Finite Group Databasebeta

← #484×D109 #28372934

Subgroups — #484×D109 #28372934

Order 13952.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D109C211yesmaximal, characteristic
2C2×C4⋊C4×D109C211yesmaximal, characteristic
3#20295363C212yesmaximal, complemented
4#20295362C211yesmaximal, characteristic, complemented
5#20295361C211yesmaximal, characteristic, complemented
6#20295360C212yesmaximal, complemented
7#20295359C211yesmaximal, characteristic, complemented
8#107×D109C212yesmaximal, complemented
9#103×D109C212yesmaximal, complemented
10C22⋊C8×D109C211yesmaximal, characteristic
11#484×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2211yescharacteristic
13C22×C4×D109—21no—
14C22×C4×D109C2211yescharacteristic
15C2×C8×D109C2212yes—
16#16812679C2211yescharacteristic
17#16812678C2211yescharacteristic
18#16812677—21no—
19#16812676—21no—
20#16812675C2211yescharacteristic
21C2×C4⋊Dic109C2211yescharacteristic
22C2×#13182008—21no—
23C4⋊C4×D109—21no—
24C4⋊C4×D109—21no—
25C4⋊C4×D109—21no—
26C4⋊C4×D109C2212yes—
27C4⋊C4×D109C2212yes—
28C4⋊C4×D109C2211yescharacteristic
29C22⋊C4×D109—21no—
30#16812633C2212yescomplemented
31#16812632C2212yescomplemented
32#16812632C2212yescomplemented
33#16812624C2212yescomplemented
34#16812623C2212yescomplemented
35#16812622C2212yescomplemented
36C22⋊#11724784C2211yescharacteristic
37#16812611C2212yes—
38C22⋊Q8×C109C2211yescharacteristic
39C2×C4⋊C4×C109C2211yescharacteristic
40#107×C109C2212yescomplemented
41#103×C109C2212yescomplemented
42C22⋊C8×C109C2211yescharacteristic
43C23×D109D411yescharacteristic
44C2×Dic218—21no—
45C2×Dic218C2311yescharacteristic
46Q8×D109—24no—
47C22×Dic109—21no—
48C22×Dic109D411yescharacteristic
49C2×C4×D109—24no—
50C2×C4×D109—21no—
51C2×C4×D109—21no—
52C2×C4×D109—22no—
53C2×C4×D109—22no—
54C2×C4×D109—44no—
55C2×C4×D109D411yescharacteristic
56C2×C4×D109D411yescharacteristic
57C2×C4×D109C2311yescharacteristic
58C2×C4×D109C2311yescharacteristic
59C2×#11724784C2312yes—
60C8×D109—24no—
61C4⋊Dic109—21no—
62C4⋊Dic109—21no—
63C4⋊Dic109C2312yes—
64C4⋊Dic109C2311yescharacteristic
65#13182008—21no—
66#13182008—22no—
67#13182008—22no—
68#13182008—21no—
69#13182008—21no—
70C22⋊Dic109—21no—
71#13182006—21no—
72#13182006—21no—
73C2×Q8×C109C2311yescharacteristic
74C22×C4×C109—21no—
75C22×C4×C109C2311yescharacteristic
76C2×C8×C109C2312yes—
77C4⋊C4×C109—21no—
78C4⋊C4×C109—21no—
79C4⋊C4×C109C2312yes—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C23×C109C2×D411yescharacteristic
83C22×D109—22no—
84C22×D109—22no—
85C22×D109Q1614yes—
86C22×D109C2×D411yescharacteristic
87C22×D109C2×D411yescharacteristic
88Q8×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—21no—
91C2×C4×C109—21no—
92C2×C4×C109—22no—
93C2×C4×C109—42no—
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2411yescharacteristic
96C2×Dic109—22no—
97C2×Dic109—21no—
98C2×Dic109—21no—
99C2×Dic109—22no—
100C2×Dic109—42no—
101C2×Dic109C2×D411yescharacteristic
102C2×Dic109C2×D411yescharacteristic
103C4×D109—22no—
104C4×D109—22no—
105C4×D109—22no—
106C4×D109—44no—
107C4×D109—42no—
108C4×D109—42no—
109C4×D109—44no—
110C4×D109#4114yes—
111Dic218—22no—
112Dic218—42no—
113C8×C109—22no—
114#11724784—22no—
115C22×C109—21no—
116C22×C109—21no—
117C22×C109C2×Q1612yes—
118C22×C109C22×D411yescharacteristic
119C2×D109—21no—
120C2×D109—21no—
121C2×D109—28no—
122C2×D109#12312yes—
123C2×D109C2×Q1612yes—
124C2×D109C2×Q1612yes—
125C2×D109AGL(1,ℤ/8)12yes—
126C4×C109—21no—
127C4×C109—41no—
128C4×C109—42no—
129C4×C109—41no—
130C4×C109C2×#4112yes—
131Dic109—21no—
132Dic109—41no—
133Dic109—42no—
134Dic109—41no—
135Dic109#12312yes—
136C2×C109—22no—
137C2×C109C2×#12311yescharacteristic
138C2×C109C22×Q1611yescharacteristic
139C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
