Finite Group Databasebeta

← #29600330

Subgroups — #29600330

Order 15616.

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

normal subgroups only

This group has 458 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#21447481C212yesmaximal, complemented
3#21443982C211yesmaximal, characteristic, complemented
4#21443981C212yesmaximal, complemented
5C61×#1408C211yesmaximal, characteristic, complemented
6C2×#14519801—22no—
7#16840310C2211yescharacteristic
8#127×D61C411yescharacteristic, complemented
9#16840150—22no—
10#16840138C2211yescharacteristic
11#16840135C411yescharacteristic, complemented
12#16840134C411yescharacteristic, complemented
13#16839734—22no—
14C4×#11758189C411yescharacteristic, complemented
15#16839556—22no—
16#16839555C2212yescomplemented
17C61×#493C2211yescharacteristic
18C61×#353C2212yescomplemented
19C2×#11758224D412yes—
20C2×D4×D61—21no—
21C2×D4×D61—41no—
22#14519801—44no—
23#14519801—44no—
24C2×#11758190—22no—
25C2×#11758200—22no—
26C2×#11758199—22no—
27C2×#11758189—21no—
28#14519777C2×C411yescharacteristic
29#14519774C2×C411yescharacteristic
30#14519773—21no—
31#14519773—41no—
32#14519764D412yes—
33#14519748—41no—
34D4×Dic61—41no—
35#14519733—42no—
36C42×D61C2×C411yescharacteristic
37#14519692—22no—
38#14519691—42no—
39#14519687—21no—
40C4×#92418C2×C411yescharacteristic
41C2×D8×C61—22no—
42#127×C61C2×C411yescharacteristic
43#127×C61C2311yescharacteristic
44#99×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C23×D61—41no—
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—22no—
55#11758224—42no—
56#11758224—42no—
57D4×D61—42no—
58D4×D61—42no—
59D4×D61—84no—
60C2×D244—21no—
61C2×D244—41no—
62C2×C4×D61—21no—
63C2×C4×D61C22⋊C411yescharacteristic
64C2×Dic122C22⋊C412yes—
65D8×C61—44no—
66C2×C8×C61—21no—
67C22⋊C4×C61—42no—
68C42×C61C22×C411yescharacteristic
69C22⋊Dic61—42no—
70C22⋊Dic61—42no—
71#11758200—44no—
72#11758199—44no—
73C4⋊Dic61—41no—
74#11758196—82no—
75C4×Dic61—21no—
76C4×Dic61—82no—
77C4×Dic61C22⋊C411yescharacteristic
78C2×#92418—21no—
79#11758190—44no—
80#11758189—24no—
81C23×C61—22no—
82C23×C61—41no—
83C22×D61—41no—
84C22×D61—81no—
85C22×D61—81no—
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—22no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—82no—
103C2×Dic61#9911yescharacteristic
104D244—42no—
105D244—82no—
106C4×D61—22no—
107C4×D61—42no—
108Dic122—42no—
109C8×C61—22no—
110#92418—22no—
111#13585—611nomaximal, Sylow
112C22×C61—42no—
113C22×C61—82no—
114C22×C61—82no—
115C22×C61—81no—
116C22×C61C2×#9911yescharacteristic
117C2×D61—21no—
118C2×D61—41no—
119C2×D61—82no—
120C2×D61—81no—
121C2×D61—162no—
122C2×D61—162no—
123C4×C61—22no—
124C4×C61—21no—
125C4×C61—162no—
126Dic61—41no—
127Dic61—42no—
128Dic61—162no—
129Dic61#35312yes—
130#3151—611no—
131#2126—612no—
132#1408—611no—
133#1408—612no—
134#1408D6111yescharacteristic, complemented
135C2×C61—21no—
136C2×C61—82no—
