Finite Group Databasebeta

← #470×D107 #28346059

Subgroups — #470×D107 #28346059

Order 13696.

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

normal subgroups only

This group has 492 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
1C2×Q16×D107C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic
3C22⋊Q8×D107C212yesmaximal
4#20292886C211yesmaximal, characteristic, complemented
5#20292885C211yesmaximal, characteristic, complemented
6#20292884C212yesmaximal, complemented
7#20292883C212yesmaximal, complemented
8#20292882C211yesmaximal, characteristic, complemented
9#106×D107C211yesmaximal, characteristic, complemented
10#103×D107C212yesmaximal, complemented
11#470×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14C2×Dic428C2211yescharacteristic, complemented
15C2×#13133808C2212yescomplemented
16Q16×D107—24no—
17C2×#13133802C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20M16×D107—24no—
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812089C2212yes—
24#16812088C2212yes—
25#16812087—22no—
26#16812086—22no—
27#16812085C2212yes—
28C4⋊C4×D107—22no—
29C4⋊C4×D107C2212yes—
30C22⋊C4×D107—22no—
31#16812041C2211yescharacteristic, complemented
32#16812040C2212yescomplemented
33#16812034C2212yescomplemented
34#16812033C2212yescomplemented
35#16812032C2212yescomplemented
36C2×Q16×C107C2211yescharacteristic, complemented
37C2×M16×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#106×C107C2211yescharacteristic, complemented
40#103×C107C2212yescomplemented
41C23×D107D411yescharacteristic
42C2×Dic214—22no—
43C2×Dic214C2312yes—
44Q8×D107—24no—
45Q8×D107—24no—
46C22×Dic107D411yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107D411yescharacteristic
52C2×C4×D107D411yescharacteristic
53C2×C4×D107C2311yescharacteristic
54C2×C4×D107C2311yescharacteristic
55Dic428—22no—
56#13133808—24no—
57#13133802—22no—
58#13133801—22no—
59#13133801—22no—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C8×D107—22no—
63C8×D107D414yes—
64C4⋊Dic107—22no—
65C4⋊Dic107C2312yes—
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68Dic107⋊C4—22no—
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—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—24no—
84C22×D107C2×D411yescharacteristic
85C22×D107C2×D411yescharacteristic
86Q8×C107—22no—
87Q8×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107C2×D411yescharacteristic
98C2×Dic107C2×D411yescharacteristic
99C4×D107—22no—
100C4×D107—21no—
101C4×D107—21no—
102C4×D107—22no—
103C4×D107—44no—
104C4×D107—44no—
105C4×D107C2×D412yes—
106C4×D107#4112yes—
107Dic214—22no—
108Dic214—22no—
109Dic214—44no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C107⋊C8—21no—
113C107⋊C8C2×D412yes—
114C22×C107—21no—
115C22×C107—22no—
116C22×C107C22×D411yescharacteristic
117C2×D107—21no—
118C2×D107—22no—
119C2×D107—44no—
120C2×D107#12112yes—
121C2×D107#13714yes—
122C4×C107—21no—
123C4×C107—42no—
124C4×C107—42no—
125C4×C107C22×D411yescharacteristic
126C4×C107C2×#4111yescharacteristic
127Dic107—21no—
128Dic107—42no—
129Dic107—42no—
130Dic107#12112yes—
131C2×C107—41no—
132C2×C107C2×#12111yescharacteristic
133C2×C107C2×#13712yes—
134D107—41no—
135D107#47014yesdirect product factor
136C2×#470—1071nomaximal, Sylow
137C107C2×#47011yescharacteristic, complemented, Sylow
138C22×Q16—1071no—
139C22×M16—1071no—
140C2×C22⋊Q8—1072no—
141#470—1071no—
142#470—1071no—
143#470—1072no—
