Finite Group Databasebeta

← #28481065

Subgroups — #28481065

Order 15104.

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

normal subgroups only

This group has 620 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#21425078C211yesmaximal, characteristic
2#21422826C212yesmaximal
3#21422734C211yesmaximal, characteristic
4#21422608C212yesmaximal
5C59×#3556C211yesmaximal, characteristic
6#16837459C2212yes—
7#16837426C2211yescharacteristic
8#16837248C2211yescharacteristic
9#16837161—22no—
10#16837151—22no—
11#16837148—22no—
12#16837144—22no—
13#16837139—24no—
14C59×#564—22no—
15C59×#557—22no—
16C59×#557—22no—
17C59×#557C414yescomplemented
18C59×#541—24no—
19C59×#541—22no—
20C59×#533C2212yes—
21C59×#533C2211yescharacteristic
22C2×#11745664—21no—
23C2×C4×Dic59—21no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27#14273638—22no—
28C59⋊#95—22no—
29C4⋊Q8×C59—22no—
30C4⋊Q8×C59—24no—
31C4⋊Q8×C59C2×C412yes—
32C4⋊Q8×C59D412yes—
33C4⋊Q8×C59D412yes—
34#126×C59—28no—
35#125×C59—22no—
36#125×C59—28no—
37#125×C59—28no—
38#125×C59—24no—
39#125×C59—28no—
40#125×C59—22no—
41#125×C59D412yes—
42C22⋊Q8×C59—22no—
43C22⋊Q8×C59—22no—
44C22⋊Q8×C59D414yes—
45C4×Q8×C59—24no—
46C4×Q8×C59—24no—
47C4×Q8×C59—24no—
48C4×Q8×C59—22no—
49C4×Q8×C59—22no—
50C4×Q8×C59C2×C414yes—
51#117×C59—22no—
52#117×C59D412yes—
53C2×C42×C59C2311yescharacteristic
54C2×Q8×C59—22no—
55C2×Q8×C59—22no—
56C2×Q8×C59C22⋊C414yes—
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D412yes—
59C22×Dic59—41no—
60C4⋊C4×C59—28no—
61C4⋊C4×C59—28no—
62C4⋊C4×C59—28no—
63C4⋊C4×C59—22no—
64C4⋊C4×C59—24no—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—24no—
69C4⋊C4×C59—24no—
70C4⋊C4×C59C22⋊C414yes—
71C4⋊C4×C59C22⋊C414yes—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—22no—
74C42×C59—28no—
75C42×C59—24no—
76C42×C59—22no—
77C42×C59C22×C411yescharacteristic
78C42×C59C2×D412yes—
79C42×C59C2×D411yescharacteristic
80C4×Dic59—22no—
81C4×Dic59—41no—
82#11745664—42no—
83C2×C59⋊C8—42no—
84C2×C59⋊C8—41no—
85C23×C59#12011yescharacteristic
86Q8×C59—44no—
87Q8×C59—44no—
88Q8×C59—42no—
89Q8×C59—42no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—24no—
93C2×C4×C59—24no—
94C2×C4×C59—24no—
95C2×C4×C59—42no—
96C2×C4×C59C2×C22⋊C412yes—
97C2×C4×C59C2×C22⋊C411yescharacteristic
98C2×C4×C59#12012yes—
99C2×C4×C59#12011yescharacteristic
100C2×Dic59—42no—
101C2×Dic59—82no—
102C59⋊C8—82no—
103C59⋊C8—81no—
104#8495—591nomaximal, Sylow
105C22×C59—21no—
106C22×C59—22no—
107C22×C59#37911yescharacteristic
108C4×C59—21no—
109C4×C59—22no—
110C4×C59—22no—
111C4×C59—44no—
112C4×C59—44no—
113C4×C59—44no—
114C4×C59#45411yescharacteristic
115C4×C59#45611yescharacteristic
116Dic59—82no—
117#3556D5911yescharacteristic
118#1995—591no—
119#1559—592no—
120#1537—591no—
121#1506—592no—
122C2×C59—41no—
123C2×C59#179311yescharacteristic
124C2×C59#179612yes—
125#564—22no—
126#557—22no—
127#557—22no—
128#557Dic5914yescomplemented
129#541—24no—
130#541—22no—
131#533C2×D5912yes—
132#533C2×D5911yescharacteristic
133#432—592no—
134#423—591no—
135#376—591no—
136#339—1182no—
137#333—1182no—
138#332—1182no—
139#330—1182no—
140#328—1184no—
141C59#849511yescharacteristic, complemented, Sylow
142C2×M16—1181no—
143C4⋊Q8—22no—
144C4⋊Q8—24no—
145C4⋊Q8C2×Dic5912yes—
146C4⋊Q8C22⋊D5912yes—
147C4⋊Q8C22⋊D5912yes—
148#126—28no—
149#125—22no—
150#125—28no—
151#125—28no—
152#125—24no—
153#125—28no—
154#125—22no—
155#125C22⋊D5912yes—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8C22⋊D5914yes—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C2×Dic5914yes—
165#117—22no—
166#117C22⋊D5912yes—
167C2×C42—1181no—
168C2×C42C22×D5911yescharacteristic
169C4⋊C8—1182no—
170C4⋊C8—1182no—
171C22⋊C8—1182no—
172#97—1182no—
173#95—1182no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8C22⋊Dic5914yes—
177C22×C4—2361no—
178C22×C4C2×C22⋊D5911yescharacteristic
179C22×C4C2×C22⋊D5912yes—
180M16—2362no—
181C2×C8—2362no—
182C2×C8—2361no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4C22⋊Dic5914yes—
194C4⋊C4C22⋊Dic5914yes—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C42—28no—
198C42—24no—
199C42—22no—
200C42—1182no—
201C42—2361no—
202C42C22×Dic5911yescharacteristic
203C42C2×C22⋊D5911yescharacteristic
204C42C2×C22⋊D5912yes—
205C23#1427369611yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—42no—
216C2×C4—4722no—
217C2×C4—2362no—
218C2×C4C2×C22⋊Dic5912yes—
219C2×C4C2×C22⋊Dic5911yescharacteristic
220C2×C4#1427369612yes—
221C2×C4#1427369611yescharacteristic
222C8—4722no—
223C8—4721no—
224C22—21no—
225C22—22no—
226C22#1683725811yescharacteristic
227C4—21no—
228C4—22no—
229C4—22no—
230C4—44no—
231C4—44no—
232C4—44no—
233C4—4722no—
234C4#1683757511yescharacteristic
235C4#1683758511yescharacteristic
236C2—41no—
237C2#2142405511yescharacteristic
238C2#2142407112yes—
subgroupG/H
centerC22#16837258
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3556C2
socleC22×C59#379
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#3556D59
hypocenterC59#8495
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
17. C59×#557C4semidirect
128. #557Dic59semidirect
141. C59#8495semidirect

Maximal subgroups

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

subgroupindex
#214250782
#214228262
#214227342
#214226082
C59×#35562
#849559