Finite Group Databasebeta

← #29590091

Subgroups — #29590091

Order 15616.

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

normal subgroups only

This group has 360 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#21454730C211yesmaximal, characteristic
2#21444122C211yesmaximal, characteristic, complemented
3#21443794C212yesmaximal, complemented
4#21443791C212yesmaximal, complemented
5C61×#2165C211yesmaximal, characteristic, complemented
6#16840562C411yescharacteristic
7#16840528—21no—
8#16839885—21no—
9#16839885C412yes—
10#16839880—21no—
11#16839880C2212yes—
12#16839811C411yescharacteristic
13C2×#14519705—21no—
14C4×#11758194—21no—
15#16839484—21no—
16#16839483C2211yescharacteristic, complemented
17C8×#92418C2211yescharacteristic, complemented
18C4×M16×C61C2211yescharacteristic
19C61×#364C2212yescomplemented
20C61×#349—21no—
21C61×#347—21no—
22C2×#11758222—21no—
23#14519810—22no—
24#14519805—22no—
25#14519799—21no—
26#14519799C2×C412yes—
27#14519785—42no—
28#14519783—22no—
29#14519783—22no—
30C2×#11758194—22no—
31C2×#11758194—21no—
32M16×D61—42no—
33#14519713—21no—
34#14519713C2×C412yes—
35#14519705—22no—
36#14519705—22no—
37#14519688—22no—
38C4×#92418—21no—
39C4×#92418—21no—
40C2×M16×C61C2×C411yescharacteristic
41C2×M16×C61D411yescharacteristic
42C2×C42×C61D411yescharacteristic
43M32×C61—22no—
44#97×C61C2×C411yescharacteristic
45#97×C61D411yescharacteristic
46C4×C8×C61D411yescharacteristic
47C4×C8×C61C2311yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61C22⋊C411yescharacteristic
50C2×C22⋊D61—41no—
51#11758222—22no—
52#11758222—22no—
53#11758222—41no—
54#11758222—41no—
55C2×D244—21no—
56C2×C4×D61—41no—
57C2×Dic122—21no—
58M16×C61—22no—
59M16×C61—22no—
60C2×C8×C61C22⋊C411yescharacteristic
61C2×C8×C61C22⋊C411yescharacteristic
62C2×C8×C61C22×C411yescharacteristic
63C2×C8×C61C2×D411yescharacteristic
64C42×C61—21no—
65C42×C61C22⋊C411yescharacteristic
66C42×C61C2×D411yescharacteristic
67#11758194—22no—
68#11758194—44no—
69#11758194—41no—
70C2×#92418—22no—
71C2×#92418—22no—
72C2×#92418—41no—
73Dic244—21no—
74Dic244M1612yescomplemented
75D488—21no—
76D488M1612yescomplemented
77#11758190—22no—
78#11758190—22no—
79#11758189—42no—
80#11758189—42no—
81C8×D61—42no—
82C8×D61—42no—
83C16×C61—42no—
84C23×C61#10411yescharacteristic
85C22×D61—41no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—42no—
91C2×C4×C61#10411yescharacteristic
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C22⋊D61—42no—
94C22⋊D61—42no—
95C2×Dic61—41no—
96D244—22no—
97D244—41no—
98C4×D61—42no—
99C4×D61—42no—
100Dic122—22no—
101Dic122—41no—
102C8×C61—21no—
103C8×C61—21no—
104C8×C61C2×M1612yes—
105C8×C61AGL(1,ℤ/8)12yes—
106#92418—24no—
107#92418—42no—
108#92418—42no—
109#11669—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—41no—
112C22×C61C2×#10411yescharacteristic
113C2×D61—42no—
114C2×D61—161no—
115C4×C61—22no—
116C4×C61—21no—
117C4×C61—41no—
118C4×C61#40711yescharacteristic
119C4×C61#41811yescharacteristic
120Dic61—42no—
121#2956—611no—
122#2165D6111yescharacteristic, complemented
123#1455—611no—
124#1336—612no—
125#1335—612no—
126C2×C61—21no—
