Finite Group Databasebeta

← #28486534

Subgroups — #28486534

Order 15104.

474 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433177C211yesmaximal, characteristic
2#21424092C211yesmaximal, characteristic
3#21422525C212yesmaximal
4#21421954C212yesmaximal
5C59×#2581C211yesmaximal, characteristic
6Q8×C59⋊C8—24no—
7#16837455—22no—
8#16837455C2212yes—
9#16837426C2212yes—
10C2×#14273646—21no—
11#16837171—21no—
12Q8⋊(C59⋊C8)—24no—
13C2×C4×Q8×C59C2211yescharacteristic
14C2×C4×Q8×C59C2211yescharacteristic
15C59×#400C412yes—
16C59×#395C412yes—
17C59×#391—21no—
18C59×#389—21no—
19C59×#387—22no—
20C59×#382C2211yescharacteristic
21C2×#11745664—22no—
22C4⋊(C59⋊C8)—24no—
23C4⋊(C59⋊C8)—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273646—42no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—24no—
29C4×C59⋊C8D414yes—
30C22×Q8×C59C2×C411yescharacteristic
31C22×Q8×C59D411yescharacteristic
32C4×Q8×C59—22no—
33C4×Q8×C59—22no—
34C4×Q8×C59—22no—
35C4×Q8×C59—22no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59C2×C412yes—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C42×C59—21no—
43C2×C42×C59D412yes—
44C2×C42×C59C2311yescharacteristic
45#95×C59—22no—
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—21no—
49#95×C59C2×C412yes—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59—22no—
53C2×Q8×C59—42no—
54C2×Q8×C59SD1612yes—
55C2×Q8×C59Q1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59C22⋊C412yes—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C4⋊C4×C59—24no—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—44no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C42×C59—22no—
72C42×C59—22no—
73C42×C59—42no—
74C42×C59C2×D412yes—
75C42×C59#4112yes—
76#11745664—42no—
77#11745664—42no—
78C2×C59⋊C8—24no—
79C2×C59⋊C8—42no—
80C23×C59C2×C22⋊C411yescharacteristic
81Q8×C59—24no—
82Q8×C59—24no—
83Q8×C59—24no—
84Q8×C59—42no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59#10314yes—
100C2×C4×C59C4×D412yes—
101C2×C4×C59#12011yescharacteristic
102C2×C4×C59#12112yes—
103C2×C4×C59#12311yescharacteristic
104C2×C4×C59C2×SD1611yescharacteristic
105C2×C4×C59C2×Q1611yescharacteristic
106C2×C4×C59#13512yes—
107C59⋊C8—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—42no—
121C4×C59—82no—
122C4×C59—81no—
123C4×C59#46512yes—
124C4×C59#46712yes—
125#2982—591no—
126#2581D5911yescharacteristic
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×Q8C2×D5911yescharacteristic
136C2×C4×Q8C2×D5911yescharacteristic
137C8×Q8—1184no—
138#431—592no—
139#431—1182no—
140#423—592no—
141C2×#101—1181no—
142#400Dic5912yes—
143#395Dic5912yes—
144#391—21no—
145#389—21no—
146#387—22no—
147#382C2×D5911yescharacteristic
148#343—1181no—
149Q8⋊C8—1184no—
150C59#948411yescharacteristic, complemented, Sylow
151C22×Q8C2×Dic5911yescharacteristic
152C22×Q8C22⋊D5911yescharacteristic
153C2×M16—1182no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4C2×Dic5911yescharacteristic
162C2×C4⋊C4C2×Dic5912yes—
163C2×C4⋊C4C22⋊D5911yescharacteristic
164C2×C42—21no—
165C2×C42C22⋊D5912yes—
166C2×C42C22×D5911yescharacteristic
167C4⋊C8—1184no—
168C4⋊C8—1182no—
169C4⋊C8—1182no—
170C4⋊C8—1182no—
171#101—2362no—
172C22⋊C8—1182no—
173C4×C8—594no—
174C4×C8—1184no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—21no—
179#95C2×Dic5912yes—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8#1174567112yes—
185C2×Q8#1174567212yes—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4C22⋊Dic5912yes—
190C22×C4C22⋊Dic5912yes—
191C22×C4C22×Dic5911yescharacteristic
192C22×C4C2×C22⋊D5911yescharacteristic
193C22×C4C2×C22⋊D5911yescharacteristic
194M16—2362no—
195M16—2362no—
196C2×C8—1184no—
197C2×C8—2362no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42D4×D5912yes—
209C42#1174568712yes—
210C23C2×C22⋊Dic5911yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—24no—
214Q8—42no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4#1427365114yes—
230C2×C4D4×Dic5912yes—
231C2×C4#1427369511yescharacteristic
232C2×C4#1427369912yes—
233C2×C4#1427370811yescharacteristic
234C2×C4C2×#1174567111yescharacteristic
235C2×C4C2×#1174567211yescharacteristic
236C2×C4#1427374712yes—
237C8—4722no—
238C8—2364no—
239C22—21no—
240C22—21no—
241C22#1683728211yescharacteristic
242C22C2×#1427365111yescharacteristic
243C22#1683739811yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4#1683763312yes—
253C4#1683764112yes—
254C2—21no—
255C2—21no—
256C2#2142451511yescharacteristic
257C2#2142452011yescharacteristic
258C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2581C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2581D59
hypocenterC59#9484
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

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
#214331772
#214240922
#214225252
#214219542
C59×#25812
#948459