Finite Group Databasebeta

← #27151347

Subgroups — #27151347

Order 12416.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#15690833C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#20261248C212yesmaximal, complemented
4#20260990C211yesmaximal, characteristic, complemented
5#20260989C211yesmaximal, characteristic, complemented
6#20260988C211yesmaximal, characteristic, complemented
7#20260984C212yesmaximal, complemented
8#20260983C212yesmaximal, complemented
9#20260770C211yesmaximal, characteristic, complemented
10#20260730C212yesmaximal, complemented
11#470×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C22×C4×D97C2211yescharacteristic
14#15690833—24no—
15C2×#13095498C2212yescomplemented
16C2×D776C2211yescharacteristic, complemented
17C2×#13095495C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20M16×D97—24no—
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690795—22no—
24#15690786C2212yes—
25#15690785C2212yes—
26#15690784C2212yes—
27#15690780—22no—
28#15690772—22no—
29#15690770—22no—
30#15690768C2212yes—
31#15690743C2212yescomplemented
32#15690742C2211yescharacteristic, complemented
33#15690734C2212yescomplemented
34#15690732C2212yescomplemented
35#15690730C2212yescomplemented
36C2×Q16×C97C2211yescharacteristic, complemented
37C2×M16×C97C2211yescharacteristic
38C22⋊Q8×C97C2212yes—
39#106×C97C2211yescharacteristic, complemented
40#103×C97C2212yescomplemented
41C23×D97D411yescharacteristic
42#13095512—24no—
43#13095512—24no—
44C2×D388—22no—
45C2×D388—22no—
46C2×D388C2312yes—
47C2×C22⋊D97—22no—
48C22×Dic97D411yescharacteristic
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97C2311yescharacteristic
55C2×C4×D97C2311yescharacteristic
56#13095498—24no—
57D776—22no—
58#13095495—22no—
59#13095494—22no—
60#13095494—22no—
61C2×#11715895C2311yescharacteristic
62C2×#11715895C2311yescharacteristic
63C8×D97—22no—
64C8×D97D414yescomplemented
65C4⋊Dic97C2312yes—
66#13095490—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—22no—
72C2×Q8×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—22no—
75M16×C97—22no—
76C2×C8×C97C2311yescharacteristic
77C2×C8×C97C2311yescharacteristic
78C4⋊C4×C97—22no—
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—42no—
85C22×D97C2×D411yescharacteristic
86C22×D97C2×D411yescharacteristic
87Q8×C97—22no—
88Q8×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2411yescharacteristic
95C22⋊D97—44no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100D388—22no—
101D388—22no—
102D388—44no—
103D388—44no—
104C4×D97—22no—
105C4×D97—21no—
106C4×D97—21no—
107C4×D97—22no—
108C4×D97—44no—
109C4×D97C2×D412yes—
110C4×D97#4112yes—
111C8×C97—21no—
112C8×C97C2×D412yes—
113#11715895—21no—
114#11715895C2×D412yes—
115C22×C97—21no—
116C22×C97—22no—
117C22×C97C22×D411yescharacteristic
118C2×D97—22no—
119C2×D97—21no—
120C2×D97—22no—
121C2×D97—44no—
122C2×D97—42no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97#12112yes—
126C4×C97—21no—
127C4×C97—42no—
128C4×C97—42no—
129C4×C97C22×D411yescharacteristic
130C4×C97C2×#4111yescharacteristic
131Dic97—21no—
132Dic97—42no—
133Dic97#12112yes—
134C2×C97—41no—
135C2×C97C2×#12111yescharacteristic
136C2×C97#57512yes—
137D97—22no—
138D97—41no—
139D97—82no—
140#3054—971nomaximal, Sylow
141C97#305411yescharacteristic, complemented, Sylow
142C2×#135—971no—
143C22×M16—971no—
144#525—972no—
145#470—971no—
146#470D9711yescharacteristic, complemented
