Finite Group Databasebeta

← #29590862

Subgroups — #29590862

Order 15616.

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

normal subgroups only

This group has 324 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#21456557C211yesmaximal, characteristic
2#21445968C211yesmaximal, characteristic, complemented
3#21443811C212yesmaximal, complemented
4#21443805C212yesmaximal, complemented
5C61×#2165C211yesmaximal, characteristic, complemented
6#16840312—21no—
7#16840160C411yescharacteristic, complemented
8#16840139C411yescharacteristic, complemented
9#16840109—21no—
10#16840015—21no—
11#16840015C2212yes—
12#16839875—21no—
13#16839875C412yescomplemented
14#16839533—21no—
15#16839522—21no—
16#16839520C2211yescharacteristic
17#16839519C2211yescharacteristic
18C4×M16×C61C2211yescharacteristic
19C61×#364—21no—
20C61×#349C2212yes—
21C61×#347—21no—
22C2×#11758222—41no—
23C2×D488—21no—
24C2×D488C2×C412yes—
25C2×#11758194—21no—
26#14519777—21no—
27#14519774—21no—
28#14519765—41no—
29#14519751—22no—
30#14519751—41no—
31C4×D244—22no—
32C4×D244—41no—
33#14519736—21no—
34#14519709—21no—
35#14519709C2×C412yes—
36#14519699—22no—
37#14519699—22no—
38C2×M16×C61D411yescharacteristic
39C2×M16×C61C2311yescharacteristic
40C2×C42×C61D411yescharacteristic
41M32×C61—22no—
42#97×C61C2×C411yescharacteristic
43#97×C61D411yescharacteristic
44C4×C8×C61C2×C411yescharacteristic
45C4×C8×C61D411yescharacteristic
46C22×C4×C61—21no—
47C22×C4×C61C2×D411yescharacteristic
48C2×C22⋊D61—42no—
49#11758222—84no—
50C2×D244—22no—
51C2×D244—41no—
52C2×D244—41no—
53C2×C4×D61—42no—
54C2×Dic122—41no—
55M16×C61—41no—
56M16×C61—41no—
57C2×C8×C61C22⋊C411yescharacteristic
58C2×C8×C61C22⋊C411yescharacteristic
59C2×C8×C61C22×C411yescharacteristic
60C2×C8×C61C2×D411yescharacteristic
61C42×C61—21no—
62C42×C61C22⋊C411yescharacteristic
63C42×C61C22⋊C411yescharacteristic
64C22⋊Dic61—41no—
65#11758198—42no—
66#11758198—42no—
67C4⋊Dic61—22no—
68C4⋊Dic61—41no—
69C4⋊Dic61—41no—
70C4×Dic61—41no—
71#11758194—41no—
72#11758194—41no—
73C2×#92418—41no—
74D488—42no—
75D488—42no—
76C16×C61—42no—
77C23×C61AGL(1,ℤ/8)11yescharacteristic
78C22×D61—42no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—42no—
84C2×C4×C61C2×C22⋊C411yescharacteristic
85C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
86C22⋊D61—84no—
87C2×Dic61—42no—
88C2×Dic61—81no—
89D244—42no—
90D244—42no—
91D244—82no—
92D244—82no—
93C4×D61—84no—
94Dic122—82no—
95C8×C61—21no—
96C8×C61—21no—
97C8×C61—21no—
98C8×C61—21no—
99#92418—81no—
100#11837—611nomaximal, Sylow
101C22×C61—21no—
102C22×C61—41no—
103C22×C61#41811yescharacteristic
104C2×D61—82no—
105C2×D61—82no—
106C2×D61—82no—
107C4×C61—22no—
108C4×C61—21no—
109C4×C61—41no—
110C4×C61#41011yescharacteristic
111C4×C61#42111yescharacteristic
112Dic61—82no—
113Dic61—161no—
114#3184—611no—
115#2165D6111yescharacteristic, complemented
116#1841—611no—
117#1341—612no—
118#1339—612no—
119C2×C61—21no—
120C2×C61—41no—
121C2×C61#151411yescharacteristic
122D61—162no—
123#532—1221no—
124#498—611no—
125#493—611no—
126#486—1221no—
127#469—612no—
128#469—1221no—
129#442—612no—
130#442—1221no—
131C4×M16C2×D6111yescharacteristic
132#364—21no—
133#349C2×D6112yes—
134#347—21no—
135#344—1221no—
136#339—1221no—
137#338—611no—
138H(3,ℤ/4)—611no—
139C61#1183711yescharacteristic, complemented, Sylow
140C2×#41—2441no—
141C2×D8—612no—
142C2×D8—1221no—
143C2×M16—1221no—
144C2×M16D24411yescharacteristic
145C2×M16C22×D6111yescharacteristic
146C4⋊Q8—1221no—
147#127—1221no—
148#124—2441no—
149#121—1222no—
150#121—2441no—
151C4×D4—1222no—
152C4×D4—2441no—
153#117—1221no—
154C2×C42D24411yescharacteristic
155M32—22no—
156#106—612no—
157#106—1221no—
158#102—1222no—
159#102—1222no—
160#97C4×D6111yescharacteristic
161#97C22⋊D6111yescharacteristic
162C4×C8C4×D6111yescharacteristic
163C4×C8C22⋊D6111yescharacteristic
164#41—4884no—
165C2×Q8—2441no—
166C2×D4—1222no—
167C2×D4—2441no—
168C2×D4—2441no—
169C2×D4—2442no—
170C22×C4—21no—
171C22×C4—2442no—
172C22×C4C2×D24411yescharacteristic
173D8—2442no—
174D8—2442no—
175M16—41no—
176M16—41no—
177M16—2441no—
178M16—2441no—
179C2×C8—2441no—
180C2×C8#1175819811yescharacteristic
181C2×C8#1175819811yescharacteristic
182C2×C8C2×C4×D6111yescharacteristic
183C2×C8C2×C22⋊D6111yescharacteristic
184C4⋊C4—1222no—
185C4⋊C4—2441no—
186C4⋊C4—2441no—
187C22⋊C4—2442no—
188C22⋊C4—2441no—
189C22⋊C4—2442no—
190C42—21no—
191C42—2441no—
192C42#1175819811yescharacteristic
193C42#1175819811yescharacteristic
194C16—42no—
195C23—2442no—
196C23#1451980511yescharacteristic
197Q8—4882no—
198D4—4882no—
199D4—4882no—
200D4—4884no—
201D4—2442no—
202D4—2442no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—4884no—
209C2×C4—4881no—
210C2×C4—2442no—
211C2×C4C2×#1175819811yescharacteristic
212C2×C4#1451980511yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—4881no—
218C22—21no—
219C22—41no—
220C22—4882no—
221C22—4882no—
222C22—4882no—
223C22#1683976811yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—9761no—
228C4—4882no—
229C4#1683973511yescharacteristic
230C4#1683977811yescharacteristic
231C2—21no—
232C2—41no—
233C2—9762no—
234C2#2144436711yescharacteristic
subgroupG/H
centerC2#21444367
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#2165C2
socleC2×C61#1514
prime radical#21456557C2
cosocleC2×M16×C61C23
hypercenter#2165D61
hypocenterC61#11837
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21445968C2semidirect
3. #21443811C2semidirect
4. #21443805C2semidirect
5. C61×#2165C2semidirect
7. #16840160C4semidirect
8. #16840139C4semidirect
13. #16839875C4semidirect
115. #2165D61semidirect
139. C61#11837semidirect

Maximal subgroups

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

subgroupindex
#214565572
#214459682
#214438112
#214438052
C61×#21652
#1183761