Finite Group Databasebeta

← #27205925

Subgroups — #27205925

Order 13184.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#20268129C212yesmaximal, complemented
3#20268121C211yesmaximal, characteristic, complemented
4#20268104C211yesmaximal, characteristic, complemented
5#20268030C212yesmaximal, complemented
6#20268022C212yesmaximal, complemented
7#20267951C212yesmaximal, complemented
8#20267947C211yesmaximal, characteristic, complemented
9#20267930C211yesmaximal, characteristic, complemented
10C2×C22⋊(C103⋊C8)C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C103C211yesmaximal, characteristic, complemented
12C2×#13110105C2211yescharacteristic
13C2×#13110095C2212yescomplemented
14C2×#13110092C2212yescomplemented
15C22×C103⋊C8C2212yes—
16#15693004—21no—
17#15693000C2211yescharacteristic
18#15693000C2211yescharacteristic
19#15692995C2211yescharacteristic
20#15692995C2211yescharacteristic
21#15692990—21no—
22C4×C22⋊D103—21no—
23C4×C22⋊D103—21no—
24#15692980C2211yescharacteristic
25#15692954C2212yescomplemented
26#15692952C2212yescomplemented
27#15692945C2212yescomplemented
28#15692945C2212yescomplemented
29#15692944C2212yescomplemented
30#15692941C2212yescomplemented
31C22⋊(C103⋊C8)C2211yescharacteristic
32C22⋊(C103⋊C8)C2211yescharacteristic
33C22⋊(C103⋊C8)C2212yes—
34C22×Q8×C103C2211yescharacteristic
35C23×C4×C103C2211yescharacteristic
36C22⋊Q8×C103—21no—
37C22⋊Q8×C103—21no—
38C22⋊Q8×C103C2212yes—
39C22⋊Q8×C103C2212yes—
40C2×C4⋊C4×C103—21no—
41C2×C4⋊C4×C103C2211yescharacteristic
42C2×C22⋊C4×C103—21no—
43#13110105—42no—
44C2×Dic206C2311yescharacteristic
45C2×D412C2311yescharacteristic
46C2×C22⋊D103—21no—
47C2×C22⋊D103—21no—
48C2×C22⋊D103—21no—
49C2×C4×D103—21no—
50#13110095—42no—
51#13110092—42no—
52C2×C103⋊C8—42no—
53C2×C103⋊C8C2312yes—
54C2×C103⋊C8C2312yes—
55C4⋊Dic103C2311yescharacteristic
56C4⋊Dic103C2311yescharacteristic
57Dic103⋊C4—21no—
58Dic103⋊C4—21no—
59C22⋊Dic103—21no—
60C22⋊Dic103—21no—
61C22⋊Dic103—21no—
62(C2×D103)⋊C4—21no—
63(C2×D103)⋊C4—21no—
64C4×Dic103—21no—
65C24×C103D411yescharacteristic
66C2×Q8×C103—21no—
67C2×Q8×C103—42no—
68C2×Q8×C103C2312yes—
69C22×C4×C103—21no—
70C22×C4×C103—21no—
71C22×C4×C103—42no—
72C22×C4×C103D411yescharacteristic
73C22×C4×C103D411yescharacteristic
74C22×C4×C103D411yescharacteristic
75C22×C4×C103C2311yescharacteristic
76C22×C4×C103C2311yescharacteristic
77C22×C4×C103C2311yescharacteristic
78C4⋊C4×C103—22no—
79C4⋊C4×C103—22no—
80C4⋊C4×C103—21no—
81C4⋊C4×C103C2312yes—
82C22⋊C4×C103—22no—
83C22⋊C4×C103—22no—
84C23×C103—22no—
85C23×C103—22no—
86C23×C103—41no—
87C23×C103C2×D411yescharacteristic
88C23×C103C2×D411yescharacteristic
89C23×C103C2×D411yescharacteristic
90C22×D103—21no—
91Q8×C103—22no—
92Q8×C103—22no—
93Q8×C103—42no—
94C2×C4×C103—22no—
95C2×C4×C103—22no—
96C2×C4×C103—22no—
97C2×C4×C103—22no—
98C2×C4×C103—22no—
99C2×C4×C103—41no—
100C2×C4×C103—41no—
101C2×C4×C103—41no—
102C2×C4×C103—42no—
103C2×C4×C103—42no—
104C2×C4×C103C2×D411yescharacteristic
105C2×C4×C103C2×D411yescharacteristic
106C2×C4×C103C2×D411yescharacteristic
107C2×C4×C103C2411yescharacteristic
108C22⋊D103—42no—
109C22⋊D103—42no—
110C22⋊D103—42no—
111C2×Dic103—21no—
112C2×Dic103—21no—
113C2×Dic103—21no—
114D412—22no—
115C4×D103—42no—
116Dic206—22no—
117C103⋊C8—22no—
118C103⋊C8—22no—
119C22×C103—21no—
120C22×C103—21no—
121C22×C103—21no—
122C22×C103—21no—
123C22×C103—21no—
124C22×C103—21no—
125C22×C103—21no—
126C22×C103—44no—
127C22×C103#13512yes—
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×C103—42no—
137C4×C103—42no—
138C4×C103H(5,𝔽2)12yes—
139Dic103—41no—
140Dic103—41no—
141Dic103—41no—
142C2×C103—22no—
143C2×C103—22no—
144C2×C103—41no—
145C2×C103#53411yescharacteristic
146C2×C103C2×#13511yescharacteristic
147C2×C103C2×#13711yescharacteristic
