Finite Group Databasebeta

← #29572579

Subgroups — #29572579

Order 15616.

656 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 115 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#21460314C211yesmaximal, characteristic
2#21446797C211yesmaximal, characteristic
3#21446795C212yesmaximal
4#21444556C212yesmaximal
5C61×#1561C211yesmaximal, characteristic
6C61⋊#564C414yescomplemented
7#16840491—22no—
8#16840490—22no—
9#16840487—22no—
10#16840374—22no—
11#16840372—24no—
12#16840319C2212yes—
13#16839786C2211yescharacteristic
14C42×Dic61C2211yescharacteristic
15#16839523—24no—
16#16839521—22no—
17#16839509—22no—
18#16839507—22no—
19C61×#533C2211yescharacteristic
20C61×#423C2212yes—
21C61×#332—22no—
22C2×#11758194—21no—
23#14519777—22no—
24#14519777D412yes—
25#14519777D412yes—
26#14519775—24no—
27#14519770—28no—
28#14519769—22no—
29#14519769C2×C414yes—
30#14519768—24no—
31#14519768—28no—
32#14519767—28no—
33#14519767—28no—
34#14519756—22no—
35#14519756—22no—
36Q8×Dic61—24no—
37C4×Dic122—24no—
38C4×Dic122—22no—
39C4×Dic122—22no—
40#14519742—24no—
41#14519736D414yes—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44C4⋊#92418—22no—
45C2×M16×C61—22no—
46C4⋊Q8×C61D412yes—
47#125×C61C2×C412yes—
48C22⋊Q8×C61—22no—
49C4×Q8×C61—22no—
50#117×C61D412yes—
51C2×C42×C61C2311yescharacteristic
52C4⋊C8×C61—22no—
53C4⋊C8×C61—22no—
54C2×Q8×C61—22no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61C2×D412yes—
57C22×Dic61—41no—
58C2×Dic122—22no—
59C2×Dic122—22no—
60M16×C61—44no—
61C2×C8×C61—42no—
62C4⋊C4×C61—24no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61C22⋊C414yes—
65C22⋊C4×C61—22no—
66C42×C61—22no—
67C42×C61C22×C411yescharacteristic
68C42×C61C2×D412yes—
69C42×C61C2×D411yescharacteristic
70C22⋊Dic61—24no—
71C4⋊Dic61—28no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61C22⋊C414yes—
75C4⋊Dic61C22⋊C414yes—
76#11758196—28no—
77#11758196—28no—
78#11758196—24no—
79#11758196—24no—
80C4×Dic61—28no—
81C4×Dic61—24no—
82C4×Dic61—22no—
83C4×Dic61—24no—
84C4×Dic61—41no—
85C4×Dic61—42no—
86#11758194—42no—
87C2×#92418—41no—
88C23×C61#12011yescharacteristic
89Q8×C61—42no—
90Q8×C61—42no—
91C2×C4×C61—21no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—24no—
96C2×C4×C61C2×C22⋊C412yes—
97C2×C4×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61#12012yes—
99C2×C4×C61#12011yescharacteristic
100C2×Dic61—24no—
101C2×Dic61—24no—
102C2×Dic61—44no—
103C2×Dic61—42no—
104Dic122—42no—
105Dic122—42no—
106Dic122—42no—
107Dic122—42no—
108C8×C61—82no—
109#92418—81no—
110#8533—611nomaximal, Sylow
111C22×C61—22no—
112C22×C61—21no—
113C22×C61#37911yescharacteristic
114C4×C61—22no—
115C4×C61—21no—
116C4×C61—44no—
117C4×C61#45411yescharacteristic
118C4×C61#45412yes—
119C4×C61#45612yes—
120C4×C61#45611yescharacteristic
121Dic61—44no—
122Dic61—44no—
123Dic61—44no—
124#3556—611no—
125#1994—612no—
126#1994—611no—
127#1561—612no—
128#1561D6111yescharacteristic
129C2×C61—41no—
130C2×C61#179312yes—
131C2×C61#179311yescharacteristic
132#564—614no—
133#557—1222no—
134#557—1222no—
135#557—1222no—
136#541—1224no—
137#541—1222no—
138#533—612no—
139#533C2×D6111yescharacteristic
140#423—611no—
141#423C2×D6112yes—
142C43—611no—
143#339—1224no—
144#339—1222no—
145#332—22no—
146#332—1222no—
147#332—1222no—
148C61#853311yescharacteristic, complemented, Sylow
149C2×M16—22no—
150C2×M16—1221no—
151C4⋊Q8—612no—
152C4⋊Q8—612no—
153C4⋊Q8—1222no—
154C4⋊Q8—1224no—
155C4⋊Q8C22⋊D6112yes—
156#126—1228no—
157#125—614no—
158#125—1222no—
159#125—1228no—
160#125—1228no—
161#125—1224no—
162#125—1228no—
163#125C4×D6112yes—
164C22⋊Q8—22no—
165C22⋊Q8—1222no—
166C22⋊Q8—1222no—
167C4×Q8—22no—
168C4×Q8—1224no—
169C4×Q8—1224no—
170C4×Q8—1224no—
171C4×Q8—1222no—
172C4×Q8—1222no—
173#117—614no—
174#117D24412yes—
175C2×C42—1221no—
176C2×C42—1222no—
177C2×C42C22×D6111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—1222no—
181C2×Q8—22no—
182C2×Q8—1222no—
183C2×Q8—1222no—
184C22×C4—2441no—
185C22×C4C2×D24411yescharacteristic
186C22×C4D4×D6112yes—
187M16—44no—
188M16—2442no—
189C2×C8—42no—
190C2×C8—2441no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—614no—
194C4⋊C4—614no—
195C4⋊C4—1228no—
196C4⋊C4—1228no—
197C4⋊C4—1228no—
198C4⋊C4—1224no—
199C4⋊C4—1224no—
200C4⋊C4—1222no—
201C4⋊C4—1222no—
202C4⋊C4#1175819814yes—
203C22⋊C4—22no—
204C22⋊C4—1224no—
205C42—22no—
206C42—1228no—
207C42—1224no—
208C42—1222no—
209C42—1224no—
210C42—2441no—
211C42—2442no—
212C42C2×C4×D6111yescharacteristic
213C42D4×D6112yes—
214C42C2×C22⋊D6111yescharacteristic
215C23#1451974711yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—2442no—
219Q8—2442no—
220Q8—2442no—
221Q8—2442no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—1224no—
228C2×C4—1224no—
229C2×C4—2444no—
230C2×C4—2442no—
231C2×C4C22⋊C4×D6112yes—
232C2×C4C2×#1175819811yescharacteristic
233C2×C4#1451974512yes—
234C2×C4#1451974711yescharacteristic
235C8—82no—
236C8—4881no—
237C22—22no—
238C22—21no—
239C22#1683961711yescharacteristic
240C4—22no—
241C4—21no—
242C4—44no—
243C4—2444no—
244C4—2444no—
245C4—2444no—
246C4#1683993312yes—
247C4#1683993711yescharacteristic
248C4#1683994312yes—
249C4#1683994711yescharacteristic
250C2—41no—
251C2#2144577612yes—
252C2#2144578011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1561C2
socleC22×C61#379
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1561D61
hypocenterC61#8533
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
6. C61⋊#564C4semidirect
148. C61#8533semidirect

Maximal subgroups

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

subgroupindex
#214603142
#214467972
#214467952
#214445562
C61×#15612
#853361