Finite Group Databasebeta

← C59×#12130 #28424239

Subgroups — C59×#12130 #28424239

Order 15104.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2811C211yesmaximal, characteristic, complemented
2C59×#2007C211yesmaximal, characteristic
3C59×#1604C212yesmaximal
4C59×#1354C212yesmaximal, complemented
5C59×#1319C211yesmaximal, characteristic, complemented
6C2×#137×C59—21no—
7C59×#537C2211yescharacteristic
8C59×#448—22no—
9C2×#104×C59—21no—
10C59×#395—22no—
11C59×#395C2212yes—
12C59×#391—21no—
13C59×#391—21no—
14C59×#391—22no—
15C59×#391C2211yescharacteristic
16C59×#343C2211yescharacteristic
17C59×#327—22no—
18C59×#324C2212yes—
19C2×#41×C59—21no—
20C22×Q8×C59—21no—
21C22×Q8×C59D411yescharacteristic
22C22×Q8×C59D412yes—
23#137×C59—42no—
24#137×C59—42no—
25C2×Q16×C59—41no—
26C2×SD16×C59—41no—
27C2×SD16×C59—42no—
28C2×M16×C59—21no—
29C22⋊Q8×C59—41no—
30#121×C59—22no—
31#121×C59—41no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59C2311yescharacteristic
36C2×C42×C59—21no—
37C2×C42×C59D411yescharacteristic
38#104×C59—42no—
39#103×C59—42no—
40C22⋊C8×C59—22no—
41#95×C59—22no—
42#95×C59—22no—
43#95×C59—22no—
44#95×C59—22no—
45#95×C59D412yes—
46#41×C59—41no—
47#41×C59—41no—
48#41×C59—41no—
49#41×C59—41no—
50C2×Q8×C59—24no—
51C2×Q8×C59—21no—
52C2×Q8×C59—21no—
53C2×Q8×C59—42no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59—82no—
57C2×Q8×C59—81no—
58C2×Q8×C59—81no—
59C2×Q8×C59SD1612yescomplemented
60C2×D4×C59—21no—
61C2×D4×C59—22no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—41no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D412yes—
70Q16×C59—44no—
71SD16×C59—44no—
72SD16×C59—84no—
73M16×C59—22no—
74M16×C59—41no—
75C2×C8×C59—41no—
76C2×C8×C59—42no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—82no—
80C4⋊C4×C59—81no—
81C4⋊C4×C59—81no—
82C22⋊C4×C59—42no—
83C42×C59—41no—
84C42×C59—81no—
85C23×C59—41no—
86C23×C59#12011yescharacteristic
87Q8×C59—42no—
88Q8×C59—41no—
89Q8×C59—41no—
90Q8×C59—44no—
91Q8×C59—81no—
92Q8×C59—84no—
93Q8×C59—82no—
94D4×C59—41no—
95D4×C59—41no—
96D4×C59—82no—
97D4×C59—81no—
98D4×C59—81no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—24no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—41no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—81no—
117C2×C4×C59—81no—
118C2×C4×C59C2×SD1611yescharacteristic
119C2×C4×C59#13711yescharacteristic
120C8×C59—42no—
121C8×C59—82no—
122#12130C5911yesmaximal, characteristic, direct product factor, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—81no—
126C22×C59—81no—
127C22×C59—81no—
128C22×C59#44811yescharacteristic
129C22×C59#45311yescharacteristic
130C22×C59B(4,𝔽2)11yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—44no—
135C4×C59—42no—
136C4×C59—42no—
137C4×C59—81no—
138C4×C59—82no—
139#2811C2×C5911yescharacteristic, complemented
140#2007C2×C5911yescharacteristic
141#1604C2×C5912yes—
142#1354C2×C5912yescomplemented
143#1319C2×C5911yescharacteristic, complemented
144C2×C59—21no—
145C2×C59—21no—
146C2×C59—161no—
147C2×C59#159611yescharacteristic
148C2×C59#219911yescharacteristic
149C2×C59#220511yescharacteristic
150C2×#137—21no—
151#537C22×C5911yescharacteristic
152#448—22no—
153C2×#104—21no—
154#395—22no—
155#395C22×C5912yes—
156#391—21no—
157#391—21no—
158#391—22no—
159#391C22×C5911yescharacteristic
160#343C22×C5911yescharacteristic
161#327—22no—
162#324C22×C5912yes—
163C59#1213011yescharacteristic, direct product factor, Sylow
164C2×#41—21no—
165C22×Q8—21no—
166C22×Q8D4×C5911yescharacteristic
167C22×Q8D4×C5912yes—
168#137—42no—
169#137—42no—
170C2×Q16—41no—
171C2×SD16—41no—
172C2×SD16—42no—
173C2×M16—21no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C23×C5911yescharacteristic
181C2×C42—21no—
182C2×C42D4×C5911yescharacteristic
183#104—42no—
184#103—42no—
185C22⋊C8—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95D4×C5912yes—
191#41—41no—
192#41—41no—
193#41—41no—
194#41—41no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8SD16×C5912yescomplemented
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4C2×D4×C5911yescharacteristic
214C22×C4C2×D4×C5912yes—
215Q16—44no—
216SD16—44no—
217SD16—84no—
218M16—22no—
219M16—41no—
220C2×C8—41no—
221C2×C8—42no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C22⋊C4—42no—
228C42—41no—
229C42—81no—
230C23—41no—
231C23#120×C5911yescharacteristic
232Q8—42no—
233Q8—41no—
234Q8—41no—
235Q8—44no—
236Q8—81no—
237Q8—84no—
238Q8—82no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×SD16×C5911yescharacteristic
264C2×C4#137×C5911yescharacteristic
265C8—42no—
266C8—82no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22C59×#44811yescharacteristic
273C22C59×#45311yescharacteristic
274C22C59×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—44no—
279C4—42no—
280C4—42no—
281C4—81no—
282C4—82no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2C59×#159611yescharacteristic
287C2C59×#219911yescharacteristic
288C2C59×#220511yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC59×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C4⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#2811C2semidirect
4. C59×#1354C2semidirect
5. C59×#1319C2semidirect
59. C2×Q8×C59SD16semidirect
122. #12130C59direct (N × H)
139. #2811C2×C59semidirect
142. #1354C2×C59semidirect
143. #1319C2×C59semidirect
163. C59#12130direct (N × H)
204. C2×Q8SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#28112
C59×#20072
C59×#16042
C59×#13542
C59×#13192
#1213059