140D109—22no—
141D109#48414yesdirect product factor
142C2×#484—1091nomaximal, Sylow
143C109C2×#48411yescharacteristic, complemented, Sylow
144C2×C22⋊Q8—1091no—
145C22×C4⋊C4—1091no—
146#484—1091no—
147#484—1091no—
148#484—1092no—
149#484—1092no—
150#484—1091no—
151#484D10911yescharacteristic, direct product factor
152C2×#107—1092no—
153C2×#103—1092no—
154C2×C22⋊C8—1091no—
155C22×Q8—1091no—
156C23×C4—1091no—
157C23×C4—2181no—
158C22×C8—1092no—
159C22⋊Q8—1091no—
160C22⋊Q8—1091no—
161C22⋊Q8—1091no—
162C22⋊Q8—2181no—
163C22⋊Q8—2181no—
164C22⋊Q8C2×D10911yescharacteristic
165C2×C4⋊C4—1091no—
166C2×C4⋊C4—1092no—
167C2×C4⋊C4—1092no—
168C2×C4⋊C4—1091no—
169C2×C4⋊C4—2181no—
170C2×C4⋊C4—2181no—
171C2×C4⋊C4—2181no—
172C2×C4⋊C4—2181no—
173C2×C4⋊C4C2×D10911yescharacteristic
174C2×C22⋊C4—2181no—
175#107—1092no—
176#107—1092no—
177#107—1092no—
178#107C2×D10912yescomplemented
179#103—1092no—
180#103—1092no—
181#103—1092no—
182#103C2×D10912yescomplemented
183C22⋊C8—1091no—
184C22⋊C8—1092no—
185C22⋊C8C2×D10911yescharacteristic
186C24—1091no—
187C2×Q8—1091no—
188C2×Q8—2181no—
189C2×Q8—2184no—
190C2×Q8C22×D10911yescharacteristic
191C22×C4—21no—
192C22×C4—1091no—
193C22×C4—1091no—
194C22×C4—1091no—
195C22×C4—1091no—
196C22×C4—1091no—
197C22×C4—4364no—
198C22×C4—2184no—
199C22×C4—2181no—
200C22×C4—2181no—
201C22×C4—2182no—
202C22×C4—2182no—
203C22×C4—2181no—
204C22×C4C22×D10911yescharacteristic
205C2×C8—1092no—
206C2×C8—2184no—
207C2×C8C22×D10912yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—1092no—
211C4⋊C4—1091no—
212C4⋊C4—2182no—
213C4⋊C4—2182no—
214C4⋊C4—2181no—
215C4⋊C4—2181no—
216C4⋊C4—2181no—
217C4⋊C4—2181no—
218C4⋊C4—2181no—
219C4⋊C4C22×D10912yes—
220C4⋊C4C22×D10911yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2181no—
223C22⋊C4—2181no—
224C22⋊C4—2181no—
225C23—1091no—
226C23—1094no—
227C23—1091no—
228C23—2182no—
229C23—2182no—
230C23D4×D10911yescharacteristic
231Q8—22no—
232Q8—4362no—
233Q8—2182no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—1091no—
240C2×C4—1094no—
241C2×C4—1091no—
242C2×C4—4364no—
243C2×C4—4362no—
244C2×C4—4362no—
245C2×C4—4364no—
246C2×C4—4362no—
247C2×C4—2182no—
248C2×C4—2182no—
249C2×C4—2182no—
250C2×C4—2181no—
251C2×C4—2181no—
252C2×C4—2182no—
253C2×C4—2182no—
254C2×C4D4×D10911yescharacteristic
255C2×C4C23×D10911yescharacteristic
256C8—22no—
257C8—2182no—
258C22—21no—
259C22—21no—
260C22—1092no—
261C22—1092no—
262C22—1092no—
263C22—1092no—
264C22—2188no—
265C22—2181no—
266C22—2181no—
267C22Q16×D10912yes—
268C22C2×D4×D10911yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—1092no—
274C4—4361no—
275C4—4362no—
276C4—4361no—
277C4—2181no—
278C4#41×D10912yes—
279C2—22no—
280C2—1094no—
281C2—2182no—
282C2#123×D10911yescharacteristic
283C2C2×Q16×D10911yescharacteristic
284C2AGL(1,ℤ/8)×D10911yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#484×C109C2
socleC22×C109C22×D4
prime radicalC23×D109D4
hypercenter#484D109
hypocenterC109C2×#484
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20295363C2semidirect
4. #20295362C2semidirect
5. #20295361C2semidirect
6. #20295360C2semidirect
7. #20295359C2semidirect
8. #107×D109C2semidirect
9. #103×D109C2semidirect
11. #484×C109C2semidirect
30. #16812633C22semidirect
31. #16812632C22semidirect
32. #16812632C22semidirect
33. #16812624C22semidirect
34. #16812623C22semidirect
35. #16812622C22semidirect
40. #107×C109C22semidirect
41. #103×C109C22semidirect
141. D109#484direct (N × H)
143. C109C2×#484semidirect
151. #484D109direct (N × H)
178. #107C2×D109semidirect
182. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1092
C2×C4⋊C4×D1092
#202953632
#202953622
#202953612
#202953602
#202953592
#107×D1092
#103×D1092
C22⋊C8×D1092
#484×C1092
C2×#484109