Finite Group Databasebeta

← #496×D109 #28373462

Subgroups — #496×D109 #28373462

Order 13952.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D109C212yesmaximal, complemented
2C2×D8×D109C212yesmaximal, complemented
3#127×D109C211yesmaximal, characteristic, complemented
4#124×D109C211yesmaximal, characteristic, complemented
5#20295416C211yesmaximal, characteristic, complemented
6#20295415C211yesmaximal, characteristic, complemented
7#20295414C212yesmaximal, complemented
8#20295413C211yesmaximal, characteristic, complemented
9#20295412C212yesmaximal, complemented
10#97×D109C211yesmaximal, characteristic, complemented
11#496×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2211yescharacteristic
13C2×D4×D109—22no—
14C2×D4×D109C2211yescharacteristic
15C2×D4×D109C2212yes—
16C2×#13182018C2212yescomplemented
17C2×#13182017C2212yescomplemented
18C2×#13182016C2212yescomplemented
19SD16×D109—28no—
20C2×D872C2212yescomplemented
21C2×#13182014C2212yescomplemented
22C2×#13182014C2212yescomplemented
23D8×D109—28no—
24C2×C8×D109C2212yes—
25#16812696C2211yescharacteristic, complemented
26#16812695—22no—
27#16812695C2212yescomplemented
28#16812688C2211yescharacteristic, complemented
29#16812687C2211yescharacteristic, complemented
30#16812686C2211yescharacteristic, complemented
31#16812685—22no—
32C22⋊C4×D109—22no—
33C42×D109C2211yescharacteristic
34#16812610C2211yescharacteristic, complemented
35#16812609C2212yescomplemented
36C2×SD16×C109C2212yescomplemented
37C2×D8×C109C2212yescomplemented
38#127×C109C2211yescharacteristic, complemented
39#124×C109C2211yescharacteristic, complemented
40#97×C109C2211yescharacteristic, complemented
41C23×D109—21no—
42C23×D109—22no—
43C2×Dic218—21no—
44C2×Dic218C2311yescharacteristic
45Q8×D109—24no—
46C2×D436—22no—
47C2×D436C2311yescharacteristic
48C2×D436C2312yes—
49C2×C22⋊D109—21no—
50C2×C22⋊D109—22no—
51C2×C22⋊D109—24no—
52D4×D109—24no—
53D4×D109—24no—
54D4×D109—24no—
55D4×D109—48no—
56C2×C4×D109—21no—
57C2×C4×D109D412yes—
58C2×C4×D109C2311yescharacteristic
59#13182018—24no—
60#13182017—24no—
61#13182016—24no—
62D872—24no—
63#13182014—24no—
64#13182014—24no—
65C2×#11724784C2312yes—
66C8×D109D418yescomplemented
67C22⋊Dic109—22no—
68#13182006—22no—
69#13182006—22no—
70C4×Dic109D412yes—
71C4×Dic109C2311yescharacteristic
72C2×Q8×C109C2311yescharacteristic
73C2×D4×C109—22no—
74C2×D4×C109C2311yescharacteristic
75C2×D4×C109C2312yes—
76SD16×C109—24no—
77D8×C109—24no—
78C2×C8×C109C2312yes—
79C22⋊C4×C109—22no—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C23×C109—22no—
83C22×D109—21no—
84C22×D109—22no—
85C22×D109—42no—
86C22×D109—44no—
87C22×D109—48no—
88C22×D109—44no—
89C22×D109C2×D411yescharacteristic
90Q8×C109—22no—
91D4×C109—22no—
92D4×C109—22no—
93D4×C109—22no—
94D4×C109—44no—
95C2×C4×C109—21no—
96C2×C4×C109C2×D412yes—
97C2×C4×C109C2411yescharacteristic
98C22⋊D109—48no—
99C22⋊D109—42no—
100C22⋊D109—44no—
101C2×Dic109—21no—
102C2×Dic109C2×D412yes—
103C2×Dic109C2×D411yescharacteristic
104D436—22no—
105D436—22no—
106D436—22no—
107D436—44no—
108C4×D109—24no—
109C4×D109—42no—
110C4×D109C2×D414yes—
111Dic218—22no—
112Dic218—42no—
113C8×C109C2×D414yescomplemented
114#11724784C2×D414yescomplemented
115C22×C109—41no—
116C22×C109—42no—
117C22×C109—44no—
118C22×C109—42no—
119C22×C109C22×D411yescharacteristic
120C2×D109—41no—
121C2×D109—42no—
122C2×D109—44no—
123C2×D109—42no—
124C2×D109—88no—
125C2×D109—84no—
126C2×D109#12712yes—
127C2×D109AGL(1,ℤ/8)14yes—
128C4×C109—22no—
129C4×C109—41no—
130C4×C109C22×D412yes—
131Dic109—22no—
132Dic109—41no—
133Dic109#12712yes—
134C2×C109—82no—
135C2×C109—81no—
136C2×C109C2×#12711yescharacteristic
137C2×C109C2×AGL(1,ℤ/8)12yes—
138D109—82no—
139D109—81no—
140D109#49614yesdirect product factor
141C2×#496—1091nomaximal, Sylow
142C109C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—1092no—
144C22×D8—1092no—
145C2×#127—1091no—
146C2×#124—1091no—
