Finite Group Databasebeta

← #28496555

Subgroups — #28496555

Order 15104.

1042 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 227 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×#16837366C212yesmaximal
2C2×#16837171C214yesmaximal, complemented
3C2×C59×#380C211yesmaximal, characteristic
4C22×#11745664C2214yes—
5#16837366—28no—
6C2×C22⋊(C59⋊C8)—22no—
7C2×C22⋊(C59⋊C8)—22no—
8#16837171—28no—
9C24×C4×C59C2211yescharacteristic
10C22×C4⋊C4×C59C2212yes—
11C22×C22⋊C4×C59C412yes—
12C59×#380—22no—
13C59×#380—22no—
14C2×#95×C59C412yes—
15C2×#11745664—24no—
16C2×#11745664—24no—
17C2×#11745664—24no—
18C2×#11745664—24no—
19C2×#11745664—24no—
20C2×#11745664D418yes—
21C22×C59⋊C8—24no—
22C22⋊(C59⋊C8)—28no—
23C22⋊(C59⋊C8)—28no—
24C25×C59C2×C411yescharacteristic
25C23×C4×C59—24no—
26C23×C4×C59—24no—
27C23×C4×C59—24no—
28C23×C4×C59C2×C411yescharacteristic
29C23×C4×C59C2×C414yes—
30C23×C4×C59D412yes—
31C23×C4×C59D412yes—
32C23×C4×C59C2311yescharacteristic
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—24no—
36C2×C4⋊C4×C59—24no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—44no—
40#95×C59—24no—
41C24×C59—22no—
42C24×C59—22no—
43C24×C59—22no—
44C24×C59—24no—
45C24×C59—42no—
46C24×C59C22⋊C411yescharacteristic
47C24×C59C22⋊C411yescharacteristic
48C24×C59C22×C411yescharacteristic
49C22×C4×C59—22no—
50C22×C4×C59—24no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59—24no—
54C22×C4×C59—24no—
55C22×C4×C59—24no—
56C22×C4×C59—24no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—24no—
60C22×C4×C59—28no—
61C22×C4×C59—44no—
62C22×C4×C59—44no—
63C22×C4×C59—44no—
64C22×C4×C59—44no—
65C22×C4×C59—44no—
66C22×C4×C59—44no—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59C2×D414yes—
69C22×C4×C59C2×D412yes—
70C22×C4×C59#4114yes—
71C22×C4×C59#4114yes—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—44no—
74C4⋊C4×C59—42no—
75C22⋊C4×C59—44no—
76C22⋊C4×C59—44no—
77#11745664—28no—
78#11745664—28no—
79#11745664—44no—
80#11745664—44no—
81C2×C59⋊C8—28no—
82C2×C59⋊C8—44no—
83C2×C59⋊C8—44no—
84C23×C59—21no—
85C23×C59—21no—
86C23×C59—22no—
87C23×C59—28no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—22no—
95C23×C59—22no—
96C23×C59—48no—
97C23×C59—48no—
98C23×C59—44no—
99C23×C59—44no—
100C23×C59#10012yes—
101C23×C59#10112yes—
102C23×C59C2×C22⋊C411yescharacteristic
103C23×C59#11712yes—
104C23×C59C4×D414yes—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—44no—
109C2×C4×C59—44no—
110C2×C4×C59—44no—
111C2×C4×C59—44no—
112C2×C4×C59—44no—
113C2×C4×C59—44no—
114C2×C4×C59—44no—
115C2×C4×C59—44no—
116C2×C4×C59—88no—
117C2×C4×C59#12012yes—
118C2×C4×C59#12114yes—
119C2×C4×C59#12114yes—
120C2×C4×C59#12312yes—
121C2×C4×C59#12314yes—
122C2×C4×C59#12314yes—
123C2×C4×C59#12414yes—
124C2×C4×C59#12614yes—
125C59⋊C8—48no—
126#11198—591nomaximal, Sylow
127C22×C59—22no—
128C22×C59—22no—
129C22×C59—24no—
130C22×C59—24no—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—24no—
134C22×C59—24no—
135C22×C59—416no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59#38111yescharacteristic
143C22×C59#38612yes—
144C22×C59#38814yes—
145C22×C59C2×#10011yescharacteristic
146C22×C59C2×#10111yescharacteristic
147C22×C59#41312yes—
148C4×C59—42no—
149C4×C59—84no—
150C4×C59#39414yes—
151C4×C59#39714yes—
152C2×#407—592no—
153C2×#380D5911yescharacteristic
154C2×#343—594no—
155C2×C59—24no—
156C2×C59—42no—
157C2×C59—42no—
158C2×C59#144011yescharacteristic
159C2×C59#182111yescharacteristic
160C2×C59#182211yescharacteristic
161C2×C59#190012yes—
162C2×C59#190112yes—
163C24×C4C2×D5911yescharacteristic
164C22×M16—594no—
165C22×C4⋊C4C2×D5912yes—
166C22×C22⋊C4Dic5912yes—
167#407—1188no—
168C2×C22⋊C8—1182no—
169C2×C22⋊C8—1182no—
170#380—22no—
171#380—22no—
172C2×#95Dic5912yes—
173#343—1188no—
174C59#1119811yescharacteristic, complemented, Sylow
175C25C2×Dic5911yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2×Dic5911yescharacteristic
180C23×C4C2×Dic5914yes—
181C23×C4C22⋊D5912yes—
182C23×C4C22⋊D5912yes—
183C23×C4C22×D5911yescharacteristic
184C2×M16—598no—
185C2×M16—1184no—
186C2×M16—1184no—
187C2×M16—1184no—
188C2×M16—1184no—
189C2×M16—1184no—
190C22×C8—1184no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C22⋊C8—1188no—
199C22⋊C8—1188no—
200#95—24no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—24no—
205C24—42no—
206C24C22⋊Dic5911yescharacteristic
207C24C22⋊Dic5911yescharacteristic
208C24C22×Dic5911yescharacteristic
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4C22⋊Dic5912yes—
228C22×C4#1174568514yes—
229C22×C4D4×D5914yes—
230C22×C4#1174568714yes—
231C22×C4C2×C22⋊D5912yes—
232M16—1188no—
233M16—1188no—
234M16—2364no—
235M16—2364no—
236C2×C8—1188no—
237C2×C8—2364no—
238C2×C8—2364no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23#1427364412yes—
261C23#1427364612yes—
262C23C2×C22⋊Dic5911yescharacteristic
263C23#1427368612yes—
264C23D4×Dic5914yes—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4#1427369512yes—
278C2×C4#1427369914yes—
279C2×C4#1427370014yes—
280C2×C4#1427370812yes—
281C2×C4#1427370914yes—
282C2×C4#1427371014yes—
283C2×C4#1427371314yes—
284C2×C4#1427372014yes—
285C8—2368no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—416no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22#1683726511yescharacteristic
302C22#1683728212yes—
303C22#1683729514yes—
304C22C2×#1427364411yescharacteristic
305C22C2×#1427364611yescharacteristic
306C22#1683738612yes—
307C4—42no—
308C4—84no—
309C4#1683731714yes—
310C4#1683733014yes—
311C2—24no—
312C2—42no—
313C2—42no—
314C2#2142234511yescharacteristic
315C2#2142417611yescharacteristic
316C2#2142417911yescharacteristic
317C2#2142456012yes—
318C2#2142456512yes—
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#380C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59C2×C4
cosocleC23×C4×C59C23
hypercenterC2×#380D59
hypocenterC59#11198
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16837171C2semidirect
174. C59#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#168373662
C2×#168371712
C2×C59×#3802
#1119859