Finite Group Databasebeta

← #27205071

Subgroups — #27205071

Order 13184.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#20268304C211yesmaximal, characteristic
2C2×C4⋊C4×D103C211yesmaximal, characteristic, complemented
3#20268124C211yesmaximal, characteristic, complemented
4#20268120C212yesmaximal, complemented
5#20268110C211yesmaximal, characteristic, complemented
6#20268109C211yesmaximal, characteristic, complemented
7#20268106C212yesmaximal, complemented
8#20267832C212yesmaximal, complemented
9#20267791C212yesmaximal, complemented
10#20267753C211yesmaximal, characteristic, complemented
11#484×C103C211yesmaximal, characteristic, complemented
12C2×#13110106C2211yescharacteristic
13C22×C4×D103—21no—
14C22×C4×D103C2211yescharacteristic
15C2×#13110088C2212yes—
16#15693006—21no—
17#15692997C2211yescharacteristic
18#15692996C2211yescharacteristic
19#15692995C2211yescharacteristic
20#15692991—21no—
21#15692983—21no—
22(C22⋊D103)⋊C4—21no—
23#15692979C2211yescharacteristic
24C2×C4⋊Dic103C2211yescharacteristic
25C2×Dic103⋊C4—21no—
26C4⋊C4×D103—21no—
27C4⋊C4×D103—21no—
28C4⋊C4×D103C2212yes—
29C4⋊C4×D103C2212yes—
30#15692954C2212yescomplemented
31#15692953C2212yescomplemented
32#15692952C2212yescomplemented
33#15692945C2212yescomplemented
34#15692943C2212yescomplemented
35#15692941C2212yescomplemented
36C22⋊(C103⋊C8)C2211yescharacteristic
37(C2×D103)⋊C8C2212yes—
38C22⋊Q8×C103C2211yescharacteristic
39C2×C4⋊C4×C103C2211yescharacteristic
40#107×C103C2212yescomplemented
41#103×C103C2212yescomplemented
42C22⋊C8×C103C2211yescharacteristic
43C23×D103D411yescharacteristic
44#13110106—24no—
45C2×D412—21no—
46C2×D412—21no—
47C2×D412C2311yescharacteristic
48C2×C22⋊D103—21no—
49C22×Dic103—21no—
50C22×Dic103D411yescharacteristic
51C2×C4×D103—24no—
52C2×C4×D103—22no—
53C2×C4×D103—22no—
54C2×C4×D103—21no—
55C2×C4×D103—44no—
56C2×C4×D103D411yescharacteristic
57C2×C4×D103D411yescharacteristic
58C2×C4×D103C2311yescharacteristic
59C2×C4×D103C2311yescharacteristic
60#13110088—24no—
61C2×C103⋊C8C2312yes—
62C4⋊Dic103—21no—
63C4⋊Dic103C2312yes—
64C4⋊Dic103C2311yescharacteristic
65Dic103⋊C4—22no—
66Dic103⋊C4—22no—
67Dic103⋊C4—21no—
68(C2×D103)⋊C4—21no—
69(C2×D103)⋊C4—21no—
70(C2×D103)⋊C4—21no—
71(C2×D103)⋊C4—21no—
72C4×Dic103—21no—
73C2×Q8×C103C2311yescharacteristic
74C22×C4×C103—21no—
75C22×C4×C103C2311yescharacteristic
76C2×C8×C103C2312yes—
77C4⋊C4×C103—21no—
78C4⋊C4×C103—21no—
79C4⋊C4×C103C2312yes—
80C4⋊C4×C103C2311yescharacteristic
81C22⋊C4×C103—21no—
82C23×C103C2×D411yescharacteristic
83C22×D103—22no—
84C22×D103—22no—
85C22×D103—22no—
86C22×D103—21no—
87C22×D103C2×D411yescharacteristic
88C22×D103C2×D411yescharacteristic
89Q8×C103—22no—
90C2×C4×C103—22no—
91C2×C4×C103—21no—
92C2×C4×C103—21no—
93C2×C4×C103—22no—
94C2×C4×C103—42no—
95C2×C4×C103C2×D411yescharacteristic
96C2×C4×C103C2411yescharacteristic
97C22⋊D103—42no—
98C2×Dic103—22no—
99C2×Dic103—21no—
100C2×Dic103—22no—
101C2×Dic103—42no—
102C2×Dic103C2×D411yescharacteristic
103C2×Dic103C2×D411yescharacteristic
104D412—22no—
105D412—42no—
106D412—42no—
107C4×D103—22no—
108C4×D103—22no—
109C4×D103—22no—
110C4×D103—44no—
111C4×D103—44no—
112C4×D103—42no—
113C4×D103#4114yes—
114C8×C103—22no—
115C103⋊C8—22no—
116C22×C103—21no—
117C22×C103—21no—
118C22×C103#13712yes—
119C22×C103C22×D411yescharacteristic
120C2×D103—28no—
121C2×D103—21no—
122C2×D103—21no—
123C2×D103—21no—
124C2×D103—21no—
125C2×D103—41no—
126C2×D103—41no—
127C2×D103—41no—
128C2×D103#12312yes—
129C2×D103AGL(1,ℤ/8)12yes—
130C4×C103—21no—
131C4×C103—41no—
132C4×C103—42no—
133C4×C103—41no—
134C4×C103C2×#4112yes—
135Dic103—21no—
136Dic103—42no—
137Dic103—41no—
138Dic103#12312yes—
139C2×C103—22no—
140C2×C103C2×#12311yescharacteristic
141C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C103C2×#13711yescharacteristic
