Finite Group Databasebeta

← #28345697

Subgroups — #28345697

Order 13696.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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×D107C211yesmaximal, characteristic, complemented
2C2×#16812142C211yesmaximal, characteristic, complemented
3C2×#16812140C211yesmaximal, characteristic, complemented
4#20293139C211yesmaximal, characteristic
5#20292784C214yesmaximal, complemented
6#20292767C214yesmaximal, complemented
7#20292635C211yesmaximal, characteristic, complemented
8#20292590C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214—21no—
12C22×Dic214C2211yescharacteristic
13C2×Q8×D107—24no—
14C2×Q8×D107—22no—
15C2×Q8×D107—24no—
16C2×Q8×D107C2214yes—
17C22×C4×D107—21no—
18C22×C4×D107C2211yescharacteristic
19#16812142—24no—
20#16812140—24no—
21C2×Dic428C2212yescomplemented
22C2×#13133808C2212yescomplemented
23C2×#13133807C2212yescomplemented
24C2×#13133805C2212yescomplemented
25C2×#13133802C2211yescharacteristic
26#16812093—21no—
27#16812089C2212yes—
28#16812084C2212yes—
29#16812079—21no—
30C4×D428—22no—
31#16812032C2214yescomplemented
32(C2×D107)⋊C8C2214yes—
33C22×Q8×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic
35#117×C107C2211yescharacteristic
36#103×C107C2214yescomplemented
37C23×D107D411yescharacteristic
38#13133818—22no—
39#13133818—22no—
40C2×Dic214—28no—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214—24no—
44C2×Dic214D414yescomplemented
45C2×Dic214C2311yescharacteristic
46C2×Dic214C2312yes—
47Q8×D107—216no—
48Q8×D107—28no—
49Q8×D107—28no—
50C2×D428—22no—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C22×Dic107—21no—
54C22×Dic107D411yescharacteristic
55C2×C4×D107—28no—
56C2×C4×D107—24no—
57C2×C4×D107—22no—
58C2×C4×D107—22no—
59C2×C4×D107—21no—
60C2×C4×D107D412yes—
61C2×C4×D107C2312yes—
62Dic428—24no—
63#13133808—24no—
64#13133807—24no—
65#13133805—24no—
66#13133802—22no—
67C2×C107⋊C8C2312yes—
68C4⋊Dic107—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C2×Q8×C107—22no—
72C2×Q8×C107—21no—
73C2×Q8×C107D414yescomplemented
74C2×Q8×C107C2312yes—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107—22no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—21no—
81C42×C107—21no—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—24no—
85C22×D107—21no—
86C22×D107C2×D412yes—
87Q8×C107—24no—
88Q8×C107—22no—
89Q8×C107C2×D414yes—
90C2×C4×C107—24no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—42no—
98C2×Dic107—24no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—21no—
102C2×Dic107C2×D412yes—
103D428—21no—
104D428—21no—
105D428—44no—
106C4×D107—216no—
107C4×D107—24no—
108C4×D107—22no—
109C4×D107—22no—
110C4×D107—42no—
111Dic214—216no—
112Dic214—24no—
113Dic214—22no—
114Dic214—21no—
115Dic214—21no—
116Dic214C2×D414yes—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—22no—
120C22×C107#12012yes—
121C22×C107C22×D411yescharacteristic
122C2×D107—28no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—42no—
127C2×D107#13714yescomplemented
128C4×C107—24no—
129C4×C107—42no—
130C4×C107#12012yes—
131C4×C107C22×D411yescharacteristic
132C4×C107C22×D411yescharacteristic
133Dic107—22no—
134Dic107—24no—
135Dic107—41no—
136C2×C107—22no—
137C2×C107C2×#12011yescharacteristic
138C2×C107C2×#13712yes—
139D107—24no—
140D107—81no—
141#3004—1071nomaximal, Sylow
142C107#300411yescharacteristic, complemented, Sylow
143C23×Q8—1071no—
