Finite Group Databasebeta

← #28477876

Subgroups — #28477876

Order 15104.

478 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 83 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#21431122C211yesmaximal, characteristic
2#21425639C211yesmaximal, characteristic, complemented
3#21425501C212yesmaximal, complemented
4#21421916C212yesmaximal, complemented
5C59×#1272C211yesmaximal, characteristic, complemented
6C2×#14273716—21no—
7C2×#14273658—21no—
8C2×#14273651—21no—
9C2×#14273649—22no—
10#16837319—21no—
11#16837312—22no—
12#16837304C2212yes—
13#16837294—22no—
14#16837272C2211yescharacteristic
15#16837165—22no—
16#16837157C2211yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×D236—21no—
20C22×D236—21no—
21C22×D236D412yes—
22C22×C59⋊C8—21no—
23#14273716—42no—
24#14273716—42no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59D412yes—
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×C4×Dic59—21no—
33#14273658—42no—
34#14273651—42no—
35#14273649—44no—
36C59⋊#95—21no—
37C22×Q8×C59D411yescharacteristic
38C22×C8×C59—22no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C42×C59C2311yescharacteristic
41#95×C59—22no—
42C23×D59—22no—
43C2×Q8×C59—21no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59SD1612yes—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59C2×D412yes—
51C22×Dic59—22no—
52C22×Dic59—41no—
53C2×D236—22no—
54C2×D236—22no—
55C2×D236—42no—
56C2×D236—42no—
57C2×D236—42no—
58C2×D236—81no—
59C2×D236—81no—
60C2×D236—81no—
61C2×D236—82no—
62C2×C8×C59—42no—
63C2×C8×C59—42no—
64C2×C8×C59—42no—
65C4⋊C4×C59—41no—
66C42×C59—41no—
67(C2×D59)⋊C4—44no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—42no—
72(C2×D59)⋊C4—82no—
73C4⋊Dic59—42no—
74C4⋊Dic59—41no—
75C4⋊Dic59—41no—
76C4⋊Dic59—42no—
77C4⋊Dic59—42no—
78C4×Dic59—42no—
79C2×C59⋊C8—22no—
80C2×C59⋊C8—41no—
81C2×C59⋊C8—41no—
82C23×C59#12011yescharacteristic
83C22×D59—42no—
84C22×D59—42no—
85C22×D59—42no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59—42no—
89C22×D59—42no—
90Q8×C59—42no—
91Q8×C59—81no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59C2×SD1611yescharacteristic
102C2×C4×C59#13512yes—
103C2×C4×C59AGL(1,ℤ/8)12yes—
104C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—82no—
109C2×Dic59—81no—
110D236—44no—
111D236—82no—
112D236—84no—
113D236—84no—
114C8×C59—82no—
115C59⋊C8—42no—
116#7810—591nomaximal, Sylow
117C22×C59#44912yes—
118C22×C59#45011yescharacteristic
119C22×C59#45311yescharacteristic
120C22×C59#45512yes—
121C22×C59#45811yescharacteristic
122C2×D59—82no—
123C2×D59—82no—
124C2×D59—82no—
125C2×D59—82no—
126C2×D59—82no—
127C2×D59—82no—
128C2×D59—82no—
129C4×C59—22no—
130C4×C59—22no—
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—81no—
135Dic59—82no—
136Dic59—161no—
137#2733—591no—
138#2096—591no—
139#2071—592no—
140#1272D5911yescharacteristic, complemented
141#1270—592no—
142C2×C59#161212yes—
143C2×C59#161311yescharacteristic
144C2×C59#162411yescharacteristic
145C2×C59#164212yes—
146C2×C59#165911yescharacteristic
147D59—162no—
148C2×#124—1181no—
149C2×#106—1181no—
150C2×#103—1181no—
151C2×#102—1182no—
152#394—1181no—
153#392—1182no—
154#391C2×D5911yescharacteristic
155#390—592no—
156#388—1182no—
157#384—591no—
158#340—1182no—
159#336—591no—
160#336C2×D5912yes—
161C59#781011yescharacteristic, complemented, Sylow
162C22×Q8D23611yescharacteristic
163C22×D4—592no—
164C22×D4—1181no—
165C22×D4—1181no—
166C22×C8—22no—
167C22×C8—1181no—
168#124—2362no—
169#124—2362no—
170C2×C4⋊C4—592no—
171C2×C4⋊C4—1181no—
172C2×C4⋊C4—1182no—
173C2×C4⋊C4D23611yescharacteristic
174C2×C22⋊C4—1182no—
175C2×C22⋊C4—1182no—
176C2×C22⋊C4—1182no—
177C2×C22⋊C4—1182no—
178C2×C42—1181no—
179C2×C42C22×D5911yescharacteristic
180#106—2362no—
181#103—2362no—
182#102—2364no—
183#95—22no—
184#95—1181no—
185C24—1182no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1174566012yes—
190C2×D4—1182no—
191C2×D4—1182no—
192C2×D4—4721no—
193C2×D4—4721no—
194C2×D4—4721no—
195C2×D4—4722no—
196C2×D4—2362no—
197C2×D4—2362no—
198C2×D4—2362no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1182no—
202C22×C4—2361no—
203C22×C4C2×D23611yescharacteristic
204C22×C4D4×D5912yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—1182no—
209C2×C8—2361no—
210C2×C8—2361no—
211C4⋊C4—41no—
212C4⋊C4—2361no—
213C4⋊C4—2361no—
214C4⋊C4—2362no—
215C4⋊C4—2362no—
216C4⋊C4—2362no—
217C22⋊C4—4722no—
218C22⋊C4—2364no—
219C22⋊C4—2362no—
220C22⋊C4—2362no—
221C22⋊C4—2362no—
222C22⋊C4—2362no—
223C42—41no—
224C42—2362no—
225C23—2362no—
226C23—2362no—
227C23—2362no—
228C23—2362no—
229C23—2362no—
230C23—2362no—
231C23—2362no—
232C23#1427369711yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—4722no—
236D4—4724no—
237D4—4724no—
238D4—2364no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—4722no—
249C2×C4—4721no—
250C2×C4—2362no—
251C2×C4—2362no—
252C2×C4—2362no—
253C2×C4C2×#1174566011yescharacteristic
254C2×C4#1427375012yes—
255C2×C4#1427375412yes—
256C2×C4#1427375511yescharacteristic
257C8—82no—
258C8—2362no—
259C22—4722no—
260C22—4722no—
261C22—4722no—
262C22—4722no—
263C22—4722no—
264C22—4722no—
265C22—4722no—
266C22#1683755012yes—
267C22#1683755511yescharacteristic
268C22#1683757211yescharacteristic
269C22#1683758112yes—
270C22#1683759311yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—9441no—
278C4—4722no—
279C2—9442no—
280C2#2142311312yes—
281C2#2142311811yescharacteristic
282C2#2142318011yescharacteristic
283C2#2142329912yes—
284C2#2142340311yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59#120
prime radical#21431122C2
hypercenter#1272D59
hypocenterC59#7810
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21425639C2semidirect
3. #21425501C2semidirect
4. #21421916C2semidirect
5. C59×#1272C2semidirect
140. #1272D59semidirect
161. C59#7810semidirect

Maximal subgroups

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

subgroupindex
#214311222
#214256392
#214255012
#214219162
C59×#12722
#781059