Finite Group Databasebeta

← #28345731

Subgroups — #28345731

Order 13696.

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

normal subgroups only

This group has 784 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×#16812155C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C22×D856C211yesmaximal, characteristic, complemented
4#20293191C211yesmaximal, characteristic, complemented
5#20292774C214yesmaximal, complemented
6#20292763C214yesmaximal, complemented
7#20292633C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic, complemented
9#408×C107C211yesmaximal, characteristic, complemented
10#16812155—24no—
11#16812155—24no—
12C2×#13133819—21no—
13C2×#13133818C2211yescharacteristic
14C2×#13133818C2212yescomplemented
15C22×D428C2211yescharacteristic
16C22×D428C2211yescharacteristic, complemented
17C22×D428C2212yescomplemented
18C2×D4×D107—22no—
19C2×D4×D107—21no—
20#16812137—24no—
21C2×#13133807C2212yescomplemented
22C2×D856—24no—
23C2×D856C2214yescomplemented
24C2×D856C2212yescomplemented
25#16812105—21no—
26#16812098—21no—
27#16812083C2214yescomplemented
28#16812077—22no—
29#16812068C2211yescharacteristic, complemented
30#16812034C2212yescomplemented
31#16812031C2212yescomplemented
32#16812031C2214yescomplemented
33C2×#41×C107C2211yescharacteristic
34C2×M16×C107C2211yescharacteristic, complemented
35C22×C8×C107C2211yescharacteristic, complemented
36C22⋊C8×C107C2214yescomplemented
37C23×D107—21no—
38C23×D107—22no—
39C23×D107—21no—
40#13133819—28no—
41#13133818—22no—
42#13133818—24no—
43#13133818—24no—
44#13133818D414yescomplemented
45C2×Dic214C2311yescharacteristic
46C2×D428—21no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—24no—
50C2×D428—24no—
51C2×D428—22no—
52C2×D428—22no—
53C2×D428—21no—
54C2×D428D414yescomplemented
55C2×D428C2312yes—
56C2×D428C2312yes—
57C2×D428C2311yescharacteristic
58C2×C22⋊D107—21no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—24no—
61D4×D107—216no—
62D4×D107—28no—
63C2×C4×D107—21no—
64C2×C4×D107—22no—
65#13133807—24no—
66D856—28no—
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—28no—
88C22×D107—24no—
89C22×D107—22no—
90C22×D107—42no—
91C22×D107—82no—
92C22×D107—84no—
93C22×D107—82no—
94Q8×C107—22no—
95D4×C107—24no—
96D4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—28no—
105C22⋊D107—24no—
106C22⋊D107—84no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109D428—28no—
110D428—24no—
111D428—21no—
112D428—22no—
113D428—22no—
114D428—22no—
115D428—42no—
116D428C2×D412yes—
117D428C2×D414yes—
118C4×D107—28no—
119C4×D107—24no—
120Dic214—21no—
121Dic214C2×D412yes—
122C8×C107—22no—
123C8×C107—22no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—24no—
127C22×C107#12012yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—24no—
130C2×D107—22no—
131C2×D107—44no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—44no—
135C2×D107—84no—
136C2×D107—88no—
137C2×D107—84no—
138C4×C107—22no—
139C4×C107#12012yes—
140C4×C107C22×D411yescharacteristic
141C4×C107C22×D411yescharacteristic
142Dic107—22no—
143Dic107—42no—
144C2×C107—22no—
145C2×C107—42no—
146C2×C107C2×#12011yescharacteristic
147C2×C107#57612yes—
148D107—44no—
149D107—42no—
150D107—82no—
151#3008—1071nomaximal, Sylow
152C107#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1071no—
154C2×AGL(1,ℤ/8)—1071no—
155C22×D8—1071no—
156#535—1071no—
157#449—1074no—
158#447—1074no—
159#418—1071no—
160#416—1071no—
161#408D10711yescharacteristic, complemented
162H(5,𝔽2)—2144no—
163H(5,𝔽2)—2144no—
164C2×#41—1071no—
165C2×#41—1072no—
166C2×#41—2141no—
