Finite Group Databasebeta

← #28508376

Subgroups — #28508376

Order 15104.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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#21424310C212yesmaximal
2#21424310C211yesmaximal, characteristic
3#21422223C211yesmaximal, characteristic
4#21422223C212yesmaximal
5C59×#2809C211yesmaximal, characteristic
6#16837880C2211yescharacteristic
7#16837442—22no—
8#16837442—21no—
9C2×#14273646C2212yes—
10C2×#14273646C2211yescharacteristic
11#16837162—22no—
12#16837162—21no—
13C59×#400—21no—
14C59×#398—22no—
15C59×#391C2211yescharacteristic
16C59×#391C2212yes—
17C59×#389—21no—
18C59×#387—22no—
19C2×#11745664—22no—
20C2×#11745664—21no—
21Q8×Dic59—41no—
22Q8×Dic59—41no—
23#14273686—22no—
24#14273686—21no—
25#14273660—42no—
26#14273660—41no—
27#14273658—42no—
28#14273658—41no—
29#14273646—22no—
30#14273646—21no—
31#14273646Q814yes—
32#14273646Q812yes—
33C22×Q8×C59C2311yescharacteristic
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59D412yes—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C42×C59D412yes—
41C2×C42×C59D411yescharacteristic
42#95×C59—21no—
43#95×C59—22no—
44#95×C59—22no—
45#95×C59—22no—
46#95×C59—22no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59C2×Q812yes—
50C2×Q8×C59C2×Q811yescharacteristic
51C2×Q8×C59#4111yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59C2×D412yes—
57C22×C4×C59C2×D411yescharacteristic
58C4⋊C4×C59—44no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—81no—
63C4⋊C4×C59—81no—
64C42×C59—41no—
65C42×C59—42no—
66C22⋊Dic59—41no—
67C4⋊Dic59—22no—
68C4⋊Dic59—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59—22no—
71C4×Dic59—42no—
72C4×Dic59—41no—
73#11745664—24no—
74#11745664—22no—
75#11745664—41no—
76#11745664—42no—
77C2×C59⋊C8—42no—
78C2×C59⋊C8—41no—
79C23×C59#12011yescharacteristic
80Q8×C59—42no—
81Q8×C59—41no—
82Q8×C59—41no—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59—41no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59C22⋊Q812yes—
99C2×C4×C59C22⋊Q811yescharacteristic
100C2×C4×C59C4⋊Q812yes—
101C2×C4×C59C4⋊Q811yescharacteristic
102C2×Dic59—41no—
103C2×Dic59—41no—
104C59⋊C8—44no—
105C59⋊C8—42no—
106#13465—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59#39511yescharacteristic
110C22×C59B(4,𝔽2)11yescharacteristic
111C22×C59#45811yescharacteristic
112C4×C59—22no—
113C4×C59—21no—
114C4×C59—22no—
115C4×C59—21no—
116C4×C59—42no—
117C4×C59—82no—
118C4×C59—81no—
119C4×C59—82no—
120Dic59—81no—
121Dic59—81no—
122#2809D5911yescharacteristic
123#1857—592no—
124#1857—591no—
125#1394—591no—
126#1394—592no—
127C2×C59—21no—
128C2×C59—21no—
129C2×C59#206211yescharacteristic
130C2×C59#220311yescharacteristic
131C2×C59#220411yescharacteristic
132#519—591no—
133#428—1182no—
134#428—1181no—
135C2×#101—592no—
136C2×#101—591no—
137#400—21no—
138#398—22no—
139#391C2×D5911yescharacteristic
140#391C2×D5912yes—
141#389—21no—
142#387—22no—
143#339—1182no—
144#339—1181no—
145C59#1346511yescharacteristic, complemented, Sylow
146C22×Q8C22×D5911yescharacteristic
147C2×M16—1182no—
148C2×M16—1181no—
149C4×Q8—2361no—
150C4×Q8—2361no—
151#117—1182no—
152#117—1181no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D5912yes—
158C2×C4⋊C4C22⋊D5911yescharacteristic
159C2×C42C22⋊D5912yes—
160C2×C42C22⋊D5911yescharacteristic
161#107—2362no—
162#107—2361no—
163#106—2362no—
164#106—2361no—
165#101—594no—
166#101—592no—
167#101—1182no—
168#101—1181no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8Q8×D5912yes—
177C2×Q8Q8×D5911yescharacteristic
178C2×Q8#1174568911yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×C22⋊D5912yes—
184C22×C4C2×C22⋊D5911yescharacteristic
185M16—1184no—
186M16—1182no—
187M16—2361no—
188M16—2362no—
189C2×C8—2362no—
190C2×C8—2361no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—1182no—
198C4⋊C4—1181no—
199C4⋊C4—1181no—
200C4⋊C4—1182no—
201C22⋊C4—2361no—
202C42—41no—
203C42—42no—
204C42—2362no—
205C42—2361no—
206C23#1427369611yescharacteristic
207Q8—42no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—2361no—
226C2×C4—2361no—
227C2×C4#1427370312yes—
228C2×C4#1427370311yescharacteristic
229C2×C4#1427372512yes—
230C2×C4#1427372511yescharacteristic
231C8—2364no—
232C8—2362no—
233C22—21no—
234C22—21no—
235C22#1683732111yescharacteristic
236C22#1683758911yescharacteristic
237C22#1683759211yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4—82no—
246C4—4721no—
247C4—4721no—
248C2—21no—
249C2—21no—
250C2#2142544811yescharacteristic
251C2#2142611711yescharacteristic
252C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC22×Q8×C59C23
FrattiniC22×Q8C22×D59
FittingC59×#2809C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2809D59
hypocenterC59#13465
LCSG ▷ C22×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

Complements

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

NH ≅ G/Ntype
145. C59#13465semidirect

Maximal subgroups

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

subgroupindex
#214243102
#214222232
C59×#28092
#1346559