Finite Group Databasebeta

← #28501772

Subgroups — #28501772

Order 15104.

330 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 59 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#21432368C211yesmaximal, characteristic
2#21423138C212yesmaximal
3#21422122C212yesmaximal
4#21422040C211yesmaximal, characteristic
5C59×#2061C211yesmaximal, characteristic
6#16837735—22no—
7#16837344—22no—
8#16837341C2212yes—
9#16837323—22no—
10#16837322—21no—
11#16837268—21no—
12#16837139—22no—
13#16837128C2212yes—
14C59×#544C2211yescharacteristic
15C59×#428—21no—
16C59×#382C2211yescharacteristic
17C59×#343C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118—21no—
20C2×C4⋊Dic59—21no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59D412yes—
23C2×Dic59⋊C4—22no—
24#14273660—42no—
25#14273651—42no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—22no—
28C59⋊#95—21no—
29C59⋊#95—22no—
30C59⋊#95—22no—
31C59⋊#95D412yes—
32C2×M16×C59—21no—
33C4⋊Q8×C59—41no—
34#125×C59—41no—
35C22⋊Q8×C59—22no—
36#117×C59—21no—
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59D411yescharacteristic
40#107×C59—41no—
41#106×C59—41no—
42#95×C59—22no—
43C2×Q8×C59—22no—
44C22×C4×C59—21no—
45C22×C4×C59—21no—
46C22×C4×C59C2×D411yescharacteristic
47C22×C4×C59C2×D412yes—
48C22×Dic59—22no—
49C22×Dic59—22no—
50C2×Dic118—82no—
51C2×Dic118—81no—
52M16×C59—22no—
53M16×C59—41no—
54C2×C8×C59—41no—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—21no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—41no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—41no—
61C22⋊C4×C59—41no—
62C42×C59—21no—
63C42×C59—41no—
64C42×C59Q1612yes—
65C4⋊Dic59—42no—
66C4⋊Dic59—81no—
67C4⋊Dic59—81no—
68Dic59⋊C4—82no—
69C2×C59⋊C8—42no—
70C23×C59#12011yescharacteristic
71Q8×C59—82no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59C2×Q1611yescharacteristic
84C2×C4×C59#13711yescharacteristic
85C2×Dic59—42no—
86C2×Dic59—42no—
87C2×Dic59—42no—
88C2×Dic59—42no—
89C2×Dic59—42no—
90C2×Dic59—42no—
91Dic118—84no—
92C8×C59—42no—
93C59⋊C8—82no—
94#12261—591nomaximal, Sylow
95C22×C59—21no—
96C22×C59—21no—
97C22×C59#45111yescharacteristic
98C22×C59#45611yescharacteristic
99C22×C59B(4,𝔽2)11yescharacteristic
100C4×C59—21no—
101C4×C59—21no—
102C4×C59—42no—
103C4×C59—42no—
104C4×C59—81no—
105C4×C59—81no—
106C4×C59—81no—
107Dic59—82no—
108Dic59—82no—
109#2844—591no—
110#2061D5911yescharacteristic
111#1617—592no—
112#1353—592no—
113#1324—591no—
114C2×C59—21no—
115C2×C59—21no—
116C2×C59#160511yescharacteristic
117C2×C59#220111yescharacteristic
118C2×C59#220211yescharacteristic
119#544C2×D5911yescharacteristic
120#484—1182no—
121#428—21no—
122#400—592no—
123#400—1182no—
124#395—1182no—
125#395—1181no—
126#382—1181no—
127#382C2×D5911yescharacteristic
128#343C2×D5911yescharacteristic
129#339—21no—
130#328—1182no—
131#324—592no—
132C59#1226111yescharacteristic, complemented, Sylow
133C22×Q8—1181no—
134C2×M16—21no—
135C4⋊Q8—41no—
136#125—41no—
137C22⋊Q8—22no—
138#117—21no—
139C2×C4⋊C4—592no—
140C2×C4⋊C4—1181no—
141C2×C4⋊C4—1181no—
142C2×C4⋊C4—1182no—
143C2×C4⋊C4C22⋊D5911yescharacteristic
144C2×C4⋊C4C22×D5911yescharacteristic
145C2×C42C22⋊D5911yescharacteristic
146#107—41no—
147#107—2362no—
148#106—41no—
149#103—2362no—
150C22⋊C8—1182no—
151#95—22no—
152#95—592no—
153#95—1182no—
154#95—1181no—
155#95—1182no—
156#95—1182no—
157C2×Q8—22no—
158C2×Q8—4722no—
159C2×Q8—4721no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—1182no—
163C22×C4—1182no—
164C22×C4D4×D5912yes—
165C22×C4C2×C22⋊D5911yescharacteristic
166M16—22no—
167M16—41no—
168C2×C8—41no—
169C2×C8—2362no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—4722no—
177C4⋊C4—4721no—
178C4⋊C4—4721no—
179C4⋊C4—2362no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42#1174567212yes—
184C23#1427369511yescharacteristic
185Q8—82no—
186Q8—4724no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—2362no—
199C2×C4—2362no—
200C2×C4—2362no—
201C2×C4—2362no—
202C2×C4—2362no—
203C2×C4—2362no—
204C2×C4C2×#1174567211yescharacteristic
205C2×C4#1427375711yescharacteristic
206C8—42no—
207C8—4722no—
208C22—21no—
209C22—21no—
210C22#1683755711yescharacteristic
211C22#1683758411yescharacteristic
212C22#1683758811yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—4722no—
221C4—4722no—
222C2—21no—
223C2—21no—
224C2#2142306911yescharacteristic
225C2#2142610811yescharacteristic
226C2#2142611211yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2061C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2061D59
hypocenterC59#12261
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
132. C59#12261semidirect

Maximal subgroups

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

subgroupindex
#214323682
#214231382
#214221222
#214220402
C59×#20612
#1226159