Finite Group Databasebeta

← #28346143

Subgroups — #28346143

Order 13696.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812134C211yesmaximal, characteristic, complemented
2C2×#16812116C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292878C212yesmaximal, complemented
5#20292875C212yesmaximal, complemented
6#20292861C212yesmaximal, complemented
7#20292858C212yesmaximal, complemented
8#20292667C211yesmaximal, characteristic, complemented
9#20292623C211yesmaximal, characteristic, complemented
10C2×#16812029C211yesmaximal, characteristic, complemented
11#537×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13#16812134—24no—
14C2×#13133805C2212yes—
15C2×#13133803C2212yes—
16#16812116—24no—
17C2×#13133802C2211yescharacteristic
18C2×#13133802C2212yescomplemented
19C22×C107⋊C8C2211yescharacteristic
20C22×C107⋊C8C2212yescomplemented
21#16812105—21no—
22#16812098—21no—
23#16812083C2212yes—
24#16812083C2212yes—
25#16812077—22no—
26#16812068C2211yescharacteristic
27#16812042C2212yescomplemented
28#16812040C2212yescomplemented
29#16812033C2211yescharacteristic, complemented
30#16812033C2211yescharacteristic, complemented
31#16812030C2211yescharacteristic, complemented
32#16812030C2211yescharacteristic, complemented
33#16812029C2212yes—
34#16812029C2212yes—
35C2×#41×C107C2211yescharacteristic
36C2×#41×C107C2211yescharacteristic
37C22⋊Q8×C107—21no—
38C22⋊Q8×C107C2212yescomplemented
39#121×C107—21no—
40#121×C107—22no—
41#121×C107C2212yescomplemented
42C2×C4⋊C4×C107C2211yescharacteristic
43C23×D107—21no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428—21no—
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50#13133805—24no—
51#13133803—24no—
52#13133802—24no—
53#13133802D414yescomplemented
54C2×C107⋊C8—22no—
55C2×C107⋊C8—24no—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61C22⋊Dic107—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107—21no—
65#41×C107—22no—
66#41×C107—22no—
67#41×C107—22no—
68#41×C107—22no—
69C2×Q8×C107D411yescharacteristic
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—21no—
72C2×D4×C107—22no—
73C2×D4×C107D412yes—
74C2×D4×C107D411yescharacteristic
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107—21no—
78C22×C4×C107C2312yes—
79C22×C4×C107C2311yescharacteristic
80C4⋊C4×C107—21no—
81C4⋊C4×C107—22no—
82C4⋊C4×C107C2312yes—
83C22⋊C4×C107—22no—
84C22⋊C4×C107—22no—
85C23×C107—21no—
86C23×C107C2×D412yes—
87C23×C107C2×D411yescharacteristic
88C22×D107—22no—
89C22×D107—42no—
90C22×D107—81no—
91C22×D107—81no—
92Q8×C107—22no—
93Q8×C107—21no—
94Q8×C107—21no—
95D4×C107—24no—
96D4×C107—22no—
97D4×C107—21no—
98D4×C107—21no—
99D4×C107—42no—
100D4×C107—82no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—21no—
105C2×C4×C107—42no—
106C2×C4×C107—81no—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D412yes—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107#4112yes—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—82no—
113C22⋊D107—82no—
114C2×Dic107—21no—
115C2×Dic107—21no—
116D428—22no—
117D428—22no—
118D428—42no—
119C107⋊C8—21no—
120C107⋊C8—21no—
121C107⋊C8C2×D414yes—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—22no—
127C22×C107—41no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107#12112yes—
131C22×C107C22×D411yescharacteristic
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—84no—
136C4×C107—22no—
137C4×C107—41no—
138C4×C107—42no—
139C4×C107#12112yes—
140C4×C107C22×D411yescharacteristic
141C4×C107C2×#4111yescharacteristic
142Dic107—41no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107—42no—
146C2×C107—81no—
147C2×C107C2×#12111yescharacteristic
148C2×C107#57611yescharacteristic
149C2×C107#57711yescharacteristic
150D107—82no—
