Finite Group Databasebeta

← #29593067

Subgroups — #29593067

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#21444793C212yesmaximal
3#21443848C212yesmaximal
4#21443765C211yesmaximal, characteristic
5C61×#2060C211yesmaximal, characteristic
6C2×#14519776—22no—
7#16839917—22no—
8#16839682—21no—
9#16839682C2212yes—
10#16839658—22no—
11C4×C4⋊Dic61—21no—
12#16839499—22no—
13#16839488C2212yes—
14C61×#544C2211yescharacteristic
15C61×#428—21no—
16C61×#384C2211yescharacteristic
17C61×#343C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×#11758202—42no—
22#14519776—48no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×#11758196—22no—
26C2×#11758196—22no—
27C2×C4×Dic61—22no—
28#14519700—42no—
29C22⋊#92418—22no—
30#14519684—21no—
31#14519684—22no—
32#14519684D412yes—
33C2×M16×C61—21no—
34C4⋊Q8×C61—41no—
35#125×C61—41no—
36C22⋊Q8×C61—22no—
37#117×C61—21no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C4⋊C4×C61C2311yescharacteristic
41C2×C42×C61D411yescharacteristic
42#107×C61—41no—
43#106×C61—41no—
44C2×Q8×C61—22no—
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61C2×D411yescharacteristic
48C22×C4×C61C2×D412yes—
49C22×Dic61—22no—
50C22×Dic61—22no—
51C2×Dic122—42no—
52C2×Dic122—44no—
53C2×Dic122—44no—
54C2×Dic122—81no—
55M16×C61—22no—
56M16×C61—41no—
57C2×C8×C61—41no—
58C4⋊C4×C61—21no—
59C4⋊C4×C61—21no—
60C4⋊C4×C61—22no—
61C4⋊C4×C61—41no—
62C4⋊C4×C61—44no—
63C4⋊C4×C61—42no—
64C4⋊C4×C61—41no—
65C22⋊C4×C61—41no—
66C42×C61—21no—
67C42×C61—41no—
68C42×C61Q1612yescomplemented
69#11758202—84no—
70C4⋊Dic61—24no—
71C4⋊Dic61—24no—
72#11758196—44no—
73#11758196—44no—
74C4×Dic61—44no—
75C2×#92418—42no—
76C23×C61#12011yescharacteristic
77Q8×C61—82no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—22no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61C2×Q1611yescharacteristic
90C2×C4×C61#13711yescharacteristic
91C2×Dic61—24no—
92C2×Dic61—24no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—42no—
96C2×Dic61—42no—
97Dic122—44no—
98Dic122—82no—
99Dic122—84no—
100C8×C61—42no—
101#92418—82no—
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—
113C4×C61—81no—
114C4×C61—81no—
115Dic61—44no—
116Dic61—44no—
117#2551—611no—
118#2060D6111yescharacteristic
119#1605—612no—
120#1354—612no—
121#1325—611no—
122C2×C61—21no—
123C2×C61—21no—
124C2×C61#160511yescharacteristic
125C2×C61#219611yescharacteristic
126C2×C61#220511yescharacteristic
127#544C2×D6111yescharacteristic
128C2×C4⋊Q8—1222no—
129#451—1222no—
130#428—21no—
131#395—612no—
132#395—1221no—
133#389—1222no—
134#384C2×D6111yescharacteristic
135C4×C4⋊C4—1221no—
136#343C2×D6111yescharacteristic
137#339—21no—
138#328—1222no—
139#324—612no—
140C61#1225611yescharacteristic, complemented, Sylow
141C22×Q8—612no—
142C22×Q8—1221no—
143C2×Q16—2442no—
144C2×M16—21no—
145C4⋊Q8—41no—
146C4⋊Q8—2448no—
147#125—41no—
148C22⋊Q8—22no—
149#117—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—1221no—
152C2×C4⋊C4—1221no—
153C2×C4⋊C4—1222no—
154C2×C4⋊C4—1222no—
155C2×C4⋊C4C22⋊D6111yescharacteristic
156C2×C4⋊C4C22×D6111yescharacteristic
157C2×C42—1222no—
158C2×C42C22⋊D6111yescharacteristic
159#107—41no—
160#106—41no—
161#103—2442no—
162C22⋊C8—1222no—
163#95—612no—
164#95—1221no—
165#95—1222no—
166C2×Q8—22no—
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×C4D4×D6112yes—
176C22×C4C2×C22⋊D6111yescharacteristic
177Q16—4884no—
178M16—22no—
179M16—41no—
180C2×C8—41no—
181C2×C8—2442no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—1224no—
190C4⋊C4—1224no—
191C4⋊C4—2444no—
192C4⋊C4—2444no—
193C22⋊C4—41no—
194C42—21no—
195C42—41no—
196C42—2444no—
197C42#1175820212yescomplemented
198C23#1451974511yescharacteristic
199Q8—82no—
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—41no—
213C2×C4—41no—
214C2×C4—1224no—
215C2×C4—1224no—
216C2×C4—2442no—
217C2×C4—2442no—
218C2×C4—2442no—
219C2×C4—2442no—
220C2×C4C2×#1175820211yescharacteristic
221C2×C4#1451980711yescharacteristic
222C8—42no—
223C8—4882no—
224C22—21no—
225C22—21no—
226C22#1683991711yescharacteristic
227C22#1683994411yescharacteristic
228C22#1683994811yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4—2444no—
237C4—2444no—
238C2—21no—
239C2—21no—
240C2#2144479211yescharacteristic
241C2#2144780211yescharacteristic
242C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2060C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#2060D61
hypocenterC61#12256
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
68. C42×C61Q16semidirect
140. C61#12256semidirect
197. C42#11758202semidirect

Maximal subgroups

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

subgroupindex
#214508482
#214447932
#214438482
#214437652
C61×#20602
#1225661