Finite Group Databasebeta

← #28491923

Subgroups — #28491923

Order 15104.

414 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 77 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#21424196C212yesmaximal
2#21424171C211yesmaximal, characteristic
3#21422004C211yesmaximal, characteristic
4#21421949C212yesmaximal
5C59×#2607C211yesmaximal, characteristic
6C2×#14273686C2211yescharacteristic
7#16837366C2212yes—
8#16837265C2211yescharacteristic
9#16837168—21no—
10#16837168—21no—
11#16837162—22no—
12#16837162—22no—
13C59×#399—21no—
14C59×#393—21no—
15C59×#391—22no—
16C59×#387—22no—
17C59×#381C2211yescharacteristic
18C4×C22⋊C4×C59C2212yes—
19C2×#11745664—22no—
20#14273686—22no—
21#14273686—22no—
22C2×C4⋊Dic59Q812yescomplemented
23C2×C22⋊Dic59—21no—
24C2×C22⋊Dic59—21no—
25C2×C4×Dic59—21no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28C59⋊#95—22no—
29C22×Q8×C59—21no—
30C23×C4×C59C2311yescharacteristic
31C2×C4⋊C4×C59—22no—
32C2×C4⋊C4×C59—21no—
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59D411yescharacteristic
35C2×C22⋊C4×C59D411yescharacteristic
36C2×C42×C59D412yes—
37C2×C42×C59D412yes—
38#95×C59—21no—
39#95×C59—22no—
40#95×C59—21no—
41#95×C59—22no—
42#95×C59—22no—
43C24×C59C2×D411yescharacteristic
44C2×Q8×C59—44no—
45C2×Q8×C59—81no—
46C22×C4×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59—21no—
50C22×C4×C59—41no—
51C22×C4×C59—41no—
52C22×C4×C59C2×D412yes—
53C22×C4×C59C2×Q811yescharacteristic
54C22×C4×C59#4112yes—
55C22×C4×C59#4111yescharacteristic
56C22×Dic59—22no—
57C22×Dic59—41no—
58C4⋊C4×C59—44no—
59C4⋊C4×C59—81no—
60C22⋊C4×C59—44no—
61C22⋊C4×C59—41no—
62C22⋊C4×C59—41no—
63C42×C59—42no—
64C42×C59—42no—
65C22⋊Dic59—41no—
66C22⋊Dic59—41no—
67C22⋊Dic59—41no—
68C22⋊Dic59—41no—
69C4⋊Dic59—24no—
70C4×Dic59—42no—
71#11745664—42no—
72#11745664—42no—
73C2×C59⋊C8—42no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—42no—
77C23×C59#12011yescharacteristic
78C23×C59C22⋊Q811yescharacteristic
79C23×C59#12411yescharacteristic
80Q8×C59—84no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—24no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59C22⋊Q812yes—
97C2×C4×C59#12412yes—
98C2×Dic59—42no—
99C2×Dic59—82no—
100C2×Dic59—81no—
101C2×Dic59—82no—
102C59⋊C8—82no—
103#10387—591nomaximal, Sylow
104C22×C59—22no—
105C22×C59—21no—
106C22×C59—21no—
107C22×C59—22no—
108C22×C59—22no—
109C22×C59—81no—
110C22×C59—81no—
111C22×C59#39311yescharacteristic
112C22×C59#45811yescharacteristic
113C22×C59#45811yescharacteristic
114C4×C59—22no—
115C4×C59—22no—
116C4×C59—44no—
117C4×C59—82no—
118C4×C59—81no—
119C4×C59—81no—
120Dic59—82no—
121Dic59—161no—
122#2607D5911yescharacteristic
123#1826—592no—
124#1820—591no—
125#1306—591no—
126#1284—592no—
127C2×C59—21no—
128C2×C59—21no—
129C2×C59—41no—
130C2×C59—41no—
131C2×C59#203011yescharacteristic
132C2×C59#203112yes—
133C2×#117—591no—
134#407—592no—
135#399—21no—
136#393—21no—
137#391—22no—
138#387—22no—
139#381—591no—
140#381C2×D5911yescharacteristic
141C4×C22⋊C4C2×D5912yes—
142#342—1181no—
143#342—1181no—
144#339—1182no—
145#339—1182no—
146C59#1038711yescharacteristic, complemented, Sylow
147C22×Q8—21no—
148C23×C4C22×D5911yescharacteristic
149C2×M16—1182no—
150#117—1182no—
151#117—1182no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—592no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—1181no—
157C2×C22⋊C4—1181no—
158C2×C22⋊C4C22⋊D5911yescharacteristic
159C2×C22⋊C4C22⋊D5911yescharacteristic
160C2×C42—1181no—
161C2×C42C22⋊D5912yes—
162C2×C42C22⋊D5912yes—
163C22⋊C8—1182no—
164C22⋊C8—1182no—
165#95—21no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—1182no—
171C24C2×C22⋊D5911yescharacteristic
172C2×Q8—44no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—1182no—
181C22×C4—2361no—
182C22×C4Q8×D5911yescharacteristic
183C22×C4#1174568912yes—
184C22×C4#1174568911yescharacteristic
185C22×C4C2×C22⋊D5912yes—
186M16—2362no—
187M16—2362no—
188C2×C8—2362no—
189C4⋊C4—44no—
190C4⋊C4—81no—
191C4⋊C4—1184no—
192C22⋊C4—44no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—2361no—
196C22⋊C4—2361no—
197C22⋊C4—2361no—
198C22⋊C4—2361no—
199C42—42no—
200C42—42no—
201C42—2362no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#1427369611yescharacteristic
206C23#1427370311yescharacteristic
207C23#1427371611yescharacteristic
208Q8—84no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—4721no—
225C2×C4—4722no—
226C2×C4—4722no—
227C2×C4—2362no—
228C2×C4#1427370312yes—
229C2×C4#1427371612yes—
230C8—4722no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22#1683731311yescharacteristic
239C22#1683759211yescharacteristic
240C22#1683759211yescharacteristic
241C4—22no—
242C4—22no—
243C4—44no—
244C4—81no—
245C4—82no—
246C4—81no—
247C4—9441no—
248C4—4722no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#2142526211yescharacteristic
254C2#2142526512yes—
subgroupG/H
centerC22#16837313
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#2607C2
socleC22×C59#393
prime radicalC24×C59C2×D4
hypercenter#2607D59
hypocenterC59#10387
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
22. C2×C4⋊Dic59Q8semidirect
146. C59#10387semidirect

Maximal subgroups

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

subgroupindex
#214241962
#214241712
#214220042
#214219492
C59×#26072
#1038759