143D103—22no—
144D103—22no—
145D103—81no—
146#3092—1031nomaximal, Sylow
147C103#309211yescharacteristic, complemented, Sylow
148#525—1031no—
149C22×C4⋊C4—1031no—
150#484—1031no—
151#484D10311yescharacteristic, complemented
152#483—1032no—
153#481—1031no—
154#481—1031no—
155#480—1032no—
156#428—1032no—
157#417—1032no—
158#407—1031no—
159C2×#41—1031no—
160C23×C4—1031no—
161C23×C4—2061no—
162C2×M16—1032no—
163#123—2061no—
164C22⋊Q8—1031no—
165C22⋊Q8C2×D10311yescharacteristic
166#121—1031no—
167#121—1031no—
168#120—2061no—
169C4×D4—2061no—
170C4×D4—2061no—
171#117—1031no—
172C2×C4⋊C4—1031no—
173C2×C4⋊C4—1032no—
174C2×C4⋊C4—1032no—
175C2×C4⋊C4—2061no—
176C2×C4⋊C4—2061no—
177C2×C4⋊C4—2061no—
178C2×C4⋊C4C2×D10311yescharacteristic
179#107—1032no—
180#107C2×D10312yescomplemented
181#106—1032no—
182#106—1032no—
183#103—1032no—
184#103C2×D10312yescomplemented
185#102—1032no—
186#102—1032no—
187C22⋊C8—1031no—
188C22⋊C8—1032no—
189C22⋊C8C2×D10311yescharacteristic
190C24—1031no—
191#41—2064no—
192C2×Q8C22×D10311yescharacteristic
193C2×D4—1031no—
194C2×D4—2061no—
195C2×D4—2061no—
196C2×D4—2061no—
197C22×C4—21no—
198C22×C4—1031no—
199C22×C4—1031no—
200C22×C4—1031no—
201C22×C4—1031no—
202C22×C4—1031no—
203C22×C4—4124no—
204C22×C4—2061no—
205C22×C4—2064no—
206C22×C4—2062no—
207C22×C4—2062no—
208C22×C4—2061no—
209C22×C4C22×D10311yescharacteristic
210M16—2064no—
211C2×C8—1032no—
212C2×C8C22×D10312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1032no—
216C4⋊C4—1031no—
217C4⋊C4—2062no—
218C4⋊C4—2062no—
219C4⋊C4—2061no—
220C4⋊C4—2061no—
221C4⋊C4C22×D10312yes—
222C4⋊C4C22×D10311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2061no—
225C22⋊C4—2061no—
226C22⋊C4—2061no—
227C22⋊C4—2061no—
228C42—2061no—
229C23—1031no—
230C23—1031no—
231C23—2062no—
232C23—2062no—
233C23—2061no—
234C23—2062no—
235C23D4×D10311yescharacteristic
236Q8—22no—
237D4—4122no—
238D4—4122no—
239D4—4122no—
240D4—2062no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—1031no—
247C2×C4—1034no—
248C2×C4—1031no—
249C2×C4—4122no—
250C2×C4—4124no—
251C2×C4—4124no—
252C2×C4—4122no—
253C2×C4—2062no—
254C2×C4—2062no—
255C2×C4—2062no—
256C2×C4—2061no—
257C2×C4—2062no—
258C2×C4—2062no—
259C2×C4D4×D10311yescharacteristic
260C2×C4C23×D10311yescharacteristic
261C8—22no—
262C8—2062no—
263C22—21no—
264C22—21no—
265C22—1032no—
266C22—1032no—
267C22—4121no—
268C22—4121no—
269C22—4121no—
270C22—2068no—
271C22—2061no—
272C22—2061no—
273C22—2061no—
274C22—2061no—
275C22#1569305312yes—
276C22C2×D4×D10311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—1032no—
282C4—4122no—
283C4—4121no—
284C4—2061no—
285C4#41×D10312yes—
286C2—22no—
287C2—8241no—
288C2—2062no—
289C2—2062no—
290C2#123×D10311yescharacteristic
291C2AGL(1,ℤ/8)×D10311yescharacteristic
292C2C2×#1569305311yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#484×C103C2
socleC22×C103C22×D4
prime radical#20268304C2
hypercenter#484D103
hypocenterC103#3092
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D103C2semidirect
3. #20268124C2semidirect
4. #20268120C2semidirect
5. #20268110C2semidirect
6. #20268109C2semidirect
7. #20268106C2semidirect
8. #20267832C2semidirect
9. #20267791C2semidirect
10. #20267753C2semidirect
11. #484×C103C2semidirect
30. #15692954C22semidirect
31. #15692953C22semidirect
32. #15692952C22semidirect
33. #15692945C22semidirect
34. #15692943C22semidirect
35. #15692941C22semidirect
40. #107×C103C22semidirect
41. #103×C103C22semidirect
147. C103#3092semidirect
151. #484D103semidirect
180. #107C2×D103semidirect
184. #103C2×D103semidirect

Maximal subgroups

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

subgroupindex
#202683042
C2×C4⋊C4×D1032
#202681242
#202681202
#202681102
#202681092
#202681062
#202678322
#202677912
#202677532
#484×C1032
#3092103