Finite Group Databasebeta

← #28502369

Subgroups — #28502369

Order 15104.

298 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 67 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#21436989C211yesmaximal, characteristic, complemented
2#21426133C211yesmaximal, characteristic
3#21423453C211yesmaximal, characteristic
4#21422151C211yesmaximal, characteristic, complemented
5#21422076C211yesmaximal, characteristic, complemented
6#21422061C211yesmaximal, characteristic, complemented
7C59×#2240C211yesmaximal, characteristic
8#16838127—21no—
9C2×#14273671—21no—
10#16837803—21no—
11#16837708—22no—
12#16837684—21no—
13#16837599—21no—
14C8×Dic118C2211yescharacteristic
15C4×Dic236C2211yescharacteristic
16#16837496—21no—
17C4⋊(C59⋊C16)C2211yescharacteristic
18#16837203—21no—
19#16837155C2211yescharacteristic
20#16837146—21no—
21C59×#500C2211yescharacteristic
22C59×#493C2211yescharacteristic
23C59×#357—22no—
24C4×C16×C59C2211yescharacteristic
25C2×Dic236D411yescharacteristic
26C2×#11745670—41no—
27#14273725—41no—
28#14273718—22no—
29#14273717—42no—
30Q8×Dic59—41no—
31C4×Dic118—21no—
32C4×Dic118D411yescharacteristic
33#14273692—22no—
34#14273671—42no—
35C2×C59⋊C16—21no—
36#14273661D411yescharacteristic
37#14273658—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40Dic59⋊C8—21no—
41#14273652—21no—
42#14273650—22no—
43#14273648—41no—
44C8×Dic59—21no—
45C2×D8×C59—21no—
46C2×D8×C59—21no—
47C4⋊Q8×C59—21no—
48#127×C59—21no—
49C2×C16×C59—21no—
50#107×C59—21no—
51#107×C59D412yes—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×Dic118—21no—
58C2×Dic118D811yescharacteristic
59D8×C59—42no—
60D8×C59—41no—
61D8×C59—41no—
62C2×C8×C59C2×D411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—22no—
65C4⋊C4×C59—41no—
66C42×C59C2×D411yescharacteristic
67#11745670—82no—
68C4⋊Dic59—21no—
69C4⋊Dic59—42no—
70C4⋊Dic59D811yescharacteristic
71Dic59⋊C4—21no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74Dic59⋊C4—41no—
75C4×Dic59—21no—
76C4×Dic59—42no—
77C4×Dic59—41no—
78C2×C59⋊C8—21no—
79C2×C59⋊C8—41no—
80Dic236—21no—
81Dic236SD1612yescomplemented
82C16×C59—41no—
83C59⋊C16—41no—
84C23×C59—81no—
85Q8×C59—81no—
86Q8×C59—81no—
87D4×C59—81no—
88D4×C59—81no—
89D4×C59—81no—
90D4×C59—81no—
91D4×C59—161no—
92C2×C4×C59—42no—
93C2×C4×C59#12011yescharacteristic
94C2×C4×C59C2×D811yescharacteristic
95C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
96C2×Dic59—21no—
97C2×Dic59—21no—
98C2×Dic59—41no—
99C2×Dic59—41no—
100Dic118—22no—
101Dic118—21no—
102Dic118—21no—
103Dic118—41no—
104C8×C59—21no—
105C8×C59C2×SD1611yescharacteristic
106C8×C59#13711yescharacteristic
107C59⋊C8—41no—
108C59⋊C8—81no—
109#12353—591nomaximal, Sylow
110C22×C59—161no—
111C22×C59—161no—
112C22×C59—161no—
113C22×C59#44711yescharacteristic
114C4×C59—21no—
115C4×C59—82no—
116C4×C59#44811yescharacteristic
117C4×C59#45411yescharacteristic
118C4×C59#50811yescharacteristic
119C4×C59#50911yescharacteristic
120Dic59—42no—
121Dic59—41no—
122Dic59—41no—
123Dic59—81no—
124#3383—591no—
125#2240D5911yescharacteristic
