Finite Group Databasebeta

← #27204940

Subgroups — #27204940

Order 13184.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#20268344C211yesmaximal, characteristic, complemented
2C2×#15692983C211yesmaximal, characteristic
3#20268094C212yesmaximal, complemented
4#20268090C212yesmaximal, complemented
5#20268080C212yesmaximal, complemented
6#20268070C212yesmaximal, complemented
7#20267831C211yesmaximal, characteristic, complemented
8#20267814C211yesmaximal, characteristic, complemented
9#20267797C211yesmaximal, characteristic
10#20267795C211yesmaximal, characteristic
11#428×C103C211yesmaximal, characteristic, complemented
12C22×D412C2211yescharacteristic
13C22×C4×D103—21no—
14C2×#13110089C2211yescharacteristic
15#15693020C2211yescharacteristic, complemented
16#15693020C2211yescharacteristic, complemented
17#15693012C2211yescharacteristic, complemented
18#15693012C2211yescharacteristic, complemented
19#15692998—21no—
20#15692998—21no—
21C4×Dic206—21no—
22C4×Dic206—21no—
23#15692983—21no—
24#15692983—21no—
25#15692983C2212yes—
26#15692983C2212yes—
27#15692980C2211yescharacteristic
28C2×(C2×D103)⋊C4—21no—
29C2×C4×Dic103C2211yescharacteristic
30#15692954C2212yescomplemented
31#15692952C2212yescomplemented
32Dic103⋊C8C2212yescomplemented
33Dic103⋊C8C2212yescomplemented
34#15692943C2212yes—
35#15692943C2212yes—
36#15692941C2212yes—
37#15692941C2212yes—
38C2×M16×C103C2211yescharacteristic
39#117×C103C2211yescharacteristic
40C2×C4⋊C4×C103C2211yescharacteristic
41#107×C103C2212yescomplemented
42#106×C103C2212yescomplemented
43C23×D103—21no—
44C2×Dic206—21no—
45C2×Dic206—21no—
46C2×D412—21no—
47C2×D412—22no—
48C2×D412Q814yescomplemented
49C2×D412C2312yes—
50C22×Dic103D411yescharacteristic
51C22×Dic103D411yescharacteristic
52C2×C4×D103—22no—
53C2×C4×D103—22no—
54C2×C4×D103—44no—
55#13110089—22no—
56C2×C103⋊C8C2312yes—
57C4⋊Dic103—21no—
58C4⋊Dic103—21no—
59C4⋊Dic103C2311yescharacteristic
60C4⋊Dic103C2311yescharacteristic
61Dic103⋊C4—21no—
62Dic103⋊C4—21no—
63Dic103⋊C4—21no—
64Dic103⋊C4—21no—
65C22⋊Dic103—21no—
66(C2×D103)⋊C4—22no—
67(C2×D103)⋊C4—22no—
68C4×Dic103—21no—
69C4×Dic103D411yescharacteristic
70C4×Dic103D411yescharacteristic
71C4×Dic103C2311yescharacteristic
72C4×Dic103C2311yescharacteristic
73C22×C4×C103—21no—
74C22×C4×C103C2311yescharacteristic
75M16×C103—22no—
76C2×C8×C103C2312yes—
77C4⋊C4×C103—21no—
78C4⋊C4×C103C2311yescharacteristic
79C4⋊C4×C103C2312yes—
80C4⋊C4×C103C2311yescharacteristic
81C22⋊C4×C103—21no—
82C42×C103—21no—
83C23×C103C2×D411yescharacteristic
84C22×D103—24no—
85C22×D103—22no—
86C22×D103—42no—
87C22×D103—42no—
88C2×C4×C103—21no—
89C2×C4×C103—21no—
90C2×C4×C103—22no—
91C2×C4×C103—42no—
92C2×C4×C103C2×D411yescharacteristic
93C2×C4×C103C2×Q812yes—
94C2×C4×C103#4112yes—
95C2×C4×C103C2411yescharacteristic
96C2×Dic103—22no—
97C2×Dic103—21no—
98C2×Dic103—21no—
99C2×Dic103—41no—
100C2×Dic103C2×D411yescharacteristic
101C2×Dic103C2×D411yescharacteristic
102C2×Dic103C2×D411yescharacteristic
103C2×Dic103C2×D411yescharacteristic
104D412—24no—
105D412—22no—
106D412C2×Q814yes—
107C4×D103—44no—
108C4×D103—44no—
109Dic206—42no—
110Dic206—42no—
111C8×C103—22no—
112C103⋊C8—22no—
113C22×C103—21no—
114C22×C103—21no—
115C22×C103C22⋊Q812yes—
116C22×C103C22×D411yescharacteristic
117C2×D103—24no—
118C2×D103—48no—
119C2×D103—42no—
120C2×D103—42no—
121C4×C103—41no—
122C4×C103—42no—
123C4×C103—41no—
124C4×C103C22⋊Q812yes—
125C4×C103C22×Q811yescharacteristic
126C4×C103C2×#4111yescharacteristic
127Dic103—21no—
128Dic103—21no—
129Dic103—21no—
130Dic103—41no—
131Dic103—41no—
132Dic103AGL(1,ℤ/8)12yes—
133C2×C103—22no—
134C2×C103C2×C22⋊Q811yescharacteristic
135C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C103#57511yescharacteristic