147#496—1092no—
148#496—1092no—
149#496—1091no—
150#496—1091no—
151#496—1091no—
152#496D10911yescharacteristic, direct product factor
153C2×#97—1091no—
154C22×Q8—1091no—
155C22×D4—1091no—
156C22×D4—1092no—
157C22×D4—2182no—
158C2×SD16—1092no—
159C2×SD16—1092no—
160C2×SD16—1092no—
161C2×SD16—2188no—
162C2×SD16C2×D10912yescomplemented
163C2×D8—1092no—
164C2×D8—1092no—
165C2×D8—1092no—
166C2×D8—2188no—
167C2×D8C2×D10912yescomplemented
168C22×C8—1092no—
169#127—1092no—
170#127—1091no—
171#127—2182no—
172#127C2×D10911yescharacteristic, complemented
173#124—1091no—
174#124—1091no—
175#124—1091no—
176#124—2182no—
177#124C2×D10911yescharacteristic, complemented
178C2×C22⋊C4—2182no—
179C2×C42—1091no—
180#97—1092no—
181#97—1091no—
182#97C2×D10911yescharacteristic, complemented
183C24—2181no—
184C24—2182no—
185C2×Q8—1091no—
186C2×Q8—2181no—
187C2×Q8—2184no—
188C2×Q8C22×D10911yescharacteristic
189C2×D4—22no—
190C2×D4—1091no—
191C2×D4—1092no—
192C2×D4—4368no—
193C2×D4—2184no—
194C2×D4—2184no—
195C2×D4—2184no—
196C2×D4—2181no—
197C2×D4—2184no—
198C2×D4—2182no—
199C2×D4—2182no—
200C2×D4C22×D10911yescharacteristic
201C2×D4C22×D10912yes—
202C22×C4—1091no—
203C22×C4—1092no—
204C22×C4—2181no—
205SD16—24no—
206SD16—2184no—
207SD16—2184no—
208SD16—2184no—
209D8—24no—
210D8—2184no—
211D8—2184no—
212D8—2184no—
213C2×C8—1092no—
214C2×C8—1098no—
215C2×C8C22×D10912yes—
216C22⋊C4—22no—
217C22⋊C4—2182no—
218C22⋊C4—2182no—
219C22⋊C4—2182no—
220C42—1092no—
221C42—1091no—
222C42C22×D10911yescharacteristic
223C23—21no—
224C23—22no—
225C23—1091no—
226C23—4368no—
227C23—4362no—
228C23—4364no—
229C23—4364no—
230C23—2181no—
231C23—2182no—
232Q8—22no—
233Q8—4362no—
234Q8—2182no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—4364no—
240D4—4362no—
241D4—4364no—
242D4—4368no—
243D4—2182no—
244D4—2182no—
245D4—2182no—
246C2×C4—21no—
247C2×C4—1091no—
248C2×C4—1094no—
249C2×C4—1092no—
250C2×C4—4362no—
251C2×C4—2184no—
252C2×C4—2181no—
253C2×C4D4×D10912yes—
254C2×C4C23×D10911yescharacteristic
255C8—1094no—
256C8D4×D10914yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—8728no—
262C22—8724no—
263C22—1094no—
264C22—1092no—
265C22—4364no—
266C22—4362no—
267C22—4361no—
268C22—4362no—
269C22C2×D4×D10911yescharacteristic
270C4—22no—
271C4—41no—
272C4—1092no—
273C4—4361no—
274C4—2182no—
275C4C2×D4×D10912yes—
276C2—82no—
277C2—81no—
278C2—8722no—
279C2—8721no—
280C2—1094no—
281C2#127×D10911yescharacteristic
282C2AGL(1,ℤ/8)×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#496×C109C2
socleC22×C109C22×D4
hypercenter#496D109
hypocenterC109C2×#496
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D109C2semidirect
2. C2×D8×D109C2semidirect
3. #127×D109C2semidirect
4. #124×D109C2semidirect
5. #20295416C2semidirect
6. #20295415C2semidirect
7. #20295414C2semidirect
8. #20295413C2semidirect
9. #20295412C2semidirect
10. #97×D109C2semidirect
11. #496×C109C2semidirect
16. C2×#13182018C22semidirect
17. C2×#13182017C22semidirect
18. C2×#13182016C22semidirect
20. C2×D872C22semidirect
21. C2×#13182014C22semidirect
22. C2×#13182014C22semidirect
25. #16812696C22semidirect
27. #16812695C22semidirect
28. #16812688C22semidirect
29. #16812687C22semidirect
30. #16812686C22semidirect
34. #16812610C22semidirect
35. #16812609C22semidirect
36. C2×SD16×C109C22semidirect
37. C2×D8×C109C22semidirect
38. #127×C109C22semidirect
39. #124×C109C22semidirect
40. #97×C109C22semidirect
66. C8×D109D4semidirect
113. C8×C109C2×D4semidirect
114. #11724784C2×D4semidirect
140. D109#496direct (N × H)
142. C109C2×#496semidirect
152. #496D109direct (N × H)
162. C2×SD16C2×D109semidirect
167. C2×D8C2×D109semidirect
172. #127C2×D109semidirect
177. #124C2×D109semidirect
182. #97C2×D109semidirect
256. C8D4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1092
C2×D8×D1092
#127×D1092
#124×D1092
#202954162
#202954152
#202954142
#202954132
#202954122
#97×D1092
#496×C1092
C2×#496109