Finite Group Databasebeta

← #28344977

Subgroups — #28344977

Order 13696.

772 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic
2C2×M16×D107C211yesmaximal, characteristic, complemented
3C2×#16812067C211yesmaximal, characteristic, complemented
4#20292634C212yesmaximal, complemented
5#20292630C212yesmaximal, complemented
6#20292629C211yesmaximal, characteristic, complemented
7#20292628C211yesmaximal, characteristic, complemented
8#20292626C211yesmaximal, characteristic, complemented
9#20292622C212yesmaximal, complemented
10#20292620C212yesmaximal, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C2×#13133817—21no—
16C2×#13133817C2211yescharacteristic
17#41×D107—24no—
18#41×D107C418yescomplemented
19C2×Q8×D107C2211yescharacteristic
20C2×D4×D107—21no—
21C2×D4×D107C2211yescharacteristic
22C22×C4×D107—21no—
23C22×C4×D107C2211yescharacteristic
24C2×#13133802C2211yescharacteristic, complemented
25C2×#13133801C2212yes—
26M16×D107—24no—
27C2×C8×D107C2212yescomplemented
28#16812067—21no—
29#16812067—21no—
30#16812067C2212yes—
31#16812067C2212yescomplemented
32C2×C4⋊Dic107C2211yescharacteristic, complemented
33C2×(C2×D107)⋊C4—21no—
34C2×C4×Dic107—21no—
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812032C2212yescomplemented
38#16812031C2212yescomplemented
39#16812030C2212yescomplemented
40#16812029C2212yescomplemented
41C2×#41×C107C2211yescharacteristic
42C2×M16×C107C2211yescharacteristic, complemented
43C2×C4⋊C4×C107C2211yescharacteristic, complemented
44#103×C107C2212yescomplemented
45#102×C107C2212yescomplemented
46C23×D107—21no—
47C23×D107D411yescharacteristic
48#13133819—22no—
49#13133819C2×C414yescomplemented
50#13133818—22no—
51#13133818—28no—
52#13133818C2×C414yes—
53#13133817—28no—
54#13133817—22no—
55#13133817C2×C414yescomplemented
56C2×Dic214—21no—
57C2×Dic214C2311yescharacteristic
58Q8×D107—22no—
59Q8×D107C2×C414yescomplemented
60C2×D428—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65D4×D107—28no—
66D4×D107—22no—
67D4×D107C2×C414yescomplemented
68C22×Dic107—21no—
69C22×Dic107—21no—
70C22×Dic107D411yescharacteristic
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—28no—
75C2×C4×D107C2×C414yes—
76C2×C4×D107D411yescharacteristic
77C2×C4×D107D411yescharacteristic
78C2×C4×D107D414yes—
79C2×C4×D107C2311yescharacteristic
80C2×C4×D107C2311yescharacteristic
81#13133802—22no—
82#13133801—24no—
83C2×C107⋊C8C2312yes—
84C8×D107—24no—
85C4⋊Dic107—21no—
86C4⋊Dic107C2312yes—
87(C2×D107)⋊C4—22no—
88(C2×D107)⋊C4—22no—
89C4×Dic107—22no—
90C4×Dic107—22no—
91#41×C107—22no—
92#41×C107C2×C414yes—
93C2×Q8×C107C2311yescharacteristic
94C2×D4×C107—21no—
95C2×D4×C107C2311yescharacteristic
96C22×C4×C107—21no—
97C22×C4×C107—21no—
98C22×C4×C107C2311yescharacteristic
99M16×C107—22no—
100C2×C8×C107C2312yes—
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2312yes—
103C23×C107—21no—
104C23×C107C2×D411yescharacteristic
105C22×D107—21no—
106C22×D107—24no—
107C22×D107—41no—
108C22×D107—41no—
109C22×D107—42no—
110C22×D107—42no—
111C22×D107C22⋊C414yes—
112C22×D107C2×D411yescharacteristic
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107C22×C412yes—
116D4×C107—24no—
