Finite Group Databasebeta

← #28346371

Subgroups — #28346371

Order 13696.

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

normal subgroups only

This group has 420 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#20293137C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292971C212yesmaximal, complemented
4#20292962C211yesmaximal, characteristic, complemented
5#20292958C211yesmaximal, characteristic, complemented
6#20292950C212yesmaximal, complemented
7#20292938C211yesmaximal, characteristic, complemented
8#20292662C212yesmaximal, complemented
9#20292620C212yesmaximal, complemented
10C22⋊C8×D107C211yesmaximal, characteristic, complemented
11#481×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812091—21no—
16#16812089C2211yescharacteristic
17#16812088C2211yescharacteristic
18#16812080C2211yescharacteristic
19#16812077—21no—
20D4×Dic107—21no—
21(C22⋊D107)⋊C4—21no—
22#16812067—21no—
23#16812067—21no—
24#16812067C2212yes—
25#16812067C2211yescharacteristic
26#16812067C2212yes—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×(C2×D107)⋊C4—21no—
29C2×C4×Dic107—21no—
30#16812042C2212yescomplemented
31#16812042C2212yescomplemented
32#16812041C2212yescomplemented
33#16812034C2212yescomplemented
34#16812032C2212yescomplemented
35#16812030C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37(C2×D107)⋊C8C2212yes—
38#121×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic
40#106×C107C2212yescomplemented
41#102×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic
43C23×D107D411yescharacteristic
44#13133817—24no—
45C2×Dic214C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C22×Dic107—21no—
50C22×Dic107D411yescharacteristic
51C2×C4×D107—24no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55C2×C4×D107C2311yescharacteristic
56C2×C107⋊C8C2312yes—
57C8×D107—24no—
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yes—
61C4⋊Dic107C2311yescharacteristic
62Dic107⋊C4—21no—
63C22⋊Dic107—21no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69C4×Dic107—22no—
70C4×Dic107—22no—
71C4×Dic107—21no—
72C2×D4×C107—21no—
73C2×D4×C107C2311yescharacteristic
74C22×C4×C107—21no—
75C22×C4×C107C2311yescharacteristic
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C23×C107—21no—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—41no—
86C22×D107C2×D411yescharacteristic
87C22×D107C2×D411yescharacteristic
88D4×C107—22no—
89D4×C107—42no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—42no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—42no—
97C22⋊D107—42no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—42no—
103C2×Dic107—42no—
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106C4×D107—22no—
107C4×D107—22no—
108C4×D107—22no—
109C4×D107#4114yes—
110Dic214—22no—
111C8×C107—22no—
112C107⋊C8—22no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107—41no—
118C22×C107#13512yes—
119C22×C107C22×D411yescharacteristic
120C2×D107—21no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—44no—
126C2×D107#12312yes—
127C4×C107—21no—
128C4×C107—41no—
129C4×C107—42no—
130C4×C107C2×#4112yes—
131Dic107—21no—
132Dic107—41no—
133Dic107—42no—
134Dic107—41no—
135Dic107#12312yes—
136C2×C107—22no—
137C2×C107—81no—
138C2×C107C2×#12311yescharacteristic
139C2×C107C2×#13511yescharacteristic
140C2×C107#57511yescharacteristic
141D107—22no—
142D107—41no—
143#3091—1071nomaximal, Sylow
144C107#309111yescharacteristic, complemented, Sylow
145#525—1071no—
146C2×#117—1071no—
147#485—1072no—
148#484—1071no—
149#483—1071no—
150#482—1072no—
151#481D10711yescharacteristic, complemented
152#480—1071no—
153#427—1072no—
154#416—1072no—
155C2×C22⋊C8—1071no—
156C2×#41—1071no—
157C23×C4—1071no—
158C22×C8—1072no—
159#123—2141no—
160C22⋊Q8—1071no—
161C22⋊Q8—1071no—
162#121—1071no—
163#121C2×D10711yescharacteristic
164#120—2141no—
165C4×D4—2141no—
166C4×D4—2141no—
167#117—1072no—
168#117—1072no—
169#117—1071no—
170#117—2141no—
171#117—2141no—
172C2×C4⋊C4—1071no—
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C22⋊C4—2141no—
175C2×C42—2141no—
176#107—1072no—
177#107—1072no—
178#106—1072no—
179#106C2×D10712yescomplemented
180#103—1072no—
181#103—1072no—
182#102—1072no—
183#102C2×D10712yescomplemented
184C22⋊C8—1071no—
185C22⋊C8—1072no—
186C22⋊C8C2×D10711yescharacteristic
187C24—1071no—
188#41—2144no—
189C2×Q8—1071no—
190C2×D4—21no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4C22×D10711yescharacteristic
194C22×C4—21no—
195C22×C4—1071no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—2144no—
201C22×C4—2141no—
202C22×C4—2141no—
203C22×C4C22×D10711yescharacteristic
204C2×C8—1072no—
205C2×C8—2144no—
206C2×C8C22×D10712yes—
207C4⋊C4—21no—
208C4⋊C4—1072no—
209C4⋊C4—1071no—
210C4⋊C4—2141no—
211C4⋊C4—2141no—
212C4⋊C4—2141no—
213C4⋊C4C22×D10712yes—
214C4⋊C4C22×D10711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—2142no—
217C22⋊C4—2142no—
218C22⋊C4—2141no—
219C22⋊C4—2141no—
220C22⋊C4—2141no—
221C22⋊C4—2141no—
222C42—2142no—
223C42—2142no—
224C42—2141no—
225C23—21no—
226C23—1071no—
227C23—1071no—
228C23—4281no—
229C23—2142no—
230C23—2142no—
231C23D4×D10711yescharacteristic
232Q8—2142no—
233D4—22no—
234D4—42no—
235D4—4282no—
236D4—4282no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—1071no—
242C2×C4—1074no—
243C2×C4—1071no—
244C2×C4—4282no—
245C2×C4—4282no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4D4×D10711yescharacteristic
254C2×C4C23×D10711yescharacteristic
255C8—22no—
256C8—2142no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—1072no—
263C22—4284no—
264C22—2141no—
265C22—2141no—
266C22—2141no—
267C22—2141no—
268C22—2141no—
269C22#1681213012yes—
270C22C2×D4×D10711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—1072no—
275C4—4281no—
276C4—4282no—
277C4—4281no—
278C4—2141no—
279C4#41×D10712yes—
280C2—22no—
281C2—81no—
282C2—4281no—
283C2—2142no—
284C2#123×D10711yescharacteristic
285C2C2×#1681213011yescharacteristic
286C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#481×C107C2
socleC22×C107C22×D4
prime radical#20293137C2
hypercenter#481D107
hypocenterC107#3091
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. #20292971C2semidirect
4. #20292962C2semidirect
5. #20292958C2semidirect
6. #20292950C2semidirect
7. #20292938C2semidirect
8. #20292662C2semidirect
9. #20292620C2semidirect
10. C22⋊C8×D107C2semidirect
11. #481×C107C2semidirect
30. #16812042C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812034C22semidirect
34. #16812032C22semidirect
35. #16812030C22semidirect
40. #106×C107C22semidirect
41. #102×C107C22semidirect
144. C107#3091semidirect
151. #481D107semidirect
179. #106C2×D107semidirect
183. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931372
C2×#168120672
#202929712
#202929622
#202929582
#202929502
#202929382
#202926622
#202926202
C22⋊C8×D1072
#481×C1072
#3091107