Finite Group Databasebeta

← #28497962

Subgroups — #28497962

Order 15104.

460 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 89 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#21426635C211yesmaximal, characteristic
2#21422418C212yesmaximal, complemented
3#21422040C211yesmaximal, characteristic, complemented
4#21422027C212yesmaximal
5C59×#1324C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×#14273692C2211yescharacteristic
8C2×#14273689C411yescharacteristic
9#16837452—22no—
10#16837307C411yescharacteristic
11#16837303C2211yescharacteristic
12#16837291—22no—
13#16837249—22no—
14C42×Dic59C411yescharacteristic
15#16837128C2212yes—
16#16837127—22no—
17C59×#382C2211yescharacteristic
18C59×#324C2212yes—
19C22×Dic118C2×C411yescharacteristic
20C22×D236C2×C411yescharacteristic
21C22×C4×D59—21no—
22#14273715—22no—
23#14273715C814yescomplemented
24#14273692—41no—
25#14273692—41no—
26#14273689—41no—
27#14273689—41no—
28C2×Dic59⋊C4D412yes—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—21no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59C2×C411yescharacteristic
35C2×C4×Dic59C2×C411yescharacteristic
36C2×C4×Dic59D412yes—
37Dic59⋊C8—42no—
38C22⋊(C59⋊C8)—22no—
39C8×Dic59—42no—
40C59⋊#95—22no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C22⋊C8×C59—22no—
45#95×C59—22no—
46C23×D59—21no—
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×D236C2×C812yes—
59C2×C4×D59—44no—
60C2×C4×D59—41no—
61C2×C4×D59—41no—
62C2×Dic118—41no—
63C2×Dic118—41no—
64C2×Dic118C2×C812yes—
65C2×Dic118M1612yes—
66C2×C8×C59—42no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—41no—
69C42×C59—21no—
70C42×C59C2×C812yes—
71(C2×D59)⋊C4—28no—
72(C2×D59)⋊C4—44no—
73(C2×D59)⋊C4—42no—
74Dic59⋊C4—42no—
75C4×Dic59—22no—
76C4×Dic59—22no—
77C4×Dic59—21no—
78C4×Dic59—21no—
79C4×Dic59—42no—
80C4×Dic59—42no—
81C4×Dic59—41no—
82C2×C59⋊C8—42no—
83C23×C59C2×C22⋊C411yescharacteristic
84C22×D59—22no—
85C22×D59—42no—
86C22×D59—42no—
87C22×D59—41no—
88C22×D59—41no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—41no—
98C2×C4×C59C22×C811yescharacteristic
99C2×C4×C59C2×M1611yescharacteristic
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—41no—
106C2×Dic59—41no—
107C2×Dic59—41no—
108C2×Dic59C22⋊C814yes—
109C2×Dic59#9911yescharacteristic
110C2×Dic59#9911yescharacteristic
111D236—42no—
112D236—42no—
113C4×D59—82no—
114C4×D59—82no—
115Dic118—42no—
116Dic118—42no—
117C8×C59—82no—
118C59⋊C8—82no—
119#11476—591nomaximal, Sylow
120C22×C59—21no—
121C22×C59—21no—
122C22×C59C2×C22⋊C811yescharacteristic
123C22×C59C2×#9911yescharacteristic
124C22×C59#41311yescharacteristic
125C2×D59—44no—
126C2×D59—42no—
127C2×D59—82no—
128C2×D59—81no—
129C2×D59—81no—
130C4×C59—21no—
131C4×C59—21no—
132C4×C59—42no—
133C4×C59—42no—
134C4×C59—81no—
135Dic59—21no—
136Dic59—24no—
137Dic59—42no—
138Dic59—41no—
139Dic59#35512yes—
140C4×#124—591no—
141#1461—592no—
142#1324—591no—
143#1324D5911yescharacteristic, complemented
144#1318—592no—
145C2×C59—21no—
146C2×C59—21no—
147C2×C59#145911yescharacteristic
