Finite Group Databasebeta

← #28477921

Subgroups — #28477921

Order 15104.

410 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 87 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#21431128C211yesmaximal, characteristic
2#21425639C211yesmaximal, characteristic
3#21425574C212yesmaximal
4#21421918C212yesmaximal
5C59×#1272C211yesmaximal, characteristic
6C2×#14273725—21no—
7C2×#14273659—22no—
8C2×#14273658—21no—
9C2×#14273651—21no—
10#16837332—21no—
11#16837323—22no—
12#16837300C2212yes—
13#16837287—22no—
14#16837272C2211yescharacteristic
15#16837165—22no—
16#16837157C2211yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×Dic118—21no—
20C22×C59⋊C8—21no—
21#14273725—42no—
22#14273725—42no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59D412yes—
27C2×C4⋊Dic59D412yes—
28C2×Dic59⋊C4—22no—
29C2×Dic59⋊C4—22no—
30C2×C4×Dic59—21no—
31#14273659—44no—
32#14273658—42no—
33#14273651—42no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C22×Q8×C59D411yescharacteristic
38C22×C8×C59—22no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C42×C59C2311yescharacteristic
41#95×C59—22no—
42C2×Q8×C59—21no—
43C2×Q8×C59—41no—
44C2×Q8×C59—41no—
45C2×Q8×C59SD1612yes—
46C22×C4×C59—21no—
47C22×C4×C59—21no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D412yes—
50C22×Dic59—22no—
51C22×Dic59—22no—
52C22×Dic59—41no—
53C2×Dic118—42no—
54C2×Dic118—81no—
55C2×Dic118—81no—
56C2×C8×C59—24no—
57C2×C8×C59—42no—
58C2×C8×C59—42no—
59C4⋊C4×C59—41no—
60C42×C59—41no—
61C4⋊Dic59—22no—
62C4⋊Dic59—42no—
63C4⋊Dic59—41no—
64C4⋊Dic59—41no—
65C4⋊Dic59—42no—
66C4⋊Dic59—42no—
67C4⋊Dic59—81no—
68C4⋊Dic59SD1614yes—
69Dic59⋊C4—44no—
70Dic59⋊C4—42no—
71Dic59⋊C4—42no—
72C4×Dic59—42no—
73C2×C59⋊C8—22no—
74C2×C59⋊C8—41no—
75C2×C59⋊C8—41no—
76C23×C59#12011yescharacteristic
77Q8×C59—42no—
78Q8×C59—81no—
79C2×C4×C59—21no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—42no—
88C2×C4×C59C2×SD1612yes—
89C2×C4×C59C2×SD1611yescharacteristic
90C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
91C2×C4×C59#13712yes—
92C2×Dic59—24no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—82no—
99C2×Dic59—81no—
100Dic118—84no—
101C8×C59—44no—
102C59⋊C8—42no—
103#7817—591nomaximal, Sylow
104C22×C59#44812yes—
105C22×C59#45011yescharacteristic
106C22×C59#45311yescharacteristic
107C22×C59#45612yes—
108C22×C59#45811yescharacteristic
109C4×C59—22no—
110C4×C59—22no—
111C4×C59—21no—
112C4×C59—21no—
113C4×C59—42no—
114C4×C59—81no—
115Dic59—44no—
116Dic59—82no—
117Dic59—161no—
118#2734—591no—
119#2096—591no—
120#2084—592no—
121#1272D5911yescharacteristic
122#1271—592no—
123C2×C59#161112yes—
124C2×C59#161311yescharacteristic
125C2×C59#162311yescharacteristic
126C2×C59#165112yes—
127C2×C59#166011yescharacteristic
128C2×C4⋊Q8—1181no—
129C2×#106—1182no—
130C2×#106—1181no—
131C2×#103—1181no—
132#398—1181no—
133#395—1182no—
134#391C2×D5911yescharacteristic
135#389—592no—
136#387—1182no—
137#384—591no—
138#340—1182no—
139#336—591no—
140#336C2×D5912yes—
141C59#781711yescharacteristic, complemented, Sylow
142C22×Q8—1181no—
143C22×Q8D23611yescharacteristic
144C22×C8—22no—
145C22×C8—1181no—
146C4⋊Q8—2362no—
147C4⋊Q8—2362no—
148C2×C4⋊C4—592no—
149C2×C4⋊C4—592no—
150C2×C4⋊C4—1181no—
151C2×C4⋊C4—1182no—
152C2×C4⋊C4—1182no—
153C2×C4⋊C4—1181no—
154C2×C4⋊C4—1182no—
155C2×C4⋊C4D23611yescharacteristic
156C2×C42—1181no—
157C2×C42C22×D5911yescharacteristic
158#106—2362no—
159#106—2364no—
160#103—2362no—
161#95—22no—
162#95—1181no—
163#95—1182no—
164#95—1182no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—4721no—
169C2×Q8—4721no—
170C2×Q8—2362no—
171C2×Q8#1174566012yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1182no—
175C22×C4—1182no—
176C22×C4—2361no—
177C22×C4C2×D23611yescharacteristic
178C22×C4D4×D5912yes—
179C2×C8—24no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—1182no—
183C2×C8—2361no—
184C2×C8—2361no—
185C4⋊C4—41no—
186C4⋊C4—594no—
187C4⋊C4—1182no—
188C4⋊C4—4721no—
189C4⋊C4—2361no—
190C4⋊C4—2361no—
191C4⋊C4—2362no—
192C4⋊C4—2364no—
193C4⋊C4—2362no—
194C4⋊C4—2362no—
195C4⋊C4—2362no—
196C4⋊C4—2362no—
197C42—41no—
198C42—2362no—
199C23#1427369711yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—4724no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—1184no—
213C2×C4—4722no—
214C2×C4—4721no—
215C2×C4—2362no—
216C2×C4—2362no—
217C2×C4—2362no—
218C2×C4—2362no—
219C2×C4—2362no—
220C2×C4SD16×D5912yes—
221C2×C4C2×#1174566011yescharacteristic
222C2×C4#1427375511yescharacteristic
223C2×C4#1427375912yes—
224C8—44no—
225C8—2362no—
226C22#1683754212yes—
227C22#1683755511yescharacteristic
228C22#1683757211yescharacteristic
229C22#1683758612yes—
230C22#1683759311yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—81no—
237C4—9441no—
238C4—4722no—
239C4—2364no—
240C2#2142310512yes—
241C2#2142311811yescharacteristic
242C2#2142317511yescharacteristic
243C2#2142336212yes—
244C2#2142340811yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socle, prime radicalC23×C59#120
hypercenter#1272D59
hypocenterC59#7817
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
141. C59#7817semidirect

Maximal subgroups

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

subgroupindex
#214311282
#214256392
#214255742
#214219182
C59×#12722
#781759