Finite Group Databasebeta

← C59×#8498 #28420607

Subgroups — C59×#8498 #28420607

Order 15104.

692 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3544C211yesmaximal, characteristic
2C59×#1960C211yesmaximal, characteristic, complemented
3C59×#1558C211yesmaximal, characteristic, complemented
4C59×#1532C212yesmaximal, complemented
5C59×#1512C212yesmaximal, complemented
6C59×#563—22no—
7Q82×C59C411yescharacteristic, complemented
8C59×#555C411yescharacteristic, complemented
9C59×#554C412yescomplemented
10C59×#552—22no—
11C59×#547—22no—
12C59×#544—22no—
13C59×#542—22no—
14C59×#536—22no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C59×#529C2212yes—
17C59×#432C2212yes—
18C2×#103×C59—22no—
19C2×#97×C59C2211yescharacteristic
20C59×#382C2211yescharacteristic
21C59×#333—22no—
22C59×#330—22no—
23C59×#329—22no—
24C59×#328—22no—
25C59×#327—22no—
26C2×#41×C59—22no—
27C22×Q8×C59D412yes—
28C22×C8×C59—21no—
29C4⋊Q8×C59—22no—
30C4⋊Q8×C59—22no—
31C4⋊Q8×C59—22no—
32C4⋊Q8×C59—22no—
33C4⋊Q8×C59C2×C411yescharacteristic
34C4⋊Q8×C59C2×C411yescharacteristic
35C4⋊Q8×C59D412yes—
36#126×C59—24no—
37#126×C59—22no—
38#125×C59—24no—
39#125×C59—22no—
40#125×C59D412yes—
41#124×C59—22no—
42#124×C59—22no—
43#124×C59D412yes—
44#123×C59—24no—
45#123×C59—24no—
46#123×C59—22no—
47C22⋊Q8×C59—24no—
48C22⋊Q8×C59—24no—
49C22⋊Q8×C59—24no—
50C22⋊Q8×C59—22no—
51C22⋊Q8×C59D412yes—
52#121×C59—22no—
53#121×C59D412yes—
54C4×Q8×C59—22no—
55C4×Q8×C59C2×C412yes—
56C4×D4×C59—22no—
57C4×D4×C59—22no—
58C4×D4×C59C2×C412yes—
59#117×C59—22no—
60#117×C59—22no—
61#117×C59—22no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—22no—
67#103×C59—44no—
68C22⋊C8×C59—22no—
69#97×C59—22no—
70#97×C59—22no—
71#95×C59—22no—
72#41×C59—82no—
73#41×C59—82no—
74C2×Q8×C59—22no—
75C2×Q8×C59—22no—
76C2×Q8×C59—42no—
77C2×Q8×C59—42no—
78C2×Q8×C59C22⋊C412yes—
79C2×Q8×C59C22⋊C412yes—
80C2×Q8×C59C22⋊C412yes—
81C2×D4×C59—22no—
82C2×D4×C59—22no—
83C2×D4×C59C22⋊C412yes—
84C22×C4×C59—22no—
85C22×C4×C59—24no—
86C22×C4×C59—22no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D412yes—
90C2×C8×C59—22no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—42no—
94C4⋊C4×C59—24no—
95C4⋊C4×C59—24no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—24no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—42no—
105C4⋊C4×C59C22⋊C412yes—
106C4⋊C4×C59C22⋊C412yes—
107C22⋊C4×C59—24no—
108C22⋊C4×C59—24no—
109C22⋊C4×C59—22no—
110C22⋊C4×C59—22no—
111C22⋊C4×C59—22no—
112C22⋊C4×C59—22no—
113C22⋊C4×C59—22no—
114C42×C59—22no—
115C42×C59—22no—
116C42×C59—22no—
117C42×C59C22×C411yescharacteristic
118C42×C59C2×D412yes—
119C42×C59C2×D411yescharacteristic
120C23×C59—22no—
121C23×C59#12011yescharacteristic
122Q8×C59—42no—
123Q8×C59—42no—
124Q8×C59—42no—
125Q8×C59—82no—
126Q8×C59—82no—
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—82no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—42no—
139C2×C4×C59—44no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—84no—
143C2×C4×C59—82no—
144C2×C4×C59—81no—
145C2×C4×C59C2×C22⋊C412yes—
146C2×C4×C59C2×C22⋊C411yescharacteristic
147C2×C4×C59#12012yes—
148C2×C4×C59#12011yescharacteristic
149C8×C59—42no—
150C8×C59—82no—
151#8498C5911yesmaximal, characteristic, direct product factor, Sylow
152C22×C59—22no—
153C22×C59—42no—
154C22×C59—42no—
155C22×C59—42no—
156C22×C59#37911yescharacteristic
157C22×C59#45311yescharacteristic
158C22×C59#45511yescharacteristic
