Finite Group Databasebeta

← #28345728

Subgroups — #28345728

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
2C22×#13133803C211yesmaximal, characteristic, complemented
3#20293194C211yesmaximal, characteristic, complemented
4#20292771C214yesmaximal, complemented
5#20292759C214yesmaximal, complemented
6#20292633C211yesmaximal, characteristic, complemented
7#20292623C211yesmaximal, characteristic, complemented
8#20292595C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
10#16812155—24no—
11#16812155—24no—
12C2×#13133819C2212yescomplemented
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C22×D428—21no—
16C22×D428C2211yescharacteristic
17C2×D4×D107—22no—
18C2×D4×D107C2212yescomplemented
19C2×#13133803—24no—
20C2×#13133803C2214yescomplemented
21C22×C107⋊C8C2211yescharacteristic, complemented
22#16812105—21no—
23#16812096—21no—
24#16812080C2214yescomplemented
25#16812078—22no—
26#16812068C2211yescharacteristic, complemented
27#16812033C2212yescomplemented
28#16812031C2214yescomplemented
29#16812030C2212yescomplemented
30(C2×D107)⋊C8C2214yescomplemented
31C2×#41×C107C2211yescharacteristic
32C22×D4×C107C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C107—24no—
34C2×SD16×C107C2212yescomplemented
35C2×D8×C107C2212yescomplemented
36C2×M16×C107C2211yescharacteristic, complemented
37C23×D107—21no—
38C23×D107—22no—
39#13133819—24no—
40#13133819—24no—
41#13133818—28no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214D411yescharacteristic
45C2×D428—28no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—22no—
49C2×D428D411yescharacteristic
50C2×D428D414yescomplemented
51C2×D428C2312yes—
52C2×C22⋊D107—24no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107D412yes—
55D4×D107—216no—
56D4×D107—24no—
57D4×D107—24no—
58C2×C4×D107—22no—
59C2×C4×D107C2312yes—
60#13133803—28no—
61C2×C107⋊C8—22no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107C2312yes—
64C22⋊Dic107—21no—
65C22⋊Dic107—24no—
66C4×Dic107—21no—
67C24×C107—21no—
68#41×C107—22no—
69#41×C107D414yescomplemented
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—21no—
72C2×D4×C107—22no—
73C2×D4×C107—22no—
74C2×D4×C107—21no—
75C2×D4×C107C2312yes—
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78C22×C4×C107C2311yescharacteristic
79SD16×C107—24no—
80D8×C107—24no—
81M16×C107—22no—
82C2×C8×C107C2312yes—
83C23×C107—21no—
84C23×C107—22no—
85C23×C107—42no—
86C23×C107—82no—
87C23×C107C2×D411yescharacteristic
88C22×D107—22no—
89C22×D107—28no—
90C22×D107—24no—
91C22×D107—84no—
92C22×D107—82no—
93C22×D107C2×D412yes—
94Q8×C107—21no—
95Q8×C107C2×D412yes—
96D4×C107—24no—
97D4×C107—21no—
98D4×C107—22no—
99D4×C107—22no—
100D4×C107—42no—
101D4×C107C2×D412yes—
102C2×C4×C107—21no—
103C2×C4×C107—24no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—28no—
109C22⋊D107—24no—
110C22⋊D107—84no—
111C2×Dic107—22no—
112C2×Dic107C2×D412yes—
113D428—28no—
114D428—22no—
115D428—24no—
116D428—22no—
117D428C2×D414yes—
118C4×D107—28no—
119C4×D107—22no—
120C4×D107—22no—
121Dic214—22no—
122C8×C107—22no—
123C107⋊C8—22no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—44no—
129C22×C107—84no—
130C22×C107#12012yes—
131C22×C107C22×D411yescharacteristic
132C2×D107—22no—
133C2×D107—24no—
134C2×D107—24no—
135C2×D107—44no—
136C2×D107—84no—
137C2×D107—88no—
138C4×C107—22no—
139C4×C107#12012yes—
140C4×C107C22×D411yescharacteristic
141C4×C107C22×D411yescharacteristic
142Dic107—22no—
143Dic107—42no—
144C2×C107—22no—
145C2×C107—42no—
146C2×C107—82no—
147C2×C107C2×#12011yescharacteristic
148C2×C107#57612yes—
149D107—42no—
150D107—44no—
151#3008—1071nomaximal, Sylow
152C107#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1071no—
