Finite Group Databasebeta

← #28374830

Subgroups — #28374830

Order 13952.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#20295831C211yesmaximal, characteristic, complemented
2#20295830C212yesmaximal, complemented
3#20295827C211yesmaximal, characteristic, complemented
4#20295401C211yesmaximal, characteristic, complemented
5#20295297C211yesmaximal, characteristic, complemented
6#20295295C212yesmaximal, complemented
7#20295196C212yesmaximal, complemented
8#20295192C212yesmaximal, complemented
9#20295142C211yesmaximal, characteristic, complemented
10#20295141C211yesmaximal, characteristic, complemented
11#472×C109C211yesmaximal, characteristic, complemented
12#16812745C2212yescomplemented
13#16812744C2212yescomplemented
14#16812727—22no—
15#16812727C2212yescomplemented
16#16812727C2212yescomplemented
17#16812726—22no—
18#16812725C2212yescomplemented
19#16812723—22no—
20#16812723—22no—
21#16812721—21no—
22#16812721C2211yescharacteristic
23#16812720—22no—
24#16812719C2211yescharacteristic
25#16812718—21no—
26C2×D872—21no—
27C2×D872C2211yescharacteristic, complemented
28C2×#13182014C2212yescomplemented
29D8×D109—21no—
30D8×D109—22no—
31#16812707C2212yescomplemented
32#16812705C2211yescharacteristic, complemented
33#16812695C2212yescomplemented
34#16812634C2211yescharacteristic, complemented
35#16812626C2211yescharacteristic, complemented
36#16812625C2212yescomplemented
37#16812625C2211yescharacteristic, complemented
38#16812615C2212yescomplemented
39C8×Dic109C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C109C2212yescomplemented
41C2×D8×C109C2211yescharacteristic, complemented
42#131×C109C2211yescharacteristic, complemented
43#108×C109C2211yescharacteristic, complemented
44#100×C109C2212yescomplemented
45#13182030—22no—
46#13182029—22no—
47#13182029C2311yescharacteristic
48#13182029C2312yes—
49#13182028—22no—
50#13182028—22no—
51C2×D436—22no—
52C2×D436C2312yes—
53C2×C22⋊D109—42no—
54C2×C22⋊D109—42no—
55D4×D109—22no—
56D4×D109—22no—
57D4×D109—22no—
58D4×D109—42no—
59Dic436D411yescharacteristic, complemented
60#13182019D411yescharacteristic, complemented
61#13182018—22no—
62#13182018—22no—
63#13182018D412yescomplemented
64#13182017—22no—
65#13182017D411yescharacteristic, complemented
66#13182016D411yescharacteristic, complemented
67D872—22no—
68D872—22no—
69D872—21no—
70D872—21no—
71D872D411yescharacteristic, complemented
72#13182014—22no—
73#13182014—22no—
74#13182014—22no—
75#13182014D411yescharacteristic, complemented
76#13182013C2311yescharacteristic
77#13182012—21no—
78#13182012—22no—
79C2×#11724784C2311yescharacteristic
80C8×D109—21no—
81C8×D109—22no—
82C4×Dic109C2311yescharacteristic
83#41×C109C2311yescharacteristic
84C2×D4×C109C2312yes—
85SD16×C109—22no—
86D8×C109—22no—
87D8×C109—21no—
88D8×C109—21no—
89M16×C109—21no—
90M16×C109C2311yescharacteristic
91M16×C109C2312yes—
92C2×C8×C109—21no—
93C2×C8×C109C2311yescharacteristic
94C23×C109—22no—
95C22×D109—22no—
96C22×D109—42no—
97C22×D109—42no—
98Q8×C109C2×D411yescharacteristic
99D4×C109—21no—
100D4×C109—22no—
101D4×C109—22no—
102D4×C109C2×D411yescharacteristic
103C2×C4×C109—21no—
104C2×C4×C109C2411yescharacteristic
105C22⋊D109—22no—
106C22⋊D109—21no—
107C22⋊D109—42no—
108C22⋊D109—42no—
109C22⋊D109—42no—
110C22⋊D109—84no—
111C2×Dic109—21no—
112C2×Dic109—42no—
113D436—22no—
114D436—22no—
115D436—42no—
116D436C2×D412yes—
117D436C2×D411yescharacteristic
118C4×D109—21no—
119C4×D109—22no—
120C4×D109—42no—
121Dic218C2×D412yes—
122Dic218C2×D411yescharacteristic
123C8×C109—21no—
124C8×C109—22no—
125C8×C109C2×D411yescharacteristic
126C8×C109C2×D411yescharacteristic
127#11724784—21no—
128#11724784C2×D411yescharacteristic
129#11724784C2×D411yescharacteristic
130C22×C109—21no—
131C22×C109—42no—
132C22×C109—82no—
133C22×C109H(5,𝔽2)11yescharacteristic
134C2×D109—22no—
135C2×D109—21no—
136C2×D109—42no—
137C2×D109—82no—
138C2×D109—162no—
139C2×D109—162no—
140C4×C109—21no—
141C4×C109C22×D411yescharacteristic
142C4×C109C22×D411yescharacteristic
143Dic109—22no—
144Dic109—22no—
145Dic109—21no—
146C2×C109—21no—
147C2×C109—41no—
148C2×C109—82no—
149C2×C109D4211yescharacteristic
150D109—42no—
151D109—41no—
152D109—82no—
153#3292—1091nomaximal, Sylow
