Finite Group Databasebeta

← #29567504

Subgroups — #29567504

Order 15616.

268 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 37 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#21446886C211yesmaximal, characteristic
2#21443755C211yesmaximal, characteristic, complemented
3C61×#1298C211yesmaximal, characteristic, complemented
4#16840562C411yescharacteristic
5#16840273—21no—
6#16839778—21no—
7#16839544—21no—
8#16839519C411yescharacteristic
9C61×#407C2211yescharacteristic
10C2×C22⋊C8×C61—21no—
11C2×#11758222—21no—
12#14519810—41no—
13#14519805—41no—
14#14519799—41no—
15#14519760—41no—
16#14519756—41no—
17#14519751—41no—
18#14519746—41no—
19C4×C22⋊D61—42no—
20#14519736—21no—
21#14519717—41no—
22#14519703—42no—
23C23×C4×C61D411yescharacteristic
24C2×M16×C61C2×C411yescharacteristic
25C2×M16×C61D411yescharacteristic
26C22×C8×C61—41no—
27C22⋊C8×C61—22no—
28C22⋊C8×C61—41no—
29C22⋊C8×C61—41no—
30C24×C61SD1611yescharacteristic
31C22×C4×C61—21no—
32C22×C4×C61—21no—
33C22×C4×C61—41no—
34C22×C4×C61—41no—
35C22×C4×C61C22⋊C411yescharacteristic
36C22×C4×C61D811yescharacteristic
37C2×C22⋊D61—42no—
38C2×C22⋊D61—41no—
39#11758222—41no—
40#11758222—41no—
41#11758222—41no—
42#11758222—41no—
43C2×D244—21no—
44C2×C4×D61—41no—
45C2×Dic122—21no—
46M16×C61—41no—
47M16×C61—41no—
48C2×C8×C61—21no—
49C2×C8×C61—21no—
50C2×C8×C61—42no—
51C2×C8×C61—81no—
52C2×C8×C61—81no—
53C22⋊Dic61—42no—
54C22⋊Dic61—41no—
55#11758198—42no—
56C4⋊Dic61—41no—
57#11758196—42no—
58C4×Dic61—22no—
59Dic244—41no—
60D488—41no—
61#11758190—42no—
62#11758185—81no—
63C23×C61—21no—
64C23×C61—41no—
65C23×C61—42no—
66C23×C61#10211yescharacteristic
67C22×D61—41no—
68C2×C4×C61—21no—
69C2×C4×C61—21no—
70C2×C4×C61—21no—
71C2×C4×C61—41no—
72C2×C4×C61—41no—
73C2×C4×C61—41no—
74C2×C4×C61—81no—
75C2×C4×C61#9911yescharacteristic
76C2×C4×C61#10411yescharacteristic
77C22⋊D61—81no—
78C22⋊D61—84no—
79C22⋊D61—81no—
80C2×Dic61—42no—
81C2×Dic61—41no—
82D244—41no—
83D244—41no—
84C4×D61—81no—
85C4×D61—81no—
86Dic122—41no—
87Dic122—41no—
88C8×C61—41no—
89C8×C61—41no—
90C8×C61—82no—
91#7397—611nomaximal, Sylow
92C22×C61—22no—
93C22×C61—21no—
94C22×C61—41no—
95C22×C61—41no—
96C22×C61—42no—
97C22×C61—41no—
98C22×C61—81no—
99C22×C61#32711yescharacteristic
100C2×D61—82no—
101C2×D61—81no—
102C4×C61—21no—
103C4×C61—41no—
104C4×C61AΣL(1,𝔽16)11yescharacteristic
105C4×C61#35311yescharacteristic
106Dic61—82no—
107Dic61—81no—
108#2010—611no—
109#1320—611no—
110#1298D6111yescharacteristic, complemented
111C2×C61—21no—
112C2×C61—41no—
113C2×C61—42no—
114C2×C61#134311yescharacteristic
115D61—161no—
116#575—611no—
117#525—1221no—
118#421—1221no—
119#407C2×D6111yescharacteristic
120C2×C22⋊C8—21no—
121#349—1221no—
122H(3,ℤ/4)—611no—
123C61#739711yescharacteristic, complemented, Sylow
124C2×#41—1221no—
125C23×C4C22⋊D6111yescharacteristic
126#137—2441no—
127AGL(1,ℤ/8)—2441no—
128#135—2441no—
129C2×M16C4×D6111yescharacteristic
130C2×M16D24411yescharacteristic
131C22×C8—41no—
132#123—2441no—
133C22⋊Q8—2441no—
134#121—2441no—
135#120—2441no—
136C4×D4—2442no—
137#117—1221no—
138M32—2441no—
139#104—2442no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C24#1175820111yescharacteristic
144#41—2441no—
145#41—2441no—
146#41—2441no—
147#41—2441no—
148C2×Q8—1221no—
149C2×D4—1221no—
150C2×D4—2442no—
151C2×D4—2441no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—2441no—
157C22×C4#1175819811yescharacteristic
158C22×C4#1175819911yescharacteristic
159Q16—2441no—
160SD16—2442no—
161D8—2441no—
162M16—41no—
163M16—41no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C4⋊C4—2441no—
170C4⋊C4—2442no—
171C22⋊C4—2442no—
172C22⋊C4—2442no—
173C22⋊C4—2441no—
174C42—1222no—
175C16—4881no—
176C23—21no—
177C23—41no—
178C23—42no—
179C23—2441no—
180C23#1451969711yescharacteristic
181Q8—2441no—
182Q8—2441no—
183D4—4881no—
184D4—4884no—
185D4—4881no—
186D4—2441no—
187D4—2441no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—4881no—
196C2×C4—4881no—
197C2×C4—2442no—
198C2×C4—2441no—
199C2×C4#1451969111yescharacteristic
200C2×C4#1451970511yescharacteristic
201C8—41no—
202C8—41no—
203C8—82no—
204C22—22no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—41no—
210C22—81no—
211C22—4882no—
212C22—4881no—
213C22#1683949511yescharacteristic
214C4—21no—
215C4—41no—
216C4—4882no—
217C4—4881no—
218C4#1683954811yescharacteristic
219C4#1683955411yescharacteristic
220C2—21no—
221C2—41no—
222C2—42no—
223C2—9761no—
224C2#2144381511yescharacteristic
subgroupG/H
centerC4#16839554
commutatorC2×M16×C61C2×C4
Frattini#407C2×D61
FittingC61×#1298C2
socleC2×C61#1343
prime radical#21446886C2
cosocleC61×#407C22
hypercenter#1298D61
hypocenterC61#7397
LCSG ▷ C2×M16×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #21443755C2semidirect
3. C61×#1298C2semidirect
110. #1298D61semidirect
123. C61#7397semidirect

Maximal subgroups

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

subgroupindex
#214468862
#214437552
C61×#12982
#739761