Finite Group Databasebeta

← #489×D109 #28373358

Subgroups — #489×D109 #28373358

Order 13952.

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

normal subgroups only

This group has 436 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
1C4⋊Q8×D109C211yesmaximal, characteristic, complemented
2#124×D109C211yesmaximal, characteristic
3#20295388C211yesmaximal, characteristic, complemented
4#20295387C212yesmaximal, complemented
5#20295386C211yesmaximal, characteristic, complemented
6#20295385C211yesmaximal, characteristic, complemented
7#20295384C212yesmaximal, complemented
8#103×D109C212yesmaximal, complemented
9#102×D109C212yesmaximal
10#97×D109C211yesmaximal, characteristic, complemented
11#489×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109—21no—
13C2×Q8×D109C2211yescharacteristic
14C2×D4×D109C2211yescharacteristic
15C2×C8×D109C2212yes—
16#16812699C2211yescharacteristic, complemented
17#16812698—21no—
18#16812698C2212yescomplemented
19#16812697—21no—
20#16812688C2211yescharacteristic
21#16812687C2211yescharacteristic
22#16812686C2211yescharacteristic
23#16812685—21no—
24#16812685—21no—
25C4⋊C4×D109—21no—
26C4⋊C4×D109C2212yes—
27C22⋊C4×D109—21no—
28C22⋊C4×D109—21no—
29C42×D109C2211yescharacteristic
30#16812624C2212yescomplemented
31#16812623C2212yescomplemented
32#16812622C2212yescomplemented
33#16812621C2212yes—
34#16812620C2212yes—
35#16812619C2212yes—
36#16812610C2211yescharacteristic, complemented
37#16812609C2212yescomplemented
38C4⋊Q8×C109C2211yescharacteristic, complemented
39#124×C109C2211yescharacteristic
40#103×C109C2212yescomplemented
41#102×C109C2212yes—
42#97×C109C2211yescharacteristic, complemented
43C23×D109—21no—
44C2×Dic218—21no—
45C2×Dic218—21no—
46C2×Dic218—22no—
47C2×Dic218C2311yescharacteristic
48Q8×D109—24no—
49Q8×D109—44no—
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—21no—
52D4×D109—24no—
53C2×C4×D109—21no—
54C2×C4×D109—22no—
55C2×C4×D109D411yescharacteristic
56C2×C4×D109D411yescharacteristic
57C2×C4×D109C2311yescharacteristic
58C2×#11724784C2312yes—
59C8×D109—24no—
60C4⋊Dic109—21no—
61C4⋊Dic109C2312yes—
62#13182008—22no—
63#13182008—22no—
64C22⋊Dic109—21no—
65C22⋊Dic109—21no—
66#13182006—21no—
67#13182006—21no—
68#13182006—21no—
69#13182006—21no—
70C4×Dic109D411yescharacteristic
71C4×Dic109D411yescharacteristic
72C4×Dic109C2311yescharacteristic
73C2×Q8×C109—21no—
74C2×Q8×C109C2311yescharacteristic
75C2×D4×C109C2311yescharacteristic
76C2×C8×C109C2312yes—
77C4⋊C4×C109—21no—
78C4⋊C4×C109C2312yes—
79C22⋊C4×C109—21no—
80C22⋊C4×C109—21no—
81C42×C109C2311yescharacteristic
82C23×C109—21no—
83C22×D109—21no—
84C22×D109—42no—
85C22×D109—42no—
86C22×D109—42no—
87C22×D109C2×D411yescharacteristic
88Q8×C109—22no—
89Q8×C109—42no—
90D4×C109—22no—
91C2×C4×C109—21no—
92C2×C4×C109—22no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—42no—
97C2×Dic109—21no—
98C2×Dic109—22no—
99C2×Dic109C2×D411yescharacteristic
100C2×Dic109C2×D411yescharacteristic
101C2×Dic109C2×D411yescharacteristic
102D436—22no—
103C4×D109—22no—
104C4×D109—22no—
105C4×D109—42no—
106C4×D109—44no—
107C4×D109#4114yes—
108Dic218—22no—
109Dic218—44no—
110Dic218—42no—
111Dic218—42no—
112C8×C109—22no—
113#11724784—22no—
114C22×C109—41no—
115C22×C109—41no—
116C22×C109—41no—
117C22×C109C22×D411yescharacteristic
118C2×D109—41no—
119C2×D109—41no—
120C2×D109—41no—
121C2×D109—84no—
122C2×D109#12412yes—
123C2×D109AGL(1,ℤ/8)12yes—
124C2×D109#13712yes—
125C4×C109—21no—
126C4×C109—21no—
127C4×C109—41no—
128C4×C109—42no—
129C4×C109C2×#4112yes—
130Dic109—21no—
131Dic109—21no—
132Dic109—41no—
133Dic109—42no—
134Dic109#12412yes—
135C2×C109—81no—
136C2×C109C2×#12411yescharacteristic
137C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C109C2×#13711yescharacteristic
139D109—81no—
