Finite Group Databasebeta

← #29590553

Subgroups — #29590553

Order 15616.

316 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 65 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#21456464C211yesmaximal, characteristic
2#21447632C211yesmaximal, characteristic, complemented
3#21445573C211yesmaximal, characteristic, complemented
4#21443811C211yesmaximal, characteristic, complemented
5#21443810C211yesmaximal, characteristic, complemented
6#21443803C211yesmaximal, characteristic, complemented
7C61×#1339C211yesmaximal, characteristic, complemented
8#16840365—21no—
9#16840162C411yescharacteristic, complemented
10#16840137C411yescharacteristic, complemented
11#16840125—21no—
12#16840015C2211yescharacteristic
13#16839996—21no—
14#16839874C411yescharacteristic, complemented
15#16839873C411yescharacteristic, complemented
16#16839842—21no—
17#16839720C2211yescharacteristic
18#16839718—21no—
19#16839582—21no—
20#16839544C2211yescharacteristic
21#16839539—21no—
22#16839520C2211yescharacteristic
23#16839518—21no—
24C61×#469C2211yescharacteristic
25C61×#349C2211yescharacteristic
26C61×H(3,ℤ/4)C2211yescharacteristic
27C2×D488C2×C411yescharacteristic
28C2×#11758199—21no—
29C2×#11758194—21no—
30C22×#92418D411yescharacteristic
31#14519773—21no—
32#14519768—21no—
33#14519761—41no—
34#14519759—41no—
35#14519751—21no—
36#14519749—41no—
37#14519748—41no—
38D4×Dic61—21no—
39#14519739—21no—
40#14519736D411yescharacteristic
41#14519717—21no—
42#14519710—21no—
43#14519709C2×C411yescharacteristic
44#14519699—21no—
45#14519698—21no—
46#14519697—21no—
47#14519688—22no—
48#14519687C2×C411yescharacteristic
49#14519687D411yescharacteristic
50C4×#92418—22no—
51C8×Dic61C2×C411yescharacteristic
52C8×Dic61D411yescharacteristic
53C2×D8×C61C2×C411yescharacteristic
54C2×M16×C61C2311yescharacteristic
55#121×C61—21no—
56#117×C61—21no—
57M32×C61—21no—
58#106×C61C2×C411yescharacteristic
59#102×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—41no—
62C22×C4×C61C2×D411yescharacteristic
63C2×C22⋊D61—41no—
64C2×C22⋊D61—41no—
65C2×C22⋊D61—41no—
66C22×Dic61—41no—
67C2×D244—21no—
68C2×C4×D61—41no—
69D8×C61—41no—
70M16×C61—41no—
71C2×C8×C61C22×C411yescharacteristic
72C2×C8×C61C2×D411yescharacteristic
73C4⋊C4×C61—21no—
74C4⋊C4×C61—41no—
75C22⋊C4×C61—41no—
76C22⋊C4×C61—41no—
77C42×C61—22no—
78C22⋊Dic61—21no—
79C22⋊Dic61—41no—
80#11758199—41no—
81#11758199—41no—
82#11758198—41no—
83#11758198—41no—
84C4⋊Dic61—21no—
85C4⋊Dic61—21no—
86C4⋊Dic61—41no—
87#11758196—41no—
88C4×Dic61C22⋊C411yescharacteristic
89C4×Dic61C22⋊C411yescharacteristic
90#11758194—41no—
91#11758194—41no—
92C2×#92418—21no—
93C2×#92418—21no—
94C2×#92418—22no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97D488—41no—
98C16×C61—41no—
99#11758185—41no—
100C23×C61—41no—
101C23×C61AGL(1,ℤ/8)11yescharacteristic
102C22×D61—41no—
103D4×C61—41no—
104D4×C61—41no—
105D4×C61—161no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
112C22⋊D61—161no—
113C22⋊D61—161no—
114C22⋊D61—161no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—41no—
118C2×Dic61—161no—
119D244—41no—
120D244—41no—
121C4×D61—161no—
122C8×C61—21no—
123C8×C61—21no—
124#92418—21no—
125#92418—42no—
126#92418#10411yescharacteristic
127#92418#10411yescharacteristic
128#11770—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—41no—
131C22×C61—81no—
132C22×C61—81no—
133C22×C61—81no—
134C22×C61#41811yescharacteristic
135C2×D61—81no—
136C2×D61—81no—
137C2×D61—81no—
138C4×C61—21no—
139C4×C61—81no—
140C4×C61—82no—
141C4×C61#41011yescharacteristic
142C4×C61C2×#10411yescharacteristic
143Dic61—41no—
