Finite Group Databasebeta

← #28346398

Subgroups — #28346398

Order 13696.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#121×D107C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292953C212yesmaximal, complemented
4#20292949C212yesmaximal, complemented
5#20292945C211yesmaximal, characteristic, complemented
6#20292941C211yesmaximal, characteristic, complemented
7#20292938C211yesmaximal, characteristic, complemented
8#20292666C212yesmaximal, complemented
9#20292630C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#481×C107C211yesmaximal, characteristic, complemented
12C2×D4×D107—21no—
13C2×D4×D107C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15C2×#13133801C2212yes—
16#16812084C2211yescharacteristic
17#16812083C2211yescharacteristic
18#16812082—21no—
19#16812081—21no—
20#16812080C2211yescharacteristic
21#16812067—21no—
22#16812067—21no—
23#16812067C2212yes—
24#16812067C2212yescomplemented
25C2×C4⋊Dic107C2211yescharacteristic, complemented
26C4⋊C4×D107C2211yescharacteristic
27C2×(C2×D107)⋊C4—21no—
28C22⋊C4×D107—21no—
29C2×C4×Dic107—21no—
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812040C2212yescomplemented
33#16812031C2212yescomplemented
34#16812030C2212yescomplemented
35#16812029C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic, complemented
37(C2×D107)⋊C8C2212yescomplemented
38#121×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic, complemented
40#106×C107C2212yescomplemented
41#102×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic, complemented
43C23×D107—21no—
44C23×D107D411yescharacteristic
45C2×D428—21no—
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C2×C22⋊D107—21no—
49C2×C22⋊D107—21no—
50D4×D107—24no—
51D4×D107—44no—
52C22×Dic107—21no—
53C22×Dic107D411yescharacteristic
54C2×C4×D107—24no—
55C2×C4×D107—21no—
56C2×C4×D107D411yescharacteristic
57C2×C4×D107D411yescharacteristic
58C2×C4×D107C2311yescharacteristic
59C2×C4×D107C2311yescharacteristic
60#13133801—24no—
61C2×C107⋊C8C2312yes—
62C4⋊Dic107—21no—
63C4⋊Dic107C2311yescharacteristic
64C4⋊Dic107C2312yes—
65Dic107⋊C4—21no—
66C22⋊Dic107—21no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4—21no—
71C4×Dic107—22no—
72C4×Dic107—22no—
73C2×D4×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77C2×C8×C107C2312yes—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2312yes—
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C23×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—21no—
87C22×D107—41no—
88C22×D107—42no—
89C22×D107—42no—
90C22×D107—42no—
91C22×D107C2×D411yescharacteristic
92C22×D107C2×D411yescharacteristic
93D4×C107—22no—
94D4×C107—42no—
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—42no—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—42no—
102C22⋊D107—42no—
103C22⋊D107—42no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—42no—
108C2×Dic107C2×D411yescharacteristic
109C2×Dic107C2×D411yescharacteristic
110D428—22no—
111D428—42no—
112C4×D107—22no—
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—42no—
116C4×D107#4114yes—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107AGL(1,ℤ/8)12yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—21no—
127C2×D107—21no—
128C2×D107—21no—
129C2×D107—21no—
130C2×D107—21no—
131C2×D107—44no—
132C2×D107—41no—
133C2×D107—41no—
134C2×D107—41no—
135C2×D107—84no—
136C2×D107#12312yes—
137C4×C107—21no—
138C4×C107—41no—
139C4×C107—42no—
140C4×C107C2×#4112yes—
141Dic107—21no—
142Dic107—41no—
143Dic107—42no—
144Dic107#12312yes—
145C2×C107—22no—
146C2×C107—81no—
147C2×C107C2×#12311yescharacteristic
148C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C107#57511yescharacteristic
150D107—22no—
151D107—41no—
152D107—81no—
153#3094—1071nomaximal, Sylow
154C107#309411yescharacteristic, complemented, Sylow
155C2×#121—1071no—
156C2×#117—1071no—
