Finite Group Databasebeta

← #28373171

Subgroups — #28373171

Order 13952.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#20295562C211yesmaximal, characteristic, complemented
2C2×D4×Dic109C211yesmaximal, characteristic, complemented
3#20295354C211yesmaximal, characteristic, complemented
4#20295340C211yesmaximal, characteristic, complemented
5#20295324C212yesmaximal, complemented
6#20295243C212yesmaximal, complemented
7#20295160C211yesmaximal, characteristic
8#20295135C212yesmaximal, complemented
9#20295127C212yesmaximal, complemented
10#20295080C211yesmaximal, characteristic, complemented
11#450×C109C211yesmaximal, characteristic, complemented
12C23×Dic109—21no—
13C2×#13182014C2212yes—
14#16812694—21no—
15#16812690C2211yescharacteristic, complemented
16#16812688C2211yescharacteristic, complemented
17#16812683—21no—
18#16812673C2211yescharacteristic
19Q8×Dic109C2211yescharacteristic, complemented
20D4×Dic109—21no—
21D4×Dic109—21no—
22D4×Dic109C2212yes—
23D4×Dic109C2212yes—
24#16812661C2211yescharacteristic, complemented
25#16812659—21no—
26#16812657—21no—
27C2×C4⋊Dic109C2211yescharacteristic
28C2×C22⋊Dic109—21no—
29C2×C4×Dic109C2211yescharacteristic
30#16812633C2212yescomplemented
31#16812630C2212yescomplemented
32#16812628C2212yescomplemented
33#16812623C2212yescomplemented
34#16812621C2212yescomplemented
35#16812620C2212yes—
36C22⋊#11724784C2211yescharacteristic
37#16812609C2212yescomplemented
38C22×D4×C109C2211yescharacteristic
39C2×SD16×C109C2212yes—
40C22⋊Q8×C109C2211yescharacteristic
41#102×C109C2212yes—
42C22⋊C8×C109C2211yescharacteristic
43C2×D436C2311yescharacteristic
44C2×C22⋊D109—21no—
45C22×Dic109—21no—
46C22×Dic109—22no—
47C22×Dic109—22no—
48C22×Dic109—44no—
49C22×Dic109D411yescharacteristic
50C22×Dic109D411yescharacteristic
51C2×C4×D109—21no—
52#13182014—24no—
53C2×#11724784C2312yes—
54C4⋊Dic109—21no—
55C4⋊Dic109—21no—
56C4⋊Dic109—21no—
57C4⋊Dic109C2311yescharacteristic
58C4⋊Dic109C2312yes—
59#13182008—21no—
60C22⋊Dic109—22no—
61C22⋊Dic109—22no—
62#13182006—21no—
63#13182006—21no—
64#13182006—21no—
65#13182006—21no—
66C4×Dic109—21no—
67C4×Dic109D411yescharacteristic
68C4×Dic109D411yescharacteristic
69C4×Dic109C2311yescharacteristic
70C4×Dic109C2311yescharacteristic
71C24×C109—21no—
72C2×Q8×C109C2311yescharacteristic
73C2×D4×C109—24no—
74C2×D4×C109—21no—
75C2×D4×C109C2312yes—
76C22×C4×C109C2311yescharacteristic
77SD16×C109—24no—
78C2×C8×C109C2312yes—
79C4⋊C4×C109—21no—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C23×C109—24no—
83C23×C109—22no—
84C23×C109—42no—
85C23×C109—42no—
86C23×C109C2×D411yescharacteristic
87C22×D109—21no—
88Q8×C109—22no—
89D4×C109—24no—
90D4×C109—22no—
91D4×C109#4114yes—
92C2×C4×C109—22no—
93C2×C4×C109—21no—
94C2×C4×C109—21no—
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2411yescharacteristic
97C22⋊D109—42no—
98C2×Dic109—22no—
99C2×Dic109—21no—
100C2×Dic109—22no—
101C2×Dic109—41no—
102C2×Dic109—44no—
103C2×Dic109—44no—
104C2×Dic109—42no—
105C2×Dic109C2×D411yescharacteristic
106C2×Dic109C2×D411yescharacteristic
107C2×Dic109C2×D411yescharacteristic
108C2×Dic109C2×D411yescharacteristic
109D436—22no—
110C4×D109—42no—
111C8×C109—22no—
112#11724784—22no—
113C22×C109—21no—
114C22×C109—21no—
115C22×C109—24no—
116C22×C109—48no—
117C22×C109—42no—
118C22×C109—42no—
119C22×C109AGL(1,ℤ/8)12yes—
120C22×C109C22×D411yescharacteristic
121C2×D109—41no—
122C2×D109—41no—
123C2×D109—41no—
124C4×C109—21no—
125C4×C109—41no—
126C4×C109—41no—
127C4×C109C2×#4112yes—
128Dic109—21no—
129Dic109—21no—
130Dic109—21no—
131Dic109—21no—
132Dic109—42no—
133Dic109—41no—
134C2×C109—22no—
135C2×C109—44no—
136C2×C109#52511yescharacteristic
