Finite Group Databasebeta

← #28345690

Subgroups — #28345690

Order 13696.

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

normal subgroups only

This group has 994 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×D4×D107C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C2×#16812135C211yesmaximal, characteristic, complemented
4#20293139C211yesmaximal, characteristic, complemented
5#20292779C214yesmaximal, complemented
6#20292762C214yesmaximal, complemented
7#20292631C211yesmaximal, characteristic, complemented
8#20292590C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C24×D107—21no—
11C2×#13133818C2211yescharacteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107—21no—
14C2×D4×D107—24no—
15C2×D4×D107—22no—
16C2×D4×D107—24no—
17C2×D4×D107C2214yes—
18C22×C4×D107C2211yescharacteristic
19#16812137—24no—
20#16812135—24no—
21C2×#13133807C2212yescomplemented
22C2×#13133806C2212yescomplemented
23C2×D856C2212yescomplemented
24C2×#13133803C2212yescomplemented
25C2×#13133802C2211yescharacteristic, complemented
26#16812093—21no—
27#16812089C2212yescomplemented
28#16812084C2212yescomplemented
29#16812079—21no—
30C4×D428—22no—
31#16812029C2214yescomplemented
32(C2×D107)⋊C8C2214yescomplemented
33C22×D4×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic, complemented
35#117×C107C2211yescharacteristic, complemented
36#102×C107C2214yescomplemented
37C23×D107—28no—
38C23×D107—21no—
39C23×D107—24no—
40C23×D107—48no—
41C23×D107D411yescharacteristic
42#13133818—22no—
43#13133818—22no—
44C2×Dic214C2311yescharacteristic, complemented
45C2×D428—21no—
46C2×D428—22no—
47C2×D428—22no—
48C2×D428D414yescomplemented
49C2×D428C2312yes—
50C2×D428C2311yescharacteristic, complemented
51C2×C22⋊D107—28no—
52C2×C22⋊D107—21no—
53C2×C22⋊D107—24no—
54D4×D107—216no—
55D4×D107—28no—
56D4×D107—28no—
57C22×Dic107D411yescharacteristic
58C2×C4×D107—22no—
59C2×C4×D107—22no—
60C2×C4×D107—21no—
61C2×C4×D107D412yes—
62C2×C4×D107C2312yes—
63#13133807—24no—
64#13133806—24no—
65D856—24no—
66#13133803—24no—
67#13133802—22no—
68C2×C107⋊C8C2312yescomplemented
69C4⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C24×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107—22no—
75C2×D4×C107D414yescomplemented
76C2×D4×C107C2312yes—
77C22×C4×C107C2311yescharacteristic
78M16×C107—22no—
79C2×C8×C107C2312yescomplemented
80C4⋊C4×C107C2312yescomplemented
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107—24no—
84C23×C107—22no—
85C23×C107—44no—
86C23×C107C2×D411yescharacteristic
87C22×D107—22no—
88C22×D107—24no—
89C22×D107—216no—
90C22×D107—24no—
91C22×D107—22no—
92C22×D107—21no—
93C22×D107—432no—
94C22×D107—48no—
95C22×D107—44no—
96C22×D107—48no—
97C22×D107C2×D412yes—
98D4×C107—24no—
99D4×C107—22no—
100D4×C107C2×D414yes—
101C2×C4×C107—22no—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—216no—
107C22⋊D107—42no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107C2×D412yes—
111D428—24no—
112D428—22no—
113D428—21no—
114D428—21no—
115D428—44no—
116D428C2×D414yes—
117C4×D107—24no—
118C4×D107—22no—
119C4×D107—22no—
120C4×D107—42no—
121Dic214—21no—
122Dic214—21no—
123C8×C107—22no—
124C107⋊C8—22no—
125C22×C107—22no—
126C22×C107—24no—
127C22×C107—48no—
128C22×C107—44no—
129C22×C107#12012yes—
130C22×C107C22×D411yescharacteristic
131C2×D107—28no—
132C2×D107—22no—
133C2×D107—22no—
134C2×D107—24no—
135C2×D107—432no—
136C2×D107—48no—
137C2×D107—44no—
138C2×D107—41no—
139C2×D107—42no—
140C2×D107AGL(1,ℤ/8)14yescomplemented
141C4×C107—42no—
142C4×C107#12012yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C22×D411yescharacteristic
145Dic107—22no—
146Dic107—41no—
147C2×C107—22no—
148C2×C107—44no—
149C2×C107C2×#12011yescharacteristic
150C2×C107C2×AGL(1,ℤ/8)12yes—
151D107—24no—
152D107—44no—
153D107—81no—
154#3003—1071nomaximal, Sylow
155C107#300311yescharacteristic, complemented, Sylow
156C23×D4—1071no—
157C2×AGL(1,ℤ/8)—1071no—
158C2×AGL(1,ℤ/8)—1071no—
159#525—1071no—
160#450—1074no—
161#447—1074no—
