#1352C212yesmaximal, complemented
5C59×#1344C212yesmaximal, complemented
6C2×C4×D4×C59C2211yescharacteristic
7C2×#117×C59C411yescharacteristic, complemented
8C59×#432—21no—
9C2×#99×C59—21no—
10C59×#400C411yescharacteristic, complemented
11C59×#399C2211yescharacteristic
12C59×#398C411yescharacteristic, complemented
13C59×#397—21no—
14C59×#397C2211yescharacteristic
15C59×#394—21no—
16C59×#380—21no—
17C4×C4⋊C4×C59—21no—
18C4×C22⋊C4×C59—21no—
19C4×C22⋊C4×C59C411yescharacteristic, complemented
20C59×#342C2212yes—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×D4×C59D411yescharacteristic
24C23×C4×C59C2×C411yescharacteristic
25C23×C4×C59D411yescharacteristic
26C4×D4×C59—41no—
27C4×D4×C59—41no—
28#117×C59—41no—
29#117×C59—41no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—41no—
33C2×C4⋊C4×C59C2×C411yescharacteristic
34C2×C4⋊C4×C59C2×C411yescharacteristic
35C2×C4⋊C4×C59C2×C411yescharacteristic
36C2×C22⋊C4×C59—21no—
37C2×C22⋊C4×C59—21no—
38C2×C22⋊C4×C59—21no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59C2311yescharacteristic
43C2×C42×C59—21no—
44C2×C42×C59—21no—
45C2×C42×C59D411yescharacteristic
46C4⋊C8×C59—41no—
47#99×C59—42no—
48C22⋊C8×C59—22no—
49#97×C59—41no—
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—41no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—42no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—82no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—42no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59—42no—
98C42×C59—41no—
99C42×C59—41no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59AGL(1,ℤ/8)11yescharacteristic
111C23×C59#13711yescharacteristic
112D4×C59—42no—
113D4×C59—42no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
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—41no—
138C2×C4×C59—41no—
139C2×C4×C59—81no—
140C2×C4×C59—82no—
141C2×C4×C59—82no—
142C2×C4×C59—82no—
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59#9911yescharacteristic
145C8×C59—82no—
146#11822C5911yesmaximal, characteristic, direct product factor, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59C2×#9911yescharacteristic
162C22×C59#41711yescharacteristic
163C22×C59#42111yescharacteristic
164C4×C59—21no—
165C4×C59—42no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—81no—
171C4×C59—81no—
172C4×C59—81no—
173C4×C59—162no—
174C4×C59#35212yes—
175#2395C2×C5911yescharacteristic
176#1799C2×C5911yescharacteristic
177#1466C2×C5911yescharacteristic
178#1352C2×C5912yescomplemented
179#1344C2×C5912yescomplemented
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—41no—
183C2×C59—41no—
184C2×C59—81no—
185C2×C59#151311yescharacteristic
186C2×C59C2×#35211yescharacteristic
187C2×C59#212611yescharacteristic
188C2×C4×D4C22×C5911yescharacteristic
189C2×#117C4×C5911yescharacteristic, complemented
190#432—21no—
191C2×#99—21no—
192#400C4×C5911yescharacteristic, complemented
193#399C22×C5911yescharacteristic
194#398C4×C5911yescharacteristic, complemented
195#397—21no—
196#397C22×C5911yescharacteristic
197#394—21no—
198#380—21no—
199C4×C4⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4C4×C5911yescharacteristic, complemented
202#342C22×C5912yes—
203#323—21no—
204#323C22×C5912yes—
205C59#1182211yescharacteristic, direct product factor, Sylow
206C22×D4D4×C5911yescharacteristic
207C23×C4C2×C4×C5911yescharacteristic
208C23×C4D4×C5911yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×C4×C5911yescharacteristic
217C2×C4⋊C4C2×C4×C5911yescharacteristic
218C2×C4⋊C4C2×C4×C5911yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4D4×C5911yescharacteristic
225C2×C22⋊C4C23×C5911yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42D4×C5911yescharacteristic
229C4⋊C8—41no—
230#99—42no—
231C22⋊C8—22no—
232#97—41no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—41no—
238#95C2×C4×C5911yescharacteristic
239#95C2×C4×C5911yescharacteristic
240C24C22⋊C4×C5911yescharacteristic
241C24C2×D4×C5911yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22⋊C4×C5911yescharacteristic
258C22×C4C22⋊C4×C5911yescharacteristic
259C22×C4C22⋊C4×C5911yescharacteristic
260C22×C4C22×C4×C5911yescharacteristic
261C22×C4C2×D4×C5911yescharacteristic
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊C4×C5911yescharacteristic
293C23AGL(1,ℤ/8)×C5911yescharacteristic
294C23#137×C5911yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#99×C5911yescharacteristic
327C2×C4#99×C5911yescharacteristic
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#99×C5911yescharacteristic
344C22C59×#41711yescharacteristic
345C22C59×#42111yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—162no—
356C4C59×#35212yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2C59×#151311yescharacteristic
363C2C2×C59×#35211yescharacteristic
364C2C59×#212611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C22⋊C4C23×C59
prime radicalC22×D4×C59D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. C59×#1352C2semidirect
5. C59×#1344C2semidirect
7. C2×#117×C59C4semidirect
10. C59×#400C4semidirect
12. C59×#398C4semidirect
19. C4×C22⋊C4×C59C4semidirect
146. #11822C59direct (N × H)
178. #1352C2×C59semidirect
179. #1344C2×C59semidirect
189. C2×#117C4×C59semidirect
192. #400C4×C59semidirect
194. #398C4×C59semidirect
201. C4×C22⋊C4C4×C59semidirect
205. C59#11822direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#23952
C59×#17992
C59×#14662
C59×#13522
C59×#13442
#1182259