Finite Group Databasebeta

← #28345339

Subgroups — #28345339

Order 13696.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812130C211yesmaximal, characteristic
2#20293109C211yesmaximal, characteristic, complemented
3#20293106C211yesmaximal, characteristic, complemented
4#20292737C211yesmaximal, characteristic, complemented
5#20292736C211yesmaximal, characteristic, complemented
6#20292732C211yesmaximal, characteristic, complemented
7#20292731C211yesmaximal, characteristic, complemented
8C4×#13133807C211yesmaximal, characteristic, complemented
9SD16×Dic107C211yesmaximal, characteristic, complemented
10#20292714C211yesmaximal, characteristic, complemented
11#106×D107C211yesmaximal, characteristic, complemented
12#20292627C211yesmaximal, characteristic, complemented
13#20292625C211yesmaximal, characteristic, complemented
14#20292581C211yesmaximal, characteristic, complemented
15C4×SD16×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18#16812130C418yescomplemented
19C2×#13133808C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21C2×#13133803C2211yescharacteristic
22C2×#13133801C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24Q8×Dic107C2211yescharacteristic
25C4×Dic214C2211yescharacteristic
26#16812074C2211yescharacteristic
27C4×D428—21no—
28C4×D428C2211yescharacteristic
29D4×Dic107C2211yescharacteristic
30#16812071—21no—
31#16812071C2211yescharacteristic
32C4×C22⋊D107—21no—
33(C22⋊D107)⋊C4—21no—
34#16812066—21no—
35#16812066C2211yescharacteristic
36#16812065—21no—
37C2×Dic107⋊C4—21no—
38C4⋊C4×D107—21no—
39C4⋊C4×D107C2211yescharacteristic
40C4⋊C4×D107C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812029C2211yescharacteristic
50#16812020C2211yescharacteristic
51#16812019C2211yescharacteristic
52C8×Dic107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C4×C8×C107C2211yescharacteristic
60#13133819—22no—
61#13133819C2×C414yescomplemented
62#13133817—22no—
63#13133817C2×C414yescomplemented
64C2×Dic214C2311yescharacteristic
65C2×D428—21no—
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C22×Dic107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107C2311yescharacteristic
76#13133808C2×C414yescomplemented
77#13133807C2×C414yescomplemented
78#13133803C2×C414yescomplemented
79#13133801—22no—
80C2×C107⋊C8C2311yescharacteristic
81C2×C107⋊C8C2311yescharacteristic
82C8×D107C2×C414yescomplemented
83C4⋊Dic107—21no—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4D411yescharacteristic
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4D411yescharacteristic
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C4×Dic107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104SD16×C107C2×C414yescomplemented
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C42×C107—21no—
112C42×C107C2311yescharacteristic
113C23×C107—21no—
114C22×D107—21no—
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107C22×C412yes—
118D4×C107—21no—
119D4×C107C22×C412yes—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—81no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—24no—
128C2×Dic107—21no—
129C2×Dic107—24no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—81no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136D428—24no—
137D428—21no—
138D428C22×C412yes—
139C4×D107—24no—
140C4×D107—81no—
141C4×D107—81no—
142C4×D107—81no—
143C4×D107—81no—
144C4×D107C22×C412yes—
145C4×D107#4112yes—
146Dic214—21no—
147Dic214C22×C412yes—
148C8×C107—21no—
149C8×C107C22×C412yes—
150C107⋊C8—21no—
151C107⋊C8C22×C412yes—
152C22×C107—22no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—22no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107C4×D412yes—
162C4×C107—22no—
163C4×C107—21no—
164C4×C107—21no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107C23×C411yescharacteristic
168C4×C107C2×#4111yescharacteristic
169Dic107—22no—
170Dic107—21no—
171Dic107—21no—
172Dic107—41no—
173Dic107—41no—
174Dic107C4×D412yes—
175C2×C107—42no—
176C2×C107C2×C4×D411yescharacteristic
177C2×C107#57711yescharacteristic
178C2×C107#57711yescharacteristic
179D107—22no—
180D107—42no—
181#2952—1071nomaximal, Sylow
182C107#295211yescharacteristic, complemented, Sylow
183C2×#135—1071no—
184#520—1071no—