140D109#48914yesdirect product factor
141C2×#489—1091nomaximal, Sylow
142C109C2×#48911yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—1091no—
144C2×#124—1091no—
145#489—1091no—
146#489—1091no—
147#489—1092no—
148#489—1092no—
149#489—1091no—
150#489D10911yescharacteristic, direct product factor
151C2×#103—1092no—
152C2×#102—1092no—
153C2×#97—1091no—
154C22×Q8—1091no—
155C22×Q8—2181no—
156C22×D4—1091no—
157C22×C8—1092no—
158C4⋊Q8—1092no—
159C4⋊Q8—1091no—
160C4⋊Q8—2181no—
161C4⋊Q8—2181no—
162C4⋊Q8C2×D10911yescharacteristic, complemented
163#124—1091no—
164#124—1091no—
165#124—1091no—
166#124—2181no—
167#124—2181no—
168#124C2×D10911yescharacteristic
169C2×C4⋊C4—1092no—
170C2×C4⋊C4—2181no—
171C2×C22⋊C4—2181no—
172C2×C22⋊C4—2181no—
173C2×C42—1091no—
174#103—1092no—
175#103—1092no—
176#103—1092no—
177#103C2×D10912yescomplemented
178#102—1092no—
179#102—1092no—
180#102—1092no—
181#102C2×D10912yes—
182#97—1092no—
183#97—1091no—
184#97C2×D10911yescharacteristic, complemented
185C24—2181no—
186C2×Q8—21no—
187C2×Q8—1091no—
188C2×Q8—4364no—
189C2×Q8—2181no—
190C2×Q8—2182no—
191C2×Q8—2184no—
192C2×Q8—2181no—
193C2×Q8C22×D10911yescharacteristic
194C2×D4—1091no—
195C2×D4—2184no—
196C2×D4—2181no—
197C2×D4C22×D10911yescharacteristic
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—1091no—
201C22×C4—2181no—
202C22×C4—2182no—
203C2×C8—1092no—
204C2×C8—2184no—
205C2×C8C22×D10912yes—
206C4⋊C4—21no—
207C4⋊C4—1092no—
208C4⋊C4—2182no—
209C4⋊C4—2182no—
210C4⋊C4—2181no—
211C4⋊C4C22×D10912yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2181no—
215C22⋊C4—2181no—
216C22⋊C4—2181no—
217C22⋊C4—2181no—
218C22⋊C4—2181no—
219C22⋊C4—2181no—
220C42—1091no—
221C42—1091no—
222C42—1091no—
223C42C22×D10911yescharacteristic
224C23—21no—
225C23—1091no—
226C23—4362no—
227C23—4362no—
228C23—4362no—
229C23—2181no—
230Q8—22no—
231Q8—42no—
232Q8—4362no—
233Q8—4362no—
234Q8—4364no—
235Q8—2182no—
236D4—22no—
237D4—4362no—
238D4—2182no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—1091no—
242C2×C4—1094no—
243C2×C4—1091no—
244C2×C4—1091no—
245C2×C4—4364no—
246C2×C4—4362no—
247C2×C4—2182no—
248C2×C4—2182no—
249C2×C4—2181no—
250C2×C4—2182no—
251C2×C4D4×D10911yescharacteristic
252C2×C4D4×D10911yescharacteristic
253C2×C4C23×D10911yescharacteristic
254C8—22no—
255C8—2182no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—8724no—
260C22—1092no—
261C22—1092no—
262C22—1092no—
263C22—4361no—
264C22—4361no—
265C22—4361no—
266C22C2×D4×D10911yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—1092no—
272C4—4361no—
273C4—4362no—
274C4—2181no—
275C4—2181no—
276C4#41×D10912yes—
277C2—81no—
278C2—8721no—
279C2—1094no—
280C2#124×D10911yescharacteristic
281C2AGL(1,ℤ/8)×D10911yescharacteristic
282C2#137×D10911yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#489×C109C2
socleC22×C109C22×D4
prime radicalC2×D4×D109C22
hypercenter#489D109
hypocenterC109C2×#489
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D109C2semidirect
3. #20295388C2semidirect
4. #20295387C2semidirect
5. #20295386C2semidirect
6. #20295385C2semidirect
7. #20295384C2semidirect
8. #103×D109C2semidirect
10. #97×D109C2semidirect
11. #489×C109C2semidirect
16. #16812699C22semidirect
18. #16812698C22semidirect
30. #16812624C22semidirect
31. #16812623C22semidirect
32. #16812622C22semidirect
36. #16812610C22semidirect
37. #16812609C22semidirect
38. C4⋊Q8×C109C22semidirect
40. #103×C109C22semidirect
42. #97×C109C22semidirect
140. D109#489direct (N × H)
142. C109C2×#489semidirect
150. #489D109direct (N × H)
162. C4⋊Q8C2×D109semidirect
177. #103C2×D109semidirect
184. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1092
#124×D1092
#202953882
#202953872
#202953862
#202953852
#202953842
#103×D1092
#102×D1092
#97×D1092
#489×C1092
C2×#489109