148C2×C59#212611yescharacteristic
149C2×C59C2×#35511yescharacteristic
150D59—82no—
151C2×#124—591no—
152C2×C4×Q8—591no—
153C2×C4×D4—591no—
154#431—1182no—
155#391—591no—
156#390—591no—
157#388—1182no—
158#382C2×D5911yescharacteristic
159C4×C22⋊C4—1182no—
160C43—591no—
161#324—592no—
162#324—1182no—
163#324C2×D5912yes—
164C59#1147611yescharacteristic, complemented, Sylow
165C22×Q8—591no—
166C22×D4—591no—
167C23×C4—1181no—
168#124—594no—
169#124—1182no—
170C4×Q8—2361no—
171C4×Q8—2361no—
172C4×D4—2361no—
173C4×D4—2361no—
174C2×C4⋊C4—592no—
175C2×C4⋊C4C4×D5911yescharacteristic
176C2×C4⋊C4C22×D5911yescharacteristic
177C2×C22⋊C4—1182no—
178C2×C22⋊C4—1181no—
179C2×C22⋊C4—1182no—
180C2×C42—591no—
181C2×C42—592no—
182C2×C42—591no—
183C2×C42—1181no—
184C2×C42—1181no—
185C2×C42C4×D5911yescharacteristic
186C4⋊C8—2362no—
187C22⋊C8—22no—
188C22⋊C8—1182no—
189C4×C8—2362no—
190#95—22no—
191#95—1182no—
192C24—1181no—
193C2×Q8—592no—
194C2×Q8—592no—
195C2×Q8—2361no—
196C2×Q8—2361no—
197C2×D4—592no—
198C2×D4—1181no—
199C2×D4—2361no—
200C2×D4—2361no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—592no—
204C22×C4—591no—
205C22×C4—591no—
206C22×C4—1181no—
207C22×C4—2361no—
208C22×C4—2361no—
209C22×C4—2364no—
210C22×C4C2×C4×D5911yescharacteristic
211C22×C4D4×D5912yes—
212C2×C8—42no—
213C2×C8—2362no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—2362no—
217C22⋊C4—1188no—
218C22⋊C4—2362no—
219C22⋊C4—2364no—
220C42—21no—
221C42—1182no—
222C42—1182no—
223C42—1181no—
224C42—1181no—
225C42—2361no—
226C42—2362no—
227C42—2362no—
228C42C8×D5912yes—
229C23—1182no—
230C23—2362no—
231C23—2362no—
232C23—2361no—
233C23—2361no—
234C23C22⋊C4×D5911yescharacteristic
235Q8—2362no—
236Q8—2362no—
237D4—2362no—
238D4—2362no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—591no—
249C2×C4—594no—
250C2×C4—591no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—1182no—
254C2×C4—1182no—
255C2×C4—1181no—
256C2×C4—4722no—
257C2×C4—4722no—
258C2×C4—2361no—
259C2×C4—2361no—
260C2×C4—2361no—
261C2×C4C2×C8×D5911yescharacteristic
262C2×C4M16×D5911yescharacteristic
263C8—82no—
264C8—4722no—
265C22—21no—
266C22—21no—
267C22—4722no—
268C22—4721no—
269C22—4721no—
270C22—2364no—
271C22—2362no—
272C22C22⋊C8×D5911yescharacteristic
273C22#99×D5911yescharacteristic
274C22#1683738511yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—592no—
281C4—1184no—
282C4—1181no—
283C4—2362no—
284C4—2361no—
285C2—21no—
286C2—21no—
287C2—4722no—
288C2#2142241311yescharacteristic
289C2#2142576011yescharacteristic
290C2#355×D5911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1324C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1324D59
hypocenterC59#11476
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
2. #21422418C2semidirect
3. #21422040C2semidirect
5. C59×#1324C2semidirect
23. #14273715C8semidirect
143. #1324D59semidirect
164. C59#11476semidirect

Maximal subgroups

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

subgroupindex
#214266352
#214224182
#214220402
#214220272
C59×#13242
#1147659