Finite Group Databasebeta

← #28502375

Subgroups — #28502375

Order 15104.

298 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 69 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#21436985C211yesmaximal, characteristic, complemented
2#21426320C211yesmaximal, characteristic
3#21426136C211yesmaximal, characteristic
4#21423457C211yesmaximal, characteristic
5#21422152C211yesmaximal, characteristic, complemented
6#21422076C211yesmaximal, characteristic, complemented
7C59×#1332C211yesmaximal, characteristic, complemented
8#16838131—21no—
9C2×#14273670—21no—
10#16837807C2211yescharacteristic
11#16837803—21no—
12#16837779C2211yescharacteristic
13#16837711—22no—
14#16837687—21no—
15#16837603—21no—
16C8×Dic118C2211yescharacteristic
17Q16×Dic59—21no—
18C4⋊(C59⋊C16)C2211yescharacteristic
19#16837204—22no—
20#16837202—21no—
21#16837145—21no—
22C4×Q16×C59C2211yescharacteristic
23C4×C16×C59C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×#11745660—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273727—21no—
29#14273727—41no—
30#14273724—21no—
31#14273718—22no—
32#14273718—42no—
33Q8×Dic59—22no—
34C4×Dic118—41no—
35C4×Dic118D411yescharacteristic
36#14273670—42no—
37C2×C59⋊C16—21no—
38#14273661—21no—
39#14273661D412yes—
40#14273658—22no—
41C4⋊(C59⋊C8)D411yescharacteristic
42Dic59⋊C8—21no—
43#14273651—22no—
44#14273649—41no—
45C8×Dic59—21no—
46C2×Q16×C59D411yescharacteristic
47C4×Q8×C59—21no—
48C2×C16×C59—21no—
49#107×C59D411yescharacteristic
50C4⋊C8×C59—21no—
51#103×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—21no—
54C2×D236—41no—
55C2×D236—41no—
56C2×D236—41no—
57C2×Dic118—41no—
58C2×Dic118D811yescharacteristic
59Q16×C59—21no—
60Q16×C59SD1612yescomplemented
61C2×C8×C59—41no—
62C2×C8×C59C2×D411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—21no—
65C4⋊C4×C59—41no—
66C42×C59—41no—
67C42×C59C2×D411yescharacteristic
68C4⋊Dic59—22no—
69C4⋊Dic59—42no—
70C4⋊Dic59—42no—
71C4⋊Dic59—41no—
72C4⋊Dic59D811yescharacteristic
73Dic59⋊C4—21no—
74Dic59⋊C4—42no—
75C4×Dic59—21no—
76C4×Dic59—42no—
77C2×C59⋊C8—21no—
78D472—42no—
79D472—41no—
80D472—41no—
81#11745660—82no—
82C16×C59—41no—
83C59⋊C16—41no—
84C22×D59—81no—
85Q8×C59—22no—
86Q8×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59#12011yescharacteristic
90C2×C4×C59C2×D811yescharacteristic
91C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
92C2×Dic59—21no—
93C2×Dic59—21no—
94C2×Dic59—42no—
95D236—81no—
96D236—81no—
97D236—81no—
98D236—81no—
99D236—161no—
100Dic118—21no—
101Dic118—21no—
102Dic118—81no—
103Dic118—81no—
104C8×C59—21no—
105C8×C59—81no—
106C8×C59C2×SD1611yescharacteristic
107C8×C59#13711yescharacteristic
108C59⋊C8—41no—
109#12353—591nomaximal, Sylow
110C22×C59#44711yescharacteristic
111C2×D59—161no—
112C2×D59—161no—
113C2×D59—161no—
114C4×C59—21no—
115C4×C59—42no—
116C4×C59—81no—
117C4×C59#44811yescharacteristic
118C4×C59#45411yescharacteristic
119C4×C59#50811yescharacteristic
120C4×C59#50911yescharacteristic
121Dic59—41no—
122Dic59—41no—
123Dic59—82no—
124#3383—591no—
125#2240—591no—
126#2207—591no—
127#1668—591no—
128#1364—591no—
129Q8⋊C16—591no—
