Finite Group Databasebeta

← #28501687

Subgroups — #28501687

Order 15104.

394 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425441C211yesmaximal, characteristic
2#21423103C212yesmaximal
3#21422102C212yesmaximal
4#21422042C211yesmaximal, characteristic
5C59×#2582C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837732—22no—
8#16837442—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C2×C4×Q8×C59—21no—
14C59×#398—21no—
15C59×#398C2212yes—
16C59×#391—22no—
17C59×#389—22no—
18C59×#384C2211yescharacteristic
19C2×#11745664—21no—
20#14273726—41no—
21#14273718—41no—
22#14273706—22no—
23#14273686—21no—
24#14273660—41no—
25#14273658—41no—
26#14273658—42no—
27#14273650—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59—21no—
30C4×Q8×C59—42no—
31C4×Q8×C59—42no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59D411yescharacteristic
37C2×C4⋊C4×C59D412yes—
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59—22no—
40C2×C42×C59D411yescharacteristic
41#95×C59—22no—
42#95×C59—21no—
43#95×C59—22no—
44#95×C59D412yes—
45C2×Q8×C59—24no—
46C2×Q8×C59—42no—
47C2×Q8×C59—42no—
48C22×C4×C59—21no—
49C22×C4×C59—22no—
50C22×C4×C59—21no—
51C22×C4×C59—22no—
52C22×C4×C59C2×D411yescharacteristic
53C22×C4×C59C2×D412yes—
54C2×Dic118—22no—
55C4⋊C4×C59—41no—
56C4⋊C4×C59—44no—
57C4⋊C4×C59—42no—
58C4⋊C4×C59—42no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—81no—
63C42×C59—21no—
64C42×C59—44no—
65C42×C59SD1612yes—
66C22⋊Dic59—41no—
67C4⋊Dic59—21no—
68C4⋊Dic59—22no—
69C4⋊Dic59—21no—
70Dic59⋊C4—42no—
71C4×Dic59—41no—
72#11745664—22no—
73#11745664—41no—
74C2×C59⋊C8—41no—
75C2×C59⋊C8—42no—
76C23×C59#12011yescharacteristic
77Q8×C59—48no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—24no—
84C2×C4×C59—22no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59C2×SD1611yescharacteristic
94C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
95C2×Dic59—41no—
96C2×Dic59—41no—
97Dic118—82no—
98C59⋊C8—42no—
99C59⋊C8—82no—
100#12247—591nomaximal, Sylow
101C22×C59—21no—
102C22×C59—21no—
103C22×C59#45011yescharacteristic
104C22×C59#45611yescharacteristic
105C22×C59B(4,𝔽2)11yescharacteristic
106C4×C59—24no—
107C4×C59—21no—
108C4×C59—21no—
109C4×C59—44no—
110C4×C59—42no—
111C4×C59—82no—
112C4×C59—81no—
113Dic59—81no—
114Dic59—81no—
115#2582D5911yescharacteristic
116#2060—591no—
117#1611—592no—
118#1346—592no—
119#1325—591no—
120C2×C59—21no—
121C2×C59—21no—
122C2×C59#160411yescharacteristic
123C2×C59#219811yescharacteristic
124C2×C59#220511yescharacteristic
125#544—591no—
126C2×C4×Q8—21no—
127#483—1182no—
128#428—1181no—
129#398—21no—
130#398C2×D5912yes—
131#391—22no—
132#389—22no—
133#384C2×D5911yescharacteristic
134#343—591no—
135#339—1181no—
136#328—1182no—
137#324—592no—
138C59#1224711yescharacteristic, complemented, Sylow
139C22×Q8—21no—
140C2×M16—1181no—
141C4⋊Q8—2361no—
142#125—2361no—
143C22⋊Q8—1182no—
144C4×Q8—42no—
145C4×Q8—42no—
146#117—1181no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C22⋊D5911yescharacteristic
152C2×C4⋊C4C22⋊D5912yes—
153C2×C4⋊C4C22×D5911yescharacteristic
154C2×C42—22no—
155C2×C42C22⋊D5911yescharacteristic
156#107—2361no—
157#106—2361no—
158#106—2362no—
159#103—2362no—
160C22⋊C8—1182no—
161#95—22no—
162#95—21no—
163#95—22no—
164#95C22⋊D5912yes—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—1182no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4C2×C22⋊D5911yescharacteristic
174C22×C4C2×C22⋊D5912yes—
175M16—1182no—
176M16—2361no—
177C2×C8—2362no—
178C2×C8—2361no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—1181no—
188C4⋊C4—1182no—
189C4⋊C4—1181no—
190C4⋊C4—2362no—
191C22⋊C4—2361no—
192C42—21no—
193C42—44no—
194C42—2361no—
195C42#1174567012yes—
196C23#1427369611yescharacteristic
197Q8—48no—
198Q8—4722no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—2361no—
215C2×C4—2361no—
216C2×C4C2×#1174567011yescharacteristic
217C2×C4#1427375311yescharacteristic
218C8—4722no—
219C8—2362no—
220C22—21no—
221C22—21no—
222C22#1683755311yescharacteristic
223C22#1683758511yescharacteristic
224C22#1683758911yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4—4721no—
233C4—4721no—
234C2—21no—
235C2—21no—
236C2#2142306511yescharacteristic
237C2#2142609211yescharacteristic
238C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2582C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2582D59
hypocenterC59#12247
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
138. C59#12247semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214231032
#214221022
#214220422
C59×#25822
#1224759