Finite Group Databasebeta

← #28501710

Subgroups — #28501710

Order 15104.

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

normal subgroups only

This group has 342 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#21432164C211yesmaximal, characteristic
2#21423104C212yesmaximal
3#21422101C212yesmaximal
4#21422040C211yesmaximal, characteristic
5C59×#2061C211yesmaximal, characteristic
6C2×#14273719—21no—
7#16837730—22no—
8#16837342—22no—
9#16837334—22no—
10#16837332—21no—
11#16837332C2212yes—
12#16837139—22no—
13#16837128C2212yes—
14C59×#544C2211yescharacteristic
15C59×#428—21no—
16C59×#382C2211yescharacteristic
17C59×#343C2211yescharacteristic
18C59×#339—21no—
19#14273719—42no—
20#14273719—42no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59D412yes—
25C2×Dic59⋊C4—22no—
26C2×Dic59⋊C4—22no—
27#14273658—42no—
28#14273651—42no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C59⋊#95D412yes—
34C2×M16×C59—21no—
35C4⋊Q8×C59—41no—
36#125×C59—41no—
37C22⋊Q8×C59—22no—
38#117×C59—21no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C4⋊C4×C59C2311yescharacteristic
41C2×C42×C59D411yescharacteristic
42#107×C59—41no—
43#106×C59—41no—
44#95×C59—22no—
45C2×Q8×C59—22no—
46C22×C4×C59—21no—
47C22×C4×C59—21no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D412yes—
50C22×Dic59—22no—
51C22×Dic59—22no—
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×C59SD1612yes—
65C4⋊Dic59—42no—
66C4⋊Dic59—42no—
67C4⋊Dic59—42no—
68C4⋊Dic59—81no—
69Dic59⋊C4—44no—
70Dic59⋊C4—42no—
71Dic59⋊C4—42no—
72C2×C59⋊C8—42no—
73C23×C59#12011yescharacteristic
74Q8×C59—82no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59C2×SD1611yescharacteristic
87C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
88C2×Dic59—42no—
89C2×Dic59—42no—
90C2×Dic59—42no—
91C2×Dic59—42no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C8×C59—42no—
95C59⋊C8—82no—
96#12251—591nomaximal, Sylow
97C22×C59—21no—
98C22×C59—21no—
99C22×C59#45011yescharacteristic
100C22×C59#45611yescharacteristic
101C22×C59B(4,𝔽2)11yescharacteristic
102C4×C59—21no—
103C4×C59—21no—
104C4×C59—42no—
105C4×C59—42no—
106C4×C59—81no—
107C4×C59—81no—
108C4×C59—81no—
109Dic59—82no—
110Dic59—82no—
111#2822—591no—
112#2061D5911yescharacteristic
113#1611—592no—
114#1346—592no—
115#1324—591no—
116C2×C59—21no—
117C2×C59—21no—
118C2×C59#160411yescharacteristic
119C2×C59#220111yescharacteristic
120C2×C59#220311yescharacteristic
121#544C2×D5911yescharacteristic
122C2×#125—1181no—
123#483—1182no—
124#428—21no—
125#400—1182no—
126#398—592no—
127#398—1182no—
128#398—1181no—
129#382C2×D5911yescharacteristic
130#343C2×D5911yescharacteristic
131#339—21no—
132#328—1182no—
133#324—592no—
134C59#1225111yescharacteristic, complemented, Sylow
135C2×M16—21no—
136C4⋊Q8—41no—
137#125—41no—
138#125—2362no—
139#125—2362no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—592no—
143C2×C4⋊C4—1181no—
144C2×C4⋊C4—1181no—
145C2×C4⋊C4—1182no—
146C2×C4⋊C4—1181no—
147C2×C4⋊C4—1182no—
148C2×C4⋊C4C22⋊D5911yescharacteristic
149C2×C4⋊C4C22×D5911yescharacteristic
150C2×C42C22⋊D5911yescharacteristic
151#107—41no—
152#106—41no—
153#106—2362no—
154#103—2362no—
155C22⋊C8—1182no—
156#95—22no—
157#95—592no—
158#95—1182no—
159#95—1181no—
160#95—1182no—
161C2×Q8—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1182no—
165C22×C4—1182no—
166C22×C4D4×D5912yes—
167C22×C4C2×C22⋊D5911yescharacteristic
168M16—22no—
169M16—41no—
170C2×C8—41no—
171C2×C8—2362no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—4721no—
179C4⋊C4—2362no—
180C4⋊C4—2364no—
181C4⋊C4—2362no—
182C4⋊C4—2362no—
183C4⋊C4—2362no—
184C4⋊C4—2362no—
185C22⋊C4—41no—
186C42—21no—
187C42—41no—
188C42#1174567112yes—
189C23#1427369511yescharacteristic
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2362no—
203C2×C4—2362no—
204C2×C4—2362no—
205C2×C4—2362no—
206C2×C4—2362no—
207C2×C4—2362no—
208C2×C4C2×#1174567111yescharacteristic
209C2×C4#1427375211yescharacteristic
210C8—42no—
211C8—4722no—
212C22—21no—
213C22—21no—
214C22#1683755211yescharacteristic
215C22#1683758411yescharacteristic
216C22#1683758811yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—4722no—
225C4—4722no—
226C2—21no—
227C2—21no—
228C2#2142306411yescharacteristic
229C2#2142610811yescharacteristic
230C2#2142611611yescharacteristic
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#12251
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
134. C59#12251semidirect

Maximal subgroups

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

subgroupindex
#214321642
#214231042
#214221012
#214220402
C59×#20612
#1225159