Finite Group Databasebeta

← #29708154

Subgroups — #29708154

Order 15616.

1118 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 291 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#21448439C213yesmaximal
2#21447171C218yesmaximal
3#21445552C213yesmaximal
4C61×#3458C211yesmaximal, characteristic
5C2×#14519736C2213yes—
6C22×C4⋊Dic61C2211yescharacteristic
7#16839660—26no—
8#16839634C22112yes—
9#16839631—23no—
10#16839627—23no—
11#16839529C22112yes—
12C61×#544—23no—
13C61×#544—23no—
14C61×#543C2214yes—
15C61×#533—26no—
16C2×#117×C61C2213yes—
17C23×Dic61C2312yes—
18#14519736—26no—
19#14519736Q8112yescomplemented
20C2×C4⋊Dic61—23no—
21C2×C4⋊Dic61—23no—
22C2×C4⋊Dic61—23no—
23C2×#11758196—212no—
24C2×C22⋊Dic61C2316yes—
25C2×C4×Dic61—23no—
26#14519684—212no—
27C23×C4×C61C2311yescharacteristic
28C4⋊Q8×C61—26no—
29C4⋊Q8×C61—26no—
30#125×C61—212no—
31C22⋊Q8×C61—212no—
32C22⋊Q8×C61—212no—
33C4×Q8×C61—24no—
34C4×Q8×C61—24no—
35#117×C61—26no—
36#117×C61—26no—
37C2×C4⋊C4×C61D413yes—
38C2×C4⋊C4×C61D413yes—
39C2×C22⋊C4×C61C2316yes—
40C2×C42×C61D416yes—
41C24×C61C2411yescharacteristic
42C2×Q8×C61—24no—
43C2×Q8×C61—24no—
44C22×C4×C61C2×D416yes—
45C22×C4×C61C2×D416yes—
46C22×C4×C61C2×D413yes—
47C22×C4×C61C2×D413yes—
48C22×C4×C61C2×Q816yes—
49C22×C4×C61#4112yes—
50C22×Dic61—26no—
51C22×Dic61—26no—
52C22×Dic61—82no—
53C4⋊C4×C61—212no—
54C4⋊C4×C61—212no—
55C4⋊C4×C61—26no—
56C4⋊C4×C61—26no—
57C22⋊C4×C61—212no—
58C42×C61—26no—
59C42×C61—26no—
60C22⋊Dic61—26no—
61C22⋊Dic61C2×Q8112yes—
62C4⋊Dic61—212no—
63C4⋊Dic61—41no—
64C4⋊Dic61—43no—
65#11758196—812no—
66C4×Dic61—212no—
67C23×C61—41no—
68C23×C61—41no—
69C23×C61C22×D413yes—
70C23×C61C22×Q813yes—
71C23×C61C2×#4111yescharacteristic
72Q8×C61—84no—
73Q8×C61—84no—
74C2×C4×C61—26no—
75C2×C4×C61—26no—
76C2×C4×C61—46no—
77C2×C4×C61—46no—
78C2×C4×C61#12011yescharacteristic
79C2×C4×C61#12013yes—
80C2×C4×C61C22⋊Q816yes—
81C2×C4×C61C22⋊Q816yes—
82C2×C4×C61C4⋊Q816yes—
83C2×C4×C61C4⋊Q816yes—
84C2×Dic61—212no—
85C2×Dic61—46no—
86C2×Dic61—46no—
87C2×Dic61—412no—
88C2×Dic61—42no—
89#23566—611nomaximal, Sylow
90C22×C61—23no—
91C22×C61—23no—
92C22×C61—43no—
93C22×C61—41no—
94C22×C61C2×#12011yescharacteristic
95C22×C61C2×C22⋊Q813yes—
96C22×C61C2×C4⋊Q813yes—
97C4×C61—46no—
98C4×C61—46no—
99C4×C61#39512yes—
100C4×C61#39516yes—
101Dic61—412no—
102Dic61—82no—
103#3458D6111yescharacteristic
104#2317—613no—
105#2062—618no—
106#1740—613no—
107C2×C61—23no—
108C2×C61—23no—
109C2×C61C2×#39511yescharacteristic
