Finite Group Databasebeta

← #28486538

Subgroups — #28486538

Order 15104.

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

normal subgroups only

This group has 626 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
1#21428993C211yesmaximal, characteristic
2#21424088C211yesmaximal, characteristic, complemented
3#21422522C212yesmaximal, complemented
4#21421955C212yesmaximal, complemented
5C59×#2972C211yesmaximal, characteristic, complemented
6C2×#14273698—21no—
7C2×C4×D236C2211yescharacteristic
8#16837426C2212yes—
9C2×#14273644—21no—
10#16837327C412yes—
11#16837311C412yes—
12#16837281—22no—
13#16837268C2211yescharacteristic
14#16837259—21no—
15#16837171—21no—
16#16837131—24no—
17C2×C4×D4×C59C2211yescharacteristic
18C8×D4×C59—24no—
19C59×#431C2212yes—
20C2×C22⋊C8×C59—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—22no—
23C22×C4×D59—21no—
24C2×#11745664—22no—
25#14273698—42no—
26#14273698—42no—
27C4×D236—22no—
28C4×D236—22no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C22⋊Dic59—22no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—22no—
34C4⋊(C59⋊C8)—22no—
35C4⋊(C59⋊C8)—22no—
36#14273644—42no—
37C59⋊#95—22no—
38C59⋊#95—21no—
39C59⋊#95C2×C412yes—
40C22×D4×C59D411yescharacteristic
41C23×C4×C59D412yes—
42C22×C8×C59—24no—
43C4×D4×C59—22no—
44C4×D4×C59—22no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59C2×C412yes—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49C22⋊C8×C59—24no—
50C22⋊C8×C59—22no—
51C4×C8×C59D414yescomplemented
52C24×C59C22⋊C412yes—
53C23×D59—21no—
54C2×D4×C59—22no—
55C2×D4×C59—22no—
56C2×D4×C59D812yes—
57C2×D4×C59SD1612yes—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—82no—
61C22×C4×C59C22⋊C412yes—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C2×C22⋊D59—44no—
66C2×C22⋊D59—44no—
67C2×C22⋊D59—82no—
68C22×Dic59—22no—
69C22×Dic59—21no—
70C2×D236—22no—
71C2×D236—42no—
72C2×C4×D59—42no—
73C2×C4×D59—41no—
74C2×C4×D59—41no—
75C2×C4×D59—42no—
76C2×C8×C59—24no—
77C2×C8×C59—84no—
78C4⋊C4×C59—22no—
79C22⋊C4×C59—24no—
80C42×C59C2×D412yes—
81C42×C59#4112yes—
82C22⋊Dic59—44no—
83(C2×D59)⋊C4—41no—
84(C2×D59)⋊C4—41no—
85(C2×D59)⋊C4—82no—
86C4⋊Dic59—22no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89#11745664—42no—
90#11745664—42no—
91C2×C59⋊C8—42no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59#10214yes—
97C23×C59C2×C22⋊C411yescharacteristic
98C22×D59—22no—
99C22×D59—41no—
100C22×D59—41no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105D4×C59—24no—
106D4×C59—24no—
107C2×C4×C59—22no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—44no—
111C2×C4×C59—42no—
112C2×C4×C59—82no—
113C2×C4×C59C4×D412yes—
114C2×C4×C59#12011yescharacteristic
115C2×C4×C59#12112yes—
116C2×C4×C59#12311yescharacteristic
117C2×C4×C59C2×D811yescharacteristic
118C2×C4×C59C2×SD1611yescharacteristic
119C2×C4×C59#13512yes—
120C22⋊D59—88no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—41no—
124C2×Dic59—42no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127D236—24no—
128D236—42no—
129C4×D59—82no—
130C4×D59—82no—
131C8×C59—44no—
132C59⋊C8—82no—
133#9485—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—24no—
137C22×C59—44no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—82no—
141C22×C59#38611yescharacteristic
142C22×C59C2×#10211yescharacteristic
143C22×C59#41611yescharacteristic
144C2×D59—42no—
145C2×D59—42no—
146C2×D59—42no—
147C2×D59—81no—
148C2×D59—81no—
149C2×D59—81no—
150C2×D59—81no—
151C4×C59—22no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—42no—
155C4×C59#46512yes—
156C4×C59#46612yes—
157Dic59—82no—
158Dic59—81no—
159#2972D5911yescharacteristic, complemented
160#2535—591no—
161#1800—591no—
162#1486—592no—
