Finite Group Databasebeta

← #28486436

Subgroups — #28486436

Order 15104.

554 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 99 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#21428584C211yesmaximal, characteristic
2#21426699C211yesmaximal, characteristic, complemented
3#21422626C212yesmaximal
4#21421950C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7#16837328—22no—
8#16837303C412yescomplemented
9#16837288—24no—
10#16837281—21no—
11#16837281—21no—
12#16837267C412yescomplemented
13#16837265C2212yes—
14#16837262—21no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C2211yescharacteristic
17#16837137—24no—
18C22×C4⋊C4×C59C2211yescharacteristic
19C59×#407C2212yes—
20C59×#343—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23#14273701—22no—
24#14273701—22no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×C4×Dic59—22no—
33C2×C4×Dic59C2×C412yes—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—21no—
39C59⋊#95—22no—
40C59⋊#95D414yes—
41C23×C4×C59D411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59C2311yescharacteristic
44C2×M16×C59—22no—
45C2×C4⋊C4×C59—22no—
46C2×C4⋊C4×C59—22no—
47C2×C4⋊C4×C59C2×C412yes—
48C2×C4⋊C4×C59D412yes—
49C22⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C24×C59C2×D411yescharacteristic
52C23×D59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—42no—
58C22×C4×C59—81no—
59C22×C4×C59—81no—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22⋊C412yes—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65C22×C4×C59#4112yes—
66C2×C22⋊D59—44no—
67C2×C22⋊D59—41no—
68C2×C22⋊D59—41no—
69C22×Dic59—21no—
70C22×Dic59—22no—
71C22×Dic59—24no—
72C2×D236—22no—
73C2×D236—42no—
74M16×C59—44no—
75C2×C8×C59—42no—
76C4⋊C4×C59—24no—
77C4⋊C4×C59—24no—
78C22⋊Dic59—44no—
79C22⋊Dic59—42no—
80(C2×D59)⋊C4—42no—
81(C2×D59)⋊C4—82no—
82(C2×D59)⋊C4—82no—
83C4⋊Dic59—22no—
84C4⋊Dic59—82no—
85C4×Dic59—42no—
86C4×Dic59—82no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—41no—
90C23×C59C2×C22⋊C411yescharacteristic
91C23×C59#12011yescharacteristic
92C23×C59#12311yescharacteristic
93C22×D59—42no—
94C22×D59—42no—
95C22×D59—41no—
96C22×D59—42no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—81no—
108C2×C4×C59—81no—
109C2×C4×C59C4×D412yes—
110C2×C4×C59#12112yes—
111C22⋊D59—82no—
112C22⋊D59—82no—
113C2×Dic59—44no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—44no—
117C2×Dic59—42no—
118C2×Dic59—41no—
119C2×Dic59—42no—
120C2×Dic59—44no—
121D236—82no—
122C8×C59—82no—
123#9465—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—44no—
132C22×C59#38611yescharacteristic
133C22×C59#41012yes—
134C22×C59#45412yes—
135C2×D59—84no—
136C2×D59—81no—
137C2×D59—81no—
138C2×D59—81no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—42no—
142C4×C59—42no—
143Dic59—84no—
144Dic59—82no—
145Dic59—81no—
146#2504—591no—
147#2313—591no—
148#1821D5911yescharacteristic, complemented
149#1510—592no—
150#1284—592no—
151C2×C59—22no—
152C2×C59—22no—
153C2×C59—41no—
154C2×C59#188811yescharacteristic
155C2×C59#189911yescharacteristic
156C2×C59#191111yescharacteristic
157D59—161no—
158C2×#121—591no—
159C22×C4⋊C4C2×D5911yescharacteristic
160#407C2×D5912yes—
161#397—1182no—
162#390—592no—
163#387—1184no—
164#386—1181no—
165#386—1181no—
166#382—592no—
167#381—592no—
168#380—1181no—
169C4×C22⋊C4—591no—
170C4×C22⋊C4—1181no—
171#343—22no—
172#327—1184no—
173C59#946511yescharacteristic, complemented, Sylow
174C22×D4—591no—
175C22×D4—1181no—
176C23×C4D23611yescharacteristic
177C23×C4D23611yescharacteristic
178C23×C4C22×D5911yescharacteristic
179C2×M16—22no—
180#121—1182no—
181#121—1182no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—1182no—
186C2×C4⋊C4C4×D5912yes—
187C2×C4⋊C4C22⋊D5912yes—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4—1182no—
190C2×C22⋊C4—1181no—
191C2×C22⋊C4—1182no—
192C2×C22⋊C4—1181no—
193C2×C42—592no—
194C2×C42—1182no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—594no—
198#95—1182no—
199#95—1182no—
200#95—1182no—
201#95—1182no—
202#95—1182no—
203#95—1181no—
204C24—1181no—
205C24C2×D23611yescharacteristic
206C2×D4—1182no—
207C2×D4—2362no—
208C2×D4—2364no—
209C2×D4—2361no—
210C2×D4—2361no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—1181no—
219C22×C4—1182no—
220C22×C4—1184no—
221C22×C4(C2×D59)⋊C412yes—
222C22×C4(C2×D59)⋊C412yes—
223C22×C4C2×C4×D5911yescharacteristic
224C22×C4D4×D5912yes—
225C22×C4#1174568712yes—
226C22×C4C2×C22⋊D5911yescharacteristic
227M16—44no—
228C2×C8—42no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—1182no—
232C4⋊C4—4722no—
233C22⋊C4—4722no—
234C22⋊C4—4722no—
235C22⋊C4—2362no—
236C22⋊C4—2364no—
237C22⋊C4—2362no—
238C42—4722no—
239C42—2362no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—2362no—
244C23—2362no—
245C23—2361no—
246C23—2362no—
247C23C2×(C2×D59)⋊C411yescharacteristic
248C23#1427369711yescharacteristic
249C23#1427371111yescharacteristic
250D4—4722no—
251D4—4722no—
252D4—4722no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—2364no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4—2364no—
269C2×C4—2362no—
270C2×C4—2361no—
271C2×C4—2364no—
272C2×C4—2362no—
273C2×C4(C22⋊D59)⋊C412yes—
274C2×C4#1427369912yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—4724no—
285C22—4721no—
286C22—4721no—
287C22—4721no—
288C22#1683727911yescharacteristic
289C22#1683737512yes—
290C22#1683757712yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—4724no—
296C4—4722no—
297C4—4721no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—9441no—
302C2#2142449711yescharacteristic
303C2#2142455211yescharacteristic
304C2#2142462611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1821C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1821D59
hypocenterC59#9465
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #21426699C2semidirect
4. #21421950C2semidirect
5. C59×#1821C2semidirect
8. #16837303C4semidirect
12. #16837267C4semidirect
148. #1821D59semidirect
173. C59#9465semidirect

Maximal subgroups

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

subgroupindex
#214285842
#214266992
#214226262
#214219502
C59×#18212
#946559