Finite Group Databasebeta

← #28345686

Subgroups — #28345686

Order 13696.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D107C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3#20293134C211yesmaximal, characteristic, complemented
4#20292776C214yesmaximal, complemented
5#20292759C214yesmaximal, complemented
6#20292633C211yesmaximal, characteristic, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292590C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
10C24×D107—21no—
11C22×D428C2211yescharacteristic
12C22×C22⋊D107—21no—
13C2×D4×D107—24no—
14C2×D4×D107—22no—
15C2×D4×D107—24no—
16C2×D4×D107C2214yes—
17C22×C4×D107C2211yescharacteristic
18#16812134—24no—
19C2×#13133805C2212yescomplemented
20C2×#13133803C2212yescomplemented
21C2×#13133802C2211yescharacteristic, complemented
22#16812090—21no—
23#16812085C2212yescomplemented
24#16812080C2212yescomplemented
25#16812077—21no—
26C4×C22⋊D107—22no—
27#16812068C2211yescharacteristic, complemented
28#16812031C2214yescomplemented
29#16812030C2214yescomplemented
30(C2×D107)⋊C8C2214yescomplemented
31C2×#41×C107C2211yescharacteristic, complemented
32C22×D4×C107C2211yescharacteristic
33AGL(1,ℤ/8)×C107—24no—
34C2×SD16×C107C2212yescomplemented
35C2×D8×C107C2212yescomplemented
36C2×M16×C107C2211yescharacteristic, complemented
37C23×D107—28no—
38C23×D107—21no—
39C23×D107—24no—
40C23×D107—48no—
41C23×D107D411yescharacteristic
42C2×D428—21no—
43C2×D428—22no—
44C2×D428D414yescomplemented
45C2×D428C2312yes—
46C2×C22⋊D107—28no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—24no—
49D4×D107—216no—
50D4×D107—28no—
51D4×D107—28no—
52C22×Dic107D411yescharacteristic
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107D412yes—
56C2×C4×D107C2312yes—
57#13133805—24no—
58#13133803—24no—
59#13133802—22no—
60C2×C107⋊C8C2312yescomplemented
61C4⋊Dic107C2312yescomplemented
62Dic107⋊C4—22no—
63C22⋊Dic107—21no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—22no—
66C4×Dic107—21no—
67C24×C107—21no—
68#41×C107—22no—
69#41×C107—22no—
70C2×Q8×C107C2311yescharacteristic, complemented
71C2×D4×C107—21no—
72C2×D4×C107—21no—
73C2×D4×C107—22no—
74C2×D4×C107D414yescomplemented
75C2×D4×C107C2312yes—
76C2×D4×C107C2311yescharacteristic, complemented
77C22×C4×C107—21no—
78C22×C4×C107C2311yescharacteristic
79SD16×C107—24no—
80D8×C107—24no—
81M16×C107—22no—
82C2×C8×C107C2312yescomplemented
83C23×C107—24no—
84C23×C107—22no—
85C23×C107—21no—
86C23×C107—44no—
87C23×C107C2×D411yescharacteristic
88C22×D107—22no—
89C22×D107—24no—
90C22×D107—216no—
91C22×D107—24no—
92C22×D107—22no—
93C22×D107—432no—
94C22×D107—48no—
95C22×D107—44no—
96C22×D107—48no—
97C22×D107C2×D412yes—
98Q8×C107—21no—
99Q8×C107—21no—
100D4×C107—24no—
101D4×C107—22no—
102D4×C107—21no—
103D4×C107—21no—
104D4×C107—42no—
105D4×C107C2×D414yes—
106C2×C4×C107—21no—
107C2×C4×C107—42no—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—216no—
113C22⋊D107—44no—
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107C2×D412yes—
117D428—24no—
118D428—22no—
119D428C2×D414yes—
120C4×D107—24no—
121C4×D107—22no—
122C4×D107—22no—
123C8×C107—22no—
124C107⋊C8—22no—
125C22×C107—22no—
126C22×C107—24no—
127C22×C107—48no—
128C22×C107—44no—
129C22×C107—41no—
130C22×C107—42no—
131C22×C107#12012yes—
132C22×C107C22×D411yescharacteristic
133C2×D107—28no—
134C2×D107—22no—
135C2×D107—22no—
136C2×D107—24no—
137C2×D107—432no—
138C2×D107—48no—
139C2×D107—44no—
140C2×D107AGL(1,ℤ/8)14yescomplemented
141C4×C107—41no—
142C4×C107#12012yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C22×D411yescharacteristic
145Dic107—22no—
146Dic107—42no—
147C2×C107—22no—
148C2×C107—44no—
149C2×C107—81no—
150C2×C107C2×#12011yescharacteristic
151C2×C107C2×AGL(1,ℤ/8)12yes—
152D107—24no—
153D107—44no—
154#3003—1071nomaximal, Sylow
155C107#300311yescharacteristic, complemented, Sylow
156C23×D4—1071no—
157C2×AGL(1,ℤ/8)—1071no—
158C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
159#525—1071no—
160#450—1074no—
161#447—1074no—
162#418—1071no—
163#418—1071no—
164#407—1071no—
