Finite Group Databasebeta

← #29600326

Subgroups — #29600326

Order 15616.

474 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 59 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#353×D61C212yesmaximal, complemented
3#21443983C212yesmaximal, complemented
4#21443982C211yesmaximal, characteristic, complemented
5C61×#1408C211yesmaximal, characteristic, complemented
6C2×#14519801—22no—
7#16840310C411yescharacteristic, complemented
8#127×D61C2211yescharacteristic
9#16840150—22no—
10#16840138C411yescharacteristic, complemented
11#16840135C411yescharacteristic, complemented
12#16840134C2211yescharacteristic
13#99×D61—22no—
14C4×#11758189C411yescharacteristic, complemented
15#16839555C2212yes—
16#16839554—22no—
17C61×#493C2211yescharacteristic
18C61×#353C2212yes—
19C2×#11758224—21no—
20C2×D4×D61—41no—
21C2×D4×D61D412yes—
22#14519801—44no—
23#14519801—44no—
24C2×#11758190—22no—
25C2×#11758200—22no—
26C2×#11758199—22no—
27C2×#11758189—21no—
28#14519777C2×C411yescharacteristic, complemented
29#14519774C2×C411yescharacteristic
30#14519773—41no—
31#14519773D412yes—
32#14519764—21no—
33#14519748—41no—
34D4×Dic61—41no—
35C22⋊C4×D61—42no—
36C42×D61C2×C411yescharacteristic
37#14519692—22no—
38#14519691—42no—
39#14519687—21no—
40C4×#92418C2×C411yescharacteristic, complemented
41C2×D8×C61—22no—
42#127×C61C2×C411yescharacteristic, complemented
43#127×C61C2311yescharacteristic
44#99×C61—22no—
45C4×C8×C61C2×C411yescharacteristic, complemented
46C23×D61—22no—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61C2×D412yes—
51C2×C22⋊D61—22no—
52C2×C22⋊D61—42no—
53C2×C22⋊D61—41no—
54C22×Dic61—41no—
55#11758224—42no—
56#11758224—42no—
57D4×D61—42no—
58D4×D61—42no—
59D4×D61—84no—
60C2×D244—41no—
61C2×D244C22⋊C412yes—
62C2×C4×D61—21no—
63C2×C4×D61—82no—
64C2×C4×D61C22⋊C411yescharacteristic
65C2×Dic122—21no—
66D8×C61—44no—
67C2×C8×C61—21no—
68C22⋊C4×C61—42no—
69C42×C61C22×C411yescharacteristic
70C22⋊Dic61—42no—
71C22⋊Dic61—42no—
72#11758200—44no—
73#11758199—44no—
74#11758198—82no—
75C4⋊Dic61—41no—
76C4×Dic61—21no—
77C4×Dic61C22⋊C411yescharacteristic
78C2×#92418—21no—
79#11758190—44no—
80#11758189—24no—
81C23×C61—22no—
82C23×C61—41no—
83C22×D61—22no—
84C22×D61—42no—
85C22×D61—42no—
86C22×D61—84no—
87C22×D61#9911yescharacteristic
88D4×C61—42no—
89D4×C61—42no—
90D4×C61—82no—
91D4×C61—82no—
92C2×C4×C61—21no—
93C2×C4×C61—82no—
94C2×C4×C61C2×C22⋊C411yescharacteristic
95C22⋊D61—82no—
96C22⋊D61—84no—
97C22⋊D61—82no—
98C2×Dic61—21no—
99C2×Dic61—41no—
100C2×Dic61—81no—
101C2×Dic61—81no—
102C2×Dic61—82no—
103C2×Dic61#9911yescharacteristic
104D244—42no—
105D244—82no—
106C4×D61—22no—
107C4×D61—42no—
108C4×D61—164no—
109Dic122—42no—
110C8×C61—22no—
111#92418—22no—
112#13584—611nomaximal, Sylow
113C22×C61—42no—
114C22×C61—82no—
115C22×C61—82no—
116C22×C61—81no—
117C22×C61C2×#9911yescharacteristic
118C2×D61—41no—
119C2×D61—42no—
120C2×D61—84no—
121C2×D61—82no—
122C2×D61—162no—
123C2×D61#35312yes—
124C4×C61—22no—
125C4×C61—21no—
126C4×C61—162no—
127Dic61—21no—
128Dic61—41no—
129Dic61—81no—
130Dic61—162no—
131#3151—611no—
132C2×#353—612no—
133#1409—612no—
134#1408—611no—
135#1408D6111yescharacteristic, complemented
136C2×C61—21no—