154C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
155C22×D8—1071no—
156#535—1071no—
157#449—1074no—
158#447—1074no—
159#418—1071no—
160#416—1071no—
161#408—1071no—
162H(5,𝔽2)—2144no—
163H(5,𝔽2)—2144no—
164C2×#41—1071no—
165C2×#41—1072no—
166C2×#41—2141no—
167C2×#41C2×D10711yescharacteristic
168C22×D4—1072no—
169C22×D4—1071no—
170C22×D4—2141no—
171C22×D4—2142no—
172C22×D4C2×D10711yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D10712yescomplemented
175C2×D8—1074no—
176C2×D8—2144no—
177C2×D8C2×D10712yescomplemented
178C2×M16C2×D10711yescharacteristic, complemented
179C22×C8—1071no—
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⋊C8—1074no—
189C24—21no—
190C24—2142no—
191C24—2141no—
192#41—22no—
193#41—2148no—
194#41—2144no—
195#41—2144no—
196#41—2142no—
197#41—2142no—
198#41C22⋊D10714yescomplemented
199C2×Q8—1071no—
200C2×Q8C22×D10711yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—1072no—
206C2×D4—1074no—
207C2×D4—1071no—
208C2×D4—1072no—
209C2×D4—2142no—
210C2×D4—2148no—
211C2×D4—2142no—
212C2×D4—2144no—
213C2×D4—21416no—
214C2×D4—2144no—
215C2×D4—2144no—
216C2×D4—2141no—
217C2×D4—2142no—
218C2×D4C22×D10712yes—
219C2×D4C22×D10711yescharacteristic
220C22×C4—21no—
221C22×C4—1072no—
222C22×C4—2142no—
223C22×C4C22×D10711yescharacteristic
224SD16—24no—
225D8—24no—
226D8—2148no—
227M16—22no—
228C2×C8—1072no—
229C2×C8—2142no—
230C2×C8C22×D10712yes—
231C4⋊C4—1072no—
232C22⋊C4—2144no—
233C22⋊C4—2141no—
234C42—2141no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—8562no—
240C23—8564no—
241C23—1072no—
242C23—2148no—
243C23—2144no—
244C23—2142no—
245C23D4×D10711yescharacteristic
246Q8—21no—
247Q8—2142no—
248Q8C2×C22⋊D10712yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—8564no—
255D4—1074no—
256D4—2148no—
257D4—2148no—
258D4—2144no—
259D4—2144no—
260D4—2142no—
261D4—2142no—
262D4C2×C22⋊D10712yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—1072no—
266C2×C4—2148no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4D4×D10711yescharacteristic
271C2×C4D4×D10712yes—
272C2×C4C2×C22⋊D10712yes—
273C2×C4C23×D10711yescharacteristic
274C8—22no—
275C8—2142no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—8564no—
283C22—8568no—
284C22—4284no—
285C22—2144no—
286C22—2144no—
287C22—2142no—
288C22#1681207712yes—
289C22C2×D4×D10711yescharacteristic
290C4—22no—
291C4—4282no—
292C4—2142no—
293C4#1681207712yes—
294C4C2×D4×D10711yescharacteristic
295C4C22×C22⋊D10711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—4284no—
300C2—4282no—
301C2C2×#1681207711yescharacteristic
302C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3008
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812155C2semidirect
2. C22×#13133803C2semidirect
3. #20293194C2semidirect
4. #20292771C2semidirect
5. #20292759C2semidirect
6. #20292633C2semidirect
7. #20292623C2semidirect
8. #20292595C2semidirect
9. C2×AGL(1,ℤ/8)×C107C2semidirect
12. C2×#13133819C22semidirect
18. C2×D4×D107C22semidirect
20. C2×#13133803C22semidirect
21. C22×C107⋊C8C22semidirect
24. #16812080C22semidirect
26. #16812068C22semidirect
27. #16812033C22semidirect
28. #16812031C22semidirect
29. #16812030C22semidirect
30. (C2×D107)⋊C8C22semidirect
32. C22×D4×C107C22semidirect
34. C2×SD16×C107C22semidirect
35. C2×D8×C107C22semidirect
36. C2×M16×C107C22semidirect
50. C2×D428D4semidirect
69. #41×C107D4semidirect
152. C107#3008semidirect
154. C2×AGL(1,ℤ/8)D107semidirect
172. C22×D4C2×D107semidirect
174. C2×SD16C2×D107semidirect
177. C2×D8C2×D107semidirect
178. C2×M16C2×D107semidirect
198. #41C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C22×#131338032
#202931942
#202927712
#202927592
#202926332
#202926232
#202925952
C2×AGL(1,ℤ/8)×C1072
#3008107