Finite Group Databasebeta

← #28346759

Subgroups — #28346759

Order 13696.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293238C211yesmaximal, characteristic
2#20293103C211yesmaximal, characteristic
3#20292967C211yesmaximal, characteristic
4#20292966C211yesmaximal, characteristic
5#20292935C211yesmaximal, characteristic, complemented
6#20292934C211yesmaximal, characteristic, complemented
7#20292834C211yesmaximal, characteristic, complemented
8#20292781C211yesmaximal, characteristic
9#20292781C211yesmaximal, characteristic
10#20292734C211yesmaximal, characteristic, complemented
11#20292731C211yesmaximal, characteristic, complemented
12Q16×Dic107C211yesmaximal, characteristic, complemented
13#20292719C211yesmaximal, characteristic, complemented
14#20292687C211yesmaximal, characteristic
15#462×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19#16812104—21no—
20#16812104—21no—
21#16812101C2211yescharacteristic
22#16812099—21no—
23#16812099—21no—
24#16812085C2211yescharacteristic
25#16812085C2211yescharacteristic
26Q8×Dic107C2211yescharacteristic
27Q8×Dic107C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29C4×Dic214—21no—
30C4×Dic214C2211yescharacteristic
31#16812074—21no—
32#16812074C2211yescharacteristic
33#16812067—21no—
34#16812067C2211yescharacteristic
35#16812066—21no—
36#16812066C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×Q16×C107C2211yescharacteristic
54C4⋊Q8×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61Q8×D107—22no—
62Q8×D107—22no—
63C2×C4×D107—21no—
64C2×C4×D107C2311yescharacteristic
65#13133808—22no—
66#13133808—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4D411yescharacteristic
87(C2×D107)⋊C4D411yescharacteristic
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98Q16×C107—22no—
99Q16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107—21no—
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C22×D107C2×D411yescharacteristic
109Q8×C107—21no—
110Q8×C107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114Q8×C107#4112yes—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C4×D107—24no—
130C4×D107—21no—
131C4×D107—21no—
132Dic214—24no—
133Dic214—21no—
134Dic214#4112yes—
135C8×C107—21no—
136C8×C107—21no—
137C107⋊C8—21no—
138C107⋊C8—21no—
139C22×C107C22×D411yescharacteristic
140C2×D107—21no—
141C2×D107—21no—
142C2×D107—21no—
143C4×C107—22no—
144C4×C107—21no—
145C4×C107—21no—
146C4×C107—41no—
147C4×C107—41no—
148C4×C107—41no—
149C4×C107C2×#4111yescharacteristic
150C4×C107C2×#4111yescharacteristic
151Dic107—21no—
152Dic107—22no—
153Dic107—21no—
154Dic107—21no—
155Dic107—41no—
156Dic107—41no—
157Dic107—41no—
158C2×C107#52511yescharacteristic
159C2×C107#57811yescharacteristic
160C2×C107#57811yescharacteristic
161D107—41no—
162#3127—1071nomaximal, Sylow
163C107#312711yescharacteristic, complemented, Sylow
164#541—1071no—
165#519—1071no—
166#485—1071no—
167#485—1071no—
168#479—1071no—
169#479—1071no—
170#462—1071no—
171#462D10711yescharacteristic, complemented
172#451—1071no—
173#451—1071no—
174#441—1071no—
175#441—1071no—
176C4×Q16—1071no—
177C4×Q16—1071no—
178#433—1071no—
179C22×Q8—1071no—
180C2×Q16—1071no—
181C2×Q16—1071no—
182C2×Q16C2×D10711yescharacteristic
183C2×Q16C2×D10711yescharacteristic
184C4⋊Q8C2×D10711yescharacteristic
185#126—2141no—
186#126—2141no—
187#125—1071no—
188#125—2141no—
189#125—2141no—
190C22⋊Q8—1071no—
191C22⋊Q8—1071no—
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×Q8—1071no—
196C4×Q8—1071no—
197C4×Q8—2141no—
198C4×Q8—2141no—
199C4×Q8C2×D10711yescharacteristic
200#117—1071no—
201#117—1071no—
202#117—2141no—
203#117—2141no—
204#107—1071no—
205#107—1071no—
206#106—1071no—
207#106—1071no—
208C4⋊C8—1071no—
209C4⋊C8C2×D10711yescharacteristic
210#103—1071no—
211#103—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#103—1071no—
216#103C2×D10711yescharacteristic
217#103C2×D10711yescharacteristic
218C22⋊C8—1071no—
219C22⋊C8—1071no—
220#97—1071no—
221C4×C8—1071no—
222C2×Q8—1071no—
223C2×Q8—2141no—
224C2×Q8—2142no—
225C2×Q8—2142no—
226C2×Q8C22×D10711yescharacteristic
227C2×Q8C22×D10711yescharacteristic
228C2×Q8C22×D10711yescharacteristic
229C22×C4—1071no—
230C22×C4—2141no—
231Q16—22no—
232Q16—22no—
233Q16—2142no—
234Q16—2142no—
235C2×C8—1071no—
236C2×C8—1071no—
237C2×C8C22×D10711yescharacteristic
238C2×C8C22×D10711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—1071no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—1071no—
246C4⋊C4—1071no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—1071no—
258C22⋊C4—1071no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C42—21no—
263C42—1071no—
264C42—2141no—
265C42—2141no—
266C42—2141no—
267C42—2141no—
268C42—2141no—
269C42—2141no—
270C42C22×D10711yescharacteristic
271C23—1071no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—1072no—
278Q8—2144no—
279Q8—2141no—
280Q8#1313381712yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—1071no—
286C2×C4—1071no—
287C2×C4—1071no—
288C2×C4—2144no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4—2141no—
292C2×C4—2141no—
293C2×C4—2141no—
294C2×C4—2141no—
295C2×C4D4×D10711yescharacteristic
296C2×C4D4×D10711yescharacteristic
297C2×C4C23×D10711yescharacteristic
298C8—21no—
299C8—21no—
300C8—2141no—
301C8—2141no—
302C22—2141no—
303C22—2141no—
304C22—2141no—
305C22C2×D4×D10711yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—4281no—
313C4—4281no—
314C4—4281no—
315C4—2141no—
316C4—2142no—
317C4—2141no—
318C4—2141no—
319C4#41×D10711yescharacteristic
320C4C2×#1313381711yescharacteristic
321C2—4281no—
322C2#2029313711yescharacteristic
323C2#2029344011yescharacteristic
324C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#462D107
hypocenterC107#3127
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
5. #20292935C2semidirect
6. #20292934C2semidirect
7. #20292834C2semidirect
10. #20292734C2semidirect
11. #20292731C2semidirect
12. Q16×Dic107C2semidirect
13. #20292719C2semidirect
15. #462×C107C2semidirect
163. C107#3127semidirect
171. #462D107semidirect

Maximal subgroups

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

subgroupindex
#202932382
#202931032
#202929672
#202929662
#202929352
#202929342
#202928342
#202927812
#202927342
#202927312
Q16×Dic1072
#202927192
#202926872
#462×C1072
#3127107