Finite Group Databasebeta

← #29589510

Subgroups — #29589510

Order 15616.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#21454509C211yesmaximal, characteristic
2#21447632C212yesmaximal, complemented
3#21443756C212yesmaximal, complemented
4#21443755C211yesmaximal, characteristic, complemented
5C61×#1320C211yesmaximal, characteristic, complemented
6C2×#14519785C411yescharacteristic
7C23×#92418C411yescharacteristic
8#16840273C411yescharacteristic
9#16839830—22no—
10#16839825—22no—
11#16839790C411yescharacteristic
12#16839730C2211yescharacteristic
13#16839720C2211yescharacteristic
14#16839582—22no—
15#16839544C2212yes—
16#16839539—22no—
17C61×#407C2211yescharacteristic
18C61×#349C2212yes—
19C2×#11758222C2×C411yescharacteristic
20#14519785—42no—
21C2×#11758189—21no—
22C22×#92418—21no—
23C22×#92418—21no—
24C22×#92418—42no—
25C22×#92418C2×C411yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C2×C8×D61—21no—
28#14519760—21no—
29#14519756—21no—
30#14519751—21no—
31#14519746—21no—
32C4×C22⋊D61—22no—
33#14519736C2×C411yescharacteristic
34#14519717—22no—
35#14519706—22no—
36#14519689—22no—
37#14519687D412yes—
38C8×Dic61D412yes—
39C23×C4×C61C2×C411yescharacteristic
40C2×M16×C61C2×C411yescharacteristic
41C2×M16×C61C2311yescharacteristic
42M32×C61—22no—
43C22⋊C8×C61—22no—
44C24×C61M1611yescharacteristic
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61—41no—
48C22×C4×C61—41no—
49C22×C4×C61M1611yescharacteristic
50C22×C4×C61C22×C411yescharacteristic
51C2×C22⋊D61—22no—
52C2×C22⋊D61—21no—
53#11758222—41no—
54#11758222—41no—
55C2×D244M1611yescharacteristic
56C2×C4×D61—21no—
57C2×Dic122M1611yescharacteristic
58M16×C61—41no—
59M16×C61—41no—
60C2×C8×C61—21no—
61C2×C8×C61C2×D412yes—
62C22⋊Dic61—22no—
63C22⋊Dic61—21no—
64#11758198—22no—
65C4⋊Dic61—21no—
66#11758196—22no—
67C4×Dic61C22⋊C412yes—
68C2×#92418—22no—
69C2×#92418—21no—
70C2×#92418—42no—
71C2×#92418—42no—
72C2×#92418—41no—
73C2×#92418C22⋊C411yescharacteristic
74C2×#92418C22⋊C411yescharacteristic
75#11758189—42no—
76C8×D61—42no—
77C16×C61—42no—
78#11758185—42no—
79C23×C61—21no—
80C23×C61—41no—
81C23×C61—42no—
82C23×C61C2×M1611yescharacteristic
83C22×D61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—42no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61C2×C22⋊C411yescharacteristic
92C2×C4×C61C2×M1611yescharacteristic
93C22⋊D61—42no—
94C22⋊D61—44no—
95C2×Dic61—22no—
96C2×Dic61—21no—
97D244—41no—
98C4×D61—41no—
99C4×D61—41no—
100Dic122—41no—
101C8×C61—22no—
102C8×C61—41no—
103#92418—21no—
104#92418—41no—
105#92418#10012yes—
106#11537—611nomaximal, Sylow
107C22×C61—22no—
108C22×C61—21no—
109C22×C61—41no—
110C22×C61—41no—
111C22×C61—42no—
112C22×C61—42no—
113C22×C61—41no—
114C22×C61#40711yescharacteristic
115C2×D61—42no—
116C2×D61—41no—
117C4×C61—21no—
118C4×C61—41no—
119C4×C61#41011yescharacteristic
120C4×C61C2×#10011yescharacteristic
121Dic61—42no—
122Dic61—41no—
123#2929—611no—
124#2166—612no—
125#1321—612no—
126#1320—611no—
127#1320D6111yescharacteristic, complemented
128C2×C61—21no—
129C2×C61—42no—
130C2×C61—41no—
131C2×C61#146611yescharacteristic
132D61—81no—
133C2×#131—611no—
134C23×C8—611no—
135#525—611no—
136#435—1222no—
137C8×D4—1222no—
138#424—611no—
139#408—611no—
140#407C2×D6111yescharacteristic
141#405—611no—
142#364—1222no—
143#349—612no—
144#349C2×D6112yes—
145#347—1222no—
146C61#1153711yescharacteristic, complemented, Sylow
147C2×#41—611no—
148C23×C4C4×D6111yescharacteristic
149#131—2442no—
150C2×M16—1221no—
151C2×M16C4×D6111yescharacteristic
152C2×M16C22×D6111yescharacteristic
153C22×C8—611no—
154C22×C8—611no—
155C22×C8—1221no—
156C22×C8—1221no—
157C22×C8—1221no—
158C22×C8—2442no—
159#123—1221no—
160C22⋊Q8—1221no—
161#121—1221no—
162#120—1221no—
163C4×D4—1222no—
164#117—611no—
165M32—22no—
166M32—1222no—
167C4⋊C8—1222no—
168C22⋊C8—22no—
169C22⋊C8—1222no—
170#97—612no—
171C4×C8—612no—
172C24#1175818911yescharacteristic
173#41—2441no—
174#41—2441no—
175C2×Q8—611no—
176C2×D4—611no—
177C2×D4—1222no—
178C2×D4—1221no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—1221no—
184C22×C4#1175818911yescharacteristic
185C22×C4C2×C4×D6111yescharacteristic
186M16—41no—
187M16—41no—
188M16—2442no—
189C2×C8—21no—
190C2×C8—611no—
191C2×C8—611no—
192C2×C8—1222no—
193C2×C8—1221no—
194C2×C8—2442no—
195C2×C8—2442no—
196C2×C8—2442no—
197C2×C8—2441no—
198C2×C8D4×D6112yes—
199C4⋊C4—1222no—
200C4⋊C4—1221no—
201C22⋊C4—1222no—
202C22⋊C4—1222no—
203C22⋊C4—1221no—
204C42—612no—
205C16—42no—
206C16—2442no—
207C23—21no—
208C23—41no—
209C23—42no—
210C23—1221no—
211C23C2×#1175818911yescharacteristic
212Q8—2441no—
213D4—2441no—
214D4—2444no—
215D4—2442no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—1222no—
224C2×C4—1221no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4C22⋊C4×D6111yescharacteristic
228C2×C4M16×D6111yescharacteristic
229C8—22no—
230C8—41no—
231C8—612no—
232C8—1221no—
233C8—2441no—
234C22—22no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—2442no—
242C22—2441no—
243C22#1683972411yescharacteristic
244C4—21no—
245C4—41no—
246C4—2442no—
247C4—2441no—
248C4#1683973411yescharacteristic
249C4#100×D6111yescharacteristic
250C2—21no—
251C2—42no—
252C2—41no—
253C2—4881no—
254C2#2144415811yescharacteristic
subgroupG/H
centerC4#16839734
commutatorC22×C4×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1320C2
socleC2×C61#1466
prime radical#16840273C4
cosocleC2×M16×C61C23
hypercenter#1320D61
hypocenterC61#11537
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #21447632C2semidirect
3. #21443756C2semidirect
4. #21443755C2semidirect
5. C61×#1320C2semidirect
127. #1320D61semidirect
146. C61#11537semidirect

Maximal subgroups

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

subgroupindex
#214545092
#214476322
#214437562
#214437552
C61×#13202
#1153761