Finite Group Databasebeta

← #29572516

Subgroups — #29572516

Order 15616.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#16840367C211yesmaximal, characteristic
2#21445896C211yesmaximal, characteristic, complemented
3#21444349C214yesmaximal, complemented
4C61×#1822C211yesmaximal, characteristic, complemented
5#16840367—28no—
6C2×#14519774C411yescharacteristic, complemented
7C2×#14519769C411yescharacteristic, complemented
8C2×#14519762—22no—
9C2×#14519751C2212yes—
10C2×#14519750—22no—
11C2×C4×D244C412yescomplemented
12#16839625C2212yes—
13#16839529—22no—
14#16839497—28no—
15C2×#117×C61C2211yescharacteristic
16C61×#407C2212yes—
17C61×#343—22no—
18C22×D244—22no—
19C22×D244C2×C412yes—
20C22×C22⋊D61—22no—
21C22×C4×D61—22no—
22#14519774—24no—
23#14519769—24no—
24#14519762—216no—
25#14519751—24no—
26#14519751D418yes—
27#14519750—216no—
28C4×D244—24no—
29C4×D244—24no—
30C2×C4⋊Dic61C2×C412yes—
31C2×#11758196—22no—
32C2×C22⋊Dic61—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35#14519684—24no—
36C23×C4×C61C2311yescharacteristic
37C2×M16×C61—22no—
38#117×C61—24no—
39C2×C4⋊C4×C61D412yes—
40C2×C22⋊C4×C61D412yes—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C22⋊C8×C61—24no—
44C24×C61C2×D411yescharacteristic
45C23×D61—22no—
46C22×C4×C61—42no—
47C22×C4×C61C22⋊C412yes—
48C22×C4×C61C22⋊C412yes—
49C22×C4×C61C22×C411yescharacteristic
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D414yes—
52C2×C22⋊D61—28no—
53C2×C22⋊D61—84no—
54C22×Dic61—22no—
55C22×Dic61—42no—
56C2×D244—22no—
57C2×D244—28no—
58C2×D244—44no—
59C2×D244—84no—
60C2×D244C22⋊C414yes—
61C2×C4×D61—28no—
62C2×C4×D61—84no—
63M16×C61—44no—
64C2×C8×C61—42no—
65C4⋊C4×C61—24no—
66C22⋊C4×C61—24no—
67C42×C61—22no—
68C42×C61—22no—
69C22⋊Dic61—44no—
70#11758198—28no—
71#11758198—28no—
72C4⋊Dic61—22no—
73C4⋊Dic61C22⋊C414yes—
74#11758196—28no—
75C23×C61—22no—
76C23×C61—42no—
77C23×C61C2×C22⋊C411yescharacteristic
78C23×C61#12012yes—
79C22×D61—24no—
80C22×D61—44no—
81C22×D61—44no—
82C22×D61—44no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—22no—
86C2×C4×C61—22no—
87C2×C4×C61—44no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61C2×C22⋊C412yes—
91C2×C4×C61#12012yes—
92C22⋊D61—88no—
93C2×Dic61—24no—
94C2×Dic61—44no—
95C2×Dic61—82no—
96C2×Dic61—84no—
97D244—44no—
98D244—44no—
99D244—88no—
100C4×D61—88no—
101C8×C61—82no—
102#8518—611nomaximal, Sylow
103C22×C61—22no—
104C22×C61—24no—
105C22×C61—22no—
106C22×C61—82no—
107C22×C61#37911yescharacteristic
108C22×C61#41012yes—
109C2×D61—44no—
110C2×D61—44no—
111C2×D61—44no—
112C2×D61—88no—
113C4×C61—22no—
114C4×C61—22no—
115C4×C61—42no—
116C4×C61—42no—
117Dic61—44no—
118Dic61—162no—
119C2×#540—611no—
120#1822D6111yescharacteristic, complemented
121#1820—611no—
122#1510—614no—
123C2×C61—22no—
124C2×C61—42no—
125C2×C61#178211yescharacteristic
126C2×C61#179512yes—
127D61—84no—
128#540—1228no—
129C2×#127—611no—
130C2×#125—611no—
131C2×#123—1222no—
132C2×#121—612no—
133C2×#121—1222no—
134C2×C4×D4—612no—
135C2×#117C2×D6111yescharacteristic
136#407C2×D6112yes—
137#381—612no—
138#343—22no—
139#342—1222no—
140#327—1228no—
141C61#851811yescharacteristic, complemented, Sylow
142C22×D4—612no—
143C22×D4—1222no—
144C22×D4—1222no—
145C23×C4—1222no—
146C23×C4C22×D6111yescharacteristic
147C2×M16—22no—
148#127—1224no—
149#125—1224no—
150#123—12216no—
151#121—618no—
152#121—12216no—
153#121—1224no—
154C4×D4—1224no—
155C4×D4—1224no—
156#117—24no—
157C2×C4⋊C4—612no—
158C2×C4⋊C4—1222no—
159C2×C4⋊C4C22⋊D6112yes—
160C2×C22⋊C4—1222no—
161C2×C22⋊C4—1222no—
162C2×C22⋊C4—1222no—
163C2×C22⋊C4D24412yes—
164C2×C42C4×D6111yescharacteristic
165C2×C42C4×D6111yescharacteristic
166C22⋊C8—24no—
167#95—1224no—
168C24—1222no—
169C24C2×D24411yescharacteristic
170C2×D4—614no—
171C2×D4—1228no—
172C2×D4—1228no—
173C2×D4—1222no—
174C2×D4—4884no—
175C2×D4—4884no—
176C2×D4—2444no—
177C22×C4—42no—
178C22×C4—1228no—
179C22×C4—1222no—
180C22×C4—4884no—
181C22×C4—2442no—
182C22×C4#1175819812yes—
183C22×C4#1175819812yes—
184C22×C4C2×C4×D6111yescharacteristic
185C22×C4D4×D6114yes—
186C22×C4C2×C22⋊D6111yescharacteristic
187M16—44no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—614no—
191C4⋊C4—1228no—
192C4⋊C4—1222no—
193C22⋊C4—24no—
194C22⋊C4—1228no—
195C22⋊C4—1228no—
196C22⋊C4—2444no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—42no—
201C23—1224no—
202C23—2444no—
203C23—2444no—
204C23—2444no—
205C23C2×#1175819811yescharacteristic
206C23#1451974712yes—
207D4—4888no—
208D4—4888no—
209D4—2444no—
210D4—2444no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—1224no—
219C2×C4—4888no—
220C2×C4—4882no—
221C2×C4—4884no—
222C2×C4—2444no—
223C2×C4C22⋊C4×D6112yes—
224C2×C4#1451974512yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—82no—
230C22—4888no—
231C22—2444no—
232C22—2444no—
233C22—2444no—
234C22#1683961711yescharacteristic
235C22#1683973512yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—9762no—
241C4—2444no—
242C2—22no—
243C2—42no—
244C2—4884no—
245C2#2144573611yescharacteristic
246C2#2144579012yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1822C2
socleC22×C61#379
prime radicalC2×#16840367C2
cosocleC23×C4×C61C23
hypercenter#1822D61
hypocenterC61#8518
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #21445896C2semidirect
3. #21444349C2semidirect
4. C61×#1822C2semidirect
6. C2×#14519774C4semidirect
7. C2×#14519769C4semidirect
11. C2×C4×D244C4semidirect
120. #1822D61semidirect
141. C61#8518semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
#214458962
#214443492
C61×#18222
#851861