Finite Group Databasebeta

← #27205949

Subgroups — #27205949

Order 13184.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22⋊Q8×D103C211yesmaximal, characteristic, complemented
2C2×#15692996C211yesmaximal, characteristic
3#20268126C211yesmaximal, characteristic, complemented
4#20268106C212yesmaximal, complemented
5#20268040C212yesmaximal, complemented
6#20268038C212yesmaximal, complemented
7#20267944C211yesmaximal, characteristic, complemented
8#20267941C211yesmaximal, characteristic, complemented
9#20267931C212yesmaximal, complemented
10#20267753C211yesmaximal, characteristic, complemented
11#484×C103C211yesmaximal, characteristic, complemented
12C2×Q8×D103C2211yescharacteristic, complemented
13C22×D412—21no—
14C22×D412C2211yescharacteristic
15C22×C4×D103C2211yescharacteristic
16C2×#13110094C2212yescomplemented
17C2×#13110092C2212yescomplemented
18C2×#13110088C2212yescomplemented
19#15693001C2211yescharacteristic, complemented
20#15693000C2211yescharacteristic
21#15692999—21no—
22#15692998—21no—
23#15692997C2211yescharacteristic, complemented
24#15692996—21no—
25#15692996—21no—
26#15692996C2212yes—
27#15692996C2212yes—
28C4⋊C4×D103—21no—
29C4⋊C4×D103C2211yescharacteristic, complemented
30C2×(C2×D103)⋊C4—21no—
31C22⋊C4×D103—21no—
32#15692954C2212yescomplemented
33#15692944C2212yescomplemented
34#15692943C2212yescomplemented
35#15692941C2212yescomplemented
36C22⋊(C103⋊C8)C2211yescharacteristic
37(C2×D103)⋊C8C2212yescomplemented
38C22⋊Q8×C103C2211yescharacteristic
39C2×C4⋊C4×C103C2211yescharacteristic
40#107×C103C2212yescomplemented
41#103×C103C2212yescomplemented
42C22⋊C8×C103C2211yescharacteristic
43C23×D103—21no—
44C23×D103D411yescharacteristic
45C2×Dic206—21no—
46C2×Dic206C2311yescharacteristic
47Q8×D103—42no—
48C2×D412—22no—
49C2×D412—22no—
50C2×D412—21no—
51C2×D412—42no—
52C2×D412—44no—
53C2×D412C2312yes—
54C22×Dic103D411yescharacteristic
55C2×C4×D103—21no—
56C2×C4×D103—21no—
57C2×C4×D103—42no—
58C2×C4×D103D411yescharacteristic
59C2×C4×D103D411yescharacteristic
60C2×C4×D103C2311yescharacteristic
61C2×C4×D103C2311yescharacteristic
62#13110094—42no—
63#13110092—42no—
64#13110088—42no—
65C2×C103⋊C8C2312yes—
66C4⋊Dic103—21no—
67C4⋊Dic103C2311yescharacteristic
68Dic103⋊C4—21no—
69Dic103⋊C4—21no—
70Dic103⋊C4—21no—
71C22⋊Dic103—21no—
72(C2×D103)⋊C4—21no—
73(C2×D103)⋊C4—21no—
74(C2×D103)⋊C4—22no—
75(C2×D103)⋊C4—22no—
76C2×Q8×C103C2311yescharacteristic
77C22×C4×C103—21no—
78C22×C4×C103C2311yescharacteristic
79C2×C8×C103C2312yes—
80C4⋊C4×C103—21no—
81C4⋊C4×C103—21no—
82C4⋊C4×C103C2312yes—
83C4⋊C4×C103C2311yescharacteristic
84C22⋊C4×C103—21no—
85C23×C103C2×D411yescharacteristic
86C22×D103—22no—
87C22×D103—22no—
88C22×D103—22no—
89C22×D103—41no—
90C22×D103—42no—
91C22×D103—42no—
92C22×D103—42no—
93C22×D103C2×D411yescharacteristic
94C22×D103C2×D411yescharacteristic
95Q8×C103—22no—
96C2×C4×C103—22no—
97C2×C4×C103—21no—
98C2×C4×C103—21no—
99C2×C4×C103—22no—
100C2×C4×C103—42no—
101C2×C4×C103C2×D411yescharacteristic
102C2×C4×C103C2411yescharacteristic
103C2×Dic103—21no—
104C2×Dic103—21no—
105C2×Dic103—41no—
106C2×Dic103C2×D411yescharacteristic
107C2×Dic103C2×D411yescharacteristic
108D412—22no—
109D412—22no—
110D412—42no—
111D412—44no—
112D412—44no—
113C4×D103—22no—
114C4×D103—22no—
115C4×D103—41no—
116C4×D103—41no—
117C4×D103—42no—
118C4×D103—42no—
119Dic206—22no—
120Dic206—42no—
121C8×C103—22no—
122C103⋊C8—22no—
123C22×C103—21no—
124C22×C103—21no—
125C22×C103#13712yes—
126C22×C103C22×D411yescharacteristic
127C2×D103—21no—
128C2×D103—21no—
129C2×D103—21no—
130C2×D103—21no—
131C2×D103—21no—
132C2×D103—44no—
133C2×D103—42no—
134C2×D103—42no—
135C2×D103—42no—
136C2×D103—84no—
137C2×D103AGL(1,ℤ/8)12yes—
138C4×C103—21no—
139C4×C103—41no—
140C4×C103—41no—
141C4×C103—42no—
142C4×C103H(5,𝔽2)12yes—
143Dic103—21no—
144Dic103—21no—
145Dic103—41no—
146Dic103—41no—
147C2×C103—22no—
148C2×C103#53411yescharacteristic
149C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C103C2×#13711yescharacteristic
151D103—22no—
152D103—41no—