162#418—1071no—
163#418D10711yescharacteristic, complemented
164#407—1071no—
165C25—2141no—
166C2×#41—1071no—
167C22×D4—1074no—
168C22×D4—1071no—
169C22×D4—2142no—
170C22×D4—2141no—
171C22×D4—2144no—
172C22×D4—2144no—
173C22×D4C2×D10711yescharacteristic
174C23×C4—1071no—
175AGL(1,ℤ/8)—2144no—
176AGL(1,ℤ/8)—2144no—
177C2×SD16—1072no—
178C2×SD16—1072no—
179C2×D8—1072no—
180C2×D8—1072no—
181C2×M16—1071no—
182C2×M16C2×D10711yescharacteristic, complemented
183#123—2141no—
184C22⋊Q8—1072no—
185#121—1072no—
186#120—2141no—
187C4×D4—2142no—
188#117C2×D10711yescharacteristic, complemented
189#102—1074no—
190#102C2×D10714yescomplemented
191C22⋊C8—1074no—
192C24—21no—
193C24—1071no—
194C24—4288no—
195C24—2144no—
196C24—2148no—
197C24—2141no—
198#41—2142no—
199#41—2142no—
200C2×Q8—1071no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—1074no—
204C2×D4—1072no—
205C2×D4—1071no—
206C2×D4—2142no—
207C2×D4—2144no—
208C2×D4—2148no—
209C2×D4—21416no—
210C2×D4—2141no—
211C2×D4—2148no—
212C2×D4—2148no—
213C2×D4—2141no—
214C2×D4—2142no—
215C2×D4D42814yescomplemented
216C2×D4C22×D10712yes—
217C22×C4—1071no—
218C22×C4—1072no—
219C22×C4—1072no—
220C22×C4—2141no—
221C22×C4—2142no—
222C22×C4—2142no—
223C22×C4C22×D10711yescharacteristic
224SD16—2144no—
225SD16—2144no—
226D8—2144no—
227D8—2144no—
228M16—22no—
229M16—2142no—
230C2×C8—1072no—
231C2×C8C22×D10712yescomplemented
232C4⋊C4—2142no—
233C4⋊C4C22×D10712yescomplemented
234C22⋊C4—21no—
235C22⋊C4—2142no—
236C22⋊C4—2142no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—1072no—
242C23—4284no—
243C23—4288no—
244C23—42832no—
245C23—4288no—
246C23—2144no—
247C23—21416no—
248C23—2141no—
249C23—2144no—
250C23—2142no—
251C23—2142no—
252C23D4×D10711yescharacteristic
253Q8—2141no—
254Q8—2141no—
255D4—24no—
256D4—22no—
257D4—1074no—
258D4—4284no—
259D4—4282no—
260D4—21416no—
261D4—2144no—
262D4—2141no—
263D4—2141no—
264D4—2142no—
265D4C2×D42814yes—
266C2×C4—22no—
267C2×C4—1072no—
268C2×C4—4282no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4—2144no—
272C2×C4—2142no—
273C2×C4—2142no—
274C2×C4D4×D10711yescharacteristic
275C2×C4D4×D10712yes—
276C2×C4C2×D42812yes—
277C2×C4C23×D10711yescharacteristic
278C8—22no—
279C8—2142no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—1074no—
285C22—4288no—
286C22—42832no—
287C22—4282no—
288C22—4281no—
289C22—4284no—
290C22—2148no—
291C22—2144no—
292C22—2142no—
293C22—2142no—
294C22#1681207912yes—
295C22C2×D4×D10711yescharacteristic
296C4—42no—
297C4—4281no—
298C4—2142no—
299C4#1681207912yes—
300C4C2×D4×D10711yescharacteristic
301C4C22×D42811yescharacteristic
302C2—22no—
303C2—44no—
304C2—8561no—
305C2—4284no—
306C2—2144no—
307C2C2×#1681207911yescharacteristic
308C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
hypercenter#418D107
hypocenterC107#3003
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812135C2semidirect
4. #20293139C2semidirect
5. #20292779C2semidirect
6. #20292762C2semidirect
7. #20292631C2semidirect
8. #20292590C2semidirect
9. #418×C107C2semidirect
11. C2×#13133818C22semidirect
21. C2×#13133807C22semidirect
22. C2×#13133806C22semidirect
23. C2×D856C22semidirect
24. C2×#13133803C22semidirect
25. C2×#13133802C22semidirect
27. #16812089C22semidirect
28. #16812084C22semidirect
31. #16812029C22semidirect
32. (C2×D107)⋊C8C22semidirect
34. C2×M16×C107C22semidirect
35. #117×C107C22semidirect
36. #102×C107C22semidirect
44. C2×Dic214C23semidirect
48. C2×D428D4semidirect
50. C2×D428C23semidirect
68. C2×C107⋊C8C23semidirect
75. C2×D4×C107D4semidirect
79. C2×C8×C107C23semidirect
80. C4⋊C4×C107C23semidirect
140. C2×D107AGL(1,ℤ/8)semidirect
155. C107#3003semidirect
163. #418D107semidirect
182. C2×M16C2×D107semidirect
188. #117C2×D107semidirect
190. #102C2×D107semidirect
215. C2×D4D428semidirect
231. C2×C8C22×D107semidirect
233. C4⋊C4C22×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168121372
C2×#168121352
#202931392
#202927792
#202927622
#202926312
#202925902
#418×C1072
#3003107