127C2×C61—41no—
128C2×C61#149011yescharacteristic
129D61—82no—
130#575—611no—
131#568—1221no—
132#444—612no—
133#444—1221no—
134#443—612no—
135#443—1221no—
136#430—611no—
137C2×#104—1221no—
138C4×M16—1221no—
139C4×M16C2×D6111yescharacteristic
140#364C2×D6112yescomplemented
141#349—21no—
142#347—21no—
143#322—611no—
144#322—1221no—
145C82—611no—
146C61#1166911yescharacteristic, complemented, Sylow
147C2×#41—1221no—
148#137—1222no—
149AGL(1,ℤ/8)—1222no—
150#135—612no—
151#135—1221no—
152#131—1222no—
153#131—1222no—
154#131—2442no—
155C2×M16—1222no—
156C2×M16—1221no—
157C2×M16—2442no—
158C2×M16C4×D6111yescharacteristic
159C2×M16C22⋊D6111yescharacteristic
160C2×C42D24411yescharacteristic
161M32—22no—
162#108—612no—
163#108—1221no—
164#104—1222no—
165#104—1222no—
166#97—1222no—
167#97C4×D6111yescharacteristic
168#97C22⋊D6111yescharacteristic
169C4×C8—1221no—
170C4×C8—1221no—
171C4×C8D24411yescharacteristic
172C4×C8C22×D6111yescharacteristic
173#41—1222no—
174#41—1222no—
175#41—2441no—
176#41—2441no—
177C2×Q8—1221no—
178C2×D4—1221no—
179C2×D4—2441no—
180C22×C4—21no—
181C22×C4—2441no—
182C22×C4#1175819811yescharacteristic
183Q16—612no—
184Q16—1221no—
185SD16—1222no—
186SD16—1222no—
187D8—612no—
188D8—1221no—
189M16—22no—
190M16—22no—
191M16—1222no—
192M16—2442no—
193M16—2444no—
194M16—2441no—
195M16—2442no—
196C2×C8—1222no—
197C2×C8—1222no—
198C2×C8—2442no—
199C2×C8—2441no—
200C2×C8—2442no—
201C2×C8#1175819811yescharacteristic
202C2×C8#1175819811yescharacteristic
203C2×C8C2×C4×D6111yescharacteristic
204C2×C8C2×C22⋊D6111yescharacteristic
205C42—21no—
206C42#1175819811yescharacteristic
207C42C2×D24411yescharacteristic
208C16—42no—
209C23—2441no—
210C23#1451970311yescharacteristic
211Q8—1222no—
212Q8—2441no—
213D4—1222no—
214D4—2441no—
215D4—2442no—
216D4—2442no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—2442no—
223C2×C4—2441no—
224C2×C4—2442no—
225C2×C4#1451970311yescharacteristic
226C2×C4C2×#1175819811yescharacteristic
227C8—21no—
228C8—21no—
229C8—1224no—
230C8—2442no—
231C8—2442no—
232C8M16×D6112yes—
233C8#1451980212yes—
234C22—21no—
235C22—41no—
236C22—9761no—
237C22—2442no—
238C22C2×#1451970311yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—2442no—
243C4#1683972511yescharacteristic
244C4#1683976611yescharacteristic
245C2—21no—
246C2—41no—
247C2—4882no—
248C2#2144426111yescharacteristic
subgroupG/H
centerC4#16839766
commutatorC2×C8×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#2165C2
socleC2×C61#1490
prime radical#16840562C4
cosocleC4×C8×C61C23
hypercenter#2165D61
hypocenterC61#11669
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C8 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21444122C2semidirect
3. #21443794C2semidirect
4. #21443791C2semidirect
5. C61×#2165C2semidirect
16. #16839483C22semidirect
17. C8×#92418C22semidirect
19. C61×#364C22semidirect
74. Dic244M16semidirect
76. D488M16semidirect
122. #2165D61semidirect
140. #364C2×D61semidirect
146. C61#11669semidirect

Maximal subgroups

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

subgroupindex
#214547302
#214441222
#214437942
#214437912
C61×#21652
#1166961