148D103—81no—
149#3190—1031nomaximal, Sylow
150C103#319011yescharacteristic, complemented, Sylow
151#525—1031no—
152C2×C22⋊Q8D10311yescharacteristic, complemented
153#485—1032no—
154#483—1031no—
155#480—1031no—
156#467—1032no—
157#465—1032no—
158#452—1032no—
159#451—1031no—
160#448—1031no—
161C2×C22⋊C8—1031no—
162C2×#41—1031no—
163C22×Q8C2×D10311yescharacteristic
164C23×C4C2×D10311yescharacteristic
165C2×Q16—1032no—
166C2×SD16—1032no—
167C22×C8—1032no—
168#123—2061no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—1031no—
172C22⋊Q8—1031no—
173C22⋊Q8C2×D10312yes—
174C22⋊Q8C2×D10312yes—
175#121—1031no—
176#121—1031no—
177#120—2061no—
178C4×D4—2061no—
179C4×D4—2061no—
180#117—1031no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D10311yescharacteristic
183C2×C22⋊C4—21no—
184#107—1032no—
185#106—1032no—
186#103—1032no—
187#103—1032no—
188#103—1032no—
189#102—1032no—
190C22⋊C8—1031no—
191C22⋊C8—1031no—
192C22⋊C8—1032no—
193C24C22⋊D10311yescharacteristic
194#41—4122no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—1031no—
198C2×Q8C22×D10312yes—
199C2×D4—1031no—
200C2×D4—2061no—
201C2×D4—2061no—
202C2×D4—2061no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—2061no—
207C22×C4C22⋊D10311yescharacteristic
208C22×C4C22⋊D10311yescharacteristic
209C22×C4C22⋊D10311yescharacteristic
210C22×C4C22×D10311yescharacteristic
211C22×C4C22×D10311yescharacteristic
212C22×C4C22×D10311yescharacteristic
213Q16—4122no—
214SD16—4122no—
215C2×C8—1032no—
216C2×C8—1032no—
217C2×C8—4122no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—1031no—
222C4⋊C4—1031no—
223C4⋊C4—2061no—
224C4⋊C4—2061no—
225C4⋊C4C22×D10312yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—2061no—
229C22⋊C4—2061no—
230C22⋊C4—2061no—
231C22⋊C4—2061no—
232C22⋊C4—2061no—
233C42—2061no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—2061no—
238C23C2×C22⋊D10311yescharacteristic
239C23C2×C22⋊D10311yescharacteristic
240C23C2×C22⋊D10311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—2062no—
245D4—4122no—
246D4—4122no—
247D4—4122no—
248D4—2062no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—4122no—
260C2×C4—2061no—
261C2×C4—2061no—
262C2×C4—2061no—
263C2×C4C2×C22⋊D10311yescharacteristic
264C2×C4C2×C22⋊D10311yescharacteristic
265C2×C4C2×C22⋊D10311yescharacteristic
266C2×C4C23×D10311yescharacteristic
267C8—2062no—
268C8—2062no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—4121no—
278C22—4121no—
279C22—4121no—
280C22#1569304112yes—
281C22#1569305212yes—
282C22C22×C22⋊D10311yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—4121no—
289C4—4121no—
290C4—4121no—
291C4#1569306612yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—8241no—
296C2#2026835011yescharacteristic
297C2C2×#1569304111yescharacteristic
298C2C2×#1569305211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC2×C22⋊Q8×C103C2
socleC22×C103C22×D4
prime radical#20268302C2
hypercenterC2×C22⋊Q8D103
hypocenterC103#3190
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20268129C2semidirect
3. #20268121C2semidirect
4. #20268104C2semidirect
5. #20268030C2semidirect
6. #20268022C2semidirect
7. #20267951C2semidirect
8. #20267947C2semidirect
9. #20267930C2semidirect
10. C2×C22⋊(C103⋊C8)C2semidirect
11. C2×C22⋊Q8×C103C2semidirect
13. C2×#13110095C22semidirect
14. C2×#13110092C22semidirect
25. #15692954C22semidirect
26. #15692952C22semidirect
27. #15692945C22semidirect
28. #15692945C22semidirect
29. #15692944C22semidirect
30. #15692941C22semidirect
150. C103#3190semidirect
152. C2×C22⋊Q8D103semidirect

Maximal subgroups

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

subgroupindex
#202683022
#202681292
#202681212
#202681042
#202680302
#202680222
#202679512
#202679472
#202679302
C2×C22⋊(C103⋊C8)2
C2×C22⋊Q8×C1032
#3190103