Finite Group Databasebeta

← #28345749

Subgroups — #28345749

Order 13696.

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

normal subgroups only

This group has 732 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
1C2×#16812156C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C22×#13133807C211yesmaximal, characteristic, complemented
4#20293191C211yesmaximal, characteristic, complemented
5#20292774C214yesmaximal, complemented
6#20292768C214yesmaximal, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic, complemented
9#408×C107C211yesmaximal, characteristic, complemented
10#16812156—24no—
11#16812156—24no—
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2212yescomplemented
14C2×#13133817—22no—
15C2×#13133817—21no—
16C22×Dic214C2211yescharacteristic
17C22×Dic214C2212yescomplemented
18C2×Q8×D107—21no—
19C22×D428C2211yescharacteristic
20#16812137—24no—
21C2×#13133807—24no—
22C2×#13133807C2214yes—
23C2×#13133807C2212yes—
24C2×D856C2212yes—
25#16812105—21no—
26#16812098—21no—
27#16812083C2214yes—
28#16812077—22no—
29#16812068C2211yescharacteristic
30#16812034C2212yescomplemented
31#16812034C2214yescomplemented
32#16812031C2212yescomplemented
33C2×#41×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic
35C22×C8×C107C2211yescharacteristic
36C22⋊C8×C107C2214yescomplemented
37C23×D107—21no—
38#13133818—22no—
39#13133818—24no—
40#13133818—24no—
41#13133818D414yescomplemented
42#13133817—216no—
43#13133817—28no—
44C2×Dic214—21no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—24no—
48C2×Dic214—24no—
49C2×Dic214D414yescomplemented
50C2×Dic214C2312yes—
51C2×Dic214C2311yescharacteristic
52Q8×D107—28no—
53C2×D428—22no—
54C2×D428—22no—
55C2×D428—21no—
56C2×D428C2312yes—
57C2×D428C2311yescharacteristic
58C2×C22⋊D107—21no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—24no—
61C22×Dic107—21no—
62C22×Dic107—22no—
63C2×C4×D107—21no—
64C2×C4×D107—22no—
65#13133807—28no—
66#13133807—24no—
67D856—24no—
68C4⋊Dic107C2312yes—
69C22⋊Dic107—21no—
70(C2×D107)⋊C4—24no—
71C4×Dic107—21no—
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107D411yescharacteristic
75C2×D4×C107D412yes—
76C2×D4×C107D411yescharacteristic
77C22×C4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79M16×C107—22no—
80C2×C8×C107—22no—
81C2×C8×C107C2312yes—
82C2×C8×C107C2312yes—
83C23×C107C2×D412yes—
84C23×C107C2×D411yescharacteristic
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—42no—
88C22×D107—82no—
89Q8×C107—22no—
90D4×C107—24no—
91D4×C107—22no—
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—28no—
100C22⋊D107—24no—
101C22⋊D107—84no—
102C2×Dic107—28no—
103C2×Dic107—24no—
104C2×Dic107—21no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107D428—21no—
108D428—22no—
109D428—22no—
110D428—42no—
111D428C2×D412yes—
112C4×D107—28no—
113C4×D107—24no—
114Dic214—28no—
115Dic214—24no—
116Dic214—22no—
117Dic214—21no—
118Dic214C2×D412yes—
119Dic214C2×D414yes—
120C8×C107—22no—
121C8×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—24no—
125C22×C107#12012yes—
126C22×C107C22×D411yescharacteristic
127C2×D107—22no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—44no—
131C2×D107—84no—
132C4×C107—22no—
133C4×C107#12012yes—
134C4×C107C22×D411yescharacteristic
135C4×C107C22×D411yescharacteristic
136Dic107—22no—
137Dic107—24no—
138Dic107—42no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107C2×#12011yescharacteristic
142C2×C107#57712yes—
143D107—42no—
144D107—82no—
145#3010—1071nomaximal, Sylow
146C107#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1071no—
