Finite Group Databasebeta

← #28477931

Subgroups — #28477931

Order 15104.

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

normal subgroups only

This group has 434 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#21425624C211yesmaximal, characteristic
2#21425572C212yesmaximal, complemented
3#21421919C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#2724C211yesmaximal, characteristic
6C2×#14273660—21no—
7C2×#14273658—22no—
8C2×#14273648—21no—
9#16837272C2211yescharacteristic
10#16837164—22no—
11#16837157C2212yes—
12#16837157C2211yescharacteristic
13C2×#123×C59—21no—
14C59×#398—21no—
15C59×#397—22no—
16C59×#390C2211yescharacteristic
17C59×#389C2212yes—
18C59×#385—22no—
19C22×C59⋊C8—21no—
20C22×C59⋊C8—22no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—22no—
23#14273660—42no—
24#14273658—44no—
25#14273648—42no—
26C22×D4×C59D411yescharacteristic
27C23×C4×C59—21no—
28#123×C59—42no—
29#123×C59—42no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59D412yes—
33C2×C4⋊C4×C59D412yes—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—21no—
37C2×C22⋊C4×C59—22no—
38C2×C42×C59C2311yescharacteristic
39#95×C59—22no—
40#95×C59—21no—
41#95×C59—22no—
42C24×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59SD1612yes—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—42no—
51C22×C4×C59—81no—
52C22×C4×C59—81no—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D412yes—
55C22×Dic59—41no—
56C4⋊C4×C59—22no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—42no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—81no—
63C22⋊C4×C59—42no—
64C22⋊C4×C59—44no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—41no—
68C4⋊Dic59—41no—
69C4⋊Dic59—41no—
70C4⋊Dic59—42no—
71C4⋊Dic59—42no—
72C2×C59⋊C8—22no—
73C2×C59⋊C8—24no—
74C2×C59⋊C8—42no—
75C2×C59⋊C8—41no—
76C2×C59⋊C8—42no—
77C2×C59⋊C8—41no—
78C23×C59—22no—
79C23×C59—42no—
80C23×C59—41no—
81C23×C59—42no—
82C23×C59—41no—
83C23×C59#12011yescharacteristic
84D4×C59—42no—
85D4×C59—81no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—84no—
99C2×C4×C59C2×SD1611yescharacteristic
100C2×C4×C59#13512yes—
101C2×C4×C59AGL(1,ℤ/8)12yes—
102C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
103C2×Dic59—82no—
104C2×Dic59—81no—
105C59⋊C8—44no—
106C59⋊C8—42no—
107#7819—591nomaximal, Sylow
108C22×C59—42no—
109C22×C59—44no—
110C22×C59—82no—
111C22×C59—81no—
112C22×C59—81no—
113C22×C59#44912yes—
114C22×C59#45011yescharacteristic
115C22×C59#45311yescharacteristic
116C22×C59#45612yes—
117C22×C59#45811yescharacteristic
118C4×C59—22no—
119C4×C59—22no—
120C4×C59—21no—
121C4×C59—21no—
122C4×C59—82no—
123C4×C59—81no—
124C4×C59—82no—
125Dic59—161no—
126#2724D5911yescharacteristic
127#2093—591no—
128#2084—592no—
129#1271—592no—
130#1270—591no—
131C2×C59—82no—
132C2×C59#160912yes—
133C2×C59#161311yescharacteristic
134C2×C59#162011yescharacteristic
135C2×C59#165412yes—
136C2×C59#166011yescharacteristic
137C2×#123—21no—
138C2×#107—1181no—
139C2×#106—1182no—
140C2×#102—1181no—
141#398—21no—
142#397—22no—
143#390C2×D5911yescharacteristic
144#389C2×D5912yes—
145#385—22no—
146#384—591no—
147#340—1182no—
148#336—592no—
149#336—591no—
150C59#781911yescharacteristic, complemented, Sylow
151C22×D4C22⋊D5911yescharacteristic
152C23×C4—21no—
153C22×C8—1182no—
154C22×C8—1181no—
155#123—42no—
156#123—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—1182no—
160C2×C4⋊C4—1181no—
161C2×C4⋊C4C22⋊D5912yes—
162C2×C4⋊C4C22⋊D5912yes—
163C2×C4⋊C4C22⋊D5911yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C42C22×D5911yescharacteristic
168#107—2362no—
169#106—2364no—
170#102—2362no—
171#95—22no—
172#95—21no—
173#95—22no—
174C24—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4#1174567012yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—2361no—
186C22×C4C2×C22⋊D5911yescharacteristic
187C22×C4C2×C22⋊D5912yes—
188C2×C8—1182no—
189C2×C8—1184no—
190C2×C8—2362no—
191C2×C8—2361no—
192C2×C8—2362no—
193C2×C8—2361no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—2361no—
202C4⋊C4—2361no—
203C4⋊C4—2362no—
204C4⋊C4—2362no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—41no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23#1427369611yescharacteristic
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—4722no—
232C2×C4—4721no—
233C2×C4C2×#1174567011yescharacteristic
234C2×C4#1427374712yes—
235C2×C4#1427375212yes—
236C2×C4#1427375311yescharacteristic
237C8—2362no—
238C8—2364no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#1683754712yes—
245C22#1683755311yescharacteristic
246C22#1683757011yescharacteristic
247C22#1683758512yes—
248C22#1683759211yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—9441no—
257C2—82no—
258C2#2142309312yes—
259C2#2142311611yescharacteristic
260C2#2142315511yescharacteristic
261C2#2142338012yes—
262C2#2142340711yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2724C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2724D59
hypocenterC59#7819
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #21425572C2semidirect
3. #21421919C2semidirect
150. C59#7819semidirect

Maximal subgroups

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

subgroupindex
#214256242
#214255722
#214219192
#214219172
C59×#27242
#781959