Finite Group Databasebeta

← #28498123

Subgroups — #28498123

Order 15104.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 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#21432732C211yesmaximal, characteristic
2#21425908C211yesmaximal, characteristic, complemented
3#21422046C212yesmaximal, complemented
4#21422031C212yesmaximal
5C59×#2165C211yesmaximal, characteristic, complemented
6C2×#14273684C411yescharacteristic
7#16837464—21no—
8#16837464—21no—
9#16837461—21no—
10#16837461—21no—
11#16837425C411yescharacteristic
12#16837425C2211yescharacteristic
13#16837365C411yescharacteristic
14#16837365C2211yescharacteristic
15C4×#11745664C411yescharacteristic
16#16837223—21no—
17#16837184C2212yes—
18#16837180—21no—
19C4×M16×C59C2211yescharacteristic
20C59×#364—21no—
21C59×#349C2212yes—
22C59×#347—21no—
23C22×C4×D59C2×C411yescharacteristic
24C2×#11745664C2×C411yescharacteristic
25C2×#11745664C2×C411yescharacteristic
26#14273684—21no—
27#14273684—21no—
28#14273684—22no—
29C2×Dic59⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×C4×Dic59C2×C411yescharacteristic
32#14273667—22no—
33Dic59⋊C8—21no—
34Dic59⋊C8—21no—
35Dic59⋊C8—21no—
36Dic59⋊C8—21no—
37(C2×D59)⋊C8—21no—
38(C2×D59)⋊C8—21no—
39(C2×D59)⋊C8—21no—
40(C2×D59)⋊C8—21no—
41#14273638—21no—
42C4×C59⋊C8—21no—
43C2×M16×C59C2×C411yescharacteristic
44C2×M16×C59C2311yescharacteristic
45C2×C42×C59C2×C411yescharacteristic
46M32×C59—22no—
47#97×C59D411yescharacteristic
48#97×C59D411yescharacteristic
49C4×C8×C59D411yescharacteristic
50C4×C8×C59D411yescharacteristic
51C23×D59M1611yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59C22×C411yescharacteristic
54C22×Dic59—21no—
55C22×Dic59M1611yescharacteristic
56C2×C4×D59—21no—
57C2×C4×D59—21no—
58C2×C4×D59—41no—
59C2×C4×D59—41no—
60M16×C59—41no—
61M16×C59—41no—
62C2×C8×C59C22⋊C411yescharacteristic
63C2×C8×C59C22⋊C411yescharacteristic
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C42×C59—21no—
67C42×C59C22⋊C411yescharacteristic
68C42×C59C22⋊C411yescharacteristic
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71Dic59⋊C4—22no—
72Dic59⋊C4—22no—
73C4×Dic59—21no—
74C4×Dic59M1612yes—
75#11745664—41no—
76#11745664—41no—
77C2×C59⋊C8—21no—
78C2×C59⋊C8—21no—
79C16×C59—42no—
80C59⋊C16—42no—
81C23×C59C2×M1611yescharacteristic
82C22×D59—21no—
83C22×D59—41no—
84C22×D59—81no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—81no—
90C2×C4×C59C2×C22⋊C411yescharacteristic
91C2×C4×C59C2×M1611yescharacteristic
92C2×Dic59—22no—
93C2×Dic59—21no—
94C2×Dic59—41no—
95C2×Dic59—81no—
96C4×D59—42no—
97C4×D59—41no—
98C4×D59—41no—
99C8×C59—21no—
100C8×C59—21no—
101C8×C59—21no—
102C8×C59—21no—
103C59⋊C8—41no—
104C59⋊C8—41no—
105#11522—591nomaximal, Sylow
106C22×C59—21no—
107C22×C59—41no—
108C22×C59#40711yescharacteristic
109C2×D59—42no—
110C2×D59—41no—
111C2×D59—82no—
112C4×C59—21no—
113C4×C59—41no—
114C4×C59—41no—
115C4×C59#41011yescharacteristic
116C4×C59#41311yescharacteristic
117Dic59—42no—
118Dic59—41no—