110C2×C61#302711yescharacteristic
111C2×C61#302811yescharacteristic
112#544—23no—
113#544—23no—
114#543C2×D6114yes—
115#533—26no—
116C2×#117—613no—
117C2×#117C2×D6113yes—
118C22×C4⋊C4—611no—
119#389—1226no—
120#385—6112no—
121#384—1223no—
122#382—1223no—
123#342—6112no—
124C61#2356611yescharacteristic, complemented, Sylow
125C23×C4—612no—
126C23×C4C22×D6111yescharacteristic
127C4⋊Q8—26no—
128C4⋊Q8—26no—
129#125—212no—
130C22⋊Q8—212no—
131C22⋊Q8—212no—
132C4×Q8—24no—
133C4×Q8—24no—
134#117—26no—
135#117—26no—
136#117—6112no—
137#117—1226no—
138C2×C4⋊C4—1223no—
139C2×C4⋊C4—12212no—
140C2×C4⋊C4—1223no—
141C2×C4⋊C4—1223no—
142C2×C4⋊C4C22⋊D6113yes—
143C2×C4⋊C4C22⋊D6113yes—
144C2×C22⋊C4—616no—
145C2×C22⋊C4C22×D6116yes—
146C2×C42—1223no—
147C2×C42C22⋊D6116yes—
148#95—12212no—
149C24C23×D6111yescharacteristic
150C2×Q8—24no—
151C2×Q8—24no—
152C22×C4—1226no—
153C22×C4—1226no—
154C22×C4—4882no—
155C22×C4Q8×D6116yes—
156C22×C4#1175822612yes—
157C22×C4C2×C22⋊D6116yes—
158C22×C4C2×C22⋊D6116yes—
159C22×C4C2×C22⋊D6113yes—
160C22×C4C2×C22⋊D6113yes—
161C4⋊C4—212no—
162C4⋊C4—212no—
163C4⋊C4—26no—
164C4⋊C4—26no—
165C4⋊C4—12212no—
166C4⋊C4—48812no—
167C4⋊C4—2441no—
168C4⋊C4—2443no—
169C22⋊C4—212no—
170C22⋊C4—6112no—
171C22⋊C4—1226no—
172C42—26no—
173C42—26no—
174C42—12212no—
175C23—41no—
176C23—41no—
177C23C22×C22⋊D6113yes—
178C23C2×Q8×D6113yes—
179C23C2×#1175822611yescharacteristic
180Q8—84no—
181Q8—84no—
182C2×C4—26no—
183C2×C4—26no—
184C2×C4—46no—
185C2×C4—46no—
186C2×C4—12212no—
187C2×C4—24412no—
188C2×C4—2442no—
189C2×C4—2446no—
190C2×C4—2446no—
191C2×C4#1451974611yescharacteristic
192C2×C4#1451974613yes—
193C2×C4#1451975316yes—
194C2×C4#1451975316yes—
195C2×C4#1451977516yes—
196C2×C4#1451977516yes—
197C22—23no—
198C22—23no—
199C22—43no—
200C22—41no—
201C22C2×#1451974611yescharacteristic
202C22C2×#1451975313yes—
203C22C2×#1451977513yes—
204C4—46no—
205C4—46no—
206C4—4882no—
207C4—24412no—
208C4#1683968112yes—
209C4#1683968116yes—
210C2—23no—
211C2—23no—
212C2C2×#1683968111yescharacteristic
213C2#2145522111yescharacteristic
214C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, prime radical, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
hypercenter#3458D61
hypocenterC61#23566
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
19. #14519736Q8semidirect
124. C61#23566semidirect

Maximal subgroups

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

subgroupindex
#214484392
#214471712
#214455522
C61×#34582
#2356661