Finite Group Databasebeta

← #28346757

Subgroups — #28346757

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#20293237C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic
3#20292968C211yesmaximal, characteristic
4#20292937C211yesmaximal, characteristic, complemented
5#20292933C211yesmaximal, characteristic, complemented
6#20292836C211yesmaximal, characteristic, complemented
7#20292836C211yesmaximal, characteristic, complemented
8#20292785C211yesmaximal, characteristic
9#20292782C211yesmaximal, characteristic
10#20292733C211yesmaximal, characteristic, complemented
11#20292731C211yesmaximal, characteristic, complemented
12#20292721C211yesmaximal, characteristic, complemented
13#20292719C211yesmaximal, characteristic, complemented
14#20292685C211yesmaximal, characteristic
15#485×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×Dic428C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133808C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812102—21no—
23#16812102—21no—
24#16812100—21no—
25#16812100C2211yescharacteristic
26#16812099—21no—
27#16812088C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29C4×Dic214—21no—
30C4×Dic214—21no—
31#16812074C2211yescharacteristic
32#16812074C2211yescharacteristic
33#16812074C2211yescharacteristic
34#16812074C2211yescharacteristic
35#16812074C2211yescharacteristic
36#16812068C2211yescharacteristic
37#16812065—21no—
38#16812065—21no—
39#16812042C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—22no—
61C2×Dic214—22no—
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64C2×Dic214C2311yescharacteristic
65C22×Dic107—21no—
66Dic428—22no—
67Dic428—22no—
68#13133808—22no—
69#13133808—22no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107D411yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107D411yescharacteristic
88C22⋊Dic107D411yescharacteristic
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96C4×Dic107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C23×C107C2×D411yescharacteristic
109Q8×C107—21no—
110Q8×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C2×Dic107—24no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128Dic214—24no—
129Dic214—21no—
130Dic214—21no—
131Dic214—21no—
132Dic214—21no—
133Dic214#4112yes—
134Dic214#4112yes—
135C8×C107—21no—
136C8×C107—21no—
137C107⋊C8—21no—
138C107⋊C8—21no—
139C22×C107—21no—
140C22×C107—21no—
141C22×C107—21no—
142C22×C107C22×D411yescharacteristic
143C4×C107—21no—
144C4×C107—41no—
145C4×C107—41no—
146C4×C107—41no—
147C4×C107—41no—
148C4×C107C2×#4111yescharacteristic
149C4×C107C2×#4111yescharacteristic
150Dic107—22no—
151Dic107—22no—
152Dic107—21no—
153Dic107—21no—
154Dic107—21no—
155Dic107—21no—
156Dic107—41no—
157Dic107—41no—
158C2×C107—41no—
159C2×C107#52511yescharacteristic
160C2×C107#57811yescharacteristic
161C2×C107#57811yescharacteristic
162#3127—1071nomaximal, Sylow
163C107#312711yescharacteristic, complemented, Sylow
164#541—1071no—
165#519—1071no—
166#485—1071no—
167#485D10711yescharacteristic
168#479—1071no—
169#479—1071no—
170#462—1071no—
171#462—1071no—
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×Q16—1071no—
183C2×Q16—1071no—
184C4⋊Q8—1071no—
185#126—2141no—
186#126—2141no—
187#125—1071no—
188#125—2141no—
189#125—2141no—
190C22⋊Q8—1071no—
191C22⋊Q8C2×D10711yescharacteristic
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×Q8—1071no—
196C4×Q8—1071no—
197C4×Q8—1071no—
198C4×Q8—2141no—
199C4×Q8—2141no—
200#117—1071no—
201#117—2141no—
202#117—2141no—
203#117C2×D10711yescharacteristic
204#107—1071no—
205#107C2×D10711yescharacteristic
206#106—1071no—
207#106C2×D10711yescharacteristic
208C4⋊C8—1071no—
209C4⋊C8—1071no—
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⋊C8C2×D10711yescharacteristic
220#97—1071no—
221C4×C8—1071no—
222C2×Q8—1071no—
223C2×Q8—1071no—
224C2×Q8—1071no—
225C2×Q8—2142no—
226C2×Q8—2142no—
227C2×Q8—2141no—
228C2×Q8C22×D10711yescharacteristic
229C22×C4—2141no—
230C22×C4C22×D10711yescharacteristic
231Q16—2142no—
232Q16—2142no—
233Q16—2142no—
234Q16—2142no—
235C2×C8—1071no—
236C2×C8—1071no—
237C2×C8C22×D10711yescharacteristic
238C2×C8C22×D10711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—1071no—
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4C22×D10711yescharacteristic
255C4⋊C4C22×D10711yescharacteristic
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—1071no—
260C22⋊C4—1071no—
261C22⋊C4—2141no—
262C42—21no—
263C42—1071no—
264C42—1071no—
265C42—2141no—
266C42—2141no—
267C42—2141no—
268C42—2141no—
269C42—2141no—
270C42—2141no—
271C23D4×D10711yescharacteristic
272Q8—21no—
273Q8—21no—
274Q8—1072no—
275Q8—1072no—
276Q8—2144no—
277Q8—2141no—
278Q8—2141no—
279Q8—2141no—
280Q8—2141no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—1071no—
288C2×C4—1071no—
289C2×C4—1071no—
290C2×C4—1071no—
291C2×C4—2141no—
292C2×C4—2141no—
293C2×C4—2144no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4D4×D10711yescharacteristic
297C2×C4C23×D10711yescharacteristic
298C8—21no—
299C8—21no—
300C8—2141no—
301C8—2141no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D10711yescharacteristic
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4281no—
312C4—4281no—
313C4—2142no—
314C4—2142no—
315C4—2141no—
316C4—2141no—
317C4—2141no—
318C4—2141no—
319C4#41×D10711yescharacteristic
320C4#41×D10711yescharacteristic
321C2—41no—
322C2#2029313511yescharacteristic
323C2#2029344111yescharacteristic
324C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#485D107
hypocenterC107#3127
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
4. #20292937C2semidirect
5. #20292933C2semidirect
6. #20292836C2semidirect
7. #20292836C2semidirect
10. #20292733C2semidirect
11. #20292731C2semidirect
12. #20292721C2semidirect
13. #20292719C2semidirect
163. C107#3127semidirect

Maximal subgroups

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

subgroupindex
#202932372
#202931042
#202929682
#202929372
#202929332
#202928362
#202927852
#202927822
#202927332
#202927312
#202927212
#202927192
#202926852
#485×C1072
#3127107