Finite Group Databasebeta

← #29593070

Subgroups — #29593070

Order 15616.

442 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 59 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#21450848C211yesmaximal, characteristic
2#21447164C211yesmaximal, characteristic
3#21444795C212yesmaximal
4#21443850C212yesmaximal
5C61×#1325C211yesmaximal, characteristic
6#16840393C2211yescharacteristic
7C2×#14519776—22no—
8#16839920—22no—
9#16839802—21no—
10#16839682—21no—
11#16839682C2212yes—
12#16839658—22no—
13C4×C4⋊Dic61—21no—
14#16839531C2211yescharacteristic
15#16839522—21no—
16#16839500—22no—
17C61×#384C2211yescharacteristic
18C61×#324C2212yes—
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×Dic244—42no—
22C2×#11758194—21no—
23#14519776—41no—
24#14519776—48no—
25#14519768—41no—
26#14519756—22no—
27#14519736—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×C4×Dic61—22no—
33#14519710—41no—
34#14519708—41no—
35#14519702—42no—
36#14519684—21no—
37#14519684—22no—
38#14519684D412yes—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C42×C61D411yescharacteristic
43C22⋊C8×C61—22no—
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61C2×D412yes—
48C22×Dic61—22no—
49C22×Dic61—22no—
50C2×Dic122—22no—
51C2×Dic122—42no—
52C2×Dic122—44no—
53C2×Dic122—44no—
54C2×Dic122—81no—
55C2×C8×C61—42no—
56C4⋊C4×C61—41no—
57C4⋊C4×C61—44no—
58C4⋊C4×C61—41no—
59C42×C61—21no—
60C42×C61Q1612yescomplemented
61C22⋊Dic61—41no—
62C4⋊Dic61—24no—
63C4⋊Dic61—24no—
64C4⋊Dic61—22no—
65C4⋊Dic61—21no—
66C4⋊Dic61—21no—
67#11758196—44no—
68#11758196—44no—
69#11758196—42no—
70C4×Dic61—44no—
71C4×Dic61—41no—
72#11758194—22no—
73#11758194—41no—
74C2×#92418—41no—
75Dic244—84no—
76C23×C61#12011yescharacteristic
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—41no—
86C2×C4×C61C2×Q1611yescharacteristic
87C2×C4×C61#13711yescharacteristic
88C2×Dic61—24no—
89C2×Dic61—24no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—41no—
95C2×Dic61—41no—
96Dic122—44no—
97Dic122—82no—
98Dic122—84no—
99Dic122—82no—
100C8×C61—82no—
101#92418—42no—
102#12256—611nomaximal, Sylow
103C22×C61—21no—
104C22×C61—21no—
105C22×C61#45111yescharacteristic
106C22×C61#45611yescharacteristic
107C22×C61B(4,𝔽2)11yescharacteristic
108C4×C61—24no—
109C4×C61—21no—
110C4×C61—21no—
111C4×C61—42no—
112C4×C61—81no—
113Dic61—44no—
114Dic61—44no—
115Dic61—81no—
116Dic61—81no—
117#2551—611no—
118#2060—611no—
119#1605—612no—
120#1354—612no—
121#1325D6111yescharacteristic
122C2×C61—21no—
123C2×C61—21no—
124C2×C61#160511yescharacteristic
125C2×C61#219611yescharacteristic
126C2×C61#220511yescharacteristic
127#544—611no—
128C2×C4⋊Q8—1222no—
129#451—1222no—
130#428—1221no—
131#395—612no—
132#395—1221no—
133#389—1222no—
134#384C2×D6111yescharacteristic
135C4×C4⋊C4—1221no—
136#343—611no—
137#339—1221no—
138#328—1222no—
139#324C2×D6112yes—
140C61#1225611yescharacteristic, complemented, Sylow
141C22×Q8—612no—
142C22×Q8—1221no—
143C2×Q16—2442no—
144C2×M16—1221no—
145C4⋊Q8—2441no—
146C4⋊Q8—2448no—
147#125—2441no—
148C22⋊Q8—1222no—
149#117—1221no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—1221no—
152C2×C4⋊C4—1221no—
153C2×C4⋊C4—1222no—
154C2×C4⋊C4—1222no—
155C2×C4⋊C4D24411yescharacteristic
156C2×C4⋊C4C22×D6111yescharacteristic
157C2×C42—1222no—
158C2×C42D24411yescharacteristic
159#107—2441no—
160#106—2441no—
161#103—2442no—
162C22⋊C8—22no—
163#95—612no—
164#95—1221no—
165#95—1222no—
166C2×Q8—1222no—
167C2×Q8—4881no—
168C2×Q8—2442no—
169C2×Q8—2444no—
170C2×Q8—2444no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—1222no—
174C22×C4—1222no—
175C22×C4C2×D24411yescharacteristic
176C22×C4D4×D6112yes—
177Q16—4884no—
178M16—1222no—
179M16—2441no—
180C2×C8—42no—
181C2×C8—2441no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—44no—
185C4⋊C4—1224no—
186C4⋊C4—1224no—
187C4⋊C4—1222no—
188C4⋊C4—1221no—
189C4⋊C4—1221no—
190C4⋊C4—2442no—
191C4⋊C4—2444no—
192C4⋊C4—2444no—
193C22⋊C4—2441no—
194C42—21no—
195C42—2444no—
196C42—2441no—
197C42Dic24412yescomplemented
198C23#1451974711yescharacteristic
199Q8—4882no—
200Q8—4882no—
201Q8—4884no—
202Q8—2444no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—1224no—
213C2×C4—1224no—
214C2×C4—2441no—
215C2×C4—2441no—
216C2×C4—2442no—
217C2×C4—2442no—
218C2×C4—2442no—
219C2×C4—2442no—
220C2×C4C2×Dic24411yescharacteristic
221C2×C4#1451981011yescharacteristic
222C8—82no—
223C8—2442no—
224C22—21no—
225C22—21no—
226C22#1683992011yescharacteristic
227C22#1683994711yescharacteristic
228C22#1683995011yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—81no—
234C4—4881no—
235C4—4881no—
236C4—2444no—
237C4—2444no—
238C2—21no—
239C2—21no—
240C2#2144479511yescharacteristic
241C2#2144780511yescharacteristic
242C2#2144785011yescharacteristic
subgroupG/H
centerC22#16839950
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1325C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#1325D61
hypocenterC61#12256
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
60. C42×C61Q16semidirect
140. C61#12256semidirect
197. C42Dic244semidirect

Maximal subgroups

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

subgroupindex
#214508482
#214471642
#214447952
#214438502
C61×#13252
#1225661