148C2×AGL(1,ℤ/8)—1071no—
149C22×SD16—1071no—
150#535—1071no—
151#449—1074no—
152#448—1074no—
153#419—1071no—
154#416—1071no—
155#408D10711yescharacteristic, complemented
1562−1+4—2144no—
1572−1+4—2144no—
158C2×#41—1071no—
159C2×#41—1072no—
160C2×#41—2142no—
161C2×#41—2141no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8—1071no—
164C22×Q8—1072no—
165C22×Q8—2141no—
166C22×D4—1071no—
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1074no—
169C2×SD16—1072no—
170C2×SD16—2144no—
171C2×D8—1072no—
172C2×M16C2×D10711yescharacteristic
173C22×C8C2×D10711yescharacteristic
174#127—2141no—
175#124—2141no—
176#121—1074no—
177#120—2142no—
178#117—1071no—
179#103—1072no—
180#103—1074no—
181#102—1072no—
182C22⋊C8C2×D10714yescomplemented
183C24—2141no—
184#41—22no—
185#41—22no—
186#41—1074no—
187#41—2144no—
188#41—2144no—
189#41—21416no—
190#41—2148no—
191#41—2142no—
192C2×Q8—1074no—
193C2×Q8—1072no—
194C2×Q8—1071no—
195C2×Q8—2144no—
196C2×Q8—2144no—
197C2×Q8—2148no—
198C2×Q8—2142no—
199C2×Q8—2142no—
200C2×Q8—2141no—
201C2×Q8D42811yescharacteristic
202C2×D4—1071no—
203C2×D4—1072no—
204C2×D4—2144no—
205C2×D4—2142no—
206C2×D4—2141no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2141no—
210C2×D4D42812yes—
211C2×D4D42811yescharacteristic
212C22×C4—2142no—
213C22×C4—2141no—
214C22×C4—2142no—
215C22×C4—2141no—
216C22×C4C22×D10712yes—
217C22×C4C22×D10711yescharacteristic
218SD16—2148no—
219SD16—2144no—
220D8—2144no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D10712yes—
224C2×C8C22×D10712yes—
225C4⋊C4—1072no—
226C22⋊C4—2141no—
227C22⋊C4—2144no—
228C42—2141no—
229C23—8562no—
230C23—4282no—
231C23—2142no—
232C23—2141no—
233C23C2×D42812yes—
234C23C2×D42811yescharacteristic
235Q8—22no—
236Q8—1074no—
237Q8—1072no—
238Q8—2148no—
239Q8—2144no—
240Q8—2142no—
241Q8—2141no—
242D4—24no—
243D4—22no—
244D4—8564no—
245D4—1072no—
246D4—4282no—
247D4—2148no—
248D4—2144no—
249D4—2142no—
250D4—2142no—
251D4—2141no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—2148no—
255C2×C4—2142no—
256C2×C4—2148no—
257C2×C4—2144no—
258C2×C4—2142no—
259C2×C4—2141no—
260C2×C4—2144no—
261C2×C4D4×D10712yes—
262C2×C4D4×D10712yes—
263C2×C4C2×D42812yes—
264C2×C4C2×D42811yescharacteristic
265C2×C4C23×D10711yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—8564no—
272C22—4284no—
273C22—4282no—
274C22—4282no—
275C22—2142no—
276C22#1681207912yes—
277C22C22×D42811yescharacteristic
278C4—22no—
279C4—4282no—
280C4—2144no—
281C4—2142no—
282C4#1681207912yes—
283C4C2×D4×D10711yescharacteristic
284C4C2×D4×D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—8562no—
288C2—4282no—
289C2C2×#1681207911yescharacteristic
290C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#408×C107C2
socleC22×C107C22×D4
hypercenter#408D107
hypocenterC107#3010
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16812156C2semidirect
2. C2×#16812137C2semidirect
3. C22×#13133807C2semidirect
4. #20293191C2semidirect
5. #20292774C2semidirect
6. #20292768C2semidirect
7. #20292637C2semidirect
8. #20292624C2semidirect
9. #408×C107C2semidirect
13. C2×#13133818C22semidirect
17. C22×Dic214C22semidirect
30. #16812034C22semidirect
31. #16812034C22semidirect
32. #16812031C22semidirect
36. C22⋊C8×C107C22semidirect
41. #13133818D4semidirect
49. C2×Dic214D4semidirect
146. C107#3010semidirect
155. #408D107semidirect
182. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168121372
C22×#131338072
#202931912
#202927742
#202927682
#202926372
#202926242
#408×C1072
#3010107