137C2×C61—161no—
138C2×C61C2×#35311yescharacteristic
139D61—41no—
140D61—161no—
141C2×AGL(1,ℤ/8)—1222no—
142#532—611no—
143C2×#127—611no—
144#496—1222no—
145#493—611no—
146#493C2×D6111yescharacteristic
147#492—611no—
148#492—611no—
149#410—1222no—
150C4×M16—611no—
151#354—1222no—
152#353—612no—
153#353C2×D6112yescomplemented
154C61#1358511yescharacteristic, complemented, Sylow
155C2×#41—612no—
156C22×D4—1221no—
157C22×D4—2441no—
158AGL(1,ℤ/8)—2444no—
159AGL(1,ℤ/8)—2444no—
160C2×SD16—1222no—
161C2×SD16—1222no—
162C2×D8—22no—
163C2×D8—1222no—
164C2×M16—1221no—
165C4⋊Q8—611no—
166#127—611no—
167#127—1221no—
168#127—2441no—
169#127C4×D6111yescharacteristic
170#127C22×D6111yescharacteristic
171#124—612no—
172#121—2441no—
173C4×D4—2441no—
174#117—2442no—
175C2×C42—611no—
176#99—22no—
177#99—1222no—
178#99—2442no—
179#97—1221no—
180C4×C8—611no—
181C4×C8C4×D6111yescharacteristic
182C24—2441no—
183#41—2442no—
184#41—2442no—
185C2×Q8—612no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—1222no—
190C2×D4—1221no—
191C2×D4—4884no—
192C2×D4—2441no—
193C2×D4—2441no—
194C2×D4—2442no—
195C2×D4—2442no—
196C2×D4—2442no—
197C2×D4D4×D6112yes—
198C22×C4—611no—
199C22×C4—1221no—
200C22×C4—1222no—
201SD16—2444no—
202SD16—2444no—
203D8—44no—
204D8—2444no—
205M16—1224no—
206C2×C8—21no—
207C2×C8—1221no—
208C4⋊C4—4882no—
209C4⋊C4—2441no—
210C22⋊C4—42no—
211C22⋊C4—2442no—
212C22⋊C4—2442no—
213C42—611no—
214C42—1221no—
215C42—4882no—
216C42C2×C4×D6111yescharacteristic
217C23—22no—
218C23—41no—
219C23—611no—
220C23—4881no—
221C23—4881no—
222C23—4884no—
223C23—2441no—
224Q8—2442no—
225D4—42no—
226D4—42no—
227D4—82no—
228D4—82no—
229D4—4882no—
230D4—4882no—
231D4—4882no—
232D4—4884no—
233D4—2442no—
234C2×C4—21no—
235C2×C4—82no—
236C2×C4—611no—
237C2×C4—1222no—
238C2×C4—1221no—
239C2×C4—1222no—
240C2×C4—4882no—
241C2×C4—2442no—
242C2×C4—2442no—
243C2×C4—2442no—
244C2×C4C22⋊C4×D6111yescharacteristic
245C8—22no—
246C8—1222no—
247C22—42no—
248C22—82no—
249C22—82no—
250C22—81no—
251C22—1221no—
252C22—9762no—
253C22—9762no—
254C22—4882no—
255C22—4881no—
256C22—2441no—
257C22#99×D6111yescharacteristic
258C4—22no—
259C4—21no—
260C4—162no—
261C4—612no—
262C4—9762no—
263C4—2441no—
264C4—2442no—
265C2—21no—
266C2—82no—
267C2—161no—
268C2—9761no—
269C2—2441no—
270C2#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#13585
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. #21447481C2semidirect
3. #21443982C2semidirect
4. #21443981C2semidirect
5. C61×#1408C2semidirect
8. #127×D61C4semidirect
11. #16840135C4semidirect
12. #16840134C4semidirect
14. C4×#11758189C4semidirect
16. #16839555C22semidirect
18. C61×#353C22semidirect
134. #1408D61semidirect
153. #353C2×D61semidirect
154. C61#13585semidirect

Maximal subgroups

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

subgroupindex
#214562702
#214474812
#214439822
#214439812
C61×#14082
#1358561