Finite Group Databasebeta

← #28501135

Subgroups — #28501135

Order 15104.

390 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426024C212yesmaximal
2#21425407C211yesmaximal, characteristic
3#21422108C212yesmaximal, complemented
4#21422036C211yesmaximal, characteristic, complemented
5C59×#3249C211yesmaximal, characteristic, complemented
6C22×D472C2211yescharacteristic
7C2×#14273670—22no—
8C2×#14273649—21no—
9#16837304—21no—
10#16837202—22no—
11C22⋊(C59⋊C16)C2212yes—
12#16837165C2211yescharacteristic
13C59×#533—21no—
14C59×#477—22no—
15C59×#467—21no—
16C59×#467C2212yes—
17C59×#462—22no—
18C2×C4⋊C8×C59C2211yescharacteristic
19C22×D236—21no—
20C2×D472—22no—
21C2×D472—42no—
22C2×C4⋊Dic59—21no—
23C2×(C2×D59)⋊C4—41no—
24C2×(C2×D59)⋊C4—41no—
25#14273670—44no—
26C2×C59⋊C16—22no—
27#14273649—42no—
28C2×Q16×C59—21no—
29C2×Q16×C59D412yes—
30C22×C8×C59D411yescharacteristic
31C22×C8×C59C2311yescharacteristic
32C4⋊Q8×C59—21no—
33C4⋊Q8×C59—21no—
34#125×C59—41no—
35C22⋊Q8×C59—22no—
36C22⋊Q8×C59—41no—
37C4×Q8×C59—22no—
38C4×Q8×C59—41no—
39#117×C59—41no—
40C2×C42×C59D411yescharacteristic
41#107×C59—21no—
42#107×C59D412yes—
43C4⋊C8×C59—22no—
44#103×C59—22no—
45#103×C59—22no—
46C23×D59—41no—
47C2×Q8×C59—22no—
48C2×Q8×C59—41no—
49C22×C4×C59D811yescharacteristic
50C22×C4×C59D811yescharacteristic
51C22×C4×C59C2×D411yescharacteristic
52C22×Dic59—41no—
53C2×D236—22no—
54C2×D236—41no—
55C2×D236—41no—
56C2×D236—41no—
57C2×D236—41no—
58C2×D236—41no—
59Q16×C59—42no—
60Q16×C59—81no—
61C2×C8×C59—21no—
62C2×C8×C59—41no—
63C2×C8×C59—41no—
64C2×C8×C59C2×D412yes—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C22⋊C4×C59—42no—
72C42×C59—21no—
73C42×C59—21no—
74C42×C59—42no—
75(C2×D59)⋊C4—82no—
76(C2×D59)⋊C4—82no—
77C4⋊Dic59—41no—
78C4⋊Dic59—41no—
79D472—24no—
80D472—42no—
81C59⋊C16—42no—
82C23×C59C2×D811yescharacteristic
83C22×D59—42no—
84C22×D59—81no—
85C22×D59—81no—
86C22×D59—81no—
87C22×D59—81no—
88C22×D59—81no—
89C22×D59—81no—
90Q8×C59—42no—
91Q8×C59—42no—
92Q8×C59—82no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59SD3212yescomplemented
102C2×C4×C59#12011yescharacteristic
103C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
104C2×Dic59—81no—
105C2×Dic59—81no—
106C2×Dic59—81no—
107D236—42no—
108D236—42no—
109D236—81no—
110D236—81no—
111C8×C59—22no—
112C8×C59—41no—
113#12159—591nomaximal, Sylow
114C22×C59—21no—
115C22×C59—21no—
116C22×C59#44711yescharacteristic
117C22×C59C2×SD3211yescharacteristic
118C22×C59#51011yescharacteristic
119C2×D59—82no—
120C2×D59—82no—
121C2×D59—82no—
122C2×D59—161no—
123C2×D59—161no—
124C2×D59—161no—
125C2×D59—161no—
126C4×C59—21no—
127C4×C59—22no—
128C4×C59—41no—
129C4×C59—82no—
130C4×C59—82no—
131C4×C59#45411yescharacteristic
132C4×C59B(4,𝔽2)11yescharacteristic
133Dic59—161no—
134#3249D5911yescharacteristic, complemented
135#2187—592no—
136#2054—591no—
137#1348—592no—
138#1322—591no—
139C2×C59—21no—
140C2×C59—21no—
141C2×C59#159811yescharacteristic
