Finite Group Databasebeta

← #27205899

Subgroups — #27205899

Order 13184.

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

normal subgroups only

This group has 468 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
1C2×#15692997C211yesmaximal, characteristic, complemented
2#121×D103C211yesmaximal, characteristic
3#20268120C212yesmaximal, complemented
4#20268110C211yesmaximal, characteristic, complemented
5#20268107C211yesmaximal, characteristic, complemented
6#20268023C212yesmaximal, complemented
7#20268017C212yesmaximal, complemented
8#20267940C212yesmaximal, complemented
9#20267930C211yesmaximal, characteristic, complemented
10C22⋊C8×D103C211yesmaximal, characteristic, complemented
11#448×C103C211yesmaximal, characteristic, complemented
12C2×D4×D103—21no—
13C2×D4×D103C2211yescharacteristic
14C22×C4×D103C2211yescharacteristic
15C2×#13110092C2212yescomplemented
16C2×C8×D103C2212yescomplemented
17#15692997—21no—
18#15692997—21no—
19#15692997C2212yes—
20#15692997C2212yescomplemented
21#15692996C2211yescharacteristic
22#15692995C2211yescharacteristic
23#15692994—21no—
24#15692993—21no—
25#15692992C2211yescharacteristic
26C2×C4⋊Dic103C2211yescharacteristic, complemented
27C2×Dic103⋊C4—21no—
28C4⋊C4×D103C2211yescharacteristic
29C2×(C2×D103)⋊C4—21no—
30C22⋊C4×D103—21no—
31#15692953C2212yescomplemented
32#15692952C2212yescomplemented
33#15692945C2212yescomplemented
34#15692943C2212yescomplemented
35#15692942C2212yescomplemented
36C22⋊(C103⋊C8)C2211yescharacteristic, complemented
37(C2×D103)⋊C8C2212yes—
38C22×Q8×C103C2211yescharacteristic, complemented
39C2×SD16×C103C2212yescomplemented
40#121×C103C2211yescharacteristic
41#103×C103C2212yescomplemented
42C22⋊C8×C103C2211yescharacteristic, complemented
43C23×D103—21no—
44C23×D103D411yescharacteristic
45C2×D412—21no—
46C2×D412C2311yescharacteristic
47C2×C22⋊D103—21no—
48C2×C22⋊D103—21no—
49C2×C22⋊D103—21no—
50D4×D103—42no—
51D4×D103—44no—
52C22×Dic103—21no—
53C22×Dic103D411yescharacteristic
54C2×C4×D103—21no—
55C2×C4×D103—42no—
56C2×C4×D103D411yescharacteristic
57C2×C4×D103D411yescharacteristic
58C2×C4×D103C2311yescharacteristic
59C2×C4×D103C2311yescharacteristic
60#13110092—42no—
61C2×C103⋊C8C2312yes—
62C8×D103—42no—
63C4⋊Dic103—21no—
64C4⋊Dic103C2312yes—
65C4⋊Dic103C2311yescharacteristic
66Dic103⋊C4—21no—
67Dic103⋊C4—22no—
68Dic103⋊C4—22no—
69C22⋊Dic103—21no—
70(C2×D103)⋊C4—22no—
71(C2×D103)⋊C4—22no—
72(C2×D103)⋊C4—21no—
73(C2×D103)⋊C4—21no—
74C2×Q8×C103—21no—
75C2×Q8×C103—42no—
76C2×Q8×C103C2312yes—
77C2×D4×C103—21no—
78C2×D4×C103C2311yescharacteristic
79C22×C4×C103—21no—
80C22×C4×C103C2311yescharacteristic
81SD16×C103—42no—
82C2×C8×C103C2312yes—
83C4⋊C4×C103C2311yescharacteristic
84C22⋊C4×C103—21no—
85C23×C103—21no—
86C23×C103C2×D411yescharacteristic
87C22×D103—21no—
