Finite Group Databasebeta

← C59×#12802 #28424911

Subgroups — C59×#12802 #28424911

Order 15104.

468 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×C4⋊Q8×C59C211yesmaximal, characteristic
2C2×C59×#356C212yesmaximal
3C59×#1801C212yesmaximal
4C59×#1394C212yesmaximal
5C2×C4⋊Q8×C59C2211yescharacteristic
6C2×C4×Q8×C59C411yescharacteristic
7C2×C4×Q8×C59C2211yescharacteristic
8C59×#423—22no—
9C2×#101×C59C2214yes—
10C59×#391C411yescharacteristic
11C59×#391C2211yescharacteristic
12C59×#389—22no—
13C59×#384C411yescharacteristic
14C4×C4⋊C4×C59—22no—
15C43×C59C411yescharacteristic
16C59×#356—24no—
17C59×#343—22no—
18C22×Q8×C59C2×C411yescharacteristic
19C22×Q8×C59C2311yescharacteristic
20C2×M16×C59—24no—
21C4⋊Q8×C59—22no—
22C4⋊Q8×C59D412yes—
23C4⋊Q8×C59Q812yes—
24C4×Q8×C59—41no—
25C4×Q8×C59—41no—
26C4×Q8×C59—41no—
27C4×Q8×C59—41no—
28C2×C4⋊C4×C59—22no—
29C2×C4⋊C4×C59—22no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59C2×C411yescharacteristic
33C2×C4⋊C4×C59D412yes—
34C2×C42×C59—21no—
35C2×C42×C59—21no—
36C2×C42×C59—21no—
37C2×C42×C59C2×C411yescharacteristic
38C2×C42×C59C2×C411yescharacteristic
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59D412yes—
41C4⋊C8×C59—42no—
42C4⋊C8×C59—42no—
43#101×C59—24no—
44#101×C59—24no—
45#95×C59—22no—
46#95×C59—22no—
47C2×Q8×C59—21no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59—41no—
52C2×Q8×C59C4⋊C412yes—
53C2×Q8×C59C2×D411yescharacteristic
54C2×Q8×C59C2×Q811yescharacteristic
55C2×Q8×C59#4112yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D412yes—
65M16×C59—44no—
66M16×C59—44no—
67C2×C8×C59—44no—
68C4⋊C4×C59—24no—
69C4⋊C4×C59—24no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—44no—
73C4⋊C4×C59—42no—
74C4⋊C4×C59—41no—
75C42×C59—24no—
76C42×C59—22no—
77C42×C59—22no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—42no—
82C42×C59—41no—
83C42×C59—41no—
84C42×C59—41no—
85C42×C59C4⋊C412yes—
86C23×C59C2×C22⋊C411yescharacteristic
87Q8×C59—41no—
88Q8×C59—41no—
89Q8×C59—42no—
90Q8×C59—42no—
91Q8×C59—42no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59C2×C4⋊C411yescharacteristic
115C2×C4×C59#11711yescharacteristic
116C2×C4×C59#12112yes—
117C2×C4×C59C22⋊Q812yes—
118C8×C59—84no—
119#12802C5911yesmaximal, characteristic, direct product factor, Sylow
120C22×C59—21no—
121C22×C59—21no—
122C22×C59#38011yescharacteristic
123C22×C59C2×#9911yescharacteristic
124C22×C59#41011yescharacteristic
125C4×C59—24no—
126C4×C59—22no—
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—22no—
130C4×C59—21no—
131C4×C59—42no—
132C4×C59—42no—
133C4×C59—41no—
134C4×C59—42no—
135C4×C59—81no—
136C4×C59#35612yescomplemented
137C4×C4⋊Q8C2×C5911yescharacteristic
138C2×#356C2×C5912yes—
139#1801C2×C5912yes—
140#1394C2×C5912yes—
141C2×C59—21no—
142C2×C59—21no—
143C2×C59#179911yescharacteristic
144C2×C59C2×#35611yescharacteristic
145C2×C59#213211yescharacteristic
146C2×C4⋊Q8C22×C5911yescharacteristic
147C2×C4×Q8C4×C5911yescharacteristic
148C2×C4×Q8C22×C5911yescharacteristic
149#423—22no—
150C2×#101C22×C5914yes—
151#391C4×C5911yescharacteristic
152#391C22×C5911yescharacteristic
153#389—22no—
154#384C4×C5911yescharacteristic
155C4×C4⋊C4—22no—
156C43C4×C5911yescharacteristic
157#356—24no—
158#343—22no—
159C59#1280211yescharacteristic, direct product factor, Sylow
160C22×Q8C2×C4×C5911yescharacteristic
161C22×Q8C23×C5911yescharacteristic
162C2×M16—24no—
163C4⋊Q8—22no—
164C4⋊Q8D4×C5912yes—
165C4⋊Q8Q8×C5912yes—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×Q8—41no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×C4×C5911yescharacteristic
174C2×C4⋊C4C2×C4×C5911yescharacteristic
175C2×C4⋊C4D4×C5912yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×C4×C5911yescharacteristic
180C2×C42C2×C4×C5911yescharacteristic
181C2×C42C2×C4×C5911yescharacteristic
182C2×C42D4×C5912yes—
183C4⋊C8—42no—
184C4⋊C8—42no—
185#101—24no—
186#101—24no—
187#95—22no—
188#95—22no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8C4⋊C4×C5912yes—
195C2×Q8C2×D4×C5911yescharacteristic
196C2×Q8C2×Q8×C5911yescharacteristic
197C2×Q8#41×C5912yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4C22⋊C4×C5911yescharacteristic
203C22×C4C22⋊C4×C5912yes—
204C22×C4C22⋊C4×C5911yescharacteristic
205C22×C4C22×C4×C5911yescharacteristic
206C22×C4C2×D4×C5912yes—
207M16—44no—
208M16—44no—
209C2×C8—44no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42C4⋊C4×C5912yes—
228C23C2×C22⋊C4×C5911yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4#99×C5911yescharacteristic
255C2×C4#99×C5911yescharacteristic
256C2×C4C2×C4⋊C4×C5911yescharacteristic
257C2×C4#117×C5911yescharacteristic
258C2×C4#121×C5912yes—
259C2×C4C22⋊Q8×C5912yes—
260C8—84no—
261C22—21no—
262C22—21no—
263C22C59×#38011yescharacteristic
264C22C2×#99×C5911yescharacteristic
265C22C59×#41011yescharacteristic
266C4—24no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4C59×#35612yescomplemented
278C2—21no—
279C2—21no—
280C2C59×#179911yescharacteristic
281C2C2×C59×#35611yescharacteristic
282C2C59×#213211yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC22×Q8C23×C59
prime radicalC23×C59C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C2×C4×Q8×C59 ◁ G

Complements

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

NH ≅ G/Ntype
119. #12802C59direct (N × H)
136. C4×C59#356semidirect
159. C59#12802direct (N × H)
277. C4C59×#356semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q8×C592
C2×C59×#3562
C59×#18012
C59×#13942
#1280259