Finite Group Databasebeta

← #28345706

Subgroups — #28345706

Order 13696.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812142C211yesmaximal, characteristic, complemented
3C2×#16812137C211yesmaximal, characteristic, complemented
4#20293138C211yesmaximal, characteristic, complemented
5#20292791C214yesmaximal, complemented
6#20292774C214yesmaximal, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292633C211yesmaximal, characteristic, complemented
9#407×C107C211yesmaximal, characteristic, complemented
10C2×#13133818—24no—
11C2×#13133818—24no—
12C2×#13133818—22no—
13C2×#13133818C2211yescharacteristic
14C2×#13133818C2214yes—
15C22×Dic214C2211yescharacteristic
16C22×D428C2211yescharacteristic
17C22×C22⋊D107—21no—
18C22×C4×D107—21no—
19#16812142—24no—
20#16812137—24no—
21C2×Dic428C2212yescomplemented
22C2×#13133807C2212yescomplemented
23C2×#13133807C2212yescomplemented
24C2×D856C2212yescomplemented
25#16812092—21no—
26#16812088C2212yescomplemented
27#16812083C2212yescomplemented
28#16812078—21no—
29C4×C22⋊D107—22no—
30#16812068C2211yescharacteristic
31#16812034C2214yescomplemented
32#16812031C2214yescomplemented
33C23×C4×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic
35C2×M16×C107C2211yescharacteristic
36C22⋊C8×C107C2214yescomplemented
37C23×D107—21no—
38#13133818—216no—
39#13133818—28no—
40#13133818—22no—
41#13133818—22no—
42#13133818—28no—
43C2×Dic214—22no—
44C2×Dic214—21no—
45C2×Dic214D414yescomplemented
46C2×Dic214C2311yescharacteristic
47C2×Dic214C2312yes—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428D414yescomplemented
51C2×D428C2311yescharacteristic
52C2×D428C2312yes—
53C2×C22⋊D107—28no—
54C2×C22⋊D107—24no—
55C2×C22⋊D107—22no—
56C2×C22⋊D107—21no—
57C22×Dic107—21no—
58C2×C4×D107—28no—
59C2×C4×D107—24no—
60C2×C4×D107—21no—
61Dic428—24no—
62#13133807—24no—
63#13133807—24no—
64D856—24no—
65C4⋊Dic107C2312yes—
66Dic107⋊C4—22no—
67C22⋊Dic107—21no—
68C22⋊Dic107—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107—21no—
71C24×C107D411yescharacteristic
72C22×C4×C107—22no—
73C22×C4×C107—22no—
74C22×C4×C107D412yes—
75C22×C4×C107D411yescharacteristic
76C22×C4×C107C2312yes—
77C22×C4×C107C2311yescharacteristic
78M16×C107—22no—
79M16×C107—22no—
80C2×C8×C107C2312yes—
81C2×C8×C107C2312yes—
82C23×C107—22no—
83C23×C107—42no—
84C23×C107C2×D412yes—
85C23×C107C2×D411yescharacteristic
86C22×D107—24no—
87C22×D107—22no—
88C22×D107—21no—
89C22×D107—44no—
90C2×C4×C107—22no—
91C2×C4×C107—24no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—216no—
100C22⋊D107—44no—
101C22⋊D107—42no—
102C2×Dic107—24no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—21no—
106D428—24no—
107D428—22no—
108D428—21no—
109D428—21no—
110D428C2×D414yes—
111C4×D107—216no—
112C4×D107—42no—
113Dic214—24no—
114Dic214—22no—
115Dic214—21no—
116Dic214—21no—
117Dic214C2×D414yes—
118C8×C107—22no—
119C8×C107—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—44no—
125C22×C107#12012yes—
126C22×C107C22×D411yescharacteristic
127C2×D107—24no—
128C2×D107—48no—
129C2×D107—44no—
130C2×D107—41no—
131C2×D107—42no—
132C4×C107—22no—
133C4×C107#12012yes—
134C4×C107C22×D411yescharacteristic
135C4×C107C22×D411yescharacteristic
136Dic107—24no—
137Dic107—41no—
138Dic107—42no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107C2×#12011yescharacteristic
142C2×C107#57512yes—
143D107—44no—
144D107—81no—
145#3005—1071nomaximal, Sylow
146C107#300511yescharacteristic, complemented, Sylow
