Finite Group Databasebeta

← #28476804

Subgroups — #28476804

Order 15104.

492 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 143 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#21424213C212yesmaximal, complemented
2#21421922C212yesmaximal, complemented
3#21421917C212yesmaximal
4C4×#124×C59C211yesmaximal, characteristic
5C2×C4⋊(C59⋊C8)—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—
18C4⋊(C59⋊C8)—44no—
19C22×Q8×C59C2×C411yescharacteristic
20C22×D4×C59C2×C411yescharacteristic
21C23×C4×C59—21no—
22#124×C59—22no—
23#124×C59C814yescomplemented
24C4×Q8×C59—42no—
25C4×D4×C59—42no—
26C2×C4⋊C4×C59—21no—
27C2×C4⋊C4×C59C2×C411yescharacteristic
28C2×C4⋊C4×C59C2×C411yescharacteristic
29C2×C22⋊C4×C59—22no—
30C2×C22⋊C4×C59—21no—
31C2×C22⋊C4×C59—22no—
32C2×C42×C59—21no—
33C2×C42×C59C2×C411yescharacteristic
34C2×C42×C59C2×C411yescharacteristic
35C2×C42×C59D412yes—
36C2×C42×C59D412yes—
37C2×C42×C59C2311yescharacteristic
38#95×C59—22no—
39#95×C59—22no—
40C24×C59—21no—
41C2×Q8×C59—21no—
42C2×Q8×C59—41no—
43C2×Q8×C59—41no—
44C2×Q8×C59C2×C812yes—
45C2×D4×C59—21no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59C2×C812yes—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—44no—
53C22×C4×C59—42no—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D412yes—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—41no—
62C22⋊C4×C59—28no—
63C22⋊C4×C59—42no—
64C22⋊C4×C59—42no—
65C22⋊C4×C59—42no—
66C42×C59—22no—
67C42×C59—22no—
68C42×C59—22no—
69C42×C59—42no—
70C42×C59—41no—
71C42×C59—42no—
72C42×C59C2×C812yes—
73C42×C59M1612yes—
74C2×C59⋊C8—44no—
75C2×C59⋊C8—44no—
76C2×C59⋊C8—44no—
77C23×C59—22no—
78C23×C59—41no—
79C23×C59—41no—
80C23×C59—42no—
81C23×C59—42no—
82C23×C59C2×C22⋊C411yescharacteristic
83Q8×C59—42no—
84Q8×C59—42no—
85D4×C59—42no—
86D4×C59—42no—
87C2×C4×C59—22no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—84no—
104C2×C4×C59C22⋊C814yes—
105C2×C4×C59#10012yes—
106C2×C4×C59C22×C811yescharacteristic
107C2×C4×C59C2×M1611yescharacteristic
108C2×C4×C59AGL(1,ℤ/8)12yes—
109C2×C4×C59#13712yes—
110C59⋊C8—84no—
111#7601—591nomaximal, Sylow
112C22×C59—42no—
113C22×C59—44no—
114C22×C59—81no—
115C22×C59—82no—
116C22×C59—81no—
117C22×C59C2×C22⋊C811yescharacteristic
118C22×C59#41011yescharacteristic
119C22×C59C2×#10011yescharacteristic
120C22×C59#41712yes—
121C22×C59#42112yes—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—24no—
125C4×C59—22no—
126C4×C59—22no—
127C4×C59—22no—
128C4×C59—41no—
129C4×C59—42no—
130C4×C59—81no—
131C4×C59—81no—
132C4×#124D5911yescharacteristic
133#1831—592no—
134#1272—592no—
135#1270—592no—
136C2×C59—82no—
137C2×C59#146011yescharacteristic
138C2×C59#147712yes—
139C2×C59#151212yes—
140C2×C59#153311yescharacteristic
141C2×C59#156011yescharacteristic
142C2×#124Dic5911yescharacteristic
143C2×C4×Q8Dic5911yescharacteristic
144C2×C4×D4Dic5911yescharacteristic
145C2×C4⋊C8—1182no—
146#391C2×D5911yescharacteristic
147#390C2×D5911yescharacteristic
148#388—22no—
149#382Dic5911yescharacteristic
150C4×C22⋊C4—22no—
151C43C2×D5911yescharacteristic
152#336—594no—
153#336—1182no—
154C59#760111yescharacteristic, complemented, Sylow
155C22×Q8C2×Dic5911yescharacteristic
156C22×D4C2×Dic5911yescharacteristic
157C23×C4—21no—
158C22×C8—1184no—
159#124—22no—
160#124C59⋊C814yescomplemented
161C4×Q8—42no—
162C4×D4—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Dic5911yescharacteristic
165C2×C4⋊C4C2×Dic5911yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42—21no—
170C2×C42C2×Dic5911yescharacteristic
171C2×C42C2×Dic5911yescharacteristic
172C2×C42C22⋊D5912yes—
173C2×C42C22⋊D5912yes—
174C2×C42C22×D5911yescharacteristic
175C4⋊C8—2364no—
176#95—22no—
177#95—22no—
178C24—21no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C2×C59⋊C812yes—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×C59⋊C812yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4C22⋊Dic5911yescharacteristic
193C22×C4C22⋊Dic5912yes—
194C22×C4C22⋊Dic5911yescharacteristic
195C22×C4C22×Dic5911yescharacteristic
196C22×C4C2×C22⋊D5912yes—
197C2×C8—2364no—
198C2×C8—2364no—
199C2×C8—2364no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C22⋊C4—28no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—42no—
211C42—41no—
212C42—42no—
213C42C2×C59⋊C812yes—
214C42#1174566412yes—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23C2×C22⋊Dic5911yescharacteristic
221Q8—42no—
222Q8—42no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—84no—
242C2×C4C22⋊(C59⋊C8)14yes—
243C2×C4#1427364412yes—
244C2×C4C22×C59⋊C811yescharacteristic
245C2×C4C2×#1174566411yescharacteristic
246C2×C4#1427375212yes—
247C2×C4#1427375712yes—
248C8—4724no—
249C22—42no—
250C22—44no—
251C22—82no—
252C22—81no—
253C22—81no—
254C22C2×C22⋊(C59⋊C8)11yescharacteristic
255C22#1683737611yescharacteristic
256C22C2×#1427364411yescharacteristic
257C22#1683740312yes—
258C22#1683741912yes—
259C4—22no—
260C4—22no—
261C4—21no—
262C4—21no—
263C4—22no—
264C4—24no—
265C4—42no—
266C4—41no—
267C4—81no—
268C4—81no—
269C2—82no—
270C2#2142241711yescharacteristic
271C2#2142248312yes—
272C2#2142263412yes—
273C2#2142272011yescharacteristic
274C2#2142283111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC4×#124×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenterC4×#124D59
hypocenterC59#7601
LCSG ▷ C2×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. #21424213C2semidirect
2. #21421922C2semidirect
23. #124×C59C8semidirect
154. C59#7601semidirect
160. #124C59⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214242132
#214219222
#214219172
C4×#124×C592
#760159