Finite Group Databasebeta

← #28489250

Subgroups — #28489250

Order 15104.

804 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16837558C212yesmaximal
2#21430210C211yesmaximal, characteristic
3C2×#16837138C212yesmaximal
4#21421939C211yesmaximal, characteristic
5C59×#1446C211yesmaximal, characteristic
6C22×#11745672—22no—
7C2×#14273706C2212yes—
8#16837558—24no—
9#16837558—24no—
10C2×#14273651—22no—
11C2×C22⋊(C59⋊C8)C2212yes—
12#16837334—22no—
13#16837323—22no—
14#16837262—22no—
15#16837138—24no—
16C23×Q8×C59C2211yescharacteristic
17C2×#95×C59—22no—
18C2×#95×C59C2211yescharacteristic
19C2×#95×C59C2211yescharacteristic
20C22×Dic118D412yes—
21C2×#11745672—28no—
22C2×#11745672—48no—
23C22×C59⋊C8—22no—
24#14273706—22no—
25#14273706—22no—
26#14273706—22no—
27#14273706—22no—
28C2×C4⋊Dic59—22no—
29C2×C4⋊Dic59—22no—
30C2×C4⋊Dic59D412yes—
31C2×Dic59⋊C4—22no—
32C2×C22⋊Dic59—22no—
33#14273651—28no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)D414yes—
36C59⋊#95—24no—
37C22×Q8×C59—21no—
38C22×Q8×C59—22no—
39C22×Q8×C59—22no—
40C22×Q8×C59—44no—
41C22×Q8×C59D412yes—
42C23×C4×C59—21no—
43C23×C4×C59—21no—
44C23×C4×C59D412yes—
45C23×C4×C59C2311yescharacteristic
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—44no—
49C24×C59C2×D411yescharacteristic
50C2×Q8×C59—28no—
51C2×Q8×C59—24no—
52C2×Q8×C59—44no—
53C2×Q8×C59—44no—
54C2×Q8×C59—44no—
55C2×Q8×C59—48no—
56C2×Q8×C59—42no—
57C22×C4×C59—24no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59—41no—
65C22×C4×C59—41no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59—42no—
72C22×C4×C59Q1614yescomplemented
73C22×C4×C59C2×D412yes—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D412yes—
76C22×C4×C59#4111yescharacteristic
77C22×Dic59—22no—
78C22×Dic59—22no—
79C2×Dic118—22no—
80C2×Dic118—22no—
81C2×Dic118—44no—
82C22⋊Dic59—42no—
83C22⋊Dic59—42no—
84#11745672—416no—
85C4⋊Dic59—22no—
86C4⋊Dic59—22no—
87C4⋊Dic59—82no—
88C4⋊Dic59—82no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91C2×C59⋊C8—24no—
92C2×C59⋊C8—44no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—22no—
96C23×C59#12011yescharacteristic
97C23×C59#12111yescharacteristic
98C23×C59#12711yescharacteristic
99C23×C59C2×Q1612yes—
100C23×C59#13712yes—
101Q8×C59—28no—
102Q8×C59—44no—
103Q8×C59—48no—
104C2×C4×C59—24no—
105C2×C4×C59—24no—
106C2×C4×C59—24no—
107C2×C4×C59—41no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—42no—
111C2×C4×C59—44no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—44no—
116C2×C4×C59—44no—
117C2×C4×C59—44no—
118C2×C4×C59—42no—
119C2×C4×C59—82no—
120C2×C4×C59—82no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59#12012yes—
124C2×C4×C59#12112yes—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—44no—
130Dic118—44no—
131Dic118—82no—
132C59⋊C8—44no—
133#9967—591nomaximal, Sylow
134C22×C59—22no—
135C22×C59—22no—
136C22×C59—22no—
137C22×C59—22no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59#39211yescharacteristic
143C22×C59#45112yes—
144C22×C59#45511yescharacteristic
145C22×C59#45611yescharacteristic
146C22×C59B(4,𝔽2)12yes—
147C22×C59#46212yes—
148C22×C59#49411yescharacteristic
149C22×C59#49711yescharacteristic
150C4×C59—22no—
151C4×C59—22no—
152C4×C59—44no—