157#482—1072no—
158#482—1072no—
159#481—1071no—
160#481D10711yescharacteristic, complemented
161#480—1071no—
162#480—1071no—
163#428—1072no—
164#418—1072no—
165#407—1071no—
166C22×D4—1071no—
167C22×D4—2141no—
168C23×C4—1071no—
169C2×M16—1072no—
170#121—1071no—
171#121—1071no—
172#121—1071no—
173#121—2141no—
174#121—2141no—
175#121C2×D10711yescharacteristic
176#117—1072no—
177#117—1072no—
178#117—2141no—
179#117—2141no—
180C2×C4⋊C4—1071no—
181C2×C4⋊C4—1071no—
182C2×C4⋊C4C2×D10711yescharacteristic, complemented
183C2×C22⋊C4—2141no—
184C2×C22⋊C4—2141no—
185C2×C42—2141no—
186#107—1072no—
187#107—1072no—
188#106—1072no—
189#106C2×D10712yescomplemented
190#102—1072no—
191#102—1072no—
192#102—1072no—
193#102C2×D10712yescomplemented
194C22⋊C8—1071no—
195C22⋊C8—1072no—
196C22⋊C8C2×D10711yescharacteristic, complemented
197C24—1071no—
198C24—2141no—
199C2×D4—21no—
200C2×D4—1071no—
201C2×D4—4284no—
202C2×D4—2141no—
203C2×D4—2144no—
204C2×D4—2141no—
205C2×D4—2141no—
206C2×D4—2141no—
207C2×D4C22×D10711yescharacteristic
208C22×C4—21no—
209C22×C4—1071no—
210C22×C4—1071no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—2144no—
215C22×C4—2141no—
216C22×C4—2141no—
217C22×C4C22×D10711yescharacteristic
218M16—2144no—
219C2×C8—1072no—
220C2×C8C22×D10712yes—
221C4⋊C4—21no—
222C4⋊C4—1072no—
223C4⋊C4—1071no—
224C4⋊C4—2141no—
225C4⋊C4—2141no—
226C4⋊C4C22×D10712yes—
227C4⋊C4C22×D10711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2141no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C22⋊C4—2141no—
233C22⋊C4—2141no—
234C42—2142no—
235C42—2142no—
236C23—21no—
237C23—1071no—
238C23—1071no—
239C23—4281no—
240C23—4282no—
241C23—4282no—
242C23—4282no—
243C23—2142no—
244C23—2141no—
245C23—2142no—
246C23D4×D10711yescharacteristic
247D4—22no—
248D4—42no—
249D4—4282no—
250D4—4282no—
251D4—4282no—
252D4—4282no—
253D4—2142no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—1071no—
259C2×C4—1074no—
260C2×C4—1071no—
261C2×C4—4282no—
262C2×C4—4282no—
263C2×C4—2142no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4—2141no—
268C2×C4—2142no—
269C2×C4D4×D10711yescharacteristic
270C2×C4C23×D10711yescharacteristic
271C8—22no—
272C8—2142no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—8564no—
279C22—1072no—
280C22—4284no—
281C22—4281no—
282C22—4281no—
283C22—4281no—
284C22—2141no—
285C22—2141no—
286C22—2141no—
287C22—2141no—
288C22—2141no—
289C22#1681213612yes—
290C22C2×D4×D10711yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—1072no—
295C4—4281no—
296C4—4282no—
297C4—2141no—
298C4#41×D10712yes—
299C2—22no—
300C2—81no—
301C2—8561no—
302C2—4281no—
303C2—2142no—
304C2#123×D10711yescharacteristic
305C2C2×#1681213611yescharacteristic
306C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#481×C107C2
socleC22×C107C22×D4
prime radical#121×D107C2
hypercenter#481D107
hypocenterC107#3094
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. #20292953C2semidirect
4. #20292949C2semidirect
5. #20292945C2semidirect
6. #20292941C2semidirect
7. #20292938C2semidirect
8. #20292666C2semidirect
9. #20292630C2semidirect
10. #20292589C2semidirect
11. #481×C107C2semidirect
24. #16812067C22semidirect
25. C2×C4⋊Dic107C22semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812031C22semidirect
34. #16812030C22semidirect
35. #16812029C22semidirect
36. C22⋊(C107⋊C8)C22semidirect
37. (C2×D107)⋊C8C22semidirect
39. C2×C4⋊C4×C107C22semidirect
40. #106×C107C22semidirect
41. #102×C107C22semidirect
42. C22⋊C8×C107C22semidirect
154. C107#3094semidirect
160. #481D107semidirect
182. C2×C4⋊C4C2×D107semidirect
189. #106C2×D107semidirect
193. #102C2×D107semidirect
196. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#121×D1072
C2×#168120672
#202929532
#202929492
#202929452
#202929412
#202929382
#202926662
#202926302
#202925892
#481×C1072
#3094107