137C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C109#57511yescharacteristic
139D109—81no—
140#3114—1091nomaximal, Sylow
141C109#311411yescharacteristic, complemented, Sylow
142#529—1091no—
143C2×C4×D4—1091no—
144#483—1091no—
145#480—1091no—
146#478—1092no—
147#463—1092no—
148#450D10911yescharacteristic, complemented
149#447—1091no—
150#442—1092no—
151#440—1092no—
152#432—1091no—
153C22×D4C2×D10911yescharacteristic
154C23×C4—2181no—
155C2×SD16C2×D10912yes—
156C2×D8—1092no—
157#126—2181no—
158#125—1091no—
159#124—1091no—
160#123—2181no—
161C22⋊Q8C2×D10911yescharacteristic
162#121—1091no—
163C4×Q8—1091no—
164C4×D4—1092no—
165C4×D4—1092no—
166C4×D4—1091no—
167C4×D4—2181no—
168C4×D4—2181no—
169C4×D4—2181no—
170#117—2181no—
171C2×C4⋊C4—1091no—
172C2×C22⋊C4—2181no—
173C2×C42—1091no—
174#107—1092no—
175#106—1092no—
176C4⋊C8—1092no—
177#103—1092no—
178#102—1092no—
179#102—1092no—
180#102C2×D10912yes—
181C22⋊C8—1091no—
182C22⋊C8C2×D10911yescharacteristic
183#97—1092no—
184C24—21no—
185C2×Q8C22×D10911yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—1091no—
189C2×D4—2181no—
190C2×D4C22×D10912yes—
191C22×C4—1091no—
192C22×C4—1091no—
193C22×C4—4364no—
194C22×C4—2181no—
195C22×C4—2182no—
196C22×C4—2182no—
197C22×C4—2181no—
198C22×C4C22×D10911yescharacteristic
199SD16—24no—
200D8—2184no—
201C2×C8—1092no—
202C2×C8C22×D10912yes—
203C4⋊C4—21no—
204C4⋊C4—1091no—
205C4⋊C4—1092no—
206C4⋊C4—2181no—
207C4⋊C4—2181no—
208C4⋊C4—2181no—
209C4⋊C4—2181no—
210C4⋊C4C22×D10911yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—2181no—
213C22⋊C4—2181no—
214C22⋊C4—2181no—
215C22⋊C4—2182no—
216C22⋊C4—2182no—
217C22⋊C4—2181no—
218C42—1091no—
219C42—1091no—
220C42—1091no—
221C42—1091no—
222C42—2181no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2181no—
228C23D4×D10911yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—4362no—
233D4—2182no—
234D4#1318202814yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—1091no—
239C2×C4—1091no—
240C2×C4—1091no—
241C2×C4—1091no—
242C2×C4—4362no—
243C2×C4—4362no—
244C2×C4—4364no—
245C2×C4—4364no—
246C2×C4—4361no—
247C2×C4—2181no—
248C2×C4—2182no—
249C2×C4—2182no—
250C2×C4D4×D10911yescharacteristic
251C2×C4C23×D10911yescharacteristic
252C8—22no—
253C8—2182no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—4361no—
261C22—4361no—
262C22—4361no—
263C22#1681272612yes—
264C22C2×D4×D10911yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—4362no—
269C4—4361no—
270C4—2181no—
271C4—2181no—
272C4—2181no—
273C4—2181no—
274C4C2×#1318202812yes—
275C2—22no—
276C2—44no—
277C2—8721no—
278C2#2029553211yescharacteristic
279C2C2×#1681272611yescharacteristic
280C2#2029582211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#450×C109C2
socleC22×C109C22×D4
prime radical#20295160C2
hypercenter#450D109
hypocenterC109#3114
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20295562C2semidirect
2. C2×D4×Dic109C2semidirect
3. #20295354C2semidirect
4. #20295340C2semidirect
5. #20295324C2semidirect
6. #20295243C2semidirect
8. #20295135C2semidirect
9. #20295127C2semidirect
10. #20295080C2semidirect
11. #450×C109C2semidirect
15. #16812690C22semidirect
16. #16812688C22semidirect
19. Q8×Dic109C22semidirect
24. #16812661C22semidirect
30. #16812633C22semidirect
31. #16812630C22semidirect
32. #16812628C22semidirect
33. #16812623C22semidirect
34. #16812621C22semidirect
37. #16812609C22semidirect
141. C109#3114semidirect
148. #450D109semidirect

Maximal subgroups

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

subgroupindex
#202955622
C2×D4×Dic1092
#202953542
#202953402
#202953242
#202952432
#202951602
#202951352
#202951272
#202950802
#450×C1092
#3114109