153C4×C59—44no—
154C4×C59—82no—
155Dic59—82no—
156Dic59—82no—
157C2×#451—592no—
158#2659—591no—
159C2×#328—592no—
160#1446D5911yescharacteristic
161#1280—591no—
162C2×C59—21no—
163C2×C59—22no—
164C2×C59—21no—
165C2×C59#160212yes—
166C2×C59#160512yes—
167C2×C59#200711yescharacteristic
168C2×C59#201611yescharacteristic
169C2×C59#202111yescharacteristic
170C23×Q8C2×D5911yescharacteristic
171C22×Q16—1182no—
172C2×C22⋊Q8—592no—
173#451—1184no—
174#451—1184no—
175C2×#103—1182no—
176C2×C22⋊C8—592no—
177#398—1182no—
178#395—1182no—
179#380—1182no—
180C2×#95—22no—
181C2×#95C2×D5911yescharacteristic
182C2×#95C2×D5911yescharacteristic
183#328—1184no—
184C59#996711yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8—592no—
190C22×Q8D23612yes—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C22⋊D5912yes—
194C23×C4C22×D5911yescharacteristic
195C2×Q16—1188no—
196C2×Q16—2368no—
197C22×C8—1182no—
198C22⋊Q8—1182no—
199C22⋊Q8—1182no—
200C22⋊Q8—1182no—
201C22⋊Q8—1182no—
202C2×C4⋊C4—592no—
203C2×C4⋊C4—1182no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4—1182no—
206C2×C22⋊C4—1182no—
207#103—1188no—
208C22⋊C8—594no—
209C22⋊C8—1182no—
210#95—22no—
211#95—22no—
212#95—44no—
213#95—1184no—
214C24C2×C22⋊D5911yescharacteristic
215C2×Q8—28no—
216C2×Q8—24no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—48no—
221C2×Q8—42no—
222C2×Q8—1182no—
223C2×Q8—1182no—
224C2×Q8—2364no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—1182no—
241C22×C4—1182no—
242C22×C4#1174567214yescomplemented
243C22×C4C2×D23611yescharacteristic
244C22×C4#1174568511yescharacteristic
245C22×C4D4×D5912yes—
246C22×C4D4×D5912yes—
247Q16—23616no—
248C2×C8—1184no—
249C2×C8—2364no—
250C4⋊C4—1182no—
251C4⋊C4—1182no—
252C4⋊C4—4722no—
253C4⋊C4—4722no—
254C4⋊C4—2362no—
255C4⋊C4—2362no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1427369511yescharacteristic
262C23#1427370111yescharacteristic
263C23#1427372311yescharacteristic
264C23C2×#1174567212yes—
265C23#1427375712yes—
266Q8—28no—
267Q8—44no—
268Q8—48no—
269Q8—4722no—
270Q8—2364no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—2362no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4—2362no—
294C2×C4—2364no—
295C2×C4#1427369712yes—
296C2×C4#1427370012yes—
297C8—2364no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#1683731111yescharacteristic
307C22#1683755712yes—
308C22#1683757911yescharacteristic
309C22#1683758411yescharacteristic
310C22#1683758812yes—
311C22#1683761212yes—
312C22#1683778111yescharacteristic
313C22#1683779711yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—82no—
319C4—4722no—
320C4—4722no—
321C2—21no—
322C2—21no—
323C2—22no—
324C2#2142305212yes—
325C2#2142306912yes—
326C2#2142514211yescharacteristic
327C2#2142519511yescharacteristic
328C2#2142522211yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1446C2
socleC23×C59#120
prime radicalC24×C59C2×D4
hypercenter#1446D59
hypocenterC59#9967
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
72. C22×C4×C59Q16semidirect
184. C59#9967semidirect
242. C22×C4#11745672semidirect

Maximal subgroups

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

subgroupindex
C2×#168375582
#214302102
C2×#168371382
#214219392
C59×#14462
#996759