163#1286—592no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—44no—
167C2×C59#189111yescharacteristic
168C2×C59#189311yescharacteristic
169C2×C59#191111yescharacteristic
170D59—82no—
171C2×#121—1181no—
172C2×C4×D4—591no—
173C2×C4×D4C2×D5911yescharacteristic
174C8×D4—24no—
175#431C2×D5912yes—
176#423—592no—
177C2×#100—1181no—
178C2×C22⋊C8—22no—
179#397—592no—
180#392—592no—
181#386—1182no—
182#382—591no—
183#380—1181no—
184#343—1181no—
185D4⋊C8—1184no—
186C59#948511yescharacteristic, complemented, Sylow
187C22×D4—591no—
188C22×D4—1182no—
189C22×D4D23611yescharacteristic
190C23×C4—1181no—
191C23×C4C22⋊D5912yes—
192C2×M16—1182no—
193C22×C8—24no—
194#121—2362no—
195#121—2362no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—1182no—
199C4×D4—1182no—
200C2×C4⋊C4—591no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4D23611yescharacteristic
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4C4×D5912yes—
207C2×C42C22×D5911yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1182no—
210C4⋊C8—1182no—
211#100—2362no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D5914yescomplemented
215#95—592no—
216#95—1182no—
217#95—1181no—
218C24—1181no—
219C24(C2×D59)⋊C412yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—1182no—
223C2×D4—4722no—
224C2×D4—2362no—
225C2×D4—2364no—
226C2×D4—2364no—
227C2×D4#1174566012yes—
228C2×D4D47212yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—1182no—
233C22×C4—1181no—
234C22×C4—2362no—
235C22×C4—2361no—
236C22×C4—2361no—
237C22×C4—2362no—
238C22×C4(C2×D59)⋊C412yes—
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C2×D23611yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—2362no—
243M16—2362no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—2362no—
247C4⋊C4—22no—
248C4⋊C4—1182no—
249C4⋊C4—2361no—
250C4⋊C4—2361no—
251C22⋊C4—24no—
252C22⋊C4—4722no—
253C22⋊C4—2364no—
254C22⋊C4—2361no—
255C22⋊C4—2361no—
256C42#1174568512yes—
257C42C2×C22⋊D5912yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—1182no—
263C23—2361no—
264C23—2361no—
265C23—2361no—
266C23—2361no—
267C23—2361no—
268C23—2361no—
269C23#1427364914yes—
270C23C2×(C2×D59)⋊C411yescharacteristic
271D4—24no—
272D4—24no—
273D4—1184no—
274D4—4728no—
275D4—2362no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—4722no—
283C2×C4—4722no—
284C2×C4—2362no—
285C2×C4—2362no—
286C2×C4—2361no—
287C2×C4—2362no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4C4×C22⋊D5912yes—
291C2×C4#1427369611yescharacteristic
292C2×C4#1427370112yes—
293C2×C4#1427371011yescharacteristic
294C2×C4C2×D47211yescharacteristic
295C2×C4C2×#1174566011yescharacteristic
296C2×C4#1427374912yes—
297C8—44no—
298C8—4722no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—4721no—
307C22—4721no—
308C22—4721no—
309C22—4721no—
310C22—2362no—
311C22—2362no—
312C22—2362no—
313C22#1683728111yescharacteristic
314C22C2×#1427364911yescharacteristic
315C22#1683739911yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—4722no—
321C4—4721no—
322C4#1683763112yes—
323C4#1683763812yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—4722no—
328C2#2142451411yescharacteristic
329C2#2142452411yescharacteristic
330C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2972C2
socleC22×C59#386
prime radical#21428993C2
cosocleC2×C42×C59C23
hypercenter#2972D59
hypocenterC59#9485
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21424088C2semidirect
3. #21422522C2semidirect
4. #21421955C2semidirect
5. C59×#2972C2semidirect
51. C4×C8×C59D4semidirect
159. #2972D59semidirect
186. C59#9485semidirect
214. C4×C8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214289932
#214240882
#214225222
#214219552
C59×#29722
#948559