Finite Group Databasebeta

← #28374829

Subgroups — #28374829

Order 13952.

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

normal subgroups only

This group has 476 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#20295829C212yesmaximal, complemented
2#20295828C211yesmaximal, characteristic, complemented
3#20295401C211yesmaximal, characteristic, complemented
4#20295296C211yesmaximal, characteristic, complemented
5#20295294C212yesmaximal, complemented
6#20295293C211yesmaximal, characteristic, complemented
7#20295194C212yesmaximal, complemented
8#20295192C212yesmaximal, complemented
9#20295143C211yesmaximal, characteristic, complemented
10#20295141C211yesmaximal, characteristic, complemented
11#576×C109C211yesmaximal, characteristic, complemented
12#16812745C2212yescomplemented
13#16812726—22no—
14#16812725—22no—
15#16812725C2212yescomplemented
16#16812724—22no—
17#16812724C2212yescomplemented
18#16812724C2212yescomplemented
19#16812722—21no—
20#16812719—22no—
21#16812719C2211yescharacteristic
22C2×D872C2211yescharacteristic, complemented
23C2×#13182014—22no—
24C2×#13182014C2212yescomplemented
25D8×D109—21no—
26#16812706C2212yescomplemented
27#16812705C2211yescharacteristic, complemented
28#16812695C2212yescomplemented
29#16812635C2211yescharacteristic, complemented
30#16812634C2211yescharacteristic, complemented
31#16812626C2212yescomplemented
32#16812626C2211yescharacteristic, complemented
33#16812625C2211yescharacteristic, complemented
34#16812616C2212yescomplemented
35#16812615C2212yescomplemented
36C8×Dic109C2211yescharacteristic, complemented
37H(5,𝔽2)×C109C2212yescomplemented
38AGL(1,ℤ/8)×C109—22no—
39AGL(1,ℤ/8)×C109C2212yescomplemented
40#135×C109—21no—
41#135×C109C2211yescharacteristic
42C2×D8×C109—21no—
43C2×D8×C109C2211yescharacteristic, complemented
44#131×C109C2211yescharacteristic, complemented
45#13182030—22no—
46#13182029—22no—
47#13182029C2311yescharacteristic
48C2×D436—22no—
49C2×D436C2312yes—
50C2×C22⋊D109—42no—
51D4×D109—22no—
52D4×D109—42no—
53#13182019D411yescharacteristic, complemented
54#13182017—22no—
55#13182017D411yescharacteristic, complemented
56#13182016—22no—
57#13182016—22no—
58#13182016D411yescharacteristic, complemented
59D872—21no—
60D872—21no—
61#13182014—22no—
62#13182014—22no—
63#13182014—22no—
64#13182014—22no—
65#13182014—22no—
66#13182014D411yescharacteristic, complemented
67#13182013—22no—
68#13182013C2312yes—
69#13182013C2311yescharacteristic
70#13182012—21no—
71C2×#11724784—22no—
72C2×#11724784C2311yescharacteristic
73C8×D109—21no—
74C4×Dic109C2311yescharacteristic
75#41×C109—22no—
76#41×C109—22no—
77#41×C109C2311yescharacteristic
78#41×C109C2312yes—
79C2×D4×C109—22no—
80C2×D4×C109—22no—
81C2×D4×C109—42no—
82C2×D4×C109C2312yes—
83Q16×C109D411yescharacteristic, complemented
84SD16×C109—22no—
85SD16×C109D412yescomplemented
86D8×C109—22no—
87D8×C109—21no—
88D8×C109—21no—
89D8×C109D411yescharacteristic, complemented
90M16×C109—21no—
91M16×C109C2311yescharacteristic
92C2×C8×C109—21no—
93C2×C8×C109C2311yescharacteristic
94C23×C109—22no—
95C23×C109—42no—
96C22×D109—22no—
97C22×D109—42no—
98Q8×C109C2×D411yescharacteristic
99Q8×C109C2×D412yes—
100D4×C109—21no—
101D4×C109—22no—
102D4×C109—22no—
103D4×C109—22no—
104D4×C109—42no—
105D4×C109—42no—
106D4×C109C2×D411yescharacteristic
107D4×C109C2×D412yes—
108C2×C4×C109—21no—
109C2×C4×C109—22no—
110C2×C4×C109—42no—
111C2×C4×C109C2411yescharacteristic
112C22⋊D109—21no—
113C22⋊D109—42no—
114C22⋊D109—84no—
115C2×Dic109—21no—
116D436—22no—
117D436—22no—
118D436—42no—
119D436C2×D411yescharacteristic
120C4×D109—21no—
121C4×D109—42no—
122Dic218C2×D411yescharacteristic
123C8×C109—21no—
124C8×C109C2×D411yescharacteristic
125C8×C109C2×D411yescharacteristic
126#11724784—21no—
127#11724784—22no—
128#11724784C2×D411yescharacteristic
129#11724784C2×D411yescharacteristic
130C22×C109—22no—
131C22×C109—21no—
132C22×C109—42no—
133C22×C109—82no—
134C22×C109—162no—
135C22×C109H(5,𝔽2)11yescharacteristic
136C2×D109—21no—
137C2×D109—42no—
138C2×D109—82no—
139C2×D109—162no—
140C4×C109—22no—
141C4×C109—21no—
142C4×C109C22×D411yescharacteristic
143C4×C109C22×D411yescharacteristic
144Dic109—22no—
145Dic109—21no—
146C2×C109—21no—
147C2×C109—42no—
148C2×C109—41no—
149C2×C109—82no—
150C2×C109D4211yescharacteristic
151D109—41no—
152D109—82no—
153#3292—1091nomaximal, Sylow
154C109#329211yescharacteristic, complemented, Sylow
