Finite Group Databasebeta

← #28477003

Subgroups — #28477003

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#21426639C211yesmaximal, characteristic
2#21424238C212yesmaximal, complemented
3#21421922C211yesmaximal, characteristic, complemented
4#21421916C212yesmaximal
5C59×#1272C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×Q8×Dic59C411yescharacteristic
8C2×#14273689C411yescharacteristic
9#16837304C2211yescharacteristic
10#16837294—22no—
11#16837267C411yescharacteristic
12#16837249—22no—
13C42×Dic59C2211yescharacteristic
14#16837157C2212yes—
15#16837156—22no—
16#16837156—22no—
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×D236C2×C411yescharacteristic
20C22×C4×D59—21no—
21C22×C59⋊C8—22no—
22#14273716—22no—
23#14273716—22no—
24Q8×Dic59—42no—
25#14273689—42no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—21no—
31C2×C4×Dic59—21no—
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59D412yes—
35C2×C4×Dic59D412yes—
36C59⋊#95—22no—
37C22×Q8×C59C2×C411yescharacteristic
38C22×C8×C59—22no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C42×C59C2311yescharacteristic
41#95×C59—22no—
42C23×D59—21no—
43C2×Q8×C59—21no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59M1612yes—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59C22×C411yescharacteristic
50C22×C4×C59C2×D412yes—
51C22×Dic59—21no—
52C22×Dic59C22⋊C412yes—
53C22×Dic59C22⋊C411yescharacteristic
54C22×Dic59C22⋊C411yescharacteristic
55C2×D236—21no—
56C2×D236—41no—
57C2×D236—41no—
58C2×D236M1612yes—
59C2×C4×D59—44no—
60C2×C4×D59—42no—
61C2×C8×C59—42no—
62C2×C8×C59—42no—
63C2×C8×C59—42no—
64C4⋊C4×C59—41no—
65C42×C59—41no—
66(C2×D59)⋊C4—28no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—42no—
71C4⋊Dic59—41no—
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×C59C2×C22⋊C411yescharacteristic
83C22×D59—22no—
84C22×D59—41no—
85C22×D59—41no—
86C22×D59—42no—
87C22×D59—42no—
88Q8×C59—42no—
89Q8×C59—42no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59C2×M1611yescharacteristic
100C2×C4×C59C2×M1611yescharacteristic
101C2×C4×C59#13512yes—
102C2×C4×C59#13512yes—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59#10012yes—
112D236—42no—
113D236—42no—
114C4×D59—84no—
115C8×C59—82no—
116C59⋊C8—82no—
117#7649—591nomaximal, Sylow
118C22×C59#40711yescharacteristic
119C22×C59#41011yescharacteristic
120C22×C59C2×#10011yescharacteristic
121C22×C59#41612yes—
122C22×C59#42112yes—
123C2×D59—42no—
124C2×D59—44no—
125C2×D59—81no—
126C2×D59—82no—
127C2×D59—81no—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—22no—
131C4×C59—22no—
132C4×C59—42no—
133C4×C59—81no—
134Dic59—22no—
135Dic59—41no—
136Dic59—42no—
137Dic59—81no—
138C4×#124—591no—
139#1839—592no—
140#1272—591no—
141#1272D5911yescharacteristic, complemented
142#1270—592no—
143C2×C59#146611yescharacteristic
144C2×C59#148812yes—
145C2×C59#152112yes—
146C2×C59#153311yescharacteristic
147C2×C59#154311yescharacteristic
148D59—82no—
149C2×#124—591no—
150C2×C4×Q8—591no—
151C2×C4×D4—591no—
152#391C2×D5911yescharacteristic
153#390—591no—
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×Q8C4×D5911yescharacteristic
164C22×D4—591no—
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—1182no—
176C2×C22⋊C4—1181no—
177C2×C22⋊C4—1182no—
178C2×C42—591no—
179C2×C42—591no—
180C2×C42—592no—
181C2×C42—592no—
182C2×C42—1181no—
183C2×C42C22×D5911yescharacteristic
184#95—22no—
185#95—1182no—
186C24—1181no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1174565912yes—
191C2×D4—592no—
192C2×D4—1181no—
193C2×D4—2361no—
194C2×D4—2361no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—591no—
198C22×C4—592no—
199C22×C4—591no—
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—1188no—
215C22⋊C4—2362no—
216C22⋊C4—2362no—
217C22⋊C4—2362no—
218C42—41no—
219C42—1182no—
220C42—1181no—
221C42—1181no—
222C42—1182no—
223C42—1182no—
224C42—2362no—
225C42—2362no—
226C23—1182no—
227C23—2361no—
228C23—2362no—
229C23—2362no—
230C23—2361no—
231C23C22⋊C4×D5911yescharacteristic
232Q8—42no—
233Q8—42no—
234D4—2362no—
235D4—2362no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—592no—
246C2×C4—1182no—
247C2×C4—1181no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—4724no—
253C2×C4—2361no—
254C2×C4—2362no—
255C2×C4M16×D5911yescharacteristic
256C2×C4C2×#1174565911yescharacteristic
257C2×C4#1427374812yes—
258C2×C4#1427375012yes—
259C8—82no—
260C8—4722no—
261C22—4722no—
262C22—4721no—
263C22—4721no—
264C22—2362no—
265C22—2364no—
266C22#1683736411yescharacteristic
267C22#1683737411yescharacteristic
268C22#100×D5911yescharacteristic
269C22#1683739712yes—
270C22#1683741712yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—1182no—
278C4—4721no—
279C4—2361no—
280C4—2362no—
281C2—4722no—
282C2#2142243511yescharacteristic
283C2#2142252912yes—
284C2#2142267212yes—
285C2#2142271811yescharacteristic
286C2#2142275711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7649
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21424238C2semidirect
3. #21421922C2semidirect
5. C59×#1272C2semidirect
141. #1272D59semidirect
162. C59#7649semidirect

Maximal subgroups

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

subgroupindex
#214266392
#214242382
#214219222
#214219162
C59×#12722
#764959