Finite Group Databasebeta

← #28346092

Subgroups — #28346092

Order 13696.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16812142C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293139C211yesmaximal, characteristic
4C2×#16812089C211yesmaximal, characteristic, complemented
5#20292886C212yesmaximal, complemented
6#20292874C212yesmaximal, complemented
7#20292869C212yesmaximal, complemented
8#20292857C212yesmaximal, complemented
9#20292635C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812142—24no—
16C2×Dic428C2212yescomplemented
17C2×#13133807C2212yescomplemented
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2212yes—
20M16×D107—24no—
21C2×C8×D107C2212yes—
22#16812093—21no—
23#16812089—21no—
24#16812089—21no—
25#16812089C2211yescharacteristic
26#16812089C2212yes—
27#16812089C2212yes—
28#16812089C2211yescharacteristic
29#16812084C2211yescharacteristic
30#16812084C2211yescharacteristic
31#16812079—21no—
32C4×D428—21no—
33C4×D428—21no—
34C2×C4⋊Dic107—21no—
35C2×(C2×D107)⋊C4—21no—
36#16812032C2212yescomplemented
37#16812032C2212yescomplemented
38#16812032C2212yescomplemented
39#16812029C2212yescomplemented
40C2×M16×C107C2211yescharacteristic
41#117×C107C2211yescharacteristic
42C2×C4⋊C4×C107C2211yescharacteristic
43#107×C107C2212yescomplemented
44#106×C107C2212yescomplemented
45C23×D107D411yescharacteristic
46#13133818—22no—
47#13133818—22no—
48C2×Dic214—22no—
49C2×Dic214—22no—
50C2×Dic214—21no—
51C2×Dic214C2312yes—
52C2×Dic214C2311yescharacteristic
53C2×D428—21no—
54C2×D428—21no—
55C2×D428C2311yescharacteristic
56C2×C22⋊D107—21no—
57C22×Dic107—21no—
58C22×Dic107D411yescharacteristic
59C2×C4×D107—22no—
60C2×C4×D107—22no—
61C2×C4×D107—21no—
62C2×C4×D107D411yescharacteristic
63C2×C4×D107D411yescharacteristic
64C2×C4×D107C2311yescharacteristic
65C2×C4×D107C2311yescharacteristic
66Dic428—24no—
67#13133807—24no—
68#13133802—22no—
69#13133801—24no—
70C2×C107⋊C8C2312yes—
71C8×D107—24no—
72C4⋊Dic107—22no—
73C4⋊Dic107—22no—
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76(C2×D107)⋊C4—22no—
77(C2×D107)⋊C4—22no—
78(C2×D107)⋊C4—21no—
79(C2×D107)⋊C4—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82C22×C4×C107—21no—
83C22×C4×C107C2311yescharacteristic
84M16×C107D414yescomplemented
85C2×C8×C107C2312yes—
86C4⋊C4×C107—21no—
87C4⋊C4×C107C2311yescharacteristic
88C4⋊C4×C107C2311yescharacteristic
89C4⋊C4×C107C2312yes—
90C22⋊C4×C107—21no—
91C42×C107—21no—
92C23×C107C2×D411yescharacteristic
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—21no—
96C22×D107—41no—
97C22×D107C2×D411yescharacteristic
98C22×D107C2×D411yescharacteristic
99C2×C4×C107—21no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yes—
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—42no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107—22no—
111C2×Dic107—42no—
112C2×Dic107C2×D411yescharacteristic
113C2×Dic107C2×D411yescharacteristic
114D428—21no—
115D428—21no—
116D428—42no—
117D428—42no—
118C4×D107—22no—
119C4×D107—22no—
120C4×D107—21no—
121C4×D107—21no—
122C4×D107—21no—
123C4×D107—21no—
124C4×D107—42no—
125Dic214—22no—
126Dic214—21no—
127Dic214—22no—
128Dic214—21no—
129Dic214—42no—
130C8×C107C2×D414yes—
131C107⋊C8—22no—
132C22×C107—21no—
133C22×C107—21no—
134C22×C107#12112yes—
135C22×C107C22×D411yescharacteristic
136C2×D107—21no—
137C2×D107—21no—
138C2×D107—21no—
139C2×D107—21no—
140C2×D107—21no—
141C2×D107—44no—
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C2×D107#13712yescomplemented
146C4×C107—41no—
147C4×C107—41no—
148C4×C107—42no—
149C4×C107#12112yes—
150C4×C107C22×D411yescharacteristic
151C4×C107C2×#4111yescharacteristic
152Dic107—21no—
153Dic107—21no—
154Dic107—42no—
155Dic107—41no—
156C2×C107—22no—
157C2×C107C2×#12111yescharacteristic
158C2×C107C2×#13711yescharacteristic
159C2×C107#57511yescharacteristic
160D107—22no—
161D107—41no—
162D107—81no—
163#3056—1071nomaximal, Sylow
164C107#305611yescharacteristic, complemented, Sylow
165C2×#137—1071no—