144Dic61—41no—
145Dic61—81no—
146#3176—611no—
147#2166—611no—
148#1747—611no—
149#1341—611no—
150#1341—611no—
151#1339—611no—
152#1339D6111yescharacteristic, complemented
153C2×C61—21no—
154C2×C61—41no—
155C2×C61—161no—
156C2×C61#150511yescharacteristic
157D61—161no—
158#540—1221no—
159#499—611no—
160#493—611no—
161#490—1221no—
162#469—611no—
163#469C2×D6111yescharacteristic
164#466—1221no—
165#442—611no—
166#442—611no—
167C4×D8—1221no—
168#405—611no—
169#405—1221no—
170#364—1221no—
171#349—611no—
172#349C2×D6111yescharacteristic
173#347—1221no—
174#338—611no—
175H(3,ℤ/4)—1221no—
176H(3,ℤ/4)C2×D6111yescharacteristic
177C61#1177011yescharacteristic, complemented, Sylow
178C2×D8—611no—
179C2×D8—1221no—
180C2×D8C4×D6111yescharacteristic
181C2×M16—1221no—
182C2×M16C22×D6111yescharacteristic
183C22×C8—611no—
184#127—1221no—
185#125—1221no—
186#123—2441no—
187#123—2441no—
188#121—21no—
189#121—1221no—
190#121—2441no—
191#121—2441no—
192C4×D4—1221no—
193C4×D4—1221no—
194#117—21no—
195#117—611no—
196M32—21no—
197M32—1221no—
198#107—1221no—
199#106—611no—
200#106C4×D6111yescharacteristic
201#102—21no—
202#102—1221no—
203#102—1221no—
204#102—1221no—
205#97—611no—
206#97—611no—
207#97—1222no—
208C4×C8—611no—
209C4×C8—611no—
210C4×C8—1222no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—1221no—
214C2×D4—2441no—
215C2×D4—2441no—
216C2×D4—2441no—
217C22×C4—2441no—
218C22×C4—2441no—
219C22×C4D4×D6111yescharacteristic
220D8—41no—
221D8—2441no—
222D8—2441no—
223D8—2441no—
224M16—41no—
225M16—2441no—
226M16—2441no—
227C2×C8—611no—
228C2×C8—611no—
229C2×C8—1221no—
230C2×C8—1221no—
231C2×C8—1222no—
232C2×C8C2×C4×D6111yescharacteristic
233C2×C8D4×D6111yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—1221no—
237C4⋊C4—1221no—
238C4⋊C4—2441no—
239C4⋊C4—2441no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—1221no—
243C22⋊C4—2441no—
244C22⋊C4—2441no—
245C22⋊C4—2441no—
246C42—22no—
247C42—611no—
248C42—611no—
249C16—41no—
250C16—2441no—
251C23—41no—
252C23—2441no—
253C23#1451980111yescharacteristic
254D4—41no—
255D4—41no—
256D4—161no—
257D4—9761no—
258D4—9761no—
259D4—9761no—
260D4—2441no—
261D4—2441no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—9761no—
269C2×C4—9761no—
270C2×C4—2441no—
271C2×C4C22⋊C4×D6111yescharacteristic
272C2×C4#1451980411yescharacteristic
273C8—21no—
274C8—21no—
275C8—611no—
276C8—611no—
277C8—1221no—
278C8—2442no—
279C22—21no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—4881no—
285C22—4881no—
286C22—4881no—
287C22#1683976511yescharacteristic
288C4—21no—
289C4—81no—
290C4—82no—
291C4—4881no—
292C4—2441no—
293C4—2441no—
294C4#1683973411yescharacteristic
295C4#104×D6111yescharacteristic
296C2—21no—
297C2—41no—
298C2—161no—
299C2—9761no—
300C2#2144432511yescharacteristic
subgroupG/H
centerC2#21444325
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1339C2
socleC2×C61#1505
prime radical#21456464C2
cosocleC2×M16×C61C23
hypercenter#1339D61
hypocenterC61#11770
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1339

Complements

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

NH ≅ G/Ntype
2. #21447632C2semidirect
3. #21445573C2semidirect
4. #21443811C2semidirect
5. #21443810C2semidirect
6. #21443803C2semidirect
7. C61×#1339C2semidirect
9. #16840162C4semidirect
10. #16840137C4semidirect
14. #16839874C4semidirect
15. #16839873C4semidirect
152. #1339D61semidirect
177. C61#11770semidirect

Maximal subgroups

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

subgroupindex
#214564642
#214476322
#214455732
#214438112
#214438102
#214438032
C61×#13392
#1177061