Finite Group Databasebeta

← #28477004

Subgroups — #28477004

Order 15104.

468 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 119 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#21424239C212yesmaximal, complemented
2#21421922C212yesmaximal, complemented
3#21421917C212yesmaximal
4C4×#124×C59C211yesmaximal, characteristic
5#16837157—22no—
6#16837157—22no—
7#16837157C2214yes—
8C2×#124×C59C411yescharacteristic
9C2×C4×Q8×C59C411yescharacteristic
10C2×C4×D4×C59C411yescharacteristic
11C59×#391C2211yescharacteristic
12C59×#390C2211yescharacteristic
13C59×#388—22no—
14C59×#382C411yescharacteristic
15C4×C22⋊C4×C59—22no—
16C43×C59C2211yescharacteristic
17C22×C59⋊C8—24no—
18C22×Q8×C59C2×C411yescharacteristic
19C22×D4×C59C2×C411yescharacteristic
20C23×C4×C59—21no—
21#124×C59—22no—
22#124×C59—22no—
23C4×Q8×C59—42no—
24C4×D4×C59—42no—
25C2×C4⋊C4×C59—21no—
26C2×C4⋊C4×C59C2×C411yescharacteristic
27C2×C4⋊C4×C59C2×C411yescharacteristic
28C2×C22⋊C4×C59—22no—
29C2×C22⋊C4×C59—21no—
30C2×C22⋊C4×C59—22no—
31C2×C42×C59—21no—
32C2×C42×C59C2×C411yescharacteristic
33C2×C42×C59C2×C411yescharacteristic
34C2×C42×C59D412yes—
35C2×C42×C59D412yes—
36C2×C42×C59C2311yescharacteristic
37#95×C59—22no—
38#95×C59—22no—
39C24×C59—21no—
40C2×Q8×C59—21no—
41C2×Q8×C59—41no—
42C2×Q8×C59—41no—
43C2×Q8×C59M1612yes—
44C2×D4×C59—21no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59M1612yes—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—44no—
52C22×C4×C59—42no—
53C22×C4×C59C22⋊C412yes—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D412yes—
58C4⋊C4×C59—42no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—41no—
61C22⋊C4×C59—28no—
62C22⋊C4×C59—42no—
63C22⋊C4×C59—44no—
64C42×C59—22no—
65C42×C59—21no—
66C42×C59—21no—
67C42×C59—24no—
68C42×C59—42no—
69C42×C59—41no—
70C42×C59—42no—
71C2×C59⋊C8—44no—
72C2×C59⋊C8—44no—
73C2×C59⋊C8—44no—
74C23×C59—22no—
75C23×C59—41no—
76C23×C59—41no—
77C23×C59—42no—
78C23×C59—42no—
79C23×C59C2×C22⋊C411yescharacteristic
80Q8×C59—42no—
81Q8×C59—42no—
82D4×C59—42no—
83D4×C59—42no—
84C2×C4×C59—22no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—84no—
102C2×C4×C59#10012yes—
103C2×C4×C59C2×M1611yescharacteristic
104C2×C4×C59C2×M1611yescharacteristic
105C2×C4×C59#13514yes—
106C59⋊C8—84no—
107#7649—591nomaximal, Sylow
108C22×C59—42no—
109C22×C59—44no—
110C22×C59—81no—
111C22×C59—82no—
112C22×C59—81no—
113C22×C59#40711yescharacteristic
114C22×C59#41011yescharacteristic
115C22×C59C2×#10011yescharacteristic
116C22×C59#41612yes—
117C22×C59#42112yes—
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—24no—
121C4×C59—22no—
122C4×C59—41no—
123C4×C59—42no—
124C4×C59—42no—
125C4×C59—81no—
126C4×C59—81no—
127C4×#124D5911yescharacteristic
128#1839—592no—
129#1272—592no—
130#1270—592no—
131C2×C59—82no—
132C2×C59#146611yescharacteristic
133C2×C59#148812yes—
134C2×C59#152112yes—
135C2×C59#153311yescharacteristic
136C2×C59#154311yescharacteristic
137C2×#124Dic5911yescharacteristic
138C2×C4×Q8Dic5911yescharacteristic
139C2×C4×D4Dic5911yescharacteristic
140#391C2×D5911yescharacteristic
141#390C2×D5911yescharacteristic
142#388—22no—
143#382Dic5911yescharacteristic
144C4×C22⋊C4—22no—
145C43C2×D5911yescharacteristic
146#336—594no—
147#336—1182no—
148#336—1182no—
149C59#764911yescharacteristic, complemented, Sylow
150C22×Q8C2×Dic5911yescharacteristic
151C22×D4C2×Dic5911yescharacteristic
152C23×C4—21no—
153C22×C8—1184no—
154#124—22no—
155#124—22no—
156C4×Q8—42no—
157C4×D4—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic5911yescharacteristic
160C2×C4⋊C4C2×Dic5911yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42—21no—
165C2×C42C2×Dic5911yescharacteristic
166C2×C42C2×Dic5911yescharacteristic
167C2×C42C22⋊D5912yes—
168C2×C42C22⋊D5912yes—
169C2×C42C22×D5911yescharacteristic
170#95—22no—
171#95—22no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#1174566412yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#1174566412yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C22⋊Dic5912yes—
187C22×C4C22⋊Dic5911yescharacteristic
188C22×C4C22⋊Dic5911yescharacteristic
189C22×C4C22×Dic5911yescharacteristic
190C22×C4C2×C22⋊D5912yes—
191C2×C8—2364no—
192C2×C8—2364no—
193C2×C8—2364no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—28no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—24no—
204C42—42no—
205C42—41no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊Dic5911yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4#1427364412yes—
236C2×C4C2×#1174566411yescharacteristic
237C2×C4C2×#1174566411yescharacteristic
238C2×C4#1427374714yes—
239C8—4724no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22#1683736611yescharacteristic
246C22#1683737611yescharacteristic
247C22C2×#1427364411yescharacteristic
248C22#1683739812yes—
249C22#1683741912yes—
250C4—24no—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—81no—
258C4—81no—
259C2—82no—
260C2#2142243711yescharacteristic
261C2#2142253012yes—
262C2#2142267312yes—
263C2#2142272011yescharacteristic
264C2#2142275811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC4×#124×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenterC4×#124D59
hypocenterC59#7649
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #21424239C2semidirect
2. #21421922C2semidirect
149. C59#7649semidirect

Maximal subgroups

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

subgroupindex
#214242392
#214219222
#214219172
C4×#124×C592
#764959