Finite Group Databasebeta

← #29592836

Subgroups — #29592836

Order 15616.

392 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 59 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#21453765C211yesmaximal, characteristic
2#21444861C212yesmaximal, complemented
3#21443845C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic
5C61×#2027C211yesmaximal, characteristic
6#16840095—22no—
7#16839701C2212yes—
8#16839674—22no—
9#16839668—21no—
10#16839642—21no—
11#16839637—22no—
12#16839499—22no—
13#16839488C2212yes—
14C61×#544C2211yescharacteristic
15C61×#418—21no—
16C61×#390C2211yescharacteristic
17C61×#343C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122—21no—
20C23×Dic61—21no—
21C2×C4⋊Dic61—21no—
22C2×C4⋊Dic61D412yes—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—21no—
25#14519710—42no—
26#14519701—42no—
27C22⋊#92418—22no—
28#14519684—22no—
29#14519684—22no—
30#14519684—22no—
31#14519684D412yes—
32C22×D4×C61D411yescharacteristic
33C2×M16×C61—21no—
34C4⋊Q8×C61—41no—
35#125×C61—41no—
36C22⋊Q8×C61—22no—
37#117×C61—21no—
38C2×C4⋊C4×C61C2311yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C42×C61D411yescharacteristic
41#102×C61—41no—
42#102×C61—41no—
43C24×C61—21no—
44C2×Q8×C61—22no—
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D4×C61Q1612yes—
49C22×C4×C61—21no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D412yes—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C22×Dic61—82no—
55C22×Dic61—81no—
56C2×Dic122—42no—
57C2×Dic122—81no—
58C2×Dic122—81no—
59M16×C61—41no—
60M16×C61—41no—
61C2×C8×C61—41no—
62C4⋊C4×C61—21no—
63C4⋊C4×C61—21no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—41no—
66C4⋊C4×C61—42no—
67C22⋊C4×C61—44no—
68C22⋊C4×C61—41no—
69C42×C61—41no—
70C42×C61—41no—
71C22⋊Dic61—82no—
72C22⋊Dic61—81no—
73C4⋊Dic61—42no—
74C4⋊Dic61—81no—
75C2×#92418—42no—
76C23×C61—22no—
77C23×C61—42no—
78C23×C61—42no—
79C23×C61—41no—
80C23×C61—41no—
81C23×C61#12011yescharacteristic
82Q8×C61—82no—
83D4×C61—42no—
84D4×C61—81no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61—42no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61C2×Q1611yescharacteristic
95C2×C4×C61#13711yescharacteristic
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—84no—
103Dic122—84no—
104C8×C61—81no—
105#92418—82no—
106#12221—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—21no—
109C22×C61—44no—
110C22×C61—42no—
111C22×C61—82no—
112C22×C61—81no—
113C22×C61—81no—
114C22×C61#45111yescharacteristic
115C22×C61#45511yescharacteristic
116C22×C61B(4,𝔽2)11yescharacteristic
117C4×C61—21no—
118C4×C61—21no—
119C4×C61—42no—
120C4×C61—42no—
121C4×C61—81no—
122C4×C61—81no—
123Dic61—82no—
124Dic61—82no—
125#2804—611no—
126#2027D6111yescharacteristic
127#1617—612no—
128#1353—612no—
129#1318—611no—
130C2×C61—21no—
131C2×C61—21no—
132C2×C61—82no—
133C2×C61#160211yescharacteristic
134C2×C61#219811yescharacteristic
135C2×C61#220211yescharacteristic
136#544C2×D6111yescharacteristic
137#484—1222no—
138#418—21no—
139#400—612no—
140#393—1222no—
141#391—1221no—
142#390C2×D6111yescharacteristic
143#386—1221no—
144#385—1222no—
145#343C2×D6111yescharacteristic
146#339—21no—
147#328—1222no—
148#324—612no—
149C61#1222111yescharacteristic, complemented, Sylow
150C22×Q8—1221no—
151C22×D4C22⋊D6111yescharacteristic
152C23×C4—1221no—
153C2×M16—21no—
154C4⋊Q8—41no—
155#125—41no—
156C22⋊Q8—22no—
157#117—21no—
158C2×C4⋊C4—612no—
159C2×C4⋊C4—1221no—
160C2×C4⋊C4C22×D6111yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1222no—
163C2×C22⋊C4—1221no—
164C2×C42C22⋊D6111yescharacteristic
165#107—2442no—
166#103—2442no—
167#102—41no—
168#102—41no—
169C22⋊C8—1222no—
170#95—612no—
171#95—1222no—
172#95—1222no—
173#95—1222no—
174C24—21no—
175C2×Q8—22no—
176C2×Q8—4881no—
177C2×Q8—4881no—
178C2×Q8—2442no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1175820212yes—
183C22×C4—21no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4—4882no—
187C22×C4—4881no—
188C22×C4D4×D6112yes—
189C22×C4C2×C22⋊D6111yescharacteristic
190M16—41no—
191M16—41no—
192C2×C8—41no—
193C2×C8—2442no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—4881no—
200C4⋊C4—2442no—
201C22⋊C4—44no—
202C22⋊C4—41no—
203C22⋊C4—4882no—
204C22⋊C4—4881no—
205C42—41no—
206C42—41no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1451974511yescharacteristic
213Q8—82no—
214Q8—4884no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—4884no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4C2×#1175820211yescharacteristic
234C2×C4#1451980711yescharacteristic
235C8—81no—
236C8—4882no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1683991711yescharacteristic
245C22#1683993911yescharacteristic
246C22#1683994811yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—81no—
253C4—4882no—
254C4—4882no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#2144477511yescharacteristic
259C2#2144781411yescharacteristic
260C2#2144783511yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2027C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
hypercenter#2027D61
hypocenterC61#12221
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2027

Complements

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

NH ≅ G/Ntype
2. #21444861C2semidirect
3. #21443845C2semidirect
149. C61#12221semidirect

Maximal subgroups

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

subgroupindex
#214537652
#214448612
#214438452
#214437512
C61×#20272
#1222161