Finite Group Databasebeta

← #29591542

Subgroups — #29591542

Order 15616.

412 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 69 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#21456270C211yesmaximal, characteristic
2#21447630C212yesmaximal, complemented
3#21443859C212yesmaximal, complemented
4#21443856C211yesmaximal, characteristic, complemented
5C61×#1357C211yesmaximal, characteristic, complemented
6C2×#14519801—22no—
7#16840310C411yescharacteristic
8#127×D61C411yescharacteristic
9#16840150—22no—
10#16840138C411yescharacteristic
11#16840135C2211yescharacteristic
12#16840134C411yescharacteristic
13C4×#11758189C2211yescharacteristic
14#16839583C2212yescomplemented
15#16839543—22no—
16#16839539—22no—
17C61×#493C2211yescharacteristic
18C61×#364C2212yescomplemented
19C2×#11758224—21no—
20C2×D4×D61—21no—
21C2×D4×D61—41no—
22#14519801—44no—
23#14519801—44no—
24C2×#11758190—22no—
25C2×#11758200—22no—
26C2×#11758199—22no—
27C2×#11758189D412yes—
28#14519777C2×C411yescharacteristic
29#14519774C2×C411yescharacteristic
30#14519773—21no—
31#14519773—41no—
32#14519764—21no—
33#14519748—41no—
34D4×Dic61—41no—
35C42×D61C2×C411yescharacteristic
36#14519717—22no—
37#14519687D412yes—
38C4×#92418C2×C411yescharacteristic
39C2×D8×C61—22no—
40#127×C61C2×C411yescharacteristic
41#127×C61C2×C411yescharacteristic
42M32×C61—22no—
43C4×C8×C61C2311yescharacteristic
44C23×D61—41no—
45C2×D4×C61—21no—
46C2×D4×C61—21no—
47C2×D4×C61—41no—
48C2×D4×C61—41no—
49C2×C22⋊D61—42no—
50C2×C22⋊D61—41no—
51C2×C22⋊D61—41no—
52C22×Dic61—41no—
53#11758224—44no—
54D4×D61—44no—
55D4×D61—84no—
56C2×D244—21no—
57C2×D244—41no—
58C2×C4×D61—21no—
59C2×C4×D61C22⋊C411yescharacteristic
60C2×Dic122—21no—
61D8×C61—42no—
62D8×C61—42no—
63C2×C8×C61C2×D412yes—
64C42×C61C22×C411yescharacteristic
65C22⋊Dic61—42no—
66#11758200—44no—
67#11758199—44no—
68C4⋊Dic61—41no—
69C4×Dic61—21no—
70C4×Dic61C22⋊C411yescharacteristic
71C2×#92418C22⋊C412yes—
72#11758190—42no—
73#11758190—42no—
74#11758189—22no—
75#11758189—22no—
76C16×C61—42no—
77#11758185—42no—
78C23×C61—41no—
79C23×C61—41no—
80C22×D61—41no—
81C22×D61—82no—
82C22×D61—84no—
83C22×D61#10011yescharacteristic
84D4×C61—42no—
85D4×C61—42no—
86D4×C61—82no—
87D4×C61—82no—
88C2×C4×C61—21no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C22⋊D61—82no—
91C22⋊D61—84no—
92C22⋊D61—82no—
93C2×Dic61—21no—
94C2×Dic61—41no—
95C2×Dic61—82no—
96C2×Dic61#10011yescharacteristic
97D244—42no—
98D244—82no—
99C4×D61—22no—
100C4×D61—42no—
101Dic122—42no—
102C8×C61AGL(1,ℤ/8)12yes—
103C8×C61AGL(1,ℤ/8)12yes—
104#92418—22no—
105#11992—611nomaximal, Sylow
106C22×C61—82no—
107C22×C61—82no—
108C22×C61—81no—
109C22×C61—81no—
110C22×C61C2×#10011yescharacteristic
111C2×D61—21no—
112C2×D61—41no—
113C2×D61—82no—
114C2×D61—81no—
115C2×D61—164no—
116C4×C61—22no—
117C4×C61#41812yes—
118Dic61—21no—
119Dic61—41no—
120Dic61—81no—
121#3151—611no—
122#2165—612no—
123#1359—612no—
124#1357—611no—
125#1357D6111yescharacteristic, complemented