137C2×C61—82no—
138C2×C61—161no—
139C2×C61C2×#35311yescharacteristic
140D61—22no—
141D61—82no—
142C2×AGL(1,ℤ/8)—1222no—
143#532—611no—
144C2×#127—611no—
145#496—1222no—
146#493—611no—
147#493C2×D6111yescharacteristic
148#492—611no—
149#492—611no—
150C2×#99—1222no—
151C4×M16—611no—
152#353—612no—
153#353—1222no—
154#353C2×D6112yes—
155C61#1358411yescharacteristic, complemented, Sylow
156C2×#41—1221no—
157C22×D4—612no—
158C22×D4—2441no—
159AGL(1,ℤ/8)—2444no—
160AGL(1,ℤ/8)—2444no—
161C2×SD16—1222no—
162C2×SD16—1222no—
163C2×D8—22no—
164C2×D8—1222no—
165C2×M16—1221no—
166C4⋊Q8—611no—
167#127—612no—
168#127—611no—
169#127—2441no—
170#127C4×D6111yescharacteristic, complemented
171#127C22×D6111yescharacteristic
172#124—1221no—
173#121—2441no—
174C4×D4—2441no—
175C2×C22⋊C4—2442no—
176C2×C42—611no—
177#99—22no—
178#99—1222no—
179#99—2442no—
180#97—1221no—
181C4×C8—611no—
182C4×C8C4×D6111yescharacteristic, complemented
183C24—1222no—
184#41—2442no—
185#41—2442no—
186C2×Q8—1221no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—612no—
191C2×D4—1222no—
192C2×D4—4884no—
193C2×D4—2441no—
194C2×D4—2442no—
195C2×D4—2442no—
196C2×D4—2442no—
197C2×D4—2441no—
198C2×D4D4×D6112yes—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4—4882no—
202C22×C4—2441no—
203SD16—2444no—
204SD16—2444no—
205D8—44no—
206D8—2444no—
207M16—1224no—
208C2×C8—21no—
209C2×C8—1221no—
210C4⋊C4—2441no—
211C22⋊C4—42no—
212C22⋊C4—4882no—
213C22⋊C4—2442no—
214C22⋊C4—2442no—
215C42—611no—
216C42—1221no—
217C42C2×C4×D6111yescharacteristic
218C23—22no—
219C23—41no—
220C23—611no—
221C23—1222no—
222C23—4884no—
223C23—2442no—
224C23—2442no—
225Q8—2442no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—4882no—
231D4—4882no—
232D4—4882no—
233D4—4884no—
234D4—2442no—
235C2×C4—21no—
236C2×C4—82no—
237C2×C4—611no—
238C2×C4—1222no—
239C2×C4—1221no—
240C2×C4—9764no—
241C2×C4—4881no—
242C2×C4—4881no—
243C2×C4—4882no—
244C2×C4—2442no—
245C2×C4—2441no—
246C2×C4C22⋊C4×D6111yescharacteristic
247C8—22no—
248C8—1222no—
249C22—42no—
250C22—82no—
251C22—82no—
252C22—81no—
253C22—612no—
254C22—9762no—
255C22—4882no—
256C22—4884no—
257C22—2441no—
258C22—2442no—
259C22#99×D6111yescharacteristic
260C4—22no—
261C4—21no—
262C4—162no—
263C4—1221no—
264C4—9762no—
265C4—4881no—
266C4—2441no—
267C2—21no—
268C2—82no—
269C2—161no—
270C2—1222no—
271C2—4882no—
272C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1408C2
socleC2×C61C2×#353
prime radical#21456270C2
cosocle#127×C61C23
hypercenter#1408D61
hypocenterC61#13584
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1408

Complements

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

NH ≅ G/Ntype
2. #353×D61C2semidirect
3. #21443983C2semidirect
4. #21443982C2semidirect
5. C61×#1408C2semidirect
7. #16840310C4semidirect
10. #16840138C4semidirect
11. #16840135C4semidirect
14. C4×#11758189C4semidirect
28. #14519777C2×C4semidirect
40. C4×#92418C2×C4semidirect
42. #127×C61C2×C4semidirect
45. C4×C8×C61C2×C4semidirect
135. #1408D61semidirect
155. C61#13584semidirect
170. #127C4×D61semidirect
182. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562702
#353×D612
#214439832
#214439822
C61×#14082
#1358461