Finite Group Databasebeta

← C59×#11791 #28423900

Subgroups — C59×#11791 #28423900

Order 15104.

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

normal subgroups only

This group has 572 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×#2409C211yesmaximal, characteristic
2C4×#100×C59C211yesmaximal, characteristic
3C59×#1465C211yesmaximal, characteristic
4C59×#1354C214yesmaximal, complemented
5C23×Q8×C59C411yescharacteristic
6C2×C22⋊Q8×C59—21no—
7C2×C4×D4×C59C2211yescharacteristic
8C2×#100×C59—21no—
9C59×#407—21no—
10C4×M16×C59—21no—
11C59×#395C2212yes—
12C59×#393C412yes—
13C59×#391—21no—
14C59×#386—21no—
15C59×#385—21no—
16C59×#380C411yescharacteristic
17C59×#343C2212yes—
18C59×#324C2212yes—
19C59×#323—21no—
20C22×Q8×C59—21no—
21C22×Q8×C59—41no—
22C22×Q8×C59—41no—
23C22×Q8×C59—44no—
24C22×Q8×C59C2×C412yes—
25C22×D4×C59D411yescharacteristic
26C23×C4×C59—21no—
27C23×C4×C59C2×C411yescharacteristic
28C23×C4×C59D411yescharacteristic
29C2×M16×C59—22no—
30C2×M16×C59—41no—
31C22⋊Q8×C59—44no—
32C4×D4×C59—41no—
33C4×D4×C59—41no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59C2311yescharacteristic
37C2×C22⋊C4×C59—21no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59C2×C411yescharacteristic
40C2×C22⋊C4×C59D411yescharacteristic
41C2×C42×C59D411yescharacteristic
42#100×C59—24no—
43C22⋊C8×C59—22no—
44C22⋊C8×C59—41no—
45#97×C59—41no—
46C4×C8×C59—41no—
47#95×C59—22no—
48#95×C59—22no—
49#95×C59C2×C412yes—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C22⋊C411yescharacteristic
52C2×Q8×C59—24no—
53C2×Q8×C59—42no—
54C2×Q8×C59—44no—
55C2×Q8×C59—42no—
56C2×Q8×C59—48no—
57C2×Q8×C59—44no—
58C2×Q8×C59—44no—
59C2×Q8×C59—41no—
60C2×Q8×C59—41no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—22no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—44no—
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×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79M16×C59—24no—
80M16×C59—42no—
81M16×C59—82no—
82C2×C8×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—44no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—44no—
90C22⋊C4×C59—42no—
91C42×C59—21no—
92C42×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59#10412yes—
103C23×C59C2×C22⋊C411yescharacteristic
104Q8×C59—44no—
105Q8×C59—44no—
106Q8×C59—44no—
107Q8×C59—44no—
108D4×C59—44no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—24no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—48no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—81no—
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#9911yescharacteristic
131C2×C4×C59#13712yes—
132C8×C59—44no—
133C8×C59—82no—
134#11791C5911yesmaximal, characteristic, direct product factor, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—22no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—81no—
147C22×C59C2×#9911yescharacteristic
148C22×C59#41911yescharacteristic
149C22×C59C2×#10411yescharacteristic
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—41no—
153C4×C59—44no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—82no—
157C4×C59AΣL(1,𝔽16)12yes—
158#2409C2×C5911yescharacteristic
159C4×#100C2×C5911yescharacteristic
160#1465C2×C5911yescharacteristic
161#1354C2×C5914yescomplemented
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—42no—
165C2×C59—81no—
166C2×C59#150811yescharacteristic
167C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
168C2×C59#213511yescharacteristic
169C23×Q8C4×C5911yescharacteristic
170C2×C22⋊Q8—21no—
171C2×C4×D4C22×C5911yescharacteristic
172C2×#100—21no—
173#407—21no—
174C4×M16—21no—
175#395C22×C5912yes—
176#393C4×C5912yes—
177#391—21no—
178#386—21no—
179#385—21no—
180#380C4×C5911yescharacteristic
181#343C22×C5912yes—
182#324C22×C5912yes—
183#323—21no—
184C59#1179111yescharacteristic, direct product factor, Sylow
185C22×Q8—21no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8C2×C4×C5912yes—
190C22×D4D4×C5911yescharacteristic
191C23×C4—21no—
192C23×C4C2×C4×C5911yescharacteristic
193C23×C4D4×C5911yescharacteristic
194C2×M16—22no—
195C2×M16—41no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C23×C5911yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×C4×C5911yescharacteristic
205C2×C22⋊C4D4×C5911yescharacteristic
206C2×C42D4×C5911yescharacteristic
207#100—24no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210#97—41no—
211C4×C8—41no—
212#95—22no—
213#95—22no—
214#95C2×C4×C5912yes—
215C24C22⋊C4×C5911yescharacteristic
216C24C22⋊C4×C5911yescharacteristic
217C2×Q8—24no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊C4×C5911yescharacteristic
240C22×C4C22⋊C4×C5911yescharacteristic
241C22×C4C22×C4×C5911yescharacteristic
242C22×C4C2×D4×C5911yescharacteristic
243C22×C4C2×D4×C5911yescharacteristic
244M16—24no—
245M16—42no—
246M16—82no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#104×C5912yes—
268C23C2×C22⋊C4×C5911yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—44no—
272Q8—44no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#99×C5911yescharacteristic
295C2×C4#99×C5911yescharacteristic
296C2×C4#137×C5912yes—
297C8—44no—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#99×C5911yescharacteristic
312C22C59×#41911yescharacteristic
313C22C2×#104×C5911yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4C59×AΣL(1,𝔽16)12yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2C59×#150811yescharacteristic
327C2C2×C59×AΣL(1,𝔽16)11yescharacteristic
328C2C59×#213511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C4⋊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×#1354C2semidirect
134. #11791C59direct (N × H)
161. #1354C2×C59semidirect
184. C59#11791direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#24092
C4×#100×C592
C59×#14652
C59×#13542
#1179159