147C22×#41—1071no—
148C2×#137—1071no—
149C2×AGL(1,ℤ/8)—1071no—
150#525—1071no—
151#452—1074no—
152#449—1074no—
153#419—1071no—
154#418—1071no—
155#407D10711yescharacteristic, complemented
156C2×#41—1071no—
157C2×#41—1074no—
158C2×#41—2144no—
159C2×#41—2144no—
160C2×#41—2142no—
161C22×Q8—1071no—
162C22×D4—1071no—
163C22×D4—2141no—
164C23×C4—2141no—
165C23×C4C2×D10711yescharacteristic
166#137—2144no—
167AGL(1,ℤ/8)—2144no—
168C2×Q16—1072no—
169C2×SD16—1072no—
170C2×SD16—1072no—
171C2×D8—1072no—
172C2×M16C2×D10711yescharacteristic
173C2×M16C2×D10711yescharacteristic
174#123—2141no—
175C22⋊Q8—1072no—
176#121—1072no—
177#120—2141no—
178C4×D4—2142no—
179#117—1071no—
180#103—1074no—
181#102—1074no—
182C22⋊C8C2×D10714yescomplemented
183C24—2141no—
184C24D42811yescharacteristic
185#41—2148no—
186#41—2148no—
187#41—21416no—
188#41—2142no—
189#41—2142no—
190C2×Q8—1074no—
191C2×Q8—1072no—
192C2×Q8—1071no—
193C2×Q8—2142no—
194C2×Q8—2141no—
195C2×D4—1074no—
196C2×D4—1072no—
197C2×D4—1071no—
198C2×D4—2142no—
199C2×D4—2141no—
200C2×D4—2148no—
201C2×D4—2144no—
202C2×D4—2142no—
203C2×D4—2141no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2148no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4—2144no—
210C22×C4D42812yes—
211C22×C4D42811yescharacteristic
212C22×C4C22×D10712yes—
213C22×C4C22×D10711yescharacteristic
214Q16—2144no—
215SD16—2144no—
216SD16—2144no—
217D8—2144no—
218M16—22no—
219M16—22no—
220C2×C8C22×D10712yes—
221C2×C8C22×D10712yes—
222C4⋊C4—1072no—
223C4⋊C4—2142no—
224C22⋊C4—2141no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C42—2141no—
228C23—22no—
229C23—42no—
230C23—4284no—
231C23—2141no—
232C23—2144no—
233C23—2142no—
234C23C2×D42812yes—
235C23C2×D42811yescharacteristic
236Q8—1074no—
237Q8—2141no—
238Q8—2141no—
239Q8—2144no—
240Q8—2142no—
241D4—1074no—
242D4—4284no—
243D4—4282no—
244D4—21416no—
245D4—2144no—
246D4—2141no—
247D4—2141no—
248D4—2142no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—4282no—
254C2×C4—21416no—
255C2×C4—2142no—
256C2×C4—2141no—
257C2×C4—2144no—
258C2×C4—2142no—
259C2×C4D4×D10712yes—
260C2×C4D4×D10712yes—
261C2×C4C2×D42812yes—
262C2×C4C2×D42811yescharacteristic
263C2×C4C23×D10711yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—4288no—
272C22—4282no—
273C22—4281no—
274C22—4284no—
275C22—2144no—
276C22#1681207912yes—
277C22C22×D42811yescharacteristic
278C4—22no—
279C4—4282no—
280C4—4281no—
281C4—2144no—
282C4#1681207912yes—
283C4C2×D4×D10711yescharacteristic
284C4C2×D4×D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—8561no—
288C2—4284no—
289C2C2×#1681207911yescharacteristic
290C2#2029342712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
hypercenter#407D107
hypocenterC107#3005
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812142C2semidirect
3. C2×#16812137C2semidirect
4. #20293138C2semidirect
5. #20292791C2semidirect
6. #20292774C2semidirect
7. #20292637C2semidirect
8. #20292633C2semidirect
9. #407×C107C2semidirect
21. C2×Dic428C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
24. C2×D856C22semidirect
26. #16812088C22semidirect
27. #16812083C22semidirect
31. #16812034C22semidirect
32. #16812031C22semidirect
36. C22⋊C8×C107C22semidirect
45. C2×Dic214D4semidirect
50. C2×D428D4semidirect
146. C107#3005semidirect
155. #407D107semidirect
182. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121422
C2×#168121372
#202931382
#202927912
#202927742
#202926372
#202926332
#407×C1072
#3005107