144#470—1071no—
145#470—1072no—
146#470D10711yescharacteristic, direct product factor
147C2×#106—1071no—
148C2×#103—1072no—
149C22×Q8—1072no—
150C23×C4—1071no—
151C2×Q16—1072no—
152C2×Q16—1071no—
153C2×Q16—2144no—
154C2×Q16C2×D10711yescharacteristic, complemented
155C2×M16—1071no—
156C2×M16—1071no—
157C2×M16—1071no—
158C2×M16—2144no—
159C2×M16C2×D10711yescharacteristic
160C22×C8—1071no—
161C22×C8—1071no—
162C22⋊Q8—1072no—
163C22⋊Q8—1072no—
164C22⋊Q8—1072no—
165C22⋊Q8—2142no—
166C22⋊Q8—2142no—
167C22⋊Q8C2×D10712yes—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—2142no—
170C2×C22⋊C4—2142no—
171#106—1071no—
172#106—1072no—
173#106C2×D10711yescharacteristic, complemented
174#103—1072no—
175#103—1072no—
176#103—1072no—
177#103C2×D10712yescomplemented
178C24—1071no—
179C2×Q8—1072no—
180C2×Q8—2142no—
181C2×Q8—2144no—
182C2×Q8—2144no—
183C2×Q8C22×D10712yes—
184C22×C4—1071no—
185C22×C4—1071no—
186C22×C4—1071no—
187C22×C4—1071no—
188C22×C4—1071no—
189C22×C4—2142no—
190C22×C4—2142no—
191C22×C4—2142no—
192C22×C4—2142no—
193C22×C4C22×D10711yescharacteristic
194Q16—22no—
195Q16—2144no—
196Q16—2142no—
197M16—22no—
198M16—2142no—
199M16—2142no—
200M16—2142no—
201C2×C8—1071no—
202C2×C8—1074no—
203C2×C8—1071no—
204C2×C8—2142no—
205C2×C8C22×D10711yescharacteristic
206C2×C8C22×D10711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—1072no—
209C4⋊C4—2142no—
210C4⋊C4—2142no—
211C4⋊C4—2142no—
212C4⋊C4—2142no—
213C4⋊C4C22×D10712yes—
214C22⋊C4—22no—
215C22⋊C4—2142no—
216C22⋊C4—2142no—
217C22⋊C4—2142no—
218C23—1071no—
219C23—1071no—
220C23—2142no—
221C23—2144no—
222C23D4×D10711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—4284no—
226Q8—2142no—
227Q8—2142no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1071no—
233C2×C4—1072no—
234C2×C4—1072no—
235C2×C4—1071no—
236C2×C4—4284no—
237C2×C4—4284no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2142no—
243C2×C4—2141no—
244C2×C4—2141no—
245C2×C4D4×D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—21no—
248C8—1072no—
249C8—2141no—
250C8D4×D10712yes—
251C22—22no—
252C22—21no—
253C22—1074no—
254C22—1072no—
255C22—4284no—
256C22—2142no—
257C22—2141no—
258C22C2×D4×D10711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—1072no—
263C4—4282no—
264C4—4282no—
265C4—2141no—
266C4C2×D4×D10711yescharacteristic
267C4#41×D10711yescharacteristic
268C2—41no—
269C2—1074no—
270C2—4281no—
271C2#121×D10711yescharacteristic
272C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
hypercenter#470D107
hypocenterC107C2×#470
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×Q16×D107C2semidirect
4. #20292886C2semidirect
5. #20292885C2semidirect
6. #20292884C2semidirect
7. #20292883C2semidirect
8. #20292882C2semidirect
9. #106×D107C2semidirect
10. #103×D107C2semidirect
11. #470×C107C2semidirect
14. C2×Dic428C22semidirect
15. C2×#13133808C22semidirect
31. #16812041C22semidirect
32. #16812040C22semidirect
33. #16812034C22semidirect
34. #16812033C22semidirect
35. #16812032C22semidirect
36. C2×Q16×C107C22semidirect
39. #106×C107C22semidirect
40. #103×C107C22semidirect
135. D107#470direct (N × H)
137. C107C2×#470semidirect
146. #470D107direct (N × H)
154. C2×Q16C2×D107semidirect
173. #106C2×D107semidirect
177. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
C2×M16×D1072
C22⋊Q8×D1072
#202928862
#202928852
#202928842
#202928832
#202928822
#106×D1072
#103×D1072
#470×C1072
C2×#470107