147#469—971no—
148#469—971no—
149#468—972no—
150#468—972no—
151#427—971no—
152#416—972no—
153C2×#41—972no—
154C23×C4—971no—
155#135—1944no—
156C2×Q16C2×D9711yescharacteristic, complemented
157C2×SD16—972no—
158C2×D8—971no—
159C2×M16—971no—
160C2×M16—971no—
161C2×M16—971no—
162C2×M16—1944no—
163C2×M16C2×D9711yescharacteristic
164C22×C8—971no—
165C22×C8—971no—
166#123—1942no—
167C22⋊Q8—972no—
168C22⋊Q8C2×D9712yes—
169#121—972no—
170#121—972no—
171#120—1942no—
172C4×D4—1942no—
173C4×D4—1942no—
174#117—972no—
175#107—972no—
176#106—971no—
177#106C2×D9711yescharacteristic, complemented
178#103—972no—
179#103C2×D9712yescomplemented
180#102—972no—
181#102—972no—
182C24—971no—
183#41—1944no—
184#41—1944no—
185C2×Q8C22×D9712yes—
186C2×D4—972no—
187C2×D4—1942no—
188C2×D4—1942no—
189C2×D4—1942no—
190C22×C4—971no—
191C22×C4—971no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—971no—
195C22×C4—1942no—
196C22×C4—1942no—
197C22×C4—1942no—
198C22×C4C22×D9711yescharacteristic
199Q16—22no—
200SD16—1944no—
201D8—1942no—
202M16—22no—
203M16—1942no—
204M16—1942no—
205M16—1942no—
206C2×C8—971no—
207C2×C8—974no—
208C2×C8—971no—
209C2×C8—1942no—
210C2×C8C22×D9711yescharacteristic
211C2×C8C22×D9711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—972no—
214C4⋊C4—1942no—
215C4⋊C4C22×D9712yes—
216C22⋊C4—22no—
217C22⋊C4—1942no—
218C22⋊C4—1942no—
219C22⋊C4—1942no—
220C22⋊C4—1942no—
221C42—1942no—
222C23—971no—
223C23—971no—
224C23—3882no—
225C23—1942no—
226C23—1942no—
227C23D4×D9711yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—3884no—
231D4—3884no—
232D4—3884no—
233D4—1942no—
234D4—1942no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—971no—
240C2×C4—972no—
241C2×C4—972no—
242C2×C4—971no—
243C2×C4—3884no—
244C2×C4—1942no—
245C2×C4—1942no—
246C2×C4—1942no—
247C2×C4—1942no—
248C2×C4—1941no—
249C2×C4—1941no—
250C2×C4D4×D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—21no—
253C8—972no—
254C8—1941no—
255C8D4×D9712yes—
256C22—21no—
257C22—22no—
258C22—972no—
259C22—3884no—
260C22—3882no—
261C22—3882no—
262C22—3882no—
263C22—1941no—
264C22—1942no—
265C22—1942no—
266C22C2×D4×D9711yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—972no—
271C4—3882no—
272C4—1941no—
273C4C2×D4×D9711yescharacteristic
274C4#41×D9711yescharacteristic
275C2—41no—
276C2—7762no—
277C2—3881no—
278C2—1942no—
279C2#121×D9711yescharacteristic
280C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#470×C97C2
socleC22×C97C22×D4
hypercenter#470D97
hypocenterC97#3054
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C2×M16×D97C2semidirect
3. #20261248C2semidirect
4. #20260990C2semidirect
5. #20260989C2semidirect
6. #20260988C2semidirect
7. #20260984C2semidirect
8. #20260983C2semidirect
9. #20260770C2semidirect
10. #20260730C2semidirect
11. #470×C97C2semidirect
15. C2×#13095498C22semidirect
16. C2×D776C22semidirect
31. #15690743C22semidirect
32. #15690742C22semidirect
33. #15690734C22semidirect
34. #15690732C22semidirect
35. #15690730C22semidirect
36. C2×Q16×C97C22semidirect
39. #106×C97C22semidirect
40. #103×C97C22semidirect
64. C8×D97D4semidirect
141. C97#3054semidirect
146. #470D97semidirect
156. C2×Q16C2×D97semidirect
177. #106C2×D97semidirect
179. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C2×M16×D972
#202612482
#202609902
#202609892
#202609882
#202609842
#202609832
#202607702
#202607302
#470×C972
#305497