166C22×M16—1071no—
167#525—1071no—
168C2×C22⋊Q8—1071no—
169#470—1072no—
170#468—1072no—
171#467—1072no—
172#465—1072no—
173#428D10711yescharacteristic, complemented
174#419—1071no—
175#417—1071no—
176C2×#41—1071no—
177C22×Q8—1071no—
178C23×C4—1071no—
179#137—2144no—
180C2×Q16—1072no—
181C2×SD16—1072no—
182C2×M16—1071no—
183C2×M16—1072no—
184C2×M16—2144no—
185C2×M16C2×D10711yescharacteristic
186C22×C8—1072no—
187#123—2141no—
188C22⋊Q8—1072no—
189C22⋊Q8—1072no—
190C22⋊Q8—1071no—
191C22⋊Q8—1071no—
192C22⋊Q8—2141no—
193C22⋊Q8—2141no—
194#121—1071no—
195#121—1071no—
196#120—2141no—
197C4×D4—2141no—
198C4×D4—2141no—
199#117C2×D10711yescharacteristic
200C2×C4⋊C4—2141no—
201C2×C4⋊C4C2×D10711yescharacteristic
202C2×C22⋊C4—2141no—
203#107C2×D10712yescomplemented
204#106C2×D10712yescomplemented
205#103—1072no—
206#103—1072no—
207#103—1072no—
208#102—1072no—
209C24—1071no—
210#41—2142no—
211#41—2142no—
212C2×Q8—1072no—
213C2×Q8—1071no—
214C2×Q8—2142no—
215C2×Q8—2142no—
216C2×Q8—2141no—
217C2×D4—1071no—
218C2×D4—2141no—
219C2×D4—2141no—
220C2×D4—2141no—
221C22×C4—21no—
222C22×C4—1071no—
223C22×C4—1071no—
224C22×C4—1071no—
225C22×C4—1071no—
226C22×C4—1071no—
227C22×C4—2141no—
228C22×C4—2141no—
229C22×C4—2142no—
230C22×C4—2142no—
231C22×C4C22×D10711yescharacteristic
232Q16—2144no—
233SD16—2144no—
234M16—2142no—
235M16—2144no—
236M16D42814yescomplemented
237C2×C8—1072no—
238C2×C8—2144no—
239C2×C8C22×D10712yes—
240C4⋊C4—21no—
241C4⋊C4—2142no—
242C4⋊C4—2142no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4C22×D10712yes—
246C4⋊C4C22×D10711yescharacteristic
247C4⋊C4C22×D10711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2141no—
250C22⋊C4—2141no—
251C22⋊C4—2142no—
252C22⋊C4—2142no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C42—21no—
256C23—1071no—
257C23—1071no—
258C23—4281no—
259C23—2142no—
260C23—2141no—
261C23—2142no—
262C23D4×D10711yescharacteristic
263Q8—4282no—
264Q8—2142no—
265Q8—2141no—
266Q8—2142no—
267Q8—2141no—
268D4—4282no—
269D4—4282no—
270D4—4282no—
271D4—2141no—
272D4—2141no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—1071no—
278C2×C4—1071no—
279C2×C4—4282no—
280C2×C4—4282no—
281C2×C4—2141no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4—2142no—
285C2×C4—2142no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4D4×D10711yescharacteristic
291C2×C4C2×D42812yes—
292C2×C4#1313381912yes—
293C2×C4C23×D10711yescharacteristic
294C8—2142no—
295C8C2×D42814yes—
296C22—21no—
297C22—21no—
298C22—1072no—
299C22—4284no—
300C22—4281no—
301C22—4281no—
302C22—4281no—
303C22—2141no—
304C22—2141no—
305C22—2141no—
306C22—2141no—
307C22—2141no—
308C22#1681208412yes—
309C22C2×D4×D10711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—4281no—
314C4—4282no—
315C4—2141no—
316C4—2141no—
317C4#1681208412yes—
318C4C22×D42811yescharacteristic
319C4C2×#1313381911yescharacteristic
320C2—22no—
321C2—8561no—
322C2—4281no—
323C2—2142no—
324C2C2×#1681208411yescharacteristic
325C2#137×D10711yescharacteristic
326C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radical#20293139C2
hypercenter#428D107
hypocenterC107#3056
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×M16×D107C2semidirect
4. C2×#16812089C2semidirect
5. #20292886C2semidirect
6. #20292874C2semidirect
7. #20292869C2semidirect
8. #20292857C2semidirect
9. #20292635C2semidirect
10. #20292626C2semidirect
11. #428×C107C2semidirect
16. C2×Dic428C22semidirect
17. C2×#13133807C22semidirect
36. #16812032C22semidirect
37. #16812032C22semidirect
38. #16812032C22semidirect
39. #16812029C22semidirect
43. #107×C107C22semidirect
44. #106×C107C22semidirect
84. M16×C107D4semidirect
145. C2×D107#137semidirect
164. C107#3056semidirect
173. #428D107semidirect
203. #107C2×D107semidirect
204. #106C2×D107semidirect
236. M16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×M16×D1072
#202931392
C2×#168120892
#202928862
#202928742
#202928692
#202928572
#202926352
#202926262
#428×C1072
#3056107