137D103—44no—
138#3077—1031nomaximal, Sylow
139C103#307711yescharacteristic, complemented, Sylow
140#533—1031no—
141C2×C4×D4—1031no—
142#478—1032no—
143#478—1032no—
144#476—1032no—
145#474—1032no—
146#428—1031no—
147#428D10311yescharacteristic, complemented
148#423—1031no—
149#418—1031no—
150#418—1031no—
151C22×D4—1031no—
152C23×C4—2061no—
153C2×M16—1031no—
154C2×M16C2×D10311yescharacteristic
155C4⋊Q8—1031no—
156C4⋊Q8—1031no—
157#125—1031no—
158#125—1031no—
159C22⋊Q8—2061no—
160C22⋊Q8—2061no—
161C4×Q8—2061no—
162C4×Q8—2061no—
163C4×D4—1032no—
164C4×D4—1032no—
165C4×D4—2061no—
166C4×D4—2061no—
167#117—1031no—
168#117C2×D10311yescharacteristic
169C2×C4⋊C4C2×D10311yescharacteristic
170C2×C22⋊C4—2061no—
171C2×C42—1031no—
172#107—1032no—
173#107C2×D10312yescomplemented
174#106—1032no—
175#106C2×D10312yescomplemented
176C4⋊C8—1032no—
177C4⋊C8—1032no—
178#102—1032no—
179#102—1032no—
180#102—1032no—
181#102—1032no—
182C24—2061no—
183C2×Q8—2061no—
184C2×Q8—2061no—
185C2×D4—1034no—
186C2×D4—1032no—
187C2×D4—2061no—
188C2×D4—2062no—
189C22×C4—21no—
190C22×C4—1031no—
191C22×C4—1031no—
192C22×C4—4124no—
193C22×C4—2062no—
194C22×C4—2062no—
195C22×C4C22×D10311yescharacteristic
196M16—22no—
197M16—2062no—
198C2×C8—1032no—
199C2×C8C22×D10312yes—
200C4⋊C4—21no—
201C4⋊C4—1031no—
202C4⋊C4—1031no—
203C4⋊C4—2061no—
204C4⋊C4—2061no—
205C4⋊C4—2061no—
206C4⋊C4—2061no—
207C4⋊C4—2061no—
208C4⋊C4—2061no—
209C4⋊C4C22×D10311yescharacteristic
210C4⋊C4C22×D10311yescharacteristic
211C4⋊C4C22×D10312yes—
212C22⋊C4—21no—
213C22⋊C4—2062no—
214C22⋊C4—2062no—
215C22⋊C4—2061no—
216C42—21no—
217C42—1031no—
218C42—1031no—
219C42—1031no—
220C42—1031no—
221C42—2061no—
222C23—4122no—
223C23—4122no—
224C23—2062no—
225C23—2064no—
226C23D4×D10311yescharacteristic
227Q8—4122no—
228Q8—4122no—
229D4—1034no—
230D4—2062no—
231D4—2064no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1031no—
237C2×C4—1031no—
238C2×C4—1031no—
239C2×C4—1031no—
240C2×C4—4124no—
241C2×C4—4124no—
242C2×C4—4121no—
243C2×C4—2062no—
244C2×C4—2061no—
245C2×C4—2061no—
246C2×C4D4×D10311yescharacteristic
247C2×C4Q8×D10312yes—
248C2×C4#1311010512yes—
249C2×C4C23×D10311yescharacteristic
250C8—22no—
251C8—2062no—
252C22—21no—
253C22—21no—
254C22—4128no—
255C22—4122no—
256C22—4122no—
257C22—2064no—
258C22#1569299912yes—
259C22C2×D4×D10311yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1032no—
264C4—4121no—
265C4—4121no—
266C4—2061no—
267C4—2061no—
268C4—2061no—
269C4#1569299912yes—
270C4C2×Q8×D10311yescharacteristic
271C4C2×#1311010511yescharacteristic
272C2—22no—
273C2—4124no—
274C2C2×#1569299911yescharacteristic
275C2AGL(1,ℤ/8)×D10311yescharacteristic
276C2#2026859011yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#428×C103C2
socleC22×C103C22×D4
prime radicalC22×D412C22
hypercenter#428D103
hypocenterC103#3077
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20268344C2semidirect
3. #20268094C2semidirect
4. #20268090C2semidirect
5. #20268080C2semidirect
6. #20268070C2semidirect
7. #20267831C2semidirect
8. #20267814C2semidirect
11. #428×C103C2semidirect
15. #15693020C22semidirect
16. #15693020C22semidirect
17. #15693012C22semidirect
18. #15693012C22semidirect
30. #15692954C22semidirect
31. #15692952C22semidirect
32. Dic103⋊C8C22semidirect
33. Dic103⋊C8C22semidirect
41. #107×C103C22semidirect
42. #106×C103C22semidirect
48. C2×D412Q8semidirect
139. C103#3077semidirect
147. #428D103semidirect
173. #107C2×D103semidirect
175. #106C2×D103semidirect

Maximal subgroups

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

subgroupindex
#202683442
C2×#156929832
#202680942
#202680902
#202680802
#202680702
#202678312
#202678142
#202677972
#202677952
#428×C1032
#3077103