Finite Group Databasebeta

← #29577645

Subgroups — #29577645

Order 15616.

456 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 95 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#21456222C211yesmaximal, characteristic
2#21448925C211yesmaximal, characteristic, complemented
3#21444328C212yesmaximal
4#21443711C212yesmaximal, complemented
5C61×#1534C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7#16840132C412yescomplemented
8#16840128C412yescomplemented
9#16840119—22no—
10#16839761—22no—
11C4×#11758194C2211yescharacteristic
12#16839648—22no—
13#16839648—22no—
14C4⋊C4×Dic61—22no—
15#16839608C2212yes—
16#16839497—22no—
17#16839497—22no—
18C2×#125×C61C2211yescharacteristic
19C61×#432C2212yes—
20C61×#330—22no—
21C2×#11758222—21no—
22C2×#11758194—21no—
23#14519777C2×C411yescharacteristic
24#14519774C2×C411yescharacteristic
25#14519765—21no—
26#14519751—22no—
27C4×D244—22no—
28C2×C4⋊Dic61—22no—
29C2×C4×Dic61—22no—
30C2×C4×Dic61—22no—
31#14519700—24no—
32#14519697—24no—
33#14519688C2×C412yes—
34C4×#92418—21no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684D412yes—
39#14519684D412yes—
40#125×C61—22no—
41#125×C61C2×C412yes—
42#125×C61D412yes—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61D412yes—
45C2×C42×C61C2311yescharacteristic
46C4⋊C8×C61—22no—
47C22⋊C8×C61—22no—
48#97×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D412yes—
52C2×C22⋊D61—41no—
53C22×Dic61—22no—
54C22×Dic61—22no—
55#11758222—44no—
56C2×D244—21no—
57C2×D244—22no—
58C2×C4×D61—41no—
59C2×Dic122—21no—
60C2×C8×C61—42no—
61C4⋊C4×C61—24no—
62C4⋊C4×C61—24no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61C22⋊C414yes—
65C42×C61C22×C411yescharacteristic
66C42×C61C2×D411yescharacteristic
67C42×C61#4112yes—
68#11758198—42no—
69C4⋊Dic61—22no—
70C4⋊Dic61—44no—
71C4×Dic61—44no—
72C4×Dic61—44no—
73#11758194—42no—
74C2×#92418—22no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77C2×C4×C61—22no—
78C2×C4×C61—24no—
79C2×C4×C61—42no—
80C2×C4×C61—42no—
81C2×C4×C61—42no—
82C2×C4×C61C2×C22⋊C411yescharacteristic
83C2×C4×C61C4×D412yes—
84C2×C4×C61#12112yes—
85C2×C4×C61#12311yescharacteristic
86C22⋊D61—82no—
87C2×Dic61—24no—
88C2×Dic61—42no—
89C2×Dic61—42no—
90C2×Dic61—42no—
91C2×Dic61—42no—
92C2×Dic61—41no—
93C2×Dic61—42no—
94D244—42no—
95D244—82no—
96C4×D61—82no—
97Dic122—42no—
98C8×C61—82no—
99#92418—42no—
100#9442—611nomaximal, Sylow
101C22×C61—22no—
102C22×C61#38611yescharacteristic
103C22×C61#45412yes—
104C2×D61—81no—
105C2×D61—82no—
106C4×C61—21no—
107C4×C61—21no—
108C4×C61—22no—
109C4×C61—22no—
110C4×C61—44no—
111Dic61—44no—
112Dic61—82no—
113Dic61—81no—
114#3142—611no—
115#2371—611no—
116#1534D6111yescharacteristic, complemented
117#1505—612no—
118#1300—612no—
119C2×C61—22no—
120C2×C61#188811yescharacteristic
121C2×C61#190612yes—
122D61—161no—
123#532—611no—
124C2×#125C2×D6111yescharacteristic
125#491—612no—
126#490—612no—
127#488—1222no—
128#432C2×D6112yes—
129#417—1222no—
130C4×M16—611no—
131#387—1222no—
132#387—1222no—
133C4×C4⋊C4—1222no—
134#376—612no—
135#330—22no—
136#327—1222no—
137#327—1222no—
138C61#944211yescharacteristic, complemented, Sylow
139C2×#41—1221no—
140C2×M16—1221no—
141C4⋊Q8—611no—
142#127—611no—
143#125—22no—
144#125C4×D6112yes—
145#125C22⋊D6112yes—
146#124—1221no—
147#121—1222no—
148C4×D4—1222no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—1222no—
151C2×C4⋊C4D24412yes—
152C2×C42—1222no—
153C2×C42—1222no—
154C2×C42C22×D6111yescharacteristic
155C4⋊C8—22no—
156#103—1224no—
157#102—1224no—
158C22⋊C8—22no—
159#97—22no—
160#97—612no—
161C4×C8—1221no—
162#95—612no—
163#95—612no—
164#95—1222no—
165#95—1222no—
166#95—1222no—
167#41—2444no—
168C2×Q8—1221no—
169C2×D4—1221no—
170C2×D4—1222no—
171C2×D4—2441no—
172C22×C4—22no—
173C22×C4—1222no—
174C22×C4—1222no—
175C22×C4—2441no—
176C22×C4C2×D24411yescharacteristic
177C22×C4D4×D6112yes—
178M16—2442no—
179C2×C8—42no—
180C2×C8—1222no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—1222no—
185C4⋊C4—2444no—
186C4⋊C4#1175819814yes—
187C22⋊C4—2442no—
188C42—2444no—
189C42—2444no—
190C42C2×C4×D6111yescharacteristic
191C42#1175822412yes—
192C42C2×C22⋊D6111yescharacteristic
193C23—2441no—
194C23#1451974711yescharacteristic
195Q8—2442no—
196D4—4882no—
197D4—4882no—
198D4—2442no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—1224no—
205C2×C4—4882no—
206C2×C4—2442no—
207C2×C4—2442no—
208C2×C4—2442no—
209C2×C4—2441no—
210C2×C4—2442no—
211C2×C4—2442no—
212C2×C4C2×#1175819811yescharacteristic
213C2×C4#1451973712yes—
214C2×C4#1451974912yes—
215C2×C4#1451976111yescharacteristic
216C8—82no—
217C8—2442no—
218C22—22no—
219C22—4881no—
220C22—4882no—
221C22#1683963911yescharacteristic
222C22#1683993712yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—4882no—
229C4—4881no—
230C4—2444no—
231C2—22no—
232C2—9761no—
233C2#2144622011yescharacteristic
234C2#2144631812yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1534C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1534D61
hypocenterC61#9442
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
2. #21448925C2semidirect
4. #21443711C2semidirect
5. C61×#1534C2semidirect
7. #16840132C4semidirect
8. #16840128C4semidirect
116. #1534D61semidirect
138. C61#9442semidirect

Maximal subgroups

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

subgroupindex
#214562222
#214489252
#214443282
#214437112
C61×#15342
#944261