117D4×C107—21no—
118D4×C107C22×C412yes—
119C2×C4×C107—21no—
120C2×C4×C107—24no—
121C2×C4×C107—22no—
122C2×C4×C107—42no—
123C2×C4×C107C22×C412yes—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D412yes—
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—24no—
128C22⋊D107—24no—
129C22⋊D107—24no—
130C2×Dic107—21no—
131C2×Dic107—24no—
132C2×Dic107—22no—
133C2×Dic107—42no—
134C2×Dic107C22⋊C414yes—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—24no—
138D428—21no—
139D428C22×C412yes—
140C4×D107—24no—
141C4×D107—24no—
142C4×D107—24no—
143C4×D107C22⋊C414yes—
144C4×D107C22⋊C414yes—
145C4×D107C22×C412yes—
146C4×D107C22×C412yes—
147C4×D107C2×D412yes—
148C4×D107C2×D412yes—
149Dic214—21no—
150Dic214—24no—
151Dic214C22×C412yes—
152C8×C107—22no—
153C107⋊C8—22no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—22no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107C2×C22⋊C412yes—
160C22×C107C22×D411yescharacteristic
161C2×D107—21no—
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—22no—
166C2×D107—44no—
167C2×D107—48no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107C2×C22⋊C412yes—
171C2×D107C2×C22⋊C412yes—
172C4×C107—22no—
173C4×C107—42no—
174C4×C107C2×C22⋊C412yes—
175C4×C107C23×C411yescharacteristic
176C4×C107C22×D411yescharacteristic
177Dic107—22no—
178Dic107—42no—
179Dic107C2×C22⋊C412yes—
180Dic107C2×C22⋊C412yes—
181C2×C107—22no—
182C2×C107—42no—
183C2×C107C22×C22⋊C411yescharacteristic
184C2×C107#57511yescharacteristic
185C2×C107#57511yescharacteristic
186D107—22no—
187D107—22no—
188D107—42no—
189#2896—1071nomaximal, Sylow
190C107#289611yescharacteristic, complemented, Sylow
191C22×#41—1071no—
192C22×M16—1071no—
193C2×#117—1071no—
194#419—1072no—
195#418—1072no—
196#417—1071no—
197#417—1071no—
198#417—1071no—
199#417D10711yescharacteristic, complemented
200#416—1072no—
201#416—1072no—
202C2×#41—1078no—
203C2×#41—1071no—
204C2×#41—1071no—
205C2×#41—1071no—
206C2×#41—2141no—
207C2×#41—2141no—
208C2×#41—2144no—
209C2×#41C2×D10711yescharacteristic
210C22×Q8—1071no—
211C22×D4—1071no—
212C22×D4—2141no—
213C23×C4—1071no—
214C23×C4—2141no—
215C2×M16—1071no—
216C2×M16—1072no—
217C2×M16—2144no—
218C2×M16C2×D10711yescharacteristic, complemented
219C22×C8—1072no—
220#117—1072no—
221#117—1072no—
222#117—2141no—
223#117—2141no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4C2×D10711yescharacteristic, complemented
226C2×C22⋊C4—2141no—
227C2×C42—2141no—
228#103—1072no—
229#103—1072no—
230#103—1072no—
231#103C2×D10712yescomplemented
232#102—1072no—
233#102—1072no—
234#102—1072no—
235#102C2×D10712yescomplemented
236C24—1071no—
237C24—2141no—
238#41—22no—
239#41—1074no—
240#41—1074no—
241#41—1074no—
242#41—2148no—
243#41—2148no—
244#41—2142no—
245#41—2142no—
246#41—2142no—
247#41C4×D10714yes—
248C2×Q8—1074no—
249C2×Q8—1071no—
250C2×Q8—2141no—
251C2×Q8—2142no—
252C2×Q8C22×D10711yescharacteristic
253C2×D4—21no—
254C2×D4—1074no—
255C2×D4—1071no—
256C2×D4—2141no—
257C2×D4—2142no—
258C2×D4—2141no—
259C2×D4—2148no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4C22×D10711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1071no—
266C22×C4—1074no—