126C2×C61—21no—
127C2×C61—161no—
128C2×C61—161no—
129C2×C61#155511yescharacteristic
130D61—41no—
131D61—161no—
132C2×AGL(1,ℤ/8)—1222no—
133#532—611no—
134C2×#127—611no—
135#496—1222no—
136#493—611no—
137#493C2×D6111yescharacteristic
138#492—611no—
139#492—611no—
140C4×M16—611no—
141#364—612no—
142#364C2×D6112yescomplemented
143#349—1222no—
144#347—1222no—
145C61#1199211yescharacteristic, complemented, Sylow
146C2×#41—1221no—
147C22×D4—1221no—
148C22×D4—2441no—
149AGL(1,ℤ/8)—2444no—
150AGL(1,ℤ/8)—2444no—
151C2×SD16—1222no—
152C2×SD16—1222no—
153C2×D8—22no—
154C2×D8—1222no—
155C2×M16—612no—
156C4⋊Q8—611no—
157#127—611no—
158#127—1221no—
159#127—2441no—
160#127C4×D6111yescharacteristic
161#127C4×D6111yescharacteristic
162#124—1221no—
163#121—2441no—
164C4×D4—2441no—
165C2×C42—611no—
166M32—22no—
167M32—1222no—
168#97—612no—
169C4×C8—611no—
170C4×C8C22×D6111yescharacteristic
171C24—2441no—
172#41—2444no—
173C2×Q8—1221no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—1221no—
179C2×D4—4884no—
180C2×D4—2441no—
181C2×D4—2441no—
182C2×D4—2444no—
183C2×D4—2442no—
184C2×D4—2441no—
185C22×C4—611no—
186C22×C4—1221no—
187C22×C4—2441no—
188SD16—2444no—
189SD16—2442no—
190SD16—2442no—
191D8—42no—
192D8—42no—
193D8—2444no—
194M16—1222no—
195M16—1222no—
196C2×C8—612no—
197C2×C8D4×D6112yes—
198C4⋊C4—2441no—
199C22⋊C4—2442no—
200C42—611no—
201C42—1221no—
202C42C2×C4×D6111yescharacteristic
203C16—42no—
204C16—2442no—
205C23—41no—
206C23—41no—
207C23—611no—
208C23—4882no—
209C23—4884no—
210C23—2441no—
211Q8—2442no—
212D4—42no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—4882no—
217D4—4882no—
218D4—4882no—
219D4—4884no—
220D4—2442no—
221C2×C4—21no—
222C2×C4—611no—
223C2×C4—1222no—
224C2×C4—1221no—
225C2×C4—4882no—
226C2×C4—2442no—
227C2×C4—2441no—
228C2×C4C22⋊C4×D6111yescharacteristic
229C8—1222no—
230C8#1451980112yes—
231C8#1451980412yes—
232C22—82no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22—1221no—
237C22—9764no—
238C22—4882no—
239C22—4881no—
240C22—2441no—
241C22#100×D6111yescharacteristic
242C4—22no—
243C4—1221no—
244C4—4881no—
245C4—2441no—
246C4#1683976512yes—
247C2—21no—
248C2—161no—
249C2—161no—
250C2—9761no—
251C2—2441no—
252C2#2144453111yescharacteristic
subgroupG/H
centerC2#21444531
commutatorC42×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#1357C2
socleC2×C61#1555
prime radical#21456270C2
cosocleC4×C8×C61C23
hypercenter#1357D61
hypocenterC61#11992
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1357

Complements

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

NH ≅ G/Ntype
2. #21447630C2semidirect
3. #21443859C2semidirect
4. #21443856C2semidirect
5. C61×#1357C2semidirect
14. #16839583C22semidirect
18. C61×#364C22semidirect
125. #1357D61semidirect
142. #364C2×D61semidirect
145. C61#11992semidirect

Maximal subgroups

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

subgroupindex
#214562702
#214476302
#214438592
#214438562
C61×#13572
#1199261