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—22no—
163C4×C59—42no—
164C4×C59—42no—
165C4×C59—42no—
166C4×C59—42no—
167C4×C59—42no—
168C4×C59—161no—
169#3544C2×C5911yescharacteristic
170#1960C2×C5911yescharacteristic, complemented
171#1558C2×C5911yescharacteristic, complemented
172#1532C2×C5912yescomplemented
173#1512C2×C5912yescomplemented
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—82no—
177C2×C59#179211yescharacteristic
178C2×C59#179612yes—
179#563—22no—
180Q82C4×C5911yescharacteristic, complemented
181#555C4×C5911yescharacteristic, complemented
182#554C4×C5912yescomplemented
183#552—22no—
184#547—22no—
185#544—22no—
186#542—22no—
187#536—22no—
188C2×C4⋊Q8C22×C5911yescharacteristic
189#529C22×C5912yes—
190#432C22×C5912yes—
191C2×#103—22no—
192C2×#97C22×C5911yescharacteristic
193#382C22×C5911yescharacteristic
194#333—22no—
195#330—22no—
196#329—22no—
197#328—22no—
198#327—22no—
199C59#849811yescharacteristic, direct product factor, Sylow
200C2×#41—22no—
201C22×Q8D4×C5912yes—
202C22×C8—21no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8C2×C4×C5911yescharacteristic
208C4⋊Q8C2×C4×C5911yescharacteristic
209C4⋊Q8D4×C5912yes—
210#126—24no—
211#126—22no—
212#125—24no—
213#125—22no—
214#125D4×C5912yes—
215#124—22no—
216#124—22no—
217#124D4×C5912yes—
218#123—24no—
219#123—24no—
220#123—22no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8D4×C5912yes—
226#121—22no—
227#121D4×C5912yes—
228C4×Q8—22no—
229C4×Q8C2×C4×C5912yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×C4×C5912yes—
233#117—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C42C23×C5911yescharacteristic
240C4⋊C8—22no—
241#103—44no—
242C22⋊C8—22no—
243#97—22no—
244#97—22no—
245#95—22no—
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊C4×C5912yes—
253C2×Q8C22⋊C4×C5912yes—
254C2×Q8C22⋊C4×C5912yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C5912yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C5911yescharacteristic
263C22×C4C2×D4×C5912yes—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4C22⋊C4×C5912yes—
280C4⋊C4C22⋊C4×C5912yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×C4×C5911yescharacteristic
292C42C2×D4×C5912yes—
293C42C2×D4×C5911yescharacteristic
294C23—22no—
295C23#120×C5911yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4C2×C22⋊C4×C5912yes—
320C2×C4C2×C22⋊C4×C5911yescharacteristic
321C2×C4#120×C5912yes—
322C2×C4#120×C5911yescharacteristic
323C8—42no—
324C8—82no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22C59×#37911yescharacteristic
330C22C59×#45311yescharacteristic
331C22C59×#45511yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—161no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2C59×#179211yescharacteristic
346C2C59×#179612yes—
subgroupG/H
center, socleC22×C59#379
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3544C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1960C2semidirect
3. C59×#1558C2semidirect
4. C59×#1532C2semidirect
5. C59×#1512C2semidirect
7. Q82×C59C4semidirect
8. C59×#555C4semidirect
9. C59×#554C4semidirect
151. #8498C59direct (N × H)
170. #1960C2×C59semidirect
171. #1558C2×C59semidirect
172. #1532C2×C59semidirect
173. #1512C2×C59semidirect
180. Q82C4×C59semidirect
181. #555C4×C59semidirect
182. #554C4×C59semidirect
199. C59#8498direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35442
C59×#19602
C59×#15582
C59×#15322
C59×#15122
#849859