153D103—82no—
154#3193—1031nomaximal, Sylow
155C103#319311yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—1031no—
157C2×#121—1031no—
158#484—1031no—
159#484D10311yescharacteristic, complemented
160#480—1032no—
161#468—1032no—
162#468—1032no—
163#450—1031no—
164#450—1031no—
165#448—1032no—
166#407—1031no—
167C22×Q8—1031no—
168C22×D4—1031no—
169C22×D4—2061no—
170C23×C4—1031no—
171C2×SD16—1032no—
172C2×SD16—1032no—
173C2×M16—1032no—
174C22⋊Q8—1031no—
175C22⋊Q8—1031no—
176C22⋊Q8—1031no—
177C22⋊Q8—2061no—
178C22⋊Q8—2061no—
179C22⋊Q8C2×D10311yescharacteristic
180#121—1032no—
181#121—1032no—
182#121—2061no—
183#121—2061no—
184C2×C4⋊C4—1031no—
185C2×C4⋊C4—2061no—
186C2×C4⋊C4C2×D10311yescharacteristic
187C2×C22⋊C4—2061no—
188C2×C22⋊C4—2061no—
189#107—1032no—
190#107C2×D10312yescomplemented
191#103—1032no—
192#103C2×D10312yescomplemented
193#102—1032no—
194#102—1032no—
195C22⋊C8—1031no—
196C22⋊C8—1032no—
197C22⋊C8C2×D10311yescharacteristic
198C24—1031no—
199C24—2061no—
200C2×Q8—1031no—
201C2×Q8—4122no—
202C2×Q8—2061no—
203C2×Q8C22×D10311yescharacteristic
204C2×D4—1032no—
205C2×D4—4122no—
206C2×D4—4124no—
207C2×D4—2062no—
208C2×D4—2062no—
209C2×D4—2061no—
210C22×C4—21no—
211C22×C4—1031no—
212C22×C4—1031no—
213C22×C4—1031no—
214C22×C4—1031no—
215C22×C4—1031no—
216C22×C4—4122no—
217C22×C4—2061no—
218C22×C4—2061no—
219C22×C4C22×D10311yescharacteristic
220SD16—4122no—
221SD16—4122no—
222M16—4122no—
223C2×C8—1032no—
224C2×C8C22×D10312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—1031no—
228C4⋊C4—2061no—
229C4⋊C4—2061no—
230C4⋊C4—2061no—
231C4⋊C4—2061no—
232C4⋊C4C22×D10311yescharacteristic
233C4⋊C4C22×D10312yes—
234C22⋊C4—21no—
235C22⋊C4—2062no—
236C22⋊C4—2062no—
237C22⋊C4—2061no—
238C22⋊C4—2061no—
239C22⋊C4—2061no—
240C23—1031no—
241C23—1031no—
242C23—4122no—
243C23—4122no—
244C23—4121no—
245C23—4122no—
246C23—2062no—
247C23—2062no—
248C23—2062no—
249C23D4×D10311yescharacteristic
250Q8—22no—
251Q8—4122no—
252Q8—2062no—
253D4—4122no—
254D4—4124no—
255D4—4124no—
256D4—2062no—
257D4—2062no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—1031no—
264C2×C4—1031no—
265C2×C4—4121no—
266C2×C4—4121no—
267C2×C4—4122no—
268C2×C4—4122no—
269C2×C4—4121no—
270C2×C4—2061no—
271C2×C4—2061no—
272C2×C4—2062no—
273C2×C4—2062no—
274C2×C4D4×D10311yescharacteristic
275C2×C4C23×D10311yescharacteristic
276C8—22no—
277C8—2062no—
278C22—21no—
279C22—21no—
280C22—8244no—
281C22—1032no—
282C22—4124no—
283C22—4122no—
284C22—4122no—
285C22—4122no—
286C22—2061no—
287C22—2061no—
288C22—2061no—
289C22—2061no—
290C22—2061no—
291C22#1569305312yes—
292C22C2×D4×D10311yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4121no—
298C4—4121no—
299C4—2061no—
300C4—2061no—
301C4#1569306712yes—
302C2—22no—
303C2—8242no—
304C2—4121no—
305C2—2062no—
306C2#2026835211yescharacteristic
307C2AGL(1,ℤ/8)×D10311yescharacteristic
308C2C2×#1569305311yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#484×C103C2
socleC22×C103C22×D4
prime radicalC2×#15692996C2
hypercenter#484D103
hypocenterC103#3193
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
1. C22⋊Q8×D103C2semidirect
3. #20268126C2semidirect
4. #20268106C2semidirect
5. #20268040C2semidirect
6. #20268038C2semidirect
7. #20267944C2semidirect
8. #20267941C2semidirect
9. #20267931C2semidirect
10. #20267753C2semidirect
11. #484×C103C2semidirect
12. C2×Q8×D103C22semidirect
16. C2×#13110094C22semidirect
17. C2×#13110092C22semidirect
18. C2×#13110088C22semidirect
19. #15693001C22semidirect
23. #15692997C22semidirect
29. C4⋊C4×D103C22semidirect
32. #15692954C22semidirect
33. #15692944C22semidirect
34. #15692943C22semidirect
35. #15692941C22semidirect
37. (C2×D103)⋊C8C22semidirect
40. #107×C103C22semidirect
41. #103×C103C22semidirect
155. C103#3193semidirect
159. #484D103semidirect
190. #107C2×D103semidirect
192. #103C2×D103semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1032
C2×#156929962
#202681262
#202681062
#202680402
#202680382
#202679442
#202679412
#202679312
#202677532
#484×C1032
#3193103