Finite Group Databasebeta

← #28501773

Subgroups — #28501773

Order 15104.

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

normal subgroups only

This group has 330 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#21425446C211yesmaximal, characteristic
2#21423137C212yesmaximal
3#21422123C212yesmaximal
4#21422040C211yesmaximal, characteristic
5C59×#2844C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837737—22no—
8#16837442—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C59×#400—22no—
14C59×#400C2212yes—
15C59×#395—22no—
16C59×#395—21no—
17C59×#382—21no—
18C59×#382C2211yescharacteristic
19C2×#11745664—21no—
20#14273726—41no—
21#14273718—41no—
22#14273706—22no—
23#14273686—21no—
24#14273660—41no—
25#14273660—42no—
26#14273658—41no—
27#14273650—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59—21no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—22no—
33C2×C4⋊C4×C59D411yescharacteristic
34C2×C4⋊C4×C59D412yes—
35C2×C4⋊C4×C59C2311yescharacteristic
36C2×C42×C59D411yescharacteristic
37#95×C59—22no—
38#95×C59—21no—
39#95×C59—22no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59D412yes—
43C2×Q8×C59—82no—
44C2×Q8×C59—81no—
45C22×C4×C59—21no—
46C22×C4×C59—22no—
47C22×C4×C59—22no—
48C22×C4×C59—21no—
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59C2×D412yes—
51C2×Dic118—22no—
52C4⋊C4×C59—41no—
53C4⋊C4×C59—42no—
54C4⋊C4×C59—41no—
55C4⋊C4×C59—81no—
56C4⋊C4×C59—81no—
57C4⋊C4×C59—82no—
58C42×C59—21no—
59C42×C59Q1612yes—
60C22⋊Dic59—41no—
61C4⋊Dic59—21no—
62C4⋊Dic59—21no—
63C4⋊Dic59—22no—
64Dic59⋊C4—42no—
65C4×Dic59—41no—
66#11745664—22no—
67#11745664—41no—
68C2×C59⋊C8—41no—
69C2×C59⋊C8—42no—
70C23×C59#12011yescharacteristic
71Q8×C59—84no—
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—42no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—41no—
87C2×C4×C59C2×Q1611yescharacteristic
88C2×C4×C59#13711yescharacteristic
89C2×Dic59—41no—
90C2×Dic59—41no—
91Dic118—82no—
92C59⋊C8—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—82no—
105C4×C59—82no—
106C4×C59—81no—
107Dic59—81no—
108Dic59—81no—
109#2844D5911yescharacteristic
110#2061—591no—
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#544—591no—
120#484—1182no—
121#428—1181no—
122#400—22no—
123#400C2×D5912yes—
124#395—22no—
125#395—21no—
126#382—21no—
127#382C2×D5911yescharacteristic
128#343—591no—
129#339—1181no—
130#328—1182no—
131#324—592no—
132C59#1226111yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C2×M16—1181no—
135C4⋊Q8—2361no—
136#125—2361no—
137C22⋊Q8—1182no—
138#117—1181no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C22⋊D5911yescharacteristic
143C2×C4⋊C4C22⋊D5912yes—
144C2×C4⋊C4C22×D5911yescharacteristic
145C2×C42C22⋊D5911yescharacteristic
146#107—2361no—
147#107—2362no—
148#106—2361no—
149#103—2362no—
150C22⋊C8—1182no—
151#95—22no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95C22⋊D5912yes—
157C2×Q8—82no—
158C2×Q8—81no—
159C2×Q8—1182no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C2×C22⋊D5911yescharacteristic
165C22×C4C2×C22⋊D5912yes—
166M16—1182no—
167M16—2361no—
168C2×C8—2362no—
169C2×C8—2361no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—82no—
176C4⋊C4—1181no—
177C4⋊C4—1181no—
178C4⋊C4—1182no—
179C4⋊C4—2362no—
180C22⋊C4—2361no—
181C42—21no—
182C42—2361no—
183C42#1174567212yes—
184C23#1427369611yescharacteristic
185Q8—84no—
186Q8—4722no—
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—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—2361no—
203C2×C4—2361no—
204C2×C4C2×#1174567211yescharacteristic
205C2×C4#1427375811yescharacteristic
206C8—4722no—
207C8—2362no—
208C22—21no—
209C22—21no—
210C22#1683755811yescharacteristic
211C22#1683758511yescharacteristic
212C22#1683758911yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4—81no—
220C4—4721no—
221C4—4721no—
222C2—21no—
223C2—21no—
224C2#2142307011yescharacteristic
225C2#2142610911yescharacteristic
226C2#2142611311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2844C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2844D59
hypocenterC59#12261
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

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
#214254462
#214231372
#214221232
#214220402
C59×#28442
#1226159