Finite Group Databasebeta

← #29583267

Subgroups — #29583267

Order 15616.

542 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 73 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#21454072C211yesmaximal, characteristic
2#21445891C212yesmaximal
3#21445891C211yesmaximal, characteristic
4#21443723C212yesmaximal
5C61×#1304C211yesmaximal, characteristic
6C22×C22⋊Dic61C2211yescharacteristic
7#16839674—24no—
8#16839674—22no—
9#16839618C2211yescharacteristic
10#16839618C2212yes—
11#16839528—24no—
12#16839528—22no—
13C61×#379C2212yes—
14C61×#379C2211yescharacteristic
15C22×Dic122—22no—
16C23×Dic61Q812yescomplemented
17C2×C22⋊Dic61—21no—
18C2×C22⋊Dic61—22no—
19C2×C22⋊Dic61—22no—
20C2×C22⋊Dic61—21no—
21C2×C22⋊Dic61—21no—
22C2×C22⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61D414yes—
26#14519684—26no—
27C25×C61C2311yescharacteristic
28C2×C22⋊C4×C61—22no—
29C2×C22⋊C4×C61—22no—
30C2×C22⋊C4×C61—22no—
31C2×C22⋊C4×C61—22no—
32C2×C22⋊C4×C61D412yes—
33C24×C61—42no—
34C24×C61—82no—
35C24×C61C2×D411yescharacteristic
36C24×C61C2×D412yes—
37C24×C61C2×Q811yescharacteristic
38C24×C61#4112yes—
39C24×C61#4111yescharacteristic
40C22×C4×C61—22no—
41C22×C4×C61—42no—
42C22×Dic61—24no—
43C22×Dic61—22no—
44C22×Dic61—22no—
45C22×Dic61—22no—
46C22×Dic61—41no—
47C22×Dic61—82no—
48C2×Dic122—84no—
49C2×Dic122—82no—
50C22⋊C4×C61—42no—
51C22⋊C4×C61—44no—
52C22⋊C4×C61—82no—
53C22⋊C4×C61—82no—
54C22⋊C4×C61—82no—
55C22⋊Dic61—41no—
56C22⋊Dic61—42no—
57C22⋊Dic61—41no—
58C22⋊Dic61—44no—
59C22⋊Dic61—42no—
60C22⋊Dic61—82no—
61C22⋊Dic61—82no—
62C22⋊Dic61—82no—
63C23×C61—22no—
64C23×C61—24no—
65C23×C61—42no—
66C23×C61—41no—
67C23×C61—42no—
68C23×C61—42no—
69C23×C61—42no—
70C23×C61—42no—
71C23×C61—42no—
72C23×C61—42no—
73C23×C61—42no—
74C23×C61—41no—
75C23×C61—88no—
76C23×C61#12011yescharacteristic
77C23×C61C22⋊Q812yes—
78C23×C61C22⋊Q811yescharacteristic
79C23×C61#12412yes—
80C23×C61#12411yescharacteristic
81C2×C4×C61—44no—
82C2×C4×C61—42no—
83C2×C4×C61—82no—
84C2×C4×C61—82no—
85C2×C4×C61—82no—
86C2×Dic61—44no—
87C2×Dic61—44no—
88C2×Dic61—42no—
89C2×Dic61—44no—
90C2×Dic61—82no—
91C2×Dic61—82no—
92C2×Dic61—82no—
93C2×Dic61—81no—
94C2×Dic61—82no—
95Dic122—88no—
96#10391—611nomaximal, Sylow
97C22×C61—22no—
98C22×C61—21no—
99C22×C61—22no—
100C22×C61—22no—
101C22×C61—21no—
102C22×C61—22no—
103C22×C61—22no—
104C22×C61—22no—
105C22×C61—22no—
106C22×C61—22no—
107C22×C61—22no—
108C22×C61—44no—
109C22×C61—82no—
110C22×C61—84no—
111C22×C61—84no—
112C22×C61—82no—
113C22×C61#39311yescharacteristic
114C22×C61#45812yes—
115C4×C61—82no—
116C4×C61—162no—
117Dic61—84no—
118Dic61—82no—
119Dic61—161no—
120#2840—611no—
121#1819—612no—
122#1819—611no—
123#1304—612no—
124#1304D6111yescharacteristic
125C2×C61—22no—
126C2×C61—41no—
127C2×C61—42no—
128C2×C61—42no—
129C2×C61—41no—
130C2×C61#203011yescharacteristic
131C2×C61#203212yes—
132C22×C22⋊C4—611no—
133#393—1224no—
134#393—1222no—
135#379—612no—
136#379—611no—
137#379C2×D6112yes—
138#379C2×D6111yescharacteristic
139#342—1224no—
140#342—1222no—
141C61#1039111yescharacteristic, complemented, Sylow
142C25C22×D6111yescharacteristic
143C22×Q8—1222no—
144C23×C4—612no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—614no—
150C2×C22⋊C4—1222no—
151C2×C22⋊C4—1222no—
152C2×C22⋊C4—1222no—
153C2×C22⋊C4—1221no—
154C2×C22⋊C4—1222no—
155C2×C22⋊C4—1222no—
156C2×C22⋊C4—1221no—
157C2×C22⋊C4—1221no—
158C2×C22⋊C4D24412yes—
159#95—1226no—
160C24—42no—
161C24—82no—
162C24C2×D24411yescharacteristic
163C24#1175822212yes—
164C24D4×D6112yes—
165C24#1175822411yescharacteristic
166C24Q8×D6111yescharacteristic
167C2×Q8—4884no—
168C2×Q8—4882no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—1222no—
172C22×C4—1222no—
173C22×C4—1222no—
174C22×C4—1224no—
175C22×C4—4882no—
176C22×C4—2441no—
177C22⋊C4—42no—
178C22⋊C4—44no—
179C22⋊C4—82no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—4882no—
183C22⋊C4—4882no—
184C22⋊C4—4882no—
185C22⋊C4—2442no—
186C22⋊C4—2441no—
187C22⋊C4—2442no—
188C22⋊C4—2441no—
189C22⋊C4—2444no—
190C23—22no—
191C23—24no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—88no—
203C23#1451974711yescharacteristic
204C23#1451975512yes—
205C23#1451975711yescharacteristic
206C23#1451976312yes—
207C23#1451976511yescharacteristic
208Q8—4888no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—4882no—
215C2×C4—4881no—
216C2×C4—4882no—
217C2×C4—4882no—
218C2×C4—4882no—
219C2×C4—2444no—
220C2×C4—2444no—
221C2×C4—2444no—
222C2×C4—2442no—
223C22—22no—
224C22—22no—
225C22—21no—
226C22—22no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—44no—
235C22—84no—
236C22—82no—
237C22—82no—
238C22—84no—
239C22#1683967511yescharacteristic
240C22#1683995312yes—
241C4—82no—
242C4—162no—
243C4—9761no—
244C4—4884no—
245C4—4882no—
246C2—22no—
247C2—42no—
248C2—41no—
249C2—42no—
250C2—41no—
251C2#2144698711yescharacteristic
252C2#2144699712yes—
subgroupG/H
centerC22#16839675
commutator, prime radical, cosocleC25×C61C23
FrattiniC25C22×D61
FittingC61×#1304C2
socleC22×C61#393
hypercenter#1304D61
hypocenterC61#10391
LCSG ▷ C25×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1304

Complements

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

NH ≅ G/Ntype
16. C23×Dic61Q8semidirect
141. C61#10391semidirect

Maximal subgroups

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

subgroupindex
#214540722
#214458912
#214437232
C61×#13042
#1039161