Finite Group Databasebeta

← #28477001

Subgroups — #28477001

Order 15104.

468 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 99 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#21426643C211yesmaximal, characteristic
2#21424238C212yesmaximal, complemented
3#21421921C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#1270C211yesmaximal, characteristic
6C2×#14273714C411yescharacteristic
7C2×#14273692C411yescharacteristic
8C2×D4×Dic59C411yescharacteristic
9#16837308C2211yescharacteristic
10#16837290—22no—
11#16837267C411yescharacteristic
12C22⋊C4×Dic59—22no—
13C42×Dic59C2211yescharacteristic
14#16837157C2212yes—
15#16837156—22no—
16#16837156—22no—
17C59×#390C2211yescharacteristic
18C59×#336C2212yes—
19C22×Dic118C2×C411yescharacteristic
20C23×Dic59—21no—
21C22×C59⋊C8—22no—
22#14273714—22no—
23#14273714—22no—
24#14273692—42no—
25D4×Dic59—42no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—21no—
30C2×C4×Dic59—21no—
31C2×C4×Dic59C2×C411yescharacteristic
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59D412yes—
34C2×C4×Dic59D412yes—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C22×D4×C59C2×C411yescharacteristic
38C22×C8×C59—22no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C42×C59C2311yescharacteristic
42C24×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59M1612yes—
47C22×C4×C59—21no—
48C22×C4×C59C22×C411yescharacteristic
49C22×C4×C59C2×D412yes—
50C22×Dic59—21no—
51C22×Dic59—21no—
52C22×Dic59—44no—
53C22×Dic59—42no—
54C22×Dic59C22⋊C412yes—
55C22×Dic59C22⋊C411yescharacteristic
56C22×Dic59C22⋊C411yescharacteristic
57C2×Dic118—21no—
58C2×Dic118—41no—
59C2×Dic118—41no—
60C2×Dic118M1612yes—
61C2×C8×C59—42no—
62C2×C8×C59—42no—
63C2×C8×C59—42no—
64C4⋊C4×C59—41no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—41no—
68C22⋊Dic59—28no—
69C22⋊Dic59—42no—
70C4⋊Dic59—41no—
71Dic59⋊C4—42no—
72C4×Dic59—21no—
73C4×Dic59—21no—
74C4×Dic59—22no—
75C4×Dic59—22no—
76C4×Dic59—22no—
77C4×Dic59—42no—
78C4×Dic59—42no—
79C2×C59⋊C8—42no—
80C2×C59⋊C8—42no—
81C2×C59⋊C8—42no—
82C23×C59—22no—
83C23×C59—41no—
84C23×C59—41no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59C2×C22⋊C411yescharacteristic
88D4×C59—42no—
89D4×C59—42no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—41no—
95C2×C4×C59—42no—
96C2×C4×C59C2×M1611yescharacteristic
97C2×C4×C59C2×M1611yescharacteristic
98C2×C4×C59#13512yes—
99C2×C4×C59#13512yes—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—21no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—41no—
108C2×Dic59—41no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59—84no—
112C2×Dic59#10012yes—
113Dic118—42no—
114Dic118—42no—
115C8×C59—82no—
116C59⋊C8—82no—
117#7649—591nomaximal, Sylow
118C22×C59—42no—
119C22×C59—44no—
120C22×C59—81no—
121C22×C59—82no—
122C22×C59—81no—
123C22×C59#40711yescharacteristic
124C22×C59#41011yescharacteristic
125C22×C59C2×#10011yescharacteristic
126C22×C59#41612yes—
127C22×C59#42112yes—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—22no—
131C4×C59—22no—
132C4×C59—81no—
133Dic59—22no—
134Dic59—41no—
135Dic59—42no—
136Dic59—42no—
137Dic59—81no—
138C4×#124—591no—
139#1839—592no—
140#1272—592no—
141#1270—591no—
142#1270D5911yescharacteristic
143C2×C59—82no—
144C2×C59#146611yescharacteristic
145C2×C59#148812yes—
146C2×C59#152112yes—
147C2×C59#153311yescharacteristic
148C2×C59#154311yescharacteristic
149C2×#124—591no—
150C2×C4×Q8—591no—
151C2×C4×D4—591no—
152#391—591no—
153#390C2×D5911yescharacteristic
154#388—1182no—
155#382—591no—
156C4×C22⋊C4—1182no—
157C43—591no—
158#336—592no—
159#336—1182no—
160#336—1182no—
161#336C2×D5912yes—
162C59#764911yescharacteristic, complemented, Sylow
163C22×Q8—591no—
164C22×D4C4×D5911yescharacteristic
165C23×C4—1181no—
166C22×C8—22no—
167C22×C8—1182no—
168#124—1182no—
169#124—1182no—
170C4×Q8—2362no—
171C4×D4—2362no—
172C2×C4⋊C4—591no—
173C2×C4⋊C4—1181no—
174C2×C4⋊C4C4×D5911yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—1182no—
177C2×C22⋊C4—1181no—
178C2×C42—591no—
179C2×C42—591no—
180C2×C42—592no—
181C2×C42—592no—
182C2×C42—1181no—
183C2×C42C22×D5911yescharacteristic
184#95—1182no—
185#95—1182no—
186C24—21no—
187C2×Q8—592no—
188C2×Q8—1181no—
189C2×Q8—2361no—
190C2×Q8—2361no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4#1174565912yes—
195C22×C4—21no—
196C22×C4—591no—
197C22×C4—592no—
198C22×C4—591no—
199C22×C4—1181no—
200C22×C4—1181no—
201C22×C4—2362no—
202C22×C4—2364no—
203C22×C4C2×C4×D5911yescharacteristic
204C22×C4D4×D5912yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—2362no—
209C2×C8—2362no—
210C2×C8—2362no—
211C4⋊C4—41no—
212C4⋊C4—2361no—
213C4⋊C4—2362no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—1188no—
217C22⋊C4—2362no—
218C42—41no—
219C42—1182no—
220C42—1181no—
221C42—1181no—
222C42—1182no—
223C42—1182no—
224C42—2362no—
225C42—2362no—
226C23—22no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23C22⋊C4×D5911yescharacteristic
232Q8—2362no—
233Q8—2362no—
234D4—42no—
235D4—42no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—592no—
243C2×C4—1182no—
244C2×C4—1181no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—4724no—
251C2×C4—2361no—
252C2×C4—2362no—
253C2×C4—2361no—
254C2×C4—2361no—
255C2×C4M16×D5911yescharacteristic
256C2×C4C2×#1174565911yescharacteristic
257C2×C4#1427374612yes—
258C2×C4#1427374812yes—
259C8—82no—
260C8—4722no—
261C22—42no—
262C22—44no—
263C22—82no—
264C22—81no—
265C22—81no—
266C22#1683736411yescharacteristic
267C22#1683737411yescharacteristic
268C22#100×D5911yescharacteristic
269C22#1683739512yes—
270C22#1683741712yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—81no—
276C4—1182no—
277C4—4721no—
278C4—2362no—
279C4—2361no—
280C4—2362no—
281C2—82no—
282C2#2142243511yescharacteristic
283C2#2142252712yes—
284C2#2142267012yes—
285C2#2142271811yescharacteristic
286C2#2142275511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1270D59
hypocenterC59#7649
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21424238C2semidirect
3. #21421921C2semidirect
162. C59#7649semidirect

Maximal subgroups

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

subgroupindex
#214266432
#214242382
#214219212
#214219172
C59×#12702
#764959