Finite Group Databasebeta

← #28346072

Subgroups — #28346072

Order 13696.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16812132C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293140C212yesmaximal, complemented
4#20292882C211yesmaximal, characteristic, complemented
5#20292881C211yesmaximal, characteristic, complemented
6#20292880C211yesmaximal, characteristic, complemented
7#20292876C212yesmaximal, complemented
8#20292875C212yesmaximal, complemented
9#20292662C211yesmaximal, characteristic, complemented
10#20292622C212yesmaximal, complemented
11#470×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812132—24no—
15C2×#13133805C2212yescomplemented
16C2×D856C2211yescharacteristic, complemented
17C2×#13133802C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20M16×D107—24no—
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812094—22no—
24#16812085C2212yes—
25#16812084C2212yes—
26#16812083C2212yes—
27#16812079—22no—
28#16812071—22no—
29(C22⋊D107)⋊C4—22no—
30#16812067C2212yes—
31#16812042C2212yescomplemented
32#16812041C2211yescharacteristic, complemented
33#16812033C2212yescomplemented
34#16812031C2212yescomplemented
35#16812029C2212yescomplemented
36C2×Q16×C107C2211yescharacteristic, complemented
37C2×M16×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#106×C107C2211yescharacteristic, complemented
40#103×C107C2212yescomplemented
41C23×D107D411yescharacteristic
42#13133819—24no—
43#13133819—24no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428C2312yes—
47C2×C22⋊D107—22no—
48C22×Dic107D411yescharacteristic
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55C2×C4×D107C2311yescharacteristic
56#13133805—24no—
57D856—22no—
58#13133802—22no—
59#13133801—22no—
60#13133801—22no—
61C2×C107⋊C8C2311yescharacteristic
62C2×C107⋊C8C2311yescharacteristic
63C8×D107—22no—
64C8×D107D414yescomplemented
65C4⋊Dic107C2312yes—
66Dic107⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—22no—
72C2×Q8×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74Q16×C107—22no—
75M16×C107—22no—
76C2×C8×C107C2311yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C4⋊C4×C107—22no—
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—42no—
85C22×D107C2×D411yescharacteristic
86C22×D107C2×D411yescharacteristic
87Q8×C107—22no—
88Q8×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—44no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100D428—22no—
101D428—22no—
102D428—44no—
103D428—44no—
104C4×D107—22no—
105C4×D107—21no—
106C4×D107—21no—
107C4×D107—22no—
108C4×D107—44no—
109C4×D107C2×D412yes—
110C4×D107#4112yes—
111C8×C107—21no—
112C8×C107C2×D412yes—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107—21no—
116C22×C107—22no—
117C22×C107C22×D411yescharacteristic
118C2×D107—22no—
119C2×D107—21no—
120C2×D107—22no—
121C2×D107—44no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107#12112yes—
126C4×C107—21no—
127C4×C107—42no—
128C4×C107—42no—
129C4×C107C22×D411yescharacteristic
130C4×C107C2×#4111yescharacteristic
131Dic107—21no—
132Dic107—42no—
133Dic107#12112yes—
134C2×C107—41no—
135C2×C107C2×#12111yescharacteristic
136C2×C107#57512yes—
137D107—22no—
138D107—41no—
139D107—82no—
140#3054—1071nomaximal, Sylow
141C107#305411yescharacteristic, complemented, Sylow
142C2×#135—1071no—
143C22×M16—1071no—
144#525—1072no—
145#470—1071no—
146#470D10711yescharacteristic, complemented
147#469—1071no—
148#469—1071no—
149#468—1072no—
150#468—1072no—
151#427—1071no—
152#416—1072no—
153C2×#41—1072no—
154C23×C4—1071no—
155#135—2144no—
156C2×Q16C2×D10711yescharacteristic, complemented
157C2×SD16—1072no—
158C2×D8—1071no—
159C2×M16—1071no—
160C2×M16—1071no—
161C2×M16—1071no—
162C2×M16—2144no—
163C2×M16C2×D10711yescharacteristic
164C22×C8—1071no—
165C22×C8—1071no—
166#123—2142no—
167C22⋊Q8—1072no—
168C22⋊Q8C2×D10712yes—
169#121—1072no—
170#121—1072no—
171#120—2142no—
172C4×D4—2142no—
173C4×D4—2142no—
174#117—1072no—
175#107—1072no—
176#106—1071no—
177#106C2×D10711yescharacteristic, complemented
178#103—1072no—
179#103C2×D10712yescomplemented
180#102—1072no—
181#102—1072no—
182C24—1071no—
183#41—2144no—
184#41—2144no—
185C2×Q8C22×D10712yes—
186C2×D4—1072no—
187C2×D4—2142no—
188C2×D4—2142no—
189C2×D4—2142no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—1071no—
194C22×C4—1071no—
195C22×C4—2142no—
196C22×C4—2142no—
197C22×C4—2142no—
198C22×C4C22×D10711yescharacteristic
199Q16—22no—
200SD16—2144no—
201D8—2142no—
202M16—22no—
203M16—2142no—
204M16—2142no—
205M16—2142no—
206C2×C8—1071no—
207C2×C8—1074no—
208C2×C8—1071no—
209C2×C8—2142no—
210C2×C8C22×D10711yescharacteristic
211C2×C8C22×D10711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—1072no—
214C4⋊C4—2142no—
215C4⋊C4C22×D10712yes—
216C22⋊C4—22no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C42—2142no—
222C23—1071no—
223C23—1071no—
224C23—4282no—
225C23—2142no—
226C23—2142no—
227C23D4×D10711yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—4284no—
231D4—4284no—
232D4—4284no—
233D4—2142no—
234D4—2142no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—1071no—
240C2×C4—1072no—
241C2×C4—1072no—
242C2×C4—1071no—
243C2×C4—4284no—
244C2×C4—2142no—
245C2×C4—2142no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4D4×D10711yescharacteristic
251C2×C4C23×D10711yescharacteristic
252C8—21no—
253C8—1072no—
254C8—2141no—
255C8D4×D10712yes—
256C22—22no—
257C22—21no—
258C22—1072no—
259C22—4284no—
260C22—4282no—
261C22—4282no—
262C22—4282no—
263C22—2142no—
264C22—2141no—
265C22—2142no—
266C22C2×D4×D10711yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—1072no—
271C4—4282no—
272C4—2141no—
273C4C2×D4×D10711yescharacteristic
274C4#41×D10711yescharacteristic
275C2—41no—
276C2—8562no—
277C2—4281no—
278C2—2142no—
279C2#121×D10711yescharacteristic
280C2#2029342612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
hypercenter#470D107
hypocenterC107#3054
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C2×M16×D107C2semidirect
3. #20293140C2semidirect
4. #20292882C2semidirect
5. #20292881C2semidirect
6. #20292880C2semidirect
7. #20292876C2semidirect
8. #20292875C2semidirect
9. #20292662C2semidirect
10. #20292622C2semidirect
11. #470×C107C2semidirect
15. C2×#13133805C22semidirect
16. C2×D856C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812033C22semidirect
34. #16812031C22semidirect
35. #16812029C22semidirect
36. C2×Q16×C107C22semidirect
39. #106×C107C22semidirect
40. #103×C107C22semidirect
64. C8×D107D4semidirect
141. C107#3054semidirect
146. #470D107semidirect
156. C2×Q16C2×D107semidirect
177. #106C2×D107semidirect
179. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C2×M16×D1072
#202931402
#202928822
#202928812
#202928802
#202928762
#202928752
#202926622
#202926222
#470×C1072
#3054107