Finite Group Databasebeta

← C59×#10213 #28422322

Subgroups — C59×#10213 #28422322

Order 15104.

498 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3253C211yesmaximal, characteristic
2C59×#3216C212yesmaximal, complemented
3C59×#1995C211yesmaximal, characteristic, complemented
4C59×#1505C212yesmaximal, complemented
5C59×#1307C211yesmaximal, characteristic, complemented
6C59×#533C2211yescharacteristic
7C59×#532C2212yes—
8C59×#497—22no—
9C59×#489—22no—
10C59×#476—22no—
11C59×#469—22no—
12C59×#462—22no—
13C59×#460—22no—
14C59×#459—22no—
15C59×#452—24no—
16C59×#432C2212yes—
17C59×#423C2211yescharacteristic
18C59×#376C2211yescharacteristic
19C59×#339—22no—
20C59×#327—24no—
21C2×#41×C59—22no—
22C2×Q16×C59—24no—
23C2×SD16×C59—24no—
24C2×D8×C59—22no—
25C2×M16×C59D412yes—
26C4⋊Q8×C59D412yes—
27C4⋊Q8×C59D412yes—
28#127×C59D412yes—
29#125×C59—21no—
30#124×C59—22no—
31C22⋊Q8×C59—22no—
32#121×C59—22no—
33#121×C59—22no—
34C4×Q8×C59—22no—
35C4×D4×C59—22no—
36C4×D4×C59—22no—
37#117×C59—21no—
38C2×C42×C59—21no—
39C2×C42×C59C2311yescharacteristic
40#106×C59—22no—
41C4⋊C8×C59—22no—
42C4⋊C8×C59—22no—
43#103×C59—24no—
44#102×C59—24no—
45#102×C59—24no—
46C22⋊C8×C59D414yes—
47#97×C59—22no—
48#95×C59—22no—
49#41×C59—84no—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—42no—
56C22×C4×C59—41no—
57C22×C4×C59—42no—
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59C2×D412yes—
60Q16×C59—84no—
61SD16×C59—84no—
62D8×C59—82no—
63M16×C59—42no—
64C2×C8×C59—24no—
65C2×C8×C59—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—42no—
71C22⋊C4×C59—41no—
72C22⋊C4×C59—42no—
73C22⋊C4×C59—42no—
74C42×C59—22no—
75C42×C59—41no—
76C42×C59—41no—
77C42×C59C2×D411yescharacteristic
78C42×C59C2×D412yes—
79C42×C59#4111yescharacteristic
80C23×C59—42no—
81C23×C59#12711yescharacteristic
82Q8×C59—42no—
83Q8×C59—42no—
84Q8×C59—44no—
85D4×C59—44no—
86D4×C59—42no—
87D4×C59—42no—
88D4×C59—82no—
89D4×C59—84no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—84no—
97C2×C4×C59—82no—
98C2×C4×C59#12012yes—
99C2×C4×C59#12011yescharacteristic
100C2×C4×C59#12111yescharacteristic
101C2×C4×C59#12112yes—
102C8×C59—44no—
103C8×C59—42no—
104#10213C5911yesmaximal, characteristic, direct product factor, Sylow
105C22×C59—21no—
106C22×C59—22no—
107C22×C59—82no—
108C22×C59—82no—
109C22×C59—82no—
110C22×C59#39211yescharacteristic
111C4×C59—21no—
112C4×C59—22no—
113C4×C59—22no—
114C4×C59—82no—
115C4×C59—82no—
116C4×C59—82no—
117C4×C59#45511yescharacteristic
118C4×C59#45611yescharacteristic
119#3253C2×C5911yescharacteristic
120#3216C2×C5912yescomplemented
121#1995C2×C5911yescharacteristic, complemented
122#1505C2×C5912yescomplemented
123#1307C2×C5911yescharacteristic, complemented
124C2×C59—41no—
125C2×C59—162no—
126C2×C59#200711yescharacteristic
127C2×C59#200812yes—
128#533C22×C5911yescharacteristic
129#532C22×C5912yes—
130#497—22no—
131#489—22no—
132#476—22no—
133#469—22no—
134#462—22no—
135#460—22no—
136#459—22no—
137#452—24no—
138#432C22×C5912yes—
139#423C22×C5911yescharacteristic
140#376C22×C5911yescharacteristic
141#339—22no—
142#327—24no—
143C59#1021311yescharacteristic, direct product factor, Sylow
144C2×#41—22no—
145C2×Q16—24no—
146C2×SD16—24no—
147C2×D8—22no—
148C2×M16D4×C5912yes—
149C4⋊Q8D4×C5912yes—
150C4⋊Q8D4×C5912yes—
151#127D4×C5912yes—
152#125—21no—
153#124—22no—
154C22⋊Q8—22no—
155#121—22no—
156#121—22no—
157C4×Q8—22no—
158C4×D4—22no—
159C4×D4—22no—
160#117—21no—
161C2×C42—21no—
162C2×C42C23×C5911yescharacteristic
163#106—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166#103—24no—
167#102—24no—
168#102—24no—
169C22⋊C8D4×C5914yes—
170#97—22no—
171#95—22no—
172#41—84no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4C2×D4×C5911yescharacteristic
182C22×C4C2×D4×C5912yes—
183Q16—84no—
184SD16—84no—
185D8—82no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—41no—
200C42C2×D4×C5911yescharacteristic
201C42C2×D4×C5912yes—
202C42#41×C5911yescharacteristic
203C23—42no—
204C23#127×C5911yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—44no—
208D4—44no—
209D4—42no—
210D4—42no—
211D4—82no—
212D4—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4#120×C5912yes—
222C2×C4#120×C5911yescharacteristic
223C2×C4#121×C5911yescharacteristic
224C2×C4#121×C5912yes—
225C8—44no—
226C8—42no—
227C22—21no—
228C22—22no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C59×#39211yescharacteristic
233C4—21no—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4C59×#45511yescharacteristic
240C4C59×#45611yescharacteristic
241C2—41no—
242C2—162no—
243C2C59×#200711yescharacteristic
244C2C59×#200812yes—
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3253C2
LCSG ▷ C2×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×#3216C2semidirect
3. C59×#1995C2semidirect
4. C59×#1505C2semidirect
5. C59×#1307C2semidirect
104. #10213C59direct (N × H)
120. #3216C2×C59semidirect
121. #1995C2×C59semidirect
122. #1505C2×C59semidirect
123. #1307C2×C59semidirect
143. C59#10213direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32532
C59×#32162
C59×#19952
C59×#15052
C59×#13072
#1021359