151#3062—1071nomaximal, Sylow
152C107#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1071no—
154C2×#131—1071no—
155#537D10711yescharacteristic, complemented
156#535—1071no—
157#469—1072no—
158#468—1072no—
159#466—1072no—
160#465—1072no—
161#428—1071no—
162#416—1071no—
163C2×#102—1071no—
164C2×#41C2×D10711yescharacteristic
165C2×#41C2×D10711yescharacteristic
166C22×D4—1071no—
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1072no—
169C2×D8—1072no—
170#131—2144no—
171C2×M16—1071no—
172C2×M16—1072no—
173C22×C8—1071no—
174C22×C8—1072no—
175#127—2141no—
176#124—2141no—
177C22⋊Q8—21no—
178C22⋊Q8C2×D10712yescomplemented
179#121—21no—
180#121—22no—
181#121—1072no—
182#121—1072no—
183#121C2×D10712yescomplemented
184#120—2142no—
185#117—1071no—
186C2×C4⋊C4C2×D10711yescharacteristic
187#107—1072no—
188#106—1072no—
189#103—1071no—
190#103—1071no—
191#102—1071no—
192#102—1071no—
193#102—1072no—
194#102—1072no—
195C24—2141no—
196#41—22no—
197#41—22no—
198#41—22no—
199#41—22no—
200C2×Q8C22⋊D10711yescharacteristic
201C2×Q8C22×D10711yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1072no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4—2142no—
208C2×D4—2141no—
209C2×D4—2142no—
210C2×D4C22⋊D10712yes—
211C2×D4C22⋊D10711yescharacteristic
212C2×D4C22×D10711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4C22×D10712yes—
216C22×C4C22×D10711yescharacteristic
217SD16—2144no—
218D8—2144no—
219M16—1074no—
220M16—2144no—
221C2×C8—1072no—
222C2×C8—1071no—
223C2×C8—1071no—
224C2×C8—2144no—
225C2×C8—2142no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—1071no—
229C4⋊C4—1071no—
230C4⋊C4C22×D10712yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—2141no—
234C22⋊C4—2142no—
235C22⋊C4—2142no—
236C42—2141no—
237C23—21no—
238C23—8561no—
239C23—8561no—
240C23—4282no—
241C23—2142no—
242C23C2×C22⋊D10712yes—
243C23C2×C22⋊D10711yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—21no—
251D4—42no—
252D4—82no—
253D4—8562no—
254D4—8562no—
255D4—4282no—
256D4—2142no—
257D4—2142no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4D4×D10712yes—
267C2×C4C2×C22⋊D10712yes—
268C2×C4C2×C22⋊D10711yescharacteristic
269C2×C4#1313381712yes—
270C2×C4C23×D10711yescharacteristic
271C8—1074no—
272C8—2141no—
273C8—2141no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—8564no—
283C22—4282no—
284C22—4282no—
285C22—4282no—
286C22#1681208112yes—
287C22C22×C22⋊D10711yescharacteristic
288C4—22no—
289C4—42no—
290C4—41no—
291C4—4281no—
292C4—4281no—
293C4#1681208112yes—
294C4C2×D4×D10711yescharacteristic
295C4C2×#1313381711yescharacteristic
296C2—22no—
297C2—42no—
298C2—81no—
299C2—8562no—
300C2C2×#1681208111yescharacteristic
301C2#2029343011yescharacteristic
302C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#537×C107C2
socleC22×C107C22×D4
hypercenter#537D107
hypocenterC107#3062
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×#16812116C2semidirect
3. #20293191C2semidirect
4. #20292878C2semidirect
5. #20292875C2semidirect
6. #20292861C2semidirect
7. #20292858C2semidirect
8. #20292667C2semidirect
9. #20292623C2semidirect
10. C2×#16812029C2semidirect
11. #537×C107C2semidirect
18. C2×#13133802C22semidirect
20. C22×C107⋊C8C22semidirect
27. #16812042C22semidirect
28. #16812040C22semidirect
29. #16812033C22semidirect
30. #16812033C22semidirect
31. #16812030C22semidirect
32. #16812030C22semidirect
38. C22⋊Q8×C107C22semidirect
41. #121×C107C22semidirect
53. #13133802D4semidirect
152. C107#3062semidirect
155. #537D107semidirect
178. C22⋊Q8C2×D107semidirect
183. #121C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×#168121162
#202931912
#202928782
#202928752
#202928612
#202928582
#202926672
#202926232
C2×#168120292
#537×C1072
#3062107