155#576—1092no—
156#576—1091no—
157#576D10911yescharacteristic, complemented
158#493—1091no—
159#472—1092no—
160#472—1091no—
161#472—1091no—
162#453—1092no—
163#453—1092no—
164#443—1091no—
165#443—1091no—
166H(5,𝔽2)—1092no—
167H(5,𝔽2)C2×D10912yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1092no—
170AGL(1,ℤ/8)—1092no—
171AGL(1,ℤ/8)—1092no—
172AGL(1,ℤ/8)—2182no—
173AGL(1,ℤ/8)—2182no—
174AGL(1,ℤ/8)—2182no—
175AGL(1,ℤ/8)C2×D10912yescomplemented
176#135—21no—
177#135—1091no—
178#135—2182no—
179#135—2181no—
180#135C2×D10911yescharacteristic
181C2×D8—21no—
182C2×D8—1092no—
183C2×D8—1091no—
184C2×D8—2182no—
185C2×D8—2181no—
186C2×D8C2×D10911yescharacteristic, complemented
187#131—1092no—
188#131—1091no—
189#131C2×D10911yescharacteristic, complemented
190#127—1092no—
191#108—1091no—
192#108—1091no—
193#104—1091no—
194#104—1092no—
195#104—1091no—
196#100—1092no—
197#100—1092no—
198C4×C8—1091no—
199#41—22no—
200#41—22no—
201#41—1091no—
202#41—2182no—
203#41—2182no—
204#41C22×D10911yescharacteristic
205#41C22×D10912yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—1092no—
210C2×D4—4362no—
211C2×D4—4362no—
212C2×D4—2182no—
213C2×D4—2182no—
214C2×D4C22×D10912yes—
215Q16—1091no—
216Q16C22⋊D10911yescharacteristic, complemented
217SD16—22no—
218SD16—1091no—
219SD16—1091no—
220SD16—2182no—
221SD16—2182no—
222SD16—2182no—
223SD16C22⋊D10912yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1091no—
228D8—2182no—
229D8—2182no—
230D8—2182no—
231D8—2182no—
232D8—2182no—
233D8—2181no—
234D8—2181no—
235D8C22⋊D10911yescharacteristic, complemented
236M16—21no—
237M16—1092no—
238M16—1091no—
239M16—2182no—
240M16—2181no—
241M16C22×D10911yescharacteristic
242C2×C8—21no—
243C2×C8—1091no—
244C2×C8—2182no—
245C2×C8—2181no—
246C2×C8C22×D10911yescharacteristic
247C42—1091no—
248C23—22no—
249C23—42no—
250C23—4362no—
251C23—2182no—
252Q8—1091no—
253Q8D4×D10911yescharacteristic
254Q8C2×C22⋊D10912yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8724no—
262D4—1091no—
263D4—4362no—
264D4—4362no—
265D4—2181no—
266D4—2182no—
267D4—2182no—
268D4D4×D10911yescharacteristic
269D4C2×C22⋊D10912yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—4362no—
274C2×C4—2181no—
275C2×C4—2181no—
276C2×C4C23×D10911yescharacteristic
277C8—21no—
278C8—1091no—
279C8—1091no—
280C8—2182no—
281C8—2181no—
282C8C2×C22⋊D10911yescharacteristic
283C8C2×C22⋊D10911yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—8722no—
290C22—4362no—
291C22—17442no—
292C22—2181no—
293C22#1681274411yescharacteristic
294C4—21no—
295C4—22no—
296C4—2182no—
297C4—2181no—
298C4C2×D4×D10911yescharacteristic
299C4C22×C22⋊D10911yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—8722no—
305C2—4361no—
306C2D4×C22⋊D10911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#576×C109C2
socleC2×C109D42
hypercenter#576D109
hypocenterC109#3292
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20295829C2semidirect
2. #20295828C2semidirect
3. #20295401C2semidirect
4. #20295296C2semidirect
5. #20295294C2semidirect
6. #20295293C2semidirect
7. #20295194C2semidirect
8. #20295192C2semidirect
9. #20295143C2semidirect
10. #20295141C2semidirect
11. #576×C109C2semidirect
12. #16812745C22semidirect
15. #16812725C22semidirect
17. #16812724C22semidirect
18. #16812724C22semidirect
22. C2×D872C22semidirect
24. C2×#13182014C22semidirect
26. #16812706C22semidirect
27. #16812705C22semidirect
28. #16812695C22semidirect
29. #16812635C22semidirect
30. #16812634C22semidirect
31. #16812626C22semidirect
32. #16812626C22semidirect
33. #16812625C22semidirect
34. #16812616C22semidirect
35. #16812615C22semidirect
36. C8×Dic109C22semidirect
37. H(5,𝔽2)×C109C22semidirect
39. AGL(1,ℤ/8)×C109C22semidirect
43. C2×D8×C109C22semidirect
44. #131×C109C22semidirect
53. #13182019D4semidirect
55. #13182017D4semidirect
58. #13182016D4semidirect
66. #13182014D4semidirect
83. Q16×C109D4semidirect
85. SD16×C109D4semidirect
89. D8×C109D4semidirect
154. C109#3292semidirect
157. #576D109semidirect
167. H(5,𝔽2)C2×D109semidirect
175. AGL(1,ℤ/8)C2×D109semidirect
186. C2×D8C2×D109semidirect
189. #131C2×D109semidirect
216. Q16C22⋊D109semidirect
223. SD16C22⋊D109semidirect
235. D8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202958292
#202958282
#202954012
#202952962
#202952942
#202952932
#202951942
#202951922
#202951432
#202951412
#576×C1092
#3292109