165C25—2141no—
166C2×#41C2×D10711yescharacteristic, complemented
167C22×D4—1074no—
168C22×D4—1071no—
169C22×D4—2142no—
170C22×D4—2141no—
171C22×D4—2144no—
172C22×D4—2144no—
173C22×D4C2×D10711yescharacteristic
174C23×C4—1071no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2144no—
177C2×SD16—1072no—
178C2×SD16C2×D10712yescomplemented
179C2×D8—1072no—
180C2×D8C2×D10712yescomplemented
181C2×M16—1071no—
182C2×M16C2×D10711yescharacteristic, complemented
183#123—2141no—
184C22⋊Q8—1072no—
185#121—1072no—
186#120—2141no—
187C4×D4—2142no—
188#117—1071no—
189#102—1074no—
190#102—1074no—
191C22⋊C8—1074no—
192C24—21no—
193C24—1071no—
194C24—4288no—
195C24—2144no—
196C24—2148no—
197C24—2141no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D10711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1074no—
205C2×D4—1072no—
206C2×D4—2144no—
207C2×D4—2148no—
208C2×D4—21416no—
209C2×D4—2142no—
210C2×D4—2148no—
211C2×D4—2148no—
212C2×D4—2141no—
213C2×D4—2142no—
214C2×D4C22⋊D10714yescomplemented
215C2×D4C22×D10712yes—
216C2×D4C22×D10711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—1071no—
219C22×C4—1072no—
220C22×C4—1072no—
221C22×C4—2142no—
222C22×C4—2142no—
223C22×C4C22×D10711yescharacteristic
224SD16—24no—
225SD16—2144no—
226D8—24no—
227D8—2144no—
228M16—22no—
229M16—2142no—
230C2×C8—1072no—
231C2×C8C22×D10712yescomplemented
232C4⋊C4—1072no—
233C4⋊C4—2142no—
234C22⋊C4—2142no—
235C22⋊C4—2142no—
236C22⋊C4—2141no—
237C42—2141no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—1072no—
243C23—4284no—
244C23—4288no—
245C23—42832no—
246C23—4288no—
247C23—2144no—
248C23—21416no—
249C23—2144no—
250C23—2142no—
251C23—2142no—
252C23D4×D10711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—1074no—
261D4—4284no—
262D4—21416no—
263D4—2144no—
264D4—2142no—
265D4C2×C22⋊D10714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—1072no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4—2144no—
272C2×C4—2142no—
273C2×C4—2142no—
274C2×C4D4×D10711yescharacteristic
275C2×C4D4×D10712yes—
276C2×C4C2×C22⋊D10712yes—
277C2×C4C23×D10711yescharacteristic
278C8—22no—
279C8—2142no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—42no—
285C22—41no—
286C22—1074no—
287C22—4288no—
288C22—42832no—
289C22—4284no—
290C22—2148no—
291C22—2144no—
292C22—2142no—
293C22—2142no—
294C22#1681207712yes—
295C22C2×D4×D10711yescharacteristic
296C4—41no—
297C4—4282no—
298C4—2142no—
299C4#1681207712yes—
300C4C2×D4×D10711yescharacteristic
301C4C22×C22⋊D10711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—4284no—
306C2—2144no—
307C2C2×#1681207711yescharacteristic
308C2AGL(1,ℤ/8)×D10712yes—
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#3003
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. C22×D4×D107C2semidirect
2. C2×#16812134C2semidirect
3. #20293134C2semidirect
4. #20292776C2semidirect
5. #20292759C2semidirect
6. #20292633C2semidirect
7. #20292632C2semidirect
8. #20292590C2semidirect
9. C2×AGL(1,ℤ/8)×C107C2semidirect
19. C2×#13133805C22semidirect
20. C2×#13133803C22semidirect
21. C2×#13133802C22semidirect
23. #16812085C22semidirect
24. #16812080C22semidirect
27. #16812068C22semidirect
28. #16812031C22semidirect
29. #16812030C22semidirect
30. (C2×D107)⋊C8C22semidirect
31. C2×#41×C107C22semidirect
34. C2×SD16×C107C22semidirect
35. C2×D8×C107C22semidirect
36. C2×M16×C107C22semidirect
44. C2×D428D4semidirect
60. C2×C107⋊C8C23semidirect
61. C4⋊Dic107C23semidirect
70. C2×Q8×C107C23semidirect
74. C2×D4×C107D4semidirect
76. C2×D4×C107C23semidirect
82. C2×C8×C107C23semidirect
140. C2×D107AGL(1,ℤ/8)semidirect
155. C107#3003semidirect
158. C2×AGL(1,ℤ/8)D107semidirect
166. C2×#41C2×D107semidirect
178. C2×SD16C2×D107semidirect
180. C2×D8C2×D107semidirect
182. C2×M16C2×D107semidirect
200. C2×Q8C22×D107semidirect
214. C2×D4C22⋊D107semidirect
216. C2×D4C22×D107semidirect
231. C2×C8C22×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168121342
#202931342
#202927762
#202927592
#202926332
#202926322
#202925902
C2×AGL(1,ℤ/8)×C1072
#3003107