Finite Group Databasebeta

← #28486533

Subgroups — #28486533

Order 15104.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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#21429458C211yesmaximal, characteristic
2#21424092C211yesmaximal, characteristic
3#21422526C212yesmaximal
4#21421955C212yesmaximal
5C59×#2982C211yesmaximal, characteristic
6C2×C4×Dic118C2211yescharacteristic
7#16837426C2212yes—
8C2×#14273646—21no—
9#16837341C412yes—
10#16837322C412yes—
11#16837309—21no—
12#16837298—21no—
13#16837286—22no—
14#16837268C2211yescharacteristic
15#16837171—21no—
16#16837134—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C8×Q8×C59—24no—
19C59×#431—22no—
20C59×#431C2212yes—
21C22×Dic118C2×C411yescharacteristic
22C2×#11745664—22no—
23C4×Dic118—22no—
24C4×Dic118—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C4×Dic59—21no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)—22no—
32#14273646—42no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C59⋊#95C2×C412yes—
38C22×Q8×C59D411yescharacteristic
39C4×Q8×C59—22no—
40C4×Q8×C59—22no—
41C2×C4⋊C4×C59C2×C412yes—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C42×C59D412yes—
44C2×C42×C59C2311yescharacteristic
45C4⋊C8×C59—24no—
46C4⋊C8×C59—22no—
47C22⋊C8×C59—22no—
48C4×C8×C59—24no—
49C4×C8×C59D414yes—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59SD1612yes—
53C2×Q8×C59Q1612yes—
54C22×C4×C59C22⋊C412yes—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—21no—
60C22×Dic59—22no—
61C22×Dic59—21no—
62C2×Dic118—22no—
63C2×Dic118—42no—
64C2×C8×C59—24no—
65C4⋊C4×C59—24no—
66C4⋊C4×C59—22no—
67C42×C59—22no—
68C42×C59—22no—
69C42×C59C2×D412yes—
70C42×C59#4112yes—
71C4⋊Dic59—22no—
72C4⋊Dic59—42no—
73Dic59⋊C4—44no—
74Dic59⋊C4—41no—
75Dic59⋊C4—41no—
76C4×Dic59—42no—
77#11745664—42no—
78#11745664—42no—
79C2×C59⋊C8—42no—
80C23×C59C2×C22⋊C411yescharacteristic
81Q8×C59—24no—
82Q8×C59—24no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59#10314yes—
90C2×C4×C59C4×D412yes—
91C2×C4×C59#12011yescharacteristic
92C2×C4×C59#12112yes—
93C2×C4×C59#12311yescharacteristic
94C2×C4×C59C2×SD1611yescharacteristic
95C2×C4×C59C2×Q1611yescharacteristic
96C2×C4×C59#13512yes—
97C2×Dic59—22no—
98C2×Dic59—41no—
99C2×Dic59—41no—
100C2×Dic59—42no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—41no—
104C2×Dic59—42no—
105Dic118—24no—
106Dic118—42no—
107C8×C59—44no—
108C59⋊C8—82no—
109#9484—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—21no—
112C22×C59#38611yescharacteristic
113C22×C59C2×#10311yescharacteristic
114C22×C59#41611yescharacteristic
115C4×C59—22no—
116C4×C59—24no—
117C4×C59—21no—
118C4×C59—21no—
119C4×C59—42no—
120C4×C59#46512yes—
121C4×C59#46712yes—
122Dic59—42no—
123Dic59—82no—
124Dic59—81no—
125#2982D5911yescharacteristic
126#2581—591no—
127#1801—591no—
128#1487—592no—
129#1286—592no—
130C2×C59—21no—
131C2×C59—21no—
132C2×C59#189111yescharacteristic
133C2×C59#189211yescharacteristic
134C2×C59#191211yescharacteristic
135C2×C4×Q8—591no—
136C2×C4×Q8C2×D5911yescharacteristic
137C8×Q8—24no—
138#431—22no—
139#431C2×D5912yes—
140#423—592no—
141C2×#101—1181no—
142#400—592no—
143#395—592no—
144#391—1181no—
145#389—1181no—
146#387—1182no—
147#382—591no—
148#343—1181no—
149Q8⋊C8—1184no—
150C59#948411yescharacteristic, complemented, Sylow
151C22×Q8—591no—
152C22×Q8D23611yescharacteristic
153C2×M16—1182no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1182no—
157C4×Q8—1182no—
158C2×C4⋊C4—591no—
159C2×C4⋊C4—1182no—
160C2×C4⋊C4—1181no—
161C2×C4⋊C4—1181no—
162C2×C4⋊C4C4×D5912yes—
163C2×C4⋊C4D23611yescharacteristic
164C2×C42—1181no—
165C2×C42C22⋊D5912yes—
166C2×C42C22×D5911yescharacteristic
167C4⋊C8—24no—
168C4⋊C8—22no—
169C4⋊C8—1182no—
170C4⋊C8—1182no—
171#101—2362no—
172C22⋊C8—22no—
173C4×C8—24no—
174C4×C8C22⋊D5914yes—
175#95—592no—
176#95—1182no—
177#95—1182no—
178#95—1182no—
179#95—1181no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—1182no—
183C2×Q8—2362no—
184C2×Q8#1174566012yes—
185C2×Q8Dic23612yes—
186C22×C4—1181no—
187C22×C4—1182no—
188C22×C4—1181no—
189C22×C4(C2×D59)⋊C412yes—
190C22×C4(C2×D59)⋊C412yes—
191C22×C4C2×C4×D5911yescharacteristic
192C22×C4C2×D23611yescharacteristic
193C22×C4C2×C22⋊D5911yescharacteristic
194M16—2362no—
195M16—2362no—
196C2×C8—24no—
197C2×C8—2362no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—1182no—
201C4⋊C4—2364no—
202C4⋊C4—2361no—
203C4⋊C4—2361no—
204C4⋊C4—2362no—
205C42—22no—
206C42—22no—
207C42—2362no—
208C42#1174568512yes—
209C42C2×C22⋊D5912yes—
210C23C2×(C2×D59)⋊C411yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—1184no—
214Q8—2362no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—1182no—
222C2×C4—2362no—
223C2×C4—2362no—
224C2×C4—2362no—
225C2×C4—2362no—
226C2×C4—2361no—
227C2×C4—2361no—
228C2×C4—2361no—
229C2×C4#1427365214yes—
230C2×C4C4×C22⋊D5912yes—
231C2×C4#1427369611yescharacteristic
232C2×C4#1427370112yes—
233C2×C4#1427371011yescharacteristic
234C2×C4C2×#1174566011yescharacteristic
235C2×C4C2×Dic23611yescharacteristic
236C2×C4#1427374912yes—
237C8—44no—
238C8—4722no—
239C22—21no—
240C22—21no—
241C22#1683728111yescharacteristic
242C22C2×#1427365211yescharacteristic
243C22#1683739911yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—4722no—
250C4—4721no—
251C4—2362no—
252C4#1683763112yes—
253C4#1683764312yes—
254C2—21no—
255C2—21no—
256C2#2142451411yescharacteristic
257C2#2142451911yescharacteristic
258C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2982C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2982D59
hypocenterC59#9484
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
150. C59#9484semidirect

Maximal subgroups

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

subgroupindex
#214294582
#214240922
#214225262
#214219552
C59×#29822
#948459