Finite Group Databasebeta

← #28483683

Subgroups — #28483683

Order 15104.

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

normal subgroups only

This group has 756 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
1C2×#16837322C211yesmaximal, characteristic
2#21424030C212yesmaximal
3C2×#16837128C211yesmaximal, characteristic
4#21421974C212yesmaximal
5C59×#2313C211yesmaximal, characteristic
6C23×Dic118C2211yescharacteristic
7C22×Dic59⋊C4—21no—
8C2×#14273650—24no—
9C2×C22⋊(C59⋊C8)C2212yes—
10#16837322—22no—
11#16837322C414yescomplemented
12C2×C59⋊#95—21no—
13C2×C59⋊#95C2211yescharacteristic
14#16837128—22no—
15C22×C4⋊C4×C59C2211yescharacteristic
16C59×#387—24no—
17C59×#381—21no—
18C59×#380C2212yes—
19C4×C22⋊C4×C59—21no—
20C22×Dic118—22no—
21C22×Dic118—22no—
22C22×Dic118—21no—
23C22×Dic118—44no—
24C22×Dic118C2×C412yes—
25C23×Dic59—21no—
26C23×Dic59—21no—
27C22×C59⋊C8—22no—
28C2×Dic59⋊C4—24no—
29C2×Dic59⋊C4—44no—
30#14273650—48no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—24no—
34C59⋊#95—21no—
35C59⋊#95C2×C412yes—
36C23×C4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59C2311yescharacteristic
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59C2×C412yes—
42C2×C4⋊C4×C59D412yes—
43C2×C4⋊C4×C59D414yes—
44C2×C22⋊C4×C59—22no—
45C2×C42×C59—22no—
46#95×C59—22no—
47#95×C59—24no—
48#95×C59—22no—
49C24×C59C2×D411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59—21no—
56C22×C4×C59—24no—
57C22×C4×C59—42no—
58C22×C4×C59—42no—
59C22×C4×C59C22⋊C412yes—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D412yes—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59#4112yes—
65C22×Dic59—24no—
66C22×Dic59—24no—
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C22×Dic59—41no—
72C22×Dic59—41no—
73C22×Dic59—42no—
74C2×Dic118—28no—
75C2×Dic118—24no—
76C2×Dic118—44no—
77C2×Dic118—48no—
78C2×Dic118—44no—
79C2×Dic118—44no—
80C2×Dic118—42no—
81C4⋊C4×C59—24no—
82C4⋊C4×C59—24no—
83C4⋊C4×C59—44no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C42×C59—82no—
87Dic59⋊C4—48no—
88C2×C59⋊C8—44no—
89C2×C59⋊C8—42no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59C2×C22⋊C411yescharacteristic
93C23×C59#12011yescharacteristic
94C23×C59#12311yescharacteristic
95C23×C59#13512yes—
96C23×C59#13712yes—
97C2×C4×C59—24no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—44no—
110C2×C4×C59—44no—
111C2×C4×C59—44no—
112C2×C4×C59—81no—
113C2×C4×C59C4×D412yes—
114C2×C4×C59#12112yes—
115C2×Dic59—24no—
116C2×Dic59—24no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—44no—
120C2×Dic59—48no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—44no—
124Dic118—28no—
125Dic118—48no—
126Dic118—44no—
127C59⋊C8—82no—
128#9014—591nomaximal, Sylow
129C22×C59—24no—
130C22×C59—24no—
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59#38611yescharacteristic
136C22×C59#41611yescharacteristic
137C22×C59#41911yescharacteristic
138C22×C59#45212yes—
139C22×C59#45612yes—
140C22×C59B(4,𝔽2)12yes—
141C22×C59#48512yes—
142C4×C59—22no—
143C4×C59—22no—
144C4×C59—42no—
145C4×C59—42no—
146C4×C59—84no—
147Dic59—44no—
148Dic59—44no—
149C2×#395—591no—
150#2313D5911yescharacteristic
151#1787—592no—
152C2×#324—591no—
153#1294—592no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—22no—
157C2×C59#160314yes—
158C2×C59#187311yescharacteristic
159C2×C59#188611yescharacteristic
160C2×C59#188911yescharacteristic
161C23×Q8—591no—
162C22×C4⋊C4—1181no—
163C22×C4⋊C4C2×D5911yescharacteristic
164C2×#103—1184no—
165C2×C22⋊C8—592no—
166#395—594no—
167#395—1182no—
168#387—24no—
169#381—21no—
170#380C2×D5912yes—
171C4×C22⋊C4—21no—
172C2×#95—591no—
173C2×#95—1181no—
174#324—1182no—
175C59#901411yescharacteristic, complemented, Sylow
176C22×Q8—592no—
177C22×Q8—1181no—
178C22×Q8—1182no—
179C22×Q8—1182no—
180C22×Q8—2364no—
181C23×C4—1181no—
182C23×C4—1181no—
183C23×C4C22⋊D5911yescharacteristic
184C23×C4C22⋊D5911yescharacteristic
185C23×C4C22×D5911yescharacteristic
186C22×C8—1182no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—1184no—
190C2×C4⋊C4—2364no—
191C2×C4⋊C4C4×D5912yes—
192C2×C4⋊C4D23612yes—
193C2×C4⋊C4C22⋊D5914yes—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196#103—2368no—
197C22⋊C8—1182no—
198C22⋊C8—1182no—
199#95—22no—
200#95—24no—
201#95—22no—
202#95—592no—
203#95—1184no—
204#95—1181no—
205C24C2×C22⋊D5911yescharacteristic
206C2×Q8—1188no—
207C2×Q8—1184no—
208C2×Q8—2364no—
209C2×Q8—2364no—
210C2×Q8—2364no—
211C2×Q8—2368no—
212C2×Q8—2362no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—1184no—
223C22×C4—1184no—
224C22×C4—1182no—
225C22×C4—1182no—
226C22×C4—2362no—
227C22×C4—2362no—
228C22×C4—2361no—
229C22×C4—2361no—
230C22×C4—2362no—
231C22×C4(C2×D59)⋊C412yes—
232C22×C4(C2×D59)⋊C412yes—
233C22×C4C2×C4×D5911yescharacteristic
234C22×C4C2×D23611yescharacteristic
235C22×C4#1174568512yes—
236C22×C4C2×C22⋊D5912yes—
237C2×C8—2364no—
238C2×C8—2362no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—2368no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×(C2×D59)⋊C411yescharacteristic
249C23#1427369611yescharacteristic
250C23#1427371011yescharacteristic
251C23#1427374712yes—
252C23#1427375712yes—
253Q8—1188no—
254Q8—2368no—
255Q8—2364no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—1184no—
273C2×C4—1184no—
274C2×C4—2362no—
275C2×C4—2368no—
276C2×C4—2362no—
277C2×C4—2364no—
278C2×C4—2362no—
279C2×C4—2362no—
280C2×C4—2364no—
281C2×C4C4×C22⋊D5912yes—
282C2×C4#1427370112yes—
283C8—4722no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1683728111yescharacteristic
291C22#1683739611yescharacteristic
292C22#1683741011yescharacteristic
293C22#1683756412yes—
294C22#1683758512yes—
295C22#1683758912yes—
296C22#1683774312yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—2364no—
303C4—2364no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2142305914yes—
308C2#2142440811yescharacteristic
309C2#2142448911yescharacteristic
310C2#2142450411yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2313C2
socleC23×C59#120
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2313D59
hypocenterC59#9014
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
11. #16837322C4semidirect
175. C59#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#168373222
#214240302
C2×#168371282
#214219742
C59×#23132
#901459