119#2921—591no—
120#2165—591no—
121#2165D5911yescharacteristic, complemented
122#1327—592no—
123#1320—592no—
124C2×C59—21no—
125C2×C59—41no—
126C2×C59#146411yescharacteristic
127D59—81no—
128C2×#117—591no—
129#433—1181no—
130#433—1181no—
131#433—1181no—
132#433—1181no—
133#423—591no—
134#423—591no—
135#407—591no—
136#407—591no—
137C4×M16—591no—
138C4×M16C2×D5911yescharacteristic
139#364—21no—
140#364—1181no—
141#349—592no—
142#349C2×D5912yes—
143#347—21no—
144#347—1181no—
145C59#1152211yescharacteristic, complemented, Sylow
146C23×C4—591no—
147C2×M16—591no—
148C2×M16—591no—
149C2×M16C4×D5911yescharacteristic
150C2×M16C22×D5911yescharacteristic
151#117—1182no—
152#117—1181no—
153#117—1181no—
154C2×C4⋊C4—1181no—
155C2×C22⋊C4—1181no—
156C2×C42—591no—
157C2×C42C4×D5911yescharacteristic
158M32—22no—
159M32—1182no—
160C4⋊C8—1181no—
161C4⋊C8—1181no—
162C4⋊C8—1181no—
163C4⋊C8—1181no—
164C22⋊C8—1181no—
165C22⋊C8—1181no—
166C22⋊C8—1181no—
167C22⋊C8—1181no—
168#97—1181no—
169#97D23611yescharacteristic
170#97C22⋊D5911yescharacteristic
171C4×C8—1181no—
172C4×C8D23611yescharacteristic
173C4×C8C22⋊D5911yescharacteristic
174C24—591no—
175C22×C4—21no—
176C22×C4—591no—
177C22×C4—1181no—
178C22×C4—1181no—
179C22×C4—1181no—
180C22×C4—2361no—
181C22×C4—2361no—
182C22×C4C2×C4×D5911yescharacteristic
183M16—41no—
184M16—41no—
185M16—2361no—
186M16—2361no—
187C2×C8—1181no—
188C2×C8—1181no—
189C2×C8(C2×D59)⋊C411yescharacteristic
190C2×C8(C2×D59)⋊C411yescharacteristic
191C2×C8C2×D23611yescharacteristic
192C2×C8C2×C22⋊D5911yescharacteristic
193C4⋊C4—1182no—
194C4⋊C4—1182no—
195C22⋊C4—1182no—
196C22⋊C4—1182no—
197C42—21no—
198C42—592no—
199C42—1181no—
200C42(C2×D59)⋊C411yescharacteristic
201C42(C2×D59)⋊C411yescharacteristic
202C16—42no—
203C16—2362no—
204C23—1181no—
205C23—4721no—
206C23—2361no—
207C23M16×D5911yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—81no—
213C2×C4—1182no—
214C2×C4—1181no—
215C2×C4—4721no—
216C2×C4—2362no—
217C2×C4—2361no—
218C2×C4—2361no—
219C2×C4—2361no—
220C2×C4C2×(C2×D59)⋊C411yescharacteristic
221C2×C4M16×D5911yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—2361no—
227C8—2361no—
228C22—21no—
229C22—41no—
230C22—4722no—
231C22—2362no—
232C22—2361no—
233C22#1683736511yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—2362no—
238C4—2361no—
239C4#1683737511yescharacteristic
240C4#1683738411yescharacteristic
241C2—21no—
242C2—41no—
243C2—4721no—
244C2#2142242811yescharacteristic
subgroupG/H
centerC4#16837375
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2165C2
socleC2×C59#1464
prime radicalC23×D59M16
cosocleC2×M16×C59C23
hypercenter#2165D59
hypocenterC59#11522
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21425908C2semidirect
3. #21422046C2semidirect
5. C59×#2165C2semidirect
121. #2165D59semidirect
145. C59#11522semidirect

Maximal subgroups

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

subgroupindex
#214327322
#214259082
#214220462
#214220312
C59×#21652
#1152259