Finite Group Databasebeta

← #27205976

Subgroups — #27205976

Order 13184.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#20268302C211yesmaximal, characteristic
2C2×#15693000C211yesmaximal, characteristic, complemented
3#20268133C212yesmaximal, complemented
4#20268127C211yesmaximal, characteristic, complemented
5#20268110C211yesmaximal, characteristic, complemented
6#20268049C212yesmaximal, complemented
7#20268037C212yesmaximal, complemented
8#20267955C212yesmaximal, complemented
9#20267949C211yesmaximal, characteristic, complemented
10#20267932C211yesmaximal, characteristic, complemented
11#407×C103C211yesmaximal, characteristic, complemented
12C2×#13110105C2211yescharacteristic
13C22×Dic206C2211yescharacteristic
14C2×Dic412C2212yescomplemented
15C2×#13110094C2212yescomplemented
16#15693004—21no—
17#15693000—21no—
18#15693000—21no—
19#15693000C2212yes—
20#15693000C2212yes—
21#15693000C2211yescharacteristic
22#15693000C2211yescharacteristic
23#15692995C2211yescharacteristic
24#15692995C2211yescharacteristic
25#15692990—21no—
26C4×C22⋊D103—21no—
27C4×C22⋊D103—21no—
28#15692980C2211yescharacteristic
29C2×C4⋊Dic103C2211yescharacteristic
30C2×Dic103⋊C4—21no—
31C2×C22⋊Dic103—21no—
32#15692955C2212yescomplemented
33#15692953C2212yescomplemented
34#15692946C2212yescomplemented
35#15692946C2212yescomplemented
36#15692946C2212yescomplemented
37#15692943C2212yescomplemented
38C23×C4×C103C2211yescharacteristic
39C2×M16×C103C2212yes—
40C22⋊C8×C103C2211yescharacteristic
41C22⋊C8×C103C2211yescharacteristic
42C22⋊C8×C103C2212yes—
43#13110105—42no—
44C2×Dic206—21no—
45C2×Dic206—42no—
46C2×Dic206C2312yes—
47C2×Dic206C2311yescharacteristic
48C2×D412C2311yescharacteristic
49C2×C22⋊D103—21no—
50C2×C22⋊D103—21no—
51C2×C22⋊D103—21no—
52C22×Dic103—21no—
53C22×Dic103—21no—
54C2×C4×D103—21no—
55Dic412—42no—
56#13110094—42no—
57C4⋊Dic103—21no—
58C4⋊Dic103C2312yes—
59C4⋊Dic103C2311yescharacteristic
60C4⋊Dic103C2311yescharacteristic
61Dic103⋊C4—22no—
62Dic103⋊C4—22no—
63Dic103⋊C4—21no—
64Dic103⋊C4—21no—
65C22⋊Dic103—22no—
66C22⋊Dic103—22no—
67C22⋊Dic103—21no—
68C22⋊Dic103—21no—
69C22⋊Dic103—21no—
70(C2×D103)⋊C4—21no—
71(C2×D103)⋊C4—21no—
72C4×Dic103—21no—
73C24×C103D411yescharacteristic
74C22×C4×C103—42no—
75C22×C4×C103D411yescharacteristic
76C22×C4×C103D411yescharacteristic
77C22×C4×C103D411yescharacteristic
78C22×C4×C103C2311yescharacteristic
79C22×C4×C103C2311yescharacteristic
80C22×C4×C103C2311yescharacteristic
81M16×C103—42no—
82C2×C8×C103C2312yes—
83C2×C8×C103C2312yes—
84C23×C103—22no—
85C23×C103—41no—
86C23×C103—41no—
87C23×C103C2×D411yescharacteristic
88C23×C103C2×D411yescharacteristic
89C23×C103C2×D411yescharacteristic
90C22×D103—21no—
91C2×C4×C103—22no—
92C2×C4×C103—22no—
93C2×C4×C103—22no—
94C2×C4×C103—41no—
95C2×C4×C103—41no—
96C2×C4×C103—41no—
97C2×C4×C103C2×D411yescharacteristic
98C2×C4×C103C2×D411yescharacteristic
99C2×C4×C103C2×D411yescharacteristic
100C2×C4×C103C2411yescharacteristic
101C22⋊D103—42no—
102C22⋊D103—42no—
103C22⋊D103—42no—
104C2×Dic103—21no—
105C2×Dic103—21no—
106C2×Dic103—22no—
107C2×Dic103—21no—
108C2×Dic103—22no—
109C2×Dic103—42no—
110C2×Dic103—42no—
111D412—22no—
112C4×D103—42no—
113Dic206—22no—
114Dic206—22no—
115Dic206—22no—
116Dic206—42no—
117C8×C103—22no—
118C8×C103—22no—
119C22×C103—21no—
120C22×C103—21no—
121C22×C103—21no—
122C22×C103—21no—
123C22×C103—21no—
124C22×C103—21no—
125C22×C103—21no—
126C22×C103—21no—
127C22×C103—44no—
128C22×C103#13712yes—
129C22×C103C22×D411yescharacteristic
130C2×D103—41no—
131C2×D103—41no—
132C2×D103—41no—
133C4×C103—21no—
134C4×C103—21no—
135C4×C103—21no—
136C4×C103H(5,𝔽2)12yes—
137Dic103—42no—
138Dic103—41no—
139Dic103—41no—
140Dic103—42no—
141Dic103—41no—
142C2×C103—22no—
143C2×C103—41no—
144C2×C103—41no—
145C2×C103#53411yescharacteristic
