Finite Group Databasebeta

← #28486553

Subgroups — #28486553

Order 15104.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2983C211yesmaximal, 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
18C59×#446—24no—
19C59×#432—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—
48#97×C59—24no—
49C4×C8×C59D414yes—
50C2×Q8×C59—22no—
51C2×Q8×C59—21no—
52C2×Q8×C59—21no—
53C2×Q8×C59—22no—
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—41no—
73C4⋊Dic59—41no—
74Dic59⋊C4—44no—
75Dic59⋊C4—41no—
76Dic59⋊C4—41no—
77C4×Dic59—42no—
78#11745664—42no—
79#11745664—42no—
80C2×C59⋊C8—42no—
81C23×C59C2×C22⋊C411yescharacteristic
82Q8×C59—42no—
83Q8×C59—42no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59C4×D412yes—
92C2×C4×C59#12011yescharacteristic
93C2×C4×C59#12112yes—
94C2×C4×C59#12311yescharacteristic
95C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C59#13712yes—
97C2×C4×C59#13711yescharacteristic
98C2×Dic59—22no—
99C2×Dic59—41no—
100C2×Dic59—41no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106Dic118—24no—
107Dic118—42no—
108C8×C59—44no—
109C59⋊C8—82no—
110#9488—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59#38611yescharacteristic
114C22×C59#41711yescharacteristic
115C22×C59#41911yescharacteristic
116C4×C59—22no—
117C4×C59—21no—
118C4×C59—21no—
119C4×C59—42no—
120C4×C59—42no—
121C4×C59#46812yes—
122C4×C59#47012yes—
123Dic59—42no—
124Dic59—82no—
125Dic59—81no—
126#2983D5911yescharacteristic
127#2581—591no—
128#1801—591no—
129#1487—592no—
130#1286—592no—
131C2×C59—21no—
132C2×C59—21no—
133C2×C59#189111yescharacteristic
134C2×C59#189411yescharacteristic
135C2×C59#191211yescharacteristic
136C2×C4×Q8—591no—
137C2×C4×Q8C2×D5911yescharacteristic
138#446—24no—
139#432—22no—
140#431C2×D5912yes—
141#423—592no—
142C2×#101—1181no—
143#400—592no—
144#395—592no—
145#391—1181no—
146#389—1181no—
147#387—1182no—
148#382—591no—
149#343—1181no—
150Q8⋊C8—1184no—
151C59#948811yescharacteristic, complemented, Sylow
152C22×Q8—591no—
153C22×Q8D23611yescharacteristic
154C2×M16—1182no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1182no—
158C4×Q8—1182no—
159C2×C4⋊C4—591no—
160C2×C4⋊C4—1182no—
161C2×C4⋊C4—1181no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4C4×D5912yes—
164C2×C4⋊C4D23611yescharacteristic
165C2×C42—1181no—
166C2×C42C22⋊D5912yes—
167C2×C42C22×D5911yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—1182no—
171C4⋊C8—1182no—
172#101—2362no—
173C22⋊C8—22no—
174#97—24no—
175C4×C8C22⋊D5914yes—
176#95—592no—
177#95—1182no—
178#95—1182no—
179#95—1182no—
180#95—1181no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—1182no—
186C2×Q8—2362no—
187C22×C4—1181no—
188C22×C4—1182no—
189C22×C4—1181no—
190C22×C4(C2×D59)⋊C412yes—
191C22×C4(C2×D59)⋊C412yes—
192C22×C4C2×C4×D5911yescharacteristic
193C22×C4C2×D23611yescharacteristic
194C22×C4C2×C22⋊D5911yescharacteristic
195M16—2362no—
196M16—2362no—
197C2×C8—24no—
198C2×C8—2362no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—1182no—
202C4⋊C4—2364no—
203C4⋊C4—2361no—
204C4⋊C4—2361no—
205C4⋊C4—2361no—
206C4⋊C4—2361no—
207C42—22no—
208C42—22no—
209C42—2362no—
210C42#1174568512yes—
211C42C2×C22⋊D5912yes—
212C23C2×(C2×D59)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—1184no—
216Q8—2362no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—1182no—
225C2×C4—2362no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2361no—
230C2×C4—2361no—
231C2×C4—2361no—
232C2×C4C4×C22⋊D5912yes—
233C2×C4#1427369611yescharacteristic
234C2×C4#1427370112yes—
235C2×C4#1427371011yescharacteristic
236C2×C4#1427375511yescharacteristic
237C2×C4#1427376012yes—
238C2×C4#1427376011yescharacteristic
239C8—44no—
240C8—4722no—
241C22—21no—
242C22—21no—
243C22#1683728111yescharacteristic
244C22#1683740411yescharacteristic
245C22#1683741211yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—4722no—
252C4—4721no—
253C4—2362no—
254C4#1683764812yes—
255C4#1683766012yes—
256C2—21no—
257C2—21no—
258C2#2142451411yescharacteristic
259C2#2142452911yescharacteristic
260C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2983C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2983D59
hypocenterC59#9488
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
151. C59#9488semidirect

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×#29832
#948859