Finite Group Databasebeta

← #29583250

Subgroups — #29583250

Order 15616.

414 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 75 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#21451484C211yesmaximal, characteristic
2#21445918C212yesmaximal
3#21443727C211yesmaximal, characteristic
4#21443671C212yesmaximal
5C61×#1820C211yesmaximal, characteristic
6#16839726C2212yes—
7#16839698—21no—
8#16839674—21no—
9#16839666—22no—
10#16839649—22no—
11C4×C22⋊Dic61C2212yes—
12#16839521—22no—
13#16839521—22no—
14C2×#117×C61C2211yescharacteristic
15C61×#381C2211yescharacteristic
16C61×#381C2211yescharacteristic
17C61×#342—21no—
18C61×#342—21no—
19C22×Dic122—21no—
20C2×#11758194—22no—
21C2×C4⋊Dic61—21no—
22C2×#11758196—22no—
23C2×C22⋊Dic61—22no—
24C2×C4×Dic61D412yes—
25C2×C4×Dic61D412yes—
26C22⋊#92418—22no—
27C22⋊#92418—22no—
28#14519684—21no—
29#14519684—21no—
30#14519684—22no—
31#14519684—22no—
32C23×C4×C61C2311yescharacteristic
33#117×C61—22no—
34#117×C61—22no—
35C2×C4⋊C4×C61Q812yescomplemented
36C2×C22⋊C4×C61—21no—
37C2×C22⋊C4×C61—21no—
38C2×C22⋊C4×C61D411yescharacteristic
39C2×C22⋊C4×C61D411yescharacteristic
40C2×C42×C61—21no—
41#95×C61—22no—
42#95×C61—22no—
43C24×C61C2×D411yescharacteristic
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61—22no—
47C22×C4×C61—41no—
48C22×C4×C61—41no—
49C22×C4×C61—41no—
50C22×C4×C61C2×D412yes—
51C22×C4×C61C2×Q811yescharacteristic
52C22×C4×C61#4112yes—
53C22×C4×C61#4111yescharacteristic
54C22×Dic61—22no—
55C22×Dic61—22no—
56C2×Dic122—44no—
57C2×Dic122—81no—
58C4⋊C4×C61—24no—
59C22⋊C4×C61—41no—
60C22⋊C4×C61—41no—
61C22⋊C4×C61—41no—
62C22⋊C4×C61—41no—
63C22⋊C4×C61—41no—
64C22⋊C4×C61—41no—
65C42×C61—42no—
66C22⋊Dic61—44no—
67C4⋊Dic61—81no—
68#11758196—44no—
69C4×Dic61—42no—
70C4×Dic61—42no—
71#11758194—42no—
72#11758194—42no—
73C2×#92418—42no—
74C23×C61—21no—
75C23×C61—21no—
76C23×C61—42no—
77C23×C61#12011yescharacteristic
78C23×C61C22⋊Q811yescharacteristic
79C23×C61#12411yescharacteristic
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—41no—
83C2×C4×C61—41no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61—82no—
91C2×C4×C61—81no—
92C2×C4×C61—82no—
93C2×C4×C61C22⋊Q812yes—
94C2×C4×C61#12412yes—
95C2×Dic61—24no—
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—42no—
101Dic122—84no—
102#92418—82no—
103#10387—611nomaximal, Sylow
104C22×C61—22no—
105C22×C61—21no—
106C22×C61—21no—
107C22×C61—22no—
108C22×C61—22no—
109C22×C61—81no—
110C22×C61—81no—
111C22×C61#39311yescharacteristic
112C22×C61#45811yescharacteristic
113C22×C61#45811yescharacteristic
114C4×C61—22no—
115C4×C61—22no—
116C4×C61—81no—
117C4×C61—81no—
118C4×C61—82no—
119C4×C61—161no—
120Dic61—44no—
121Dic61—82no—
122#2607—611no—
123#1826—612no—
124#1820D6111yescharacteristic
125#1306—611no—
126#1284—612no—
127C2×C61—21no—
128C2×C61—21no—
129C2×C61—41no—
130C2×C61—41no—
131C2×C61#203011yescharacteristic
132C2×C61#203112yes—
133C2×#117C2×D6111yescharacteristic
134#407—612no—
135#399—1221no—
136#393—1221no—
137#391—1222no—
138#387—1222no—
139#381C2×D6111yescharacteristic
140#381C2×D6111yescharacteristic
141C4×C22⋊C4—612no—
142#342—21no—
143#342—21no—
144#339—1222no—
145#339—1222no—
146C61#1038711yescharacteristic, complemented, Sylow
147C22×Q8—1221no—
148C23×C4C22×D6111yescharacteristic
149C2×M16—1222no—
150#117—22no—
151#117—22no—
152C2×C4⋊C4—1221no—
153C2×C4⋊C4—1222no—
154C2×C4⋊C4Dic12212yescomplemented
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—1222no—
158C2×C22⋊C4C22⋊D6111yescharacteristic
159C2×C22⋊C4C22⋊D6111yescharacteristic
160C2×C42—21no—
161C2×C42—612no—
162C2×C42—612no—
163C22⋊C8—1222no—
164C22⋊C8—1222no—
165#95—22no—
166#95—22no—
167#95—1221no—
168#95—1222no—
169#95—1221no—
170#95—1222no—
171C24C2×C22⋊D6111yescharacteristic
172C2×Q8—4881no—
173C2×Q8—2444no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—1222no—
181C22×C4—1222no—
182C22×C4C2×Dic12211yescharacteristic
183C22×C4#1175822211yescharacteristic
184C22×C4D4×D6112yes—
185C22×C4#1175822412yes—
186M16—2442no—
187M16—2442no—
188C2×C8—2442no—
189C4⋊C4—24no—
190C4⋊C4—4881no—
191C4⋊C4—2444no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—2444no—
199C42—42no—
200C42—2442no—
201C42—2442no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#1451974511yescharacteristic
206C23#1451975611yescharacteristic
207C23#1451976411yescharacteristic
208Q8—4884no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—1224no—
223C2×C4—2442no—
224C2×C4—2442no—
225C2×C4—2442no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4#1451975412yes—
229C2×C4#1451976312yes—
230C8—4882no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22#1683967411yescharacteristic
239C22#1683995111yescharacteristic
240C22#1683995111yescharacteristic
241C4—22no—
242C4—22no—
243C4—81no—
244C4—81no—
245C4—82no—
246C4—161no—
247C4—4882no—
248C4—2444no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#2144698611yescharacteristic
254C2#2144699312yes—
subgroupG/H
centerC22#16839674
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1820C2
socleC22×C61#393
prime radicalC24×C61C2×D4
hypercenter#1820D61
hypocenterC61#10387
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
35. C2×C4⋊C4×C61Q8semidirect
146. C61#10387semidirect
154. C2×C4⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214514842
#214459182
#214437272
#214436712
C61×#18202
#1038761