154C109#329211yescharacteristic, complemented, Sylow
155#576—1092no—
156#576—1091no—
157#576—1091no—
158#493—1091no—
159#472—1091no—
160#472—1092no—
161#472D10911yescharacteristic, complemented
162#453—1092no—
163#453—1092no—
164#443—1091no—
165#443—1091no—
166H(5,𝔽2)—1092no—
167H(5,𝔽2)—1092no—
168AGL(1,ℤ/8)—1092no—
169AGL(1,ℤ/8)—1092no—
170AGL(1,ℤ/8)—1092no—
171AGL(1,ℤ/8)—2182no—
172AGL(1,ℤ/8)—2182no—
173AGL(1,ℤ/8)—2182no—
174AGL(1,ℤ/8)—2182no—
175AGL(1,ℤ/8)C2×D10912yescomplemented
176#135—1091no—
177#135—1091no—
178#135—2181no—
179#135—2181no—
180#135—2182no—
181C2×D8—1092no—
182C2×D8—1091no—
183C2×D8—2181no—
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#108C2×D10911yescharacteristic, complemented
193#104—1091no—
194#104—1092no—
195#104—1091no—
196#100—1092no—
197#100C2×D10912yescomplemented
198C4×C8—1091no—
199#41—1091no—
200#41—1092no—
201#41—2182no—
202#41—2182no—
203#41—2182no—
204#41—2182no—
205#41C22×D10911yescharacteristic
206C2×D4—1092no—
207C2×D4—4362no—
208C2×D4—4362no—
209C2×D4—4362no—
210C2×D4—2182no—
211C2×D4—2182no—
212C2×D4—2182no—
213C2×D4—2182no—
214C2×D4C22×D10912yes—
215Q16—1091no—
216Q16—1091no—
217SD16—22no—
218SD16—1091no—
219SD16—1091no—
220SD16—1092no—
221SD16—2182no—
222SD16—2182no—
223SD16—2182no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1091no—
228D8—1091no—
229D8—2182no—
230D8—2182no—
231D8—2182no—
232D8—2182no—
233D8—2182no—
234D8—2181no—
235D8—2181no—
236M16—21no—
237M16—1091no—
238M16—2182no—
239M16—2181no—
240M16C22×D10912yes—
241M16C22×D10911yescharacteristic
242C2×C8—21no—
243C2×C8—1091no—
244C2×C8—2182no—
245C2×C8—2181no—
246C2×C8C22×D10911yescharacteristic
247C42—1091no—
248C23—22no—
249C23—4362no—
250C23—4362no—
251C23—2182no—
252Q8—1091no—
253Q8—1092no—
254Q8D4×D10911yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—8724no—
259D4—1091no—
260D4—1092no—
261D4—4362no—
262D4—4362no—
263D4—4362no—
264D4—4362no—
265D4—2182no—
266D4—2181no—
267D4—2182no—
268D4—2182no—
269D4D4×D10911yescharacteristic
270C2×C4—21no—
271C2×C4—4362no—
272C2×C4—4362no—
273C2×C4—2181no—
274C2×C4—2182no—
275C2×C4—2181no—
276C2×C4C23×D10911yescharacteristic
277C8—21no—
278C8—22no—
279C8—1091no—
280C8—1091no—
281C8—2181no—
282C8D4×D10911yescharacteristic
283C8D4×D10911yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—8722no—
288C22—4362no—
289C22—17442no—
290C22—17442no—
291C22—2182no—
292C22—2181no—
293C22#1681274511yescharacteristic
294C4—21no—
295C4—2182no—
296C4—2182no—
297C4—2181no—
298C4C2×D4×D10911yescharacteristic
299C4C2×D4×D10911yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—8722no—
304C2—4362no—
305C2—4361no—
306C2#2029566011yescharacteristic
subgroupG/H
centerC2#20295660
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#472×C109C2
socleC2×C109D42
hypercenter#472D109
hypocenterC109#3292
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20295831C2semidirect
2. #20295830C2semidirect
3. #20295827C2semidirect
4. #20295401C2semidirect
5. #20295297C2semidirect
6. #20295295C2semidirect
7. #20295196C2semidirect
8. #20295192C2semidirect
9. #20295142C2semidirect
10. #20295141C2semidirect
11. #472×C109C2semidirect
12. #16812745C22semidirect
13. #16812744C22semidirect
15. #16812727C22semidirect
16. #16812727C22semidirect
18. #16812725C22semidirect
27. C2×D872C22semidirect
28. C2×#13182014C22semidirect
31. #16812707C22semidirect
32. #16812705C22semidirect
33. #16812695C22semidirect
34. #16812634C22semidirect
35. #16812626C22semidirect
36. #16812625C22semidirect
37. #16812625C22semidirect
38. #16812615C22semidirect
39. C8×Dic109C22semidirect
40. AGL(1,ℤ/8)×C109C22semidirect
41. C2×D8×C109C22semidirect
42. #131×C109C22semidirect
43. #108×C109C22semidirect
44. #100×C109C22semidirect
59. Dic436D4semidirect
60. #13182019D4semidirect
63. #13182018D4semidirect
65. #13182017D4semidirect
66. #13182016D4semidirect
71. D872D4semidirect
75. #13182014D4semidirect
154. C109#3292semidirect
161. #472D109semidirect
175. AGL(1,ℤ/8)C2×D109semidirect
186. C2×D8C2×D109semidirect
189. #131C2×D109semidirect
192. #108C2×D109semidirect
197. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958312
#202958302
#202958272
#202954012
#202952972
#202952952
#202951962
#202951922
#202951422
#202951412
#472×C1092
#3292109