146C2×C103C2×#13711yescharacteristic
147C2×C103#57511yescharacteristic
148D103—81no—
149#3196—1031nomaximal, Sylow
150C103#319611yescharacteristic, complemented, Sylow
151#525—1031no—
152C2×C22⋊Q8—1031no—
153#485—1032no—
154#484—1031no—
155#481—1031no—
156#470—1032no—
157#468—1032no—
158#452—1032no—
159#451—1031no—
160#448—1031no—
161#407D10311yescharacteristic, complemented
162C2×#41—1031no—
163C22×Q8—1031no—
164C23×C4C2×D10311yescharacteristic
165C2×Q16—1032no—
166C2×SD16—1032no—
167C2×M16C2×D10312yes—
168#123—2061no—
169C22⋊Q8—1032no—
170C22⋊Q8—1032no—
171C22⋊Q8—1031no—
172C22⋊Q8—1031no—
173C22⋊Q8—2061no—
174C22⋊Q8—2061no—
175#121—1031no—
176#121—1031no—
177#120—2061no—
178C4×D4—2061no—
179C4×D4—2061no—
180#117—1031no—
181C2×C4⋊C4—1031no—
182C2×C4⋊C4—2061no—
183C2×C22⋊C4—2061no—
184#107—1032no—
185#106—1032no—
186#103—1032no—
187#103—1032no—
188#103—1032no—
189#102—1032no—
190C22⋊C8C2×D10311yescharacteristic
191C22⋊C8C2×D10311yescharacteristic
192C22⋊C8C2×D10312yes—
193C24D41211yescharacteristic
194#41—4122no—
195C2×Q8—1032no—
196C2×Q8—1031no—
197C2×Q8—4122no—
198C2×Q8—2061no—
199C2×D4—1031no—
200C2×D4—2061no—
201C2×D4—2061no—
202C2×D4—2061no—
203C22×C4—42no—
204C22×C4—2061no—
205C22×C4—2061no—
206C22×C4—2061no—
207C22×C4D41211yescharacteristic
208C22×C4D41211yescharacteristic
209C22×C4D41211yescharacteristic
210C22×C4C22×D10311yescharacteristic
211C22×C4C22×D10311yescharacteristic
212C22×C4C22×D10311yescharacteristic
213Q16—4122no—
214SD16—4122no—
215M16—42no—
216C2×C8C22×D10312yes—
217C2×C8C22×D10312yes—
218C4⋊C4—1032no—
219C4⋊C4—1031no—
220C4⋊C4—1031no—
221C4⋊C4—2062no—
222C4⋊C4—2062no—
223C4⋊C4—2061no—
224C4⋊C4—2061no—
225C4⋊C4—2061no—
226C22⋊C4—2062no—
227C22⋊C4—2062no—
228C22⋊C4—2061no—
229C22⋊C4—2061no—
230C22⋊C4—2061no—
231C22⋊C4—2061no—
232C22⋊C4—2061no—
233C42—2061no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—2061no—
238C23C2×D41211yescharacteristic
239C23C2×D41211yescharacteristic
240C23C2×D41211yescharacteristic
241Q8—4122no—
242Q8—2062no—
243Q8—2062no—
244Q8—2062no—
245D4—4122no—
246D4—4122no—
247D4—4122no—
248D4—2062no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—4122no—
256C2×C4—4122no—
257C2×C4—4122no—
258C2×C4—2061no—
259C2×C4—2061no—
260C2×C4—2062no—
261C2×C4—2061no—
262C2×C4—2062no—
263C2×C4C2×D41211yescharacteristic
264C2×C4C2×D41211yescharacteristic
265C2×C4C2×D41211yescharacteristic
266C2×C4C23×D10311yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—4121no—
279C22—4121no—
280C22—4121no—
281C22#1569305412yes—
282C22C22×D41211yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—4121no—
287C4—4122no—
288C4—4121no—
289C4—4121no—
290C4—4122no—
291C4#1569306612yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—8241no—
296C2#2026835311yescharacteristic
297C2C2×#1569305411yescharacteristic
298C2#2026859111yescharacteristic
subgroupG/H
centerC22C22×D412
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#407×C103C2
socleC22×C103C22×D4
prime radical#20268302C2
hypercenter#407D103
hypocenterC103#3196
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#15693000C2semidirect
3. #20268133C2semidirect
4. #20268127C2semidirect
5. #20268110C2semidirect
6. #20268049C2semidirect
7. #20268037C2semidirect
8. #20267955C2semidirect
9. #20267949C2semidirect
10. #20267932C2semidirect
11. #407×C103C2semidirect
14. C2×Dic412C22semidirect
15. C2×#13110094C22semidirect
32. #15692955C22semidirect
33. #15692953C22semidirect
34. #15692946C22semidirect
35. #15692946C22semidirect
36. #15692946C22semidirect
37. #15692943C22semidirect
150. C103#3196semidirect
161. #407D103semidirect

Maximal subgroups

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

subgroupindex
#202683022
C2×#156930002
#202681332
#202681272
#202681102
#202680492
#202680372
#202679552
#202679492
#202679322
#407×C1032
#3196103