130#1332D5911yescharacteristic, complemented
131C2×C59#162711yescharacteristic
132C2×C59#218611yescharacteristic
133C2×C59#219311yescharacteristic
134D59—321no—
135#557—1181no—
136C2×SD32—1181no—
137#500—591no—
138#499—1181no—
139#493—591no—
140#479—1182no—
141#475—1181no—
142#460—1181no—
143C8×Q8—591no—
144C4×Q16—1181no—
145C4×Q16C2×D5911yescharacteristic
146C4⋊C16—591no—
147#357—1182no—
148#357—1181no—
149C4×C16C2×D5911yescharacteristic
150#335C2×D5911yescharacteristic
151#331—1181no—
152C59#1235311yescharacteristic, complemented, Sylow
153C2×Q16D23611yescharacteristic
154C2×SD16—2361no—
155C2×D8—1181no—
156C2×D8—1181no—
157C4⋊Q8—1181no—
158C4⋊Q8—2361no—
159#127—1181no—
160#125—1182no—
161#125—2362no—
162C4×Q8—21no—
163C4×Q8—591no—
164C4×Q8—1182no—
165C4×Q8—2361no—
166SD32—2362no—
167C2×C16—21no—
168C2×C16—1181no—
169#107—592no—
170#107—1181no—
171#107D23611yescharacteristic
172#106—1182no—
173C4⋊C8—21no—
174C4⋊C8—591no—
175C4⋊C8—1181no—
176#103—21no—
177#103—1182no—
178#102—2361no—
179C4×C8—1181no—
180C4×C8C22×D5911yescharacteristic
181C2×Q8—21no—
182C2×Q8—591no—
183C2×Q8—2361no—
184C2×D4—2361no—
185C2×D4—2361no—
186C2×D4—2361no—
187Q16—21no—
188Q16#1174566012yescomplemented
189SD16—4722no—
190D8—2362no—
191D8—2361no—
192D8—2361no—
193C2×C8—41no—
194C2×C8—1181no—
195C2×C8C2×D23611yescharacteristic
196C2×C8D4×D5911yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—591no—
200C4⋊C4—1181no—
201C4⋊C4—1182no—
202C4⋊C4—2362no—
203C4⋊C4—2362no—
204C4⋊C4—2362no—
205C4⋊C4—2361no—
206C42—41no—
207C42—1181no—
208C42—2362no—
209C42D4×D5911yescharacteristic
210C16—41no—
211C16—2361no—
212C23—4721no—
213Q8—22no—
214Q8—41no—
215Q8—1181no—
216Q8—1181no—
217Q8—4721no—
218Q8—4721no—
219D4—9441no—
220D4—4721no—
221D4—4721no—
222D4—4721no—
223D4—4721no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—1181no—
227C2×C4—1181no—
228C2×C4—2362no—
229C2×C4#1427369711yescharacteristic
230C2×C4D8×D5911yescharacteristic
231C2×C4#1427375411yescharacteristic
232C8—21no—
233C8—81no—
234C8—2361no—
235C8C2×#1174566011yescharacteristic
236C8#1427376011yescharacteristic
237C22—9441no—
238C22—9441no—
239C22—9441no—
240C22#1683753711yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—4722no—
245C4—2361no—
246C4—2361no—
247C4#1683754311yescharacteristic
248C4#1683757611yescharacteristic
249C4#1683783511yescharacteristic
250C4#1683783911yescharacteristic
251C2—18881no—
252C2#2142320211yescharacteristic
253C2#2142602211yescharacteristic
254C2#2142606111yescharacteristic
subgroupG/H
centerC22#16837537
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1332C2
socleC22×C59#447
prime radical#16837779C22
hypercenter#1332D59
hypocenterC59#12353
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
1. #21436985C2semidirect
5. #21422152C2semidirect
6. #21422076C2semidirect
7. C59×#1332C2semidirect
60. Q16×C59SD16semidirect
130. #1332D59semidirect
152. C59#12353semidirect
188. Q16#11745660semidirect

Maximal subgroups

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

subgroupindex
#214369852
#214263202
#214261362
#214234572
#214221522
#214220762
C59×#13322
#1235359