Finite Group Databasebeta

← #28504958

Subgroups — #28504958

Order 15104.

642 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#21424024C212yesmaximal
2#21422223C214yesmaximal, complemented
3C59×#2297C211yesmaximal, characteristic
4C2×#14273646C2214yes—
5C2×#14273644—22no—
6#16837366—22no—
7C22×#41×C59C2211yescharacteristic
8C2×#117×C59C411yescharacteristic
9C22×C42×C59C411yescharacteristic
10C59×#391C2212yes—
11C59×#390C412yes—
12C59×#386—22no—
13C4×C22⋊C4×C59—22no—
14C2×#11745664—24no—
15#14273646—28no—
16#14273644—44no—
17C22⋊(C59⋊C8)—44no—
18C2×#41×C59—24no—
19C2×#41×C59—42no—
20C2×#41×C59—42no—
21C22×Q8×C59C2311yescharacteristic
22C22×D4×C59C2×C411yescharacteristic
23C22×D4×C59D412yes—
24C23×C4×C59—21no—
25C23×C4×C59C2×C411yescharacteristic
26C23×C4×C59D412yes—
27#117×C59—42no—
28C2×C4⋊C4×C59C2×C412yes—
29C2×C22⋊C4×C59—24no—
30C2×C22⋊C4×C59—21no—
31C2×C42×C59—21no—
32C2×C42×C59—21no—
33C2×C42×C59—42no—
34C2×C42×C59C2×C412yes—
35#95×C59—24no—
36C24×C59C22⋊C412yes—
37C24×C59C22⋊C411yescharacteristic
38#41×C59—416no—
39C2×Q8×C59—41no—
40C2×Q8×C59—41no—
41C2×Q8×C59#4114yes—
42C2×D4×C59—24no—
43C2×D4×C59—22no—
44C2×D4×C59—24no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59—42no—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—24no—
53C22×C4×C59—24no—
54C22×C4×C59—22no—
55C22×C4×C59—42no—
56C22×C4×C59—44no—
57C22×C4×C59—42no—
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D412yes—
61C4⋊C4×C59—42no—
62C22⋊C4×C59—48no—
63C22⋊C4×C59—42no—
64C42×C59—42no—
65C42×C59—42no—
66C42×C59—42no—
67#11745664—44no—
68#11745664—44no—
69C2×C59⋊C8—44no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—42no—
76C23×C59—44no—
77C23×C59#9912yes—
78C23×C59C2×C22⋊C411yescharacteristic
79Q8×C59—44no—
80D4×C59—48no—
81D4×C59—44no—
82C2×C4×C59—28no—
83C2×C4×C59—21no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—24no—
90C2×C4×C59—48no—
91C2×C4×C59—42no—
92C2×C4×C59—48no—
93C2×C4×C59—42no—
94C2×C4×C59—44no—
95C2×C4×C59—42no—
96C2×C4×C59#9912yes—
97C2×C4×C59#11712yes—
98C2×C4×C59#12314yes—
99C59⋊C8—84no—
100#12861—591nomaximal, Sylow
101C22×C59—22no—
102C22×C59—22no—
103C22×C59—42no—
104C22×C59—42no—
105C22×C59—44no—
106C22×C59—42no—
107C22×C59—42no—
108C22×C59—44no—
109C22×C59—82no—
110C22×C59#38111yescharacteristic
111C22×C59C2×#9911yescharacteristic
112C22×C59C2×#9911yescharacteristic
113C4×C59—24no—
114C4×C59—22no—
115C4×C59—22no—
116C4×C59—44no—
117C4×C59—82no—
118#2297D5911yescharacteristic
119#1785—592no—
120#1394—594no—
121C2×C59—21no—
122C2×C59—21no—
123C2×C59—42no—
124C2×C59—82no—
125C2×C59#181911yescharacteristic
126C2×C59#213212yes—
127C22×#41C2×D5911yescharacteristic
128C2×#117Dic5911yescharacteristic
129C22×C42Dic5911yescharacteristic
130C2×#101—594no—
131C2×#100—1182no—
132#407—1182no—
133#391C2×D5912yes—
134#390Dic5912yes—
135#386—22no—
136C4×C22⋊C4—22no—
137C59#1286111yescharacteristic, complemented, Sylow
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8C22×D5911yescharacteristic
142C22×D4C2×Dic5911yescharacteristic
143C22×D4C22⋊D5912yes—
144C23×C4—21no—
145C23×C4C2×Dic5911yescharacteristic
146C23×C4C22⋊D5912yes—
147C2×M16—1184no—
148#117—42no—
149C2×C4⋊C4C2×Dic5912yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2×Dic5912yes—
156#101—1188no—
157#100—2364no—
158C22⋊C8—2364no—
159#95—24no—
160C24C22⋊Dic5912yes—
161C24C22⋊Dic5911yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1174568514yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊Dic5911yescharacteristic
183C22×C4C22×Dic5911yescharacteristic
184C22×C4C2×C22⋊D5912yes—
185M16—2364no—
186M16—2364no—
187C2×C8—2364no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23#1427364212yes—
202C23C2×C22⋊Dic5911yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4#1427364212yes—
221C2×C4#1427368612yes—
222C2×C4#1427371014yes—
223C8—4724no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22#1683726511yescharacteristic
234C22C2×#1427364211yescharacteristic
235C22C2×#1427364211yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2#2142417011yescharacteristic
246C2#2142578112yes—
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#2297C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×Q8×C59C23
hypercenter#2297D59
hypocenterC59#12861
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #21422223C2semidirect
137. C59#12861semidirect

Maximal subgroups

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

subgroupindex
#214240242
#214222232
C59×#22972
#1286159