185#520—1071no—
186#442—1071no—
187#442—1071no—
188#441—1071no—
189#441—1071no—
190C4×SD16—1071no—
191C4×SD16—1071no—
192C4×SD16—1071no—
193C4×SD16D10711yescharacteristic, complemented
194C2×#106—1071no—
195#417—1071no—
196#417—1071no—
197#405—1071no—
198C2×#41—1071no—
199C2×#41—1071no—
200#135—1078no—
201C2×Q16—1071no—
202C2×SD16—1071no—
203C2×SD16C2×D10711yescharacteristic
204C2×D8—1071no—
205C2×M16—1071no—
206C22×C8—1071no—
207C4×Q8—1071no—
208C4×Q8—1071no—
209C4×Q8—1071no—
210C4×Q8C2×D10711yescharacteristic
211C4×D4—1071no—
212C4×D4—1071no—
213C4×D4—1071no—
214C4×D4—2141no—
215C4×D4—2141no—
216C4×D4—2141no—
217C4×D4—2141no—
218C4×D4C2×D10711yescharacteristic
219#117—1071no—
220#117—2141no—
221#117—2141no—
222C2×C4⋊C4—1071no—
223C2×C4⋊C4—1071no—
224C2×C4⋊C4—2141no—
225C2×C4⋊C4—2141no—
226#106—1071no—
227#106—1071no—
228#106—1071no—
229#106C2×D10711yescharacteristic
230#103—1071no—
231#103—1071no—
232#103—1071no—
233#103C2×D10711yescharacteristic
234#102—1071no—
235#102—1071no—
236#102—1071no—
237#102C2×D10711yescharacteristic
238#97—1071no—
239#97—1071no—
240C4×C8—1071no—
241C4×C8C2×D10711yescharacteristic
242#41—1074no—
243#41—1074no—
244#41—2142no—
245#41—2142no—
246C2×Q8—1071no—
247C2×Q8C22×D10711yescharacteristic
248C2×D4—1071no—
249C2×D4—2141no—
250C2×D4—2141no—
251C2×D4C22×D10711yescharacteristic
252C22×C4—21no—
253C22×C4—1071no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4—2141no—
257C22×C4—2141no—
258C22×C4—2141no—
259C22×C4—2141no—
260C22×C4—2141no—
261Q16—1074no—
262SD16—1074no—
263SD16C4×D10714yescomplemented
264D8—1074no—
265M16—2142no—
266C2×C8—1071no—
267C2×C8—1074no—
268C2×C8—1071no—
269C2×C8C22×D10711yescharacteristic
270C2×C8C22×D10711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—1071no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1071no—
285C22⋊C4—1071no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C42—21no—
290C42—1071no—
291C42—2141no—
292C42—2141no—
293C42—2141no—
294C42C22×D10711yescharacteristic
295C23—21no—
296C23—1071no—
297C23—2141no—
298Q8—21no—
299Q8—1072no—
300Q8—2141no—
301Q8C2×C4×D10712yes—
302D4—21no—
303D4—1072no—
304D4—2144no—
305D4—2144no—
306D4—2141no—
307D4C2×C4×D10712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—8561no—
313C2×C4—8561no—
314C2×C4—8561no—
315C2×C4—8561no—
316C2×C4—8561no—
317C2×C4—1071no—
318C2×C4—1072no—
319C2×C4—1072no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—2144no—
323C2×C4—2144no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4D4×D10711yescharacteristic
328C2×C4D4×D10711yescharacteristic
329C2×C4C23×D10711yescharacteristic
330C8—21no—
331C8—1072no—
332C8—2141no—
333C8C2×C4×D10712yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—1072no—
338C22—4281no—
339C22—4281no—
340C22—2142no—
341C22—2141no—
342C22—2141no—
343C22C2×D4×D10711yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—1072no—
350C4—4281no—
351C4—4281no—
352C4—2141no—
353C4—2141no—
354C4—2142no—
355C4C22×C4×D10711yescharacteristic
356C4#41×D10711yescharacteristic
357C2—42no—
358C2—4282no—
359C2—2142no—
360C2C4×D4×D10711yescharacteristic
361C2#2029343311yescharacteristic
362C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812130C2
cosocleC2×C4×C107C24
hypercenterC4×SD16D107
hypocenterC107#2952
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20293109C2semidirect
3. #20293106C2semidirect
4. #20292737C2semidirect
5. #20292736C2semidirect
6. #20292732C2semidirect
7. #20292731C2semidirect
8. C4×#13133807C2semidirect
9. SD16×Dic107C2semidirect
10. #20292714C2semidirect
11. #106×D107C2semidirect
12. #20292627C2semidirect
13. #20292625C2semidirect
14. #20292581C2semidirect
15. C4×SD16×C107C2semidirect
18. #16812130C4semidirect
61. #13133819C2×C4semidirect
63. #13133817C2×C4semidirect
76. #13133808C2×C4semidirect
77. #13133807C2×C4semidirect
78. #13133803C2×C4semidirect
82. C8×D107C2×C4semidirect
104. SD16×C107C2×C4semidirect
182. C107#2952semidirect
193. C4×SD16D107semidirect
263. SD16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202931092
#202931062
#202927372
#202927362
#202927322
#202927312
C4×#131338072
SD16×Dic1072
#202927142
#106×D1072
#202926272
#202926252
#202925812
C4×SD16×C1072
#2952107