Finite Group Databasebeta

← #28484187

Subgroups — #28484187

Order 15104.

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

normal subgroups only

This group has 498 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#21426812C211yesmaximal, characteristic
3#21422490C212yesmaximal
4#21421924C212yesmaximal
5C59×#1461C211yesmaximal, characteristic
6C2×#14273725C411yescharacteristic
7C2×C4×Dic118C2211yescharacteristic
8Q8×C59⋊C8—24no—
9#16837455—22no—
10#16837455C2212yes—
11#16837309C411yescharacteristic
12#16837307C411yescharacteristic
13#16837286—22no—
14#16837283—22no—
15#16837267C411yescharacteristic
16#16837248C2211yescharacteristic
17#16837132—24no—
18C2×C4×Q8×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#324—22no—
21C22×Dic118C2×C411yescharacteristic
22#14273725—22no—
23#14273725C814yescomplemented
24C4×Dic118—22no—
25C4×Dic118—22no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C4×Dic59—21no—
30C2×C4×Dic59C2×C411yescharacteristic
31C2×C4×Dic59C2×C411yescharacteristic
32C4⋊(C59⋊C8)—24no—
33C4⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35C4×C59⋊C8—24no—
36C4×C59⋊C8D414yes—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×Q8×C59C2×C411yescharacteristic
43C4×Q8×C59—22no—
44C4×Q8×C59—22no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C4⋊C4×C59D412yes—
47C2×C42×C59D412yes—
48C2×C42×C59C2311yescharacteristic
49C4⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C4×C8×C59—22no—
52C2×Q8×C59—22no—
53C2×Q8×C59—22no—
54C2×Q8×C59C2×C812yes—
55C2×Q8×C59M1612yes—
56C22×C4×C59C22⋊C412yes—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×Dic59—21no—
62C22×Dic59—21no—
63C22×Dic59—21no—
64C22×Dic59—21no—
65C2×Dic118—22no—
66C2×Dic118—22no—
67C2×Dic118—21no—
68C2×Dic118C2×C812yes—
69C2×C8×C59—42no—
70C4⋊C4×C59—22no—
71C4⋊C4×C59—24no—
72C42×C59—22no—
73C42×C59—22no—
74C42×C59C2×D412yes—
75C42×C59#4112yes—
76C4⋊Dic59—22no—
77Dic59⋊C4—28no—
78Dic59⋊C4—41no—
79Dic59⋊C4—41no—
80C4×Dic59—21no—
81C4×Dic59—22no—
82C4×Dic59—42no—
83C4×Dic59C2×C812yes—
84C2×C59⋊C8—24no—
85C23×C59C2×C22⋊C411yescharacteristic
86Q8×C59—24no—
87Q8×C59—24no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59C22⋊C814yes—
95C2×C4×C59C4×D412yes—
96C2×C4×C59#12011yescharacteristic
97C2×C4×C59#12112yes—
98C2×C4×C59#12311yescharacteristic
99C2×C4×C59C22×C811yescharacteristic
100C2×C4×C59C2×M1611yescharacteristic
101C2×C4×C59#13112yes—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—41no—
105C2×Dic59—41no—
106C2×Dic59—41no—
107C2×Dic59—41no—
108C2×Dic59—41no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59—41no—
112C2×Dic59—41no—
113Dic118—24no—
114Dic118—42no—
115C8×C59—82no—
116C59⋊C8—44no—
117#9100—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59#38611yescharacteristic
121C22×C59C2×C22⋊C811yescharacteristic
122C22×C59#40811yescharacteristic
123C4×C59—24no—
124C4×C59—22no—
125C4×C59—21no—
126C4×C59—21no—
127C4×C59—42no—
128C4×C59C8×D412yes—
129C4×C59#43512yes—
130Dic59—42no—
131Dic59—42no—
132Dic59—81no—
133Dic59—81no—
134#2982—591no—
135#2327—591no—
136#1479—592no—
137#1461D5911yescharacteristic
138#1274—592no—