267C22×C4—1071no—
268C22×C4—1071no—
269C22×C4—1074no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—2141no—
273C22×C4—2148no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4C22×D10711yescharacteristic
279M16—22no—
280M16—2144no—
281M16—2142no—
282C2×C8—1072no—
283C2×C8—2144no—
284C2×C8C22×D10712yes—
285C4⋊C4—21no—
286C4⋊C4—1072no—
287C4⋊C4—2141no—
288C4⋊C4C22×D10712yes—
289C22⋊C4—2142no—
290C22⋊C4—2142no—
291C42—2142no—
292C42—2142no—
293C23—21no—
294C23—1074no—
295C23—1071no—
296C23—1071no—
297C23—4281no—
298C23—4281no—
299C23—4282no—
300C23—4282no—
301C23—2144no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—1072no—
306Q8—2144no—
307Q8—2141no—
308Q8C2×C4×D10712yes—
309D4—24no—
310D4—21no—
311D4—1072no—
312D4—2144no—
313D4—2144no—
314D4—2144no—
315D4—2144no—
316D4—2141no—
317D4C2×C4×D10712yes—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—1074no—
323C2×C4—1071no—
324C2×C4—1071no—
325C2×C4—1074no—
326C2×C4—1074no—
327C2×C4—1072no—
328C2×C4—1072no—
329C2×C4—1072no—
330C2×C4—1072no—
331C2×C4—4282no—
332C2×C4—2144no—
333C2×C4—2144no—
334C2×C4—2144no—
335C2×C4—2142no—
336C2×C4—2141no—
337C2×C4—2144no—
338C2×C4C2×C4×D10712yes—
339C2×C4D4×D10711yescharacteristic
340C2×C4D4×D10712yes—
341C2×C4C23×D10711yescharacteristic
342C8—22no—
343C8—2142no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—1072no—
350C22—1072no—
351C22—4284no—
352C22—4288no—
353C22—4281no—
354C22—4281no—
355C22—2142no—
356C22—2141no—
357C22—2141no—
358C22—2141no—
359C22—2141no—
360C22C22⋊C4×D10712yes—
361C22C2×D4×D10711yescharacteristic
362C4—22no—
363C4—42no—
364C4—1072no—
365C4—1072no—
366C4—4282no—
367C4—2142no—
368C4C22⋊C4×D10712yes—
369C4C22×C4×D10711yescharacteristic
370C4C2×D4×D10711yescharacteristic
371C2—22no—
372C2—42no—
373C2—4282no—
374C2—2142no—
375C2—2142no—
376C2C2×C22⋊C4×D10711yescharacteristic
377C2#2029342311yescharacteristic
378C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×D107C2
cosocleC2×C4×C107C24
hypercenter#417D107
hypocenterC107#2896
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×M16×D107C2semidirect
3. C2×#16812067C2semidirect
4. #20292634C2semidirect
5. #20292630C2semidirect
6. #20292629C2semidirect
7. #20292628C2semidirect
8. #20292626C2semidirect
9. #20292622C2semidirect
10. #20292620C2semidirect
11. #417×C107C2semidirect
18. #41×D107C4semidirect
24. C2×#13133802C22semidirect
27. C2×C8×D107C22semidirect
31. #16812067C22semidirect
32. C2×C4⋊Dic107C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812032C22semidirect
38. #16812031C22semidirect
39. #16812030C22semidirect
40. #16812029C22semidirect
42. C2×M16×C107C22semidirect
43. C2×C4⋊C4×C107C22semidirect
44. #103×C107C22semidirect
45. #102×C107C22semidirect
49. #13133819C2×C4semidirect
55. #13133817C2×C4semidirect
59. Q8×D107C2×C4semidirect
67. D4×D107C2×C4semidirect
190. C107#2896semidirect
199. #417D107semidirect
218. C2×M16C2×D107semidirect
225. C2×C4⋊C4C2×D107semidirect
231. #103C2×D107semidirect
235. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1072
C2×M16×D1072
C2×#168120672
#202926342
#202926302
#202926292
#202926282
#202926262
#202926222
#202926202
#417×C1072
#2896107