144C2×#137—1071no—
145C2×#137—1071no—
146#525—1071no—
147#451—1074no—
148#448—1074no—
149#419—1071no—
150#419D10711yescharacteristic, complemented
151#407—1071no—
152C2×#41—1071no—
153C22×Q8—1074no—
154C22×Q8—1071no—
155C22×Q8—2141no—
156C22×Q8—2142no—
157C22×Q8—2144no—
158C22×Q8—2144no—
159C22×Q8C2×D10711yescharacteristic
160C23×C4—1071no—
161C23×C4—2141no—
162#137—2144no—
163#137—2144no—
164C2×Q16—1072no—
165C2×Q16—1072no—
166C2×SD16—1072no—
167C2×SD16—1072no—
168C2×M16—1071no—
169C2×M16C2×D10711yescharacteristic
170#123—2141no—
171C22⋊Q8—1072no—
172#121—1072no—
173#120—2141no—
174C4×D4—2142no—
175#117C2×D10711yescharacteristic
176#103—1074no—
177#103C2×D10714yescomplemented
178C22⋊C8—1074no—
179C24—1071no—
180#41—2142no—
181#41—2142no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—1074no—
185C2×Q8—1072no—
186C2×Q8—1071no—
187C2×Q8—2144no—
188C2×Q8—2148no—
189C2×Q8—2148no—
190C2×Q8—21416no—
191C2×Q8—2148no—
192C2×Q8—2141no—
193C2×Q8—2142no—
194C2×Q8D42814yescomplemented
195C2×Q8C22×D10712yes—
196C2×D4—1071no—
197C2×D4—2142no—
198C2×D4—2141no—
199C22×C4—21no—
200C22×C4—1071no—
201C22×C4—1072no—
202C22×C4—1072no—
203C22×C4—2144no—
204C22×C4—2148no—
205C22×C4—2141no—
206C22×C4—2141no—
207C22×C4—2142no—
208C22×C4—2142no—
209C22×C4C22×D10711yescharacteristic
210Q16—2144no—
211Q16—2144no—
212SD16—2144no—
213SD16—2144no—
214M16—22no—
215M16—2142no—
216C2×C8—1072no—
217C2×C8C22×D10712yes—
218C4⋊C4—2142no—
219C4⋊C4C22×D10712yes—
220C22⋊C4—21no—
221C22⋊C4—2142no—
222C22⋊C4—2142no—
223C42—21no—
224C23—1072no—
225C23—2144no—
226C23—2141no—
227C23—2142no—
228C23D4×D10711yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—1074no—
232Q8—21416no—
233Q8—2141no—
234Q8—2141no—
235Q8—2144no—
236Q8—2142no—
237Q8C2×D42814yes—
238D4—4284no—
239D4—4282no—
240D4—2141no—
241D4—2141no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1072no—
246C2×C4—4282no—
247C2×C4—21416no—
248C2×C4—2141no—
249C2×C4—2142no—
250C2×C4—2144no—
251C2×C4—2142no—
252C2×C4—2144no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4D4×D10712yes—
257C2×C4C2×D42812yes—
258C2×C4C23×D10711yescharacteristic
259C8—22no—
260C8—2142no—
261C22—22no—
262C22—1074no—
263C22—4282no—
264C22—4281no—
265C22—2148no—
266C22—2142no—
267C22—2142no—
268C22#1681207912yes—
269C22C2×D4×D10711yescharacteristic
270C4—24no—
271C4—42no—
272C4—4281no—
273C4—2142no—
274C4—2144no—
275C4#1681207912yes—
276C4C2×D4×D10711yescharacteristic
277C4C22×D42811yescharacteristic
278C2—22no—
279C2—8561no—
280C2—2144no—
281C2C2×#1681207911yescharacteristic
282C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radical#20293139C2
hypercenter#419D107
hypocenterC107#3004
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812142C2semidirect
3. C2×#16812140C2semidirect
5. #20292784C2semidirect
6. #20292767C2semidirect
7. #20292635C2semidirect
8. #20292590C2semidirect
9. #419×C107C2semidirect
21. C2×Dic428C22semidirect
22. C2×#13133808C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133805C22semidirect
31. #16812032C22semidirect
36. #103×C107C22semidirect
44. C2×Dic214D4semidirect
73. C2×Q8×C107D4semidirect
127. C2×D107#137semidirect
142. C107#3004semidirect
150. #419D107semidirect
177. #103C2×D107semidirect
194. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168121422
C2×#168121402
#202931392
#202927842
#202927672
#202926352
#202925902
#419×C1072
#3004107