88C22×D103—41no—
89C22×D103—41no—
90C22×D103—42no—
91C22×D103—42no—
92C22×D103—42no—
93C22×D103SD1614yescomplemented
94C22×D103C2×D411yescharacteristic
95C22×D103C2×D411yescharacteristic
96Q8×C103—22no—
97Q8×C103—22no—
98Q8×C103—42no—
99D4×C103—22no—
100D4×C103—42no—
101C2×C4×C103—22no—
102C2×C4×C103—21no—
103C2×C4×C103—22no—
104C2×C4×C103—42no—
105C2×C4×C103C2×D411yescharacteristic
106C2×C4×C103C2411yescharacteristic
107C22⋊D103—42no—
108C22⋊D103—42no—
109C22⋊D103—42no—
110C2×Dic103—22no—
111C2×Dic103—21no—
112C2×Dic103—41no—
113C2×Dic103—42no—
114C2×Dic103C2×D411yescharacteristic
115C2×Dic103C2×D411yescharacteristic
116D412—22no—
117D412—42no—
118C4×D103—22no—
119C4×D103—22no—
120C4×D103—41no—
121C4×D103—41no—
122C4×D103—42no—
123C8×C103—22no—
124C103⋊C8—22no—
125C22×C103—21no—
126C22×C103—21no—
127C22×C103—41no—
128C22×C103—41no—
129C22×C103—41no—
130C22×C103C2×SD1612yes—
131C22×C103C22×D411yescharacteristic
132C2×D103—21no—
133C2×D103—21no—
134C2×D103—21no—
135C2×D103—21no—
136C2×D103—44no—
137C2×D103—41no—
138C2×D103—41no—
139C2×D103—41no—
140C2×D103—84no—
141C2×D103C2×SD1612yes—
142C2×D103C2×SD1612yes—
143C4×C103—21no—
144C4×C103—42no—
145C4×C103—41no—
146C4×C103H(5,𝔽2)12yes—
147Dic103—21no—
148Dic103—21no—
149Dic103—41no—
150Dic103—42no—
151C2×C103—22no—
152C2×C103—81no—
153C2×C103#53411yescharacteristic
154C2×C103C22×SD1611yescharacteristic
155C2×C103#57511yescharacteristic
156D103—22no—
157D103—22no—
158D103—81no—
159#3187—1031nomaximal, Sylow
160C103#318711yescharacteristic, complemented, Sylow
161C2×C22⋊Q8—1031no—
162C2×#121—1031no—
163#483—1032no—
164#481—1031no—
165#481—1031no—
166#465—1032no—
167#465—1032no—
168#450—1032no—
169#448—1031no—
170#448D10311yescharacteristic, complemented
171C2×C22⋊C8—1031no—
172C22×Q8C2×D10311yescharacteristic, complemented
173C22×D4—1031no—
174C22×D4—2061no—
175C23×C4—1031no—
176C2×SD16—1032no—
177C2×SD16C2×D10312yescomplemented
178C22×C8—1032no—
179C22⋊Q8—1032no—
180C22⋊Q8—1032no—
181C22⋊Q8—2061no—
182C22⋊Q8—2061no—
183#121—1031no—
184#121—1031no—
185#121—1031no—
186#121—2061no—
187#121—2061no—
188#121C2×D10311yescharacteristic
189C2×C4⋊C4—1031no—
190C2×C4⋊C4—1031no—
191C2×C4⋊C4—2061no—
192C2×C22⋊C4—2061no—
193C2×C22⋊C4—2061no—
194#106—1032no—
195#106—1032no—
196#103—1032no—
197#103C2×D10312yescomplemented
198#102—1032no—
199#102—1032no—
200C22⋊C8—1031no—
201C22⋊C8—1032no—
202C22⋊C8C2×D10311yescharacteristic, complemented
203C24—1031no—
204C24—2061no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D10312yes—
208C2×D4—21no—
209C2×D4—1031no—
210C2×D4—4124no—
211C2×D4—4122no—
212C2×D4—2061no—
213C2×D4—2061no—
214C2×D4—2061no—
215C2×D4—2061no—
216C2×D4C22×D10311yescharacteristic
217C22×C4—21no—
218C22×C4—1031no—
219C22×C4—1031no—
220C22×C4—1031no—
221C22×C4—1031no—
222C22×C4—1031no—