126#2207—591no—
127#1668—591no—
128#1364—591no—
129Q8⋊C16—591no—
130#1332—591no—
131C2×C59—321no—
132C2×C59#162711yescharacteristic
133C2×C59#218611yescharacteristic
134C2×C59#219311yescharacteristic
135#557—1181no—
136C2×SD32—1181no—
137#500C2×D5911yescharacteristic
138#499—1181no—
139#493C2×D5911yescharacteristic
140#479—1182no—
141#475—1181no—
142#460—1181no—
143C8×Q8—591no—
144C4×Q16—591no—
145C4×Q16—1181no—
146C4⋊C16—591no—
147#357—22no—
148#357—1181no—
149C4×C16C2×D5911yescharacteristic
150#335—591no—
151#331—1181no—
152C59#1235311yescharacteristic, complemented, Sylow
153C2×Q16—591no—
154C2×SD16—2361no—
155C2×D8—21no—
156C2×D8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—2361no—
159#127—21no—
160#125—1182no—
161#125—2362no—
162C4×Q8—591no—
163C4×Q8—1182no—
164C4×Q8—1181no—
165C4×Q8—2361no—
166SD32—2362no—
167C2×C16—21no—
168C2×C16—1181no—
169#107—21no—
170#107—591no—
171#107C22⋊D5912yes—
172#106—1182no—
173C4⋊C8—591no—
174C4⋊C8—1181no—
175C4⋊C8—1181no—
176#103—1182no—
177#103—1181no—
178#102—2361no—
179C4×C8—1181no—
180C4×C8C22×D5911yescharacteristic
181C2×Q8—41no—
182C2×Q8—591no—
183C2×Q8—1181no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187Q16—592no—
188Q16—1181no—
189SD16—4722no—
190D8—42no—
191D8—41no—
192D8—41no—
193C2×C8—1181no—
194C2×C8—2361no—
195C2×C8D4×D5911yescharacteristic
196C2×C8C2×C22⋊D5911yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—591no—
200C4⋊C4—1181no—
201C4⋊C4—1181no—
202C4⋊C4—2362no—
203C4⋊C4—2362no—
204C4⋊C4—2362no—
205C4⋊C4—2361no—
206C42—1181no—
207C42—2362no—
208C42—2361no—
209C42D4×D5911yescharacteristic
210C16—41no—
211C16—2361no—
212C23—81no—
213Q8—81no—
214Q8—81no—
215Q8—1182no—
216Q8—1181no—
217Q8—1181no—
218Q8—2361no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—81no—
223D4—161no—
224C2×C4—42no—
225C2×C4—1181no—
226C2×C4—1181no—
227C2×C4—2361no—
228C2×C4—2361no—
229C2×C4#1427369511yescharacteristic
230C2×C4D8×D5911yescharacteristic
231C2×C4#1427375111yescharacteristic
232C8—21no—
233C8—4721no—
234C8—2361no—
235C8SD16×D5911yescharacteristic
236C8#1427375611yescharacteristic
237C22—161no—
238C22—161no—
239C22—161no—
240C22#1683753411yescharacteristic
241C4—21no—
242C4—82no—
243C4—4721no—
244C4—2362no—
245C4—2361no—
246C4—2361no—
247C4#1683753911yescharacteristic
248C4#1683757411yescharacteristic
249C4#1683783111yescharacteristic
250C4#1683783611yescharacteristic
251C2—321no—
252C2#2142319611yescharacteristic
253C2#2142601611yescharacteristic
254C2#2142605911yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2240C2
socleC22×C59#447
prime radicalC59×#493C22
hypercenter#2240D59
hypocenterC59#12353
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. #21436989C2semidirect
4. #21422151C2semidirect
5. #21422076C2semidirect
6. #21422061C2semidirect
81. Dic236SD16semidirect
152. C59#12353semidirect

Maximal subgroups

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

subgroupindex
#214369892
#214261332
#214234532
#214221512
#214220762
#214220612
C59×#22402
#1235359