142C2×C59#218511yescharacteristic
143C2×C59#219011yescharacteristic
144D59—162no—
145C22×D8—591no—
146#533—21no—
147C2×SD32—1182no—
148#477—22no—
149#467—21no—
150#467C2×D5912yes—
151#462—22no—
152C2×C4⋊C8C2×D5911yescharacteristic
153C2×#102—1181no—
154#390—1181no—
155#357—1182no—
156C22⋊C16—592no—
157#340—591no—
158C59#1215911yescharacteristic, complemented, Sylow
159C22×D4—1181no—
160C2×Q16—21no—
161C2×Q16C22⋊D5912yes—
162C2×D8—1182no—
163C2×D8—2362no—
164C22×C8C22⋊D5911yescharacteristic
165C22×C8C22×D5911yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168#125—41no—
169C22⋊Q8—22no—
170C22⋊Q8—41no—
171C4×Q8—22no—
172C4×Q8—41no—
173#117—41no—
174C2×C4⋊C4—1181no—
175C2×C22⋊C4—2361no—
176C2×C22⋊C4—2361no—
177C2×C42C22⋊D5911yescharacteristic
178SD32—2364no—
179C2×C16—1182no—
180#107—21no—
181#107C22⋊D5912yes—
182C4⋊C8—22no—
183#103—22no—
184#103—22no—
185#102—2362no—
186C24—2361no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×D4—1182no—
190C2×D4—2361no—
191C2×D4—2361no—
192C2×D4—2361no—
193C2×D4—2361no—
194C2×D4—2361no—
195C22×C4—2361no—
196C22×C4#1174566911yescharacteristic
197C22×C4#1174566911yescharacteristic
198C22×C4C2×C22⋊D5911yescharacteristic
199Q16—42no—
200Q16—81no—
201D8—1184no—
202D8—2362no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C2×C22⋊D5912yes—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—2361no—
214C4⋊C4—2361no—
215C22⋊C4—42no—
216C22⋊C4—4722no—
217C22⋊C4—4722no—
218C42—21no—
219C42—21no—
220C42—42no—
221C16—2362no—
222C23—4721no—
223C23—4721no—
224C23—4721no—
225C23—4721no—
226C23—4721no—
227C23—4721no—
228C23—2362no—
229C23C2×#1174566911yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233D4—4721no—
234D4—4721no—
235D4—2362no—
236D4—2362no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—4721no—
246C2×C4—4721no—
247C2×C4—4721no—
248C2×C4#1427367012yescomplemented
249C2×C4#1427369611yescharacteristic
250C2×C4#1427375311yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—9441no—
256C22—9441no—
257C22—9441no—
258C22—9441no—
259C22—4722no—
260C22—4722no—
261C22—4722no—
262C22#1683753611yescharacteristic
263C22C2×#1427367011yescharacteristic
264C22#1683784311yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—82no—
269C4—82no—
270C4—9441no—
271C4#1683757511yescharacteristic
272C4#1683758911yescharacteristic
273C2—21no—
274C2—21no—
275C2—9442no—
276C2#2142303111yescharacteristic
277C2#2142601311yescharacteristic
278C2#2142604211yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3249C2
socleC22×C59#447
prime radicalC22×D472C22
hypercenter#3249D59
hypocenterC59#12159
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #21422108C2semidirect
4. #21422036C2semidirect
5. C59×#3249C2semidirect
101. C2×C4×C59SD32semidirect
134. #3249D59semidirect
158. C59#12159semidirect
248. C2×C4#14273670semidirect

Maximal subgroups

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

subgroupindex
#214260242
#214254072
#214221082
#214220362
C59×#32492
#1215959