223C22×C4—4122no—
224C22×C4—2061no—
225C22×C4—2061no—
226C22×C4C22×D10311yescharacteristic
227SD16—42no—
228SD16—4122no—
229C2×C8—1032no—
230C2×C8—4122no—
231C2×C8C22×D10312yes—
232C4⋊C4—1032no—
233C4⋊C4—1031no—
234C4⋊C4—2061no—
235C4⋊C4—2062no—
236C4⋊C4—2062no—
237C4⋊C4—2061no—
238C4⋊C4C22×D10311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—2061no—
241C22⋊C4—2062no—
242C22⋊C4—2062no—
243C22⋊C4—2061no—
244C22⋊C4—2061no—
245C23—21no—
246C23—1034no—
247C23—1031no—
248C23—1031no—
249C23—4121no—
250C23—4121no—
251C23—4122no—
252C23—4122no—
253C23—4122no—
254C23—2061no—
255C23D4×D10311yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—4122no—
262D4—4122no—
263D4—4122no—
264D4—4122no—
265D4—2062no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—1031no—
271C2×C4—1031no—
272C2×C4—4121no—
273C2×C4—4121no—
274C2×C4—4122no—
275C2×C4—4122no—
276C2×C4—4121no—
277C2×C4—2062no—
278C2×C4—2061no—
279C2×C4—2062no—
280C2×C4—2062no—
281C2×C4D4×D10311yescharacteristic
282C2×C4C23×D10311yescharacteristic
283C8—22no—
284C8—2062no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—8244no—
291C22—1032no—
292C22—1032no—
293C22—4124no—
294C22—4121no—
295C22—4121no—
296C22—4121no—
297C22—2061no—
298C22—2061no—
299C22—2061no—
300C22—2061no—
301C22SD16×D10312yes—
302C22C2×D4×D10311yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—4121no—
307C4—4122no—
308C4—2061no—
309C4—2061no—
310C4#1569306612yes—
311C2—22no—
312C2—81no—
313C2—8241no—
314C2—2062no—
315C2—2062no—
316C2#2026835111yescharacteristic
317C2C2×SD16×D10311yescharacteristic
318C2#2026859011yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#448×C103C2
socleC22×C103C22×D4
prime radical#121×D103C2
hypercenter#448D103
hypocenterC103#3187
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#15692997C2semidirect
3. #20268120C2semidirect
4. #20268110C2semidirect
5. #20268107C2semidirect
6. #20268023C2semidirect
7. #20268017C2semidirect
8. #20267940C2semidirect
9. #20267930C2semidirect
10. C22⋊C8×D103C2semidirect
11. #448×C103C2semidirect
15. C2×#13110092C22semidirect
16. C2×C8×D103C22semidirect
20. #15692997C22semidirect
26. C2×C4⋊Dic103C22semidirect
31. #15692953C22semidirect
32. #15692952C22semidirect
33. #15692945C22semidirect
34. #15692943C22semidirect
35. #15692942C22semidirect
36. C22⋊(C103⋊C8)C22semidirect
38. C22×Q8×C103C22semidirect
39. C2×SD16×C103C22semidirect
41. #103×C103C22semidirect
42. C22⋊C8×C103C22semidirect
93. C22×D103SD16semidirect
160. C103#3187semidirect
170. #448D103semidirect
172. C22×Q8C2×D103semidirect
177. C2×SD16C2×D103semidirect
197. #103C2×D103semidirect
202. C22⋊C8C2×D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156929972
#121×D1032
#202681202
#202681102
#202681072
#202680232
#202680172
#202679402
#202679302
C22⋊C8×D1032
#448×C1032
#3187103