139C2×C59—21no—
140C2×C59—21no—
141C2×C59#186911yescharacteristic
142C2×C59#190911yescharacteristic
143C2×C59#191211yescharacteristic
144C2×C4⋊Q8—591no—
145C2×C4×Q8—591no—
146C2×C4×Q8C2×D5911yescharacteristic
147C8×Q8—1184no—
148#431—592no—
149#431—1182no—
150#431C2×D5912yes—
151#391—591no—
152#391—591no—
153#387—1182no—
154#387—1182no—
155#382—591no—
156#376—591no—
157Q8⋊C8—1184no—
158#324—22no—
159C59#910011yescharacteristic, complemented, Sylow
160C22×Q8—591no—
161C22×Q8C4×D5911yescharacteristic
162C4⋊Q8—594no—
163C4⋊Q8—1182no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—1182no—
167C4×Q8—1182no—
168C2×C4⋊C4—591no—
169C2×C4⋊C4—1182no—
170C2×C4⋊C4—1181no—
171C2×C4⋊C4C4×D5911yescharacteristic
172C2×C4⋊C4C22⋊D5912yes—
173C2×C42—591no—
174C2×C42—591no—
175C2×C42—1181no—
176C2×C42D23612yes—
177C2×C42C22×D5911yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—1184no—
180C4⋊C8—1182no—
181C22⋊C8—22no—
182C22⋊C8—1182no—
183C4×C8—22no—
184C4×C8—594no—
185C4×C8—1184no—
186#95—1182no—
187#95—1182no—
188#95—1182no—
189#95—1181no—
190#95—1181no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—592no—
194C2×Q8—1181no—
195C2×Q8—1182no—
196C2×Q8—1182no—
197C2×Q8C8×D5912yes—
198C2×Q8#1174565912yes—
199C22×C4—1181no—
200C22×C4—1181no—
201C22×C4—1181no—
202C22×C4—1181no—
203C22×C4(C2×D59)⋊C412yes—
204C22×C4(C2×D59)⋊C412yes—
205C22×C4C2×C4×D5911yescharacteristic
206C22×C4C2×D23611yescharacteristic
207C22×C4C2×C22⋊D5911yescharacteristic
208C2×C8—42no—
209C2×C8—1184no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—1188no—
213C4⋊C4—1182no—
214C4⋊C4—2361no—
215C4⋊C4—2361no—
216C42—22no—
217C42—22no—
218C42—592no—
219C42—1181no—
220C42—1182no—
221C42—2362no—
222C42D4×D5912yes—
223C42#1174568712yes—
224C23C2×(C2×D59)⋊C411yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—1184no—
228Q8—2362no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—2361no—
238C2×C4—2361no—
239C2×C4—2362no—
240C2×C4—2361no—
241C2×C4—2361no—
242C2×C4—2361no—
243C2×C4—2361no—
244C2×C4—2361no—
245C2×C4—2361no—
246C2×C4(C2×D59)⋊C814yes—
247C2×C4(C22⋊D59)⋊C412yes—
248C2×C4#1427369711yescharacteristic
249C2×C4#1427369912yes—
250C2×C4#1427371111yescharacteristic
251C2×C4C2×C8×D5911yescharacteristic
252C2×C4C2×#1174565911yescharacteristic
253C2×C4#1427373512yes—
254C8—82no—
255C8—2364no—
256C22—21no—
257C22—21no—
258C22#1683727911yescharacteristic
259C22C2×(C2×D59)⋊C811yescharacteristic
260C22#1683737011yescharacteristic
261C4—24no—
262C4—22no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—4721no—
267C4—4721no—
268C4—2362no—
269C4—2362no—
270C4(C22⋊D59)⋊C812yes—
271C4#1683747012yes—
272C2—21no—
273C2—21no—
274C2#2142438611yescharacteristic
275C2#2142461611yescharacteristic
276C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1461C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1461D59
hypocenterC59#9100
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
23. #14273725C8semidirect
159. C59#9100semidirect

Maximal subgroups

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

subgroupindex
#214331772
#214268122
#214224902
#214219242
C59×#14612
#910059