167C2×#41C2×D10711yescharacteristic
168C22×D4—1071no—
169C22×D4—1071no—
170C22×D4—1072no—
171C22×D4—2142no—
172C22×D4—2141no—
173AGL(1,ℤ/8)—2144no—
174C2×SD16—1072no—
175C2×D8—1074no—
176C2×D8—1072no—
177C2×D8—2144no—
178C2×M16C2×D10711yescharacteristic, complemented
179C22×C8C2×D10711yescharacteristic, complemented
180#127—2141no—
181#124—2141no—
182#121—1074no—
183#120—2142no—
184#117—1071no—
185#103—1072no—
186#102—1072no—
187#102—1074no—
188C22⋊C8C2×D10714yescomplemented
189C24—2141no—
190C24—2142no—
191C24—2141no—
192#41—22no—
193#41—22no—
194#41—1074no—
195#41—2144no—
196#41—2144no—
197#41—2148no—
198#41—2142no—
199C2×Q8—1071no—
200C2×Q8D42811yescharacteristic
201C2×D4—1074no—
202C2×D4—1072no—
203C2×D4—1071no—
204C2×D4—1072no—
205C2×D4—2144no—
206C2×D4—2144no—
207C2×D4—21416no—
208C2×D4—2148no—
209C2×D4—2144no—
210C2×D4—2142no—
211C2×D4—2141no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4—2142no—
215C2×D4—2141no—
216C2×D4—2142no—
217C2×D4—2141no—
218C2×D4D42812yes—
219C2×D4D42811yescharacteristic
220C22×C4—2142no—
221C22×C4—2141no—
222C22×C4C22×D10712yes—
223C22×C4C22×D10711yescharacteristic
224SD16—2144no—
225D8—2148no—
226D8—2144no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D10712yes—
230C2×C8C22×D10712yes—
231C4⋊C4—1072no—
232C22⋊C4—2141no—
233C22⋊C4—2144no—
234C42—2141no—
235C23—8562no—
236C23—8562no—
237C23—8564no—
238C23—4282no—
239C23—2148no—
240C23—2144no—
241C23—2142no—
242C23—2142no—
243C23—2141no—
244C23C2×D42812yes—
245C23C2×D42811yescharacteristic
246Q8—22no—
247Q8—1072no—
248Q8—2141no—
249D4—24no—
250D4—22no—
251D4—8564no—
252D4—1074no—
253D4—1072no—
254D4—4282no—
255D4—2148no—
256D4—2148no—
257D4—2144no—
258D4—2144no—
259D4—2142no—
260D4—2142no—
261D4—2141no—
262D4—2142no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—2148no—
266C2×C4—2142no—
267C2×C4—2141no—
268C2×C4—2144no—
269C2×C4D4×D10712yes—
270C2×C4D4×D10712yes—
271C2×C4C2×D42812yes—
272C2×C4C2×D42811yescharacteristic
273C2×C4C23×D10711yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—8564no—
280C22—8564no—
281C22—8568no—
282C22—4284no—
283C22—4282no—
284C22—4284no—
285C22—4282no—
286C22—2144no—
287C22—2142no—
288C22#1681207912yes—
289C22C22×D42811yescharacteristic
290C4—22no—
291C4—4282no—
292C4—2142no—
293C4#1681207912yes—
294C4C2×D4×D10711yescharacteristic
295C4C2×D4×D10711yescharacteristic
296C2—22no—
297C2—42no—
298C2—8562no—
299C2—4284no—
300C2—4282no—
301C2C2×#1681207911yescharacteristic
302C2#2029343212yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#408×C107C2
socleC22×C107C22×D4
hypercenter#408D107
hypocenterC107#3008
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×#16812155C2semidirect
2. C2×#16812137C2semidirect
3. C22×D856C2semidirect
4. #20293191C2semidirect
5. #20292774C2semidirect
6. #20292763C2semidirect
7. #20292633C2semidirect
8. #20292624C2semidirect
9. #408×C107C2semidirect
14. C2×#13133818C22semidirect
16. C22×D428C22semidirect
17. C22×D428C22semidirect
21. C2×#13133807C22semidirect
23. C2×D856C22semidirect
24. C2×D856C22semidirect
27. #16812083C22semidirect
29. #16812068C22semidirect
30. #16812034C22semidirect
31. #16812031C22semidirect
32. #16812031C22semidirect
34. C2×M16×C107C22semidirect
35. C22×C8×C107C22semidirect
36. C22⋊C8×C107C22semidirect
44. #13133818D4semidirect
54. C2×D428D4semidirect
152. C107#3008semidirect
161. #408D107semidirect
178. C2×M16C2×D107semidirect
179. C22×C8C2×D107semidirect
188. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168121372
C22×D8562
#202931912
#202927742
#202927632
#202926332
#202926242
#408×C1072
#3008107