Finite Group Databasebeta

← #28346639

Subgroups — #28346639

Order 13696.

406 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
1#20293101C211yesmaximal, characteristic, complemented
2#20292965C211yesmaximal, characteristic, complemented
3#20292948C211yesmaximal, characteristic, complemented
4#20292914C211yesmaximal, characteristic, complemented
5#20292904C211yesmaximal, characteristic, complemented
6#20292839C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292777C211yesmaximal, characteristic, complemented
9#20292760C211yesmaximal, characteristic
10#20292737C211yesmaximal, characteristic, complemented
11#20292725C211yesmaximal, characteristic, complemented
12C4×#13133805C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14#20292688C211yesmaximal, characteristic, complemented
15#542×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C4×D428C2211yescharacteristic
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812071—21no—
26#16812071—21no—
27#16812068C2211yescharacteristic
28C22⋊C4×D107—21no—
29C22⋊C4×D107—21no—
30#16812042C2211yescharacteristic, complemented
31#16812042C2211yescharacteristic, complemented
32#16812040C2211yescharacteristic, complemented
33#16812040C2211yescharacteristic, complemented
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35C4⋊(C107⋊C8)C2211yescharacteristic, complemented
36#16812033C2211yescharacteristic, complemented
37#16812033C2211yescharacteristic, complemented
38#16812030C2211yescharacteristic, complemented
39#16812030C2211yescharacteristic, complemented
40#16812029C2211yescharacteristic
41#16812029C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44C22⋊(C107⋊C8)C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46#16812020C2211yescharacteristic, complemented
47C4×C107⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C107C2211yescharacteristic, complemented
49#126×C107—21no—
50#124×C107—21no—
51#124×C107C2211yescharacteristic, complemented
52#123×C107—21no—
53C22⋊Q8×C107—21no—
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58#117×C107C2211yescharacteristic
59C23×D107—21no—
60C2×D428—21no—
61C2×D428—22no—
62C2×D428—22no—
63C2×D428C2311yescharacteristic
64C2×D428C2311yescharacteristic
65C2×C4×D107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69#13133805—22no—
70#13133805—22no—
71#13133803—22no—
72#13133803—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84C4×Dic107—21no—
85C2×Q8×C107—21no—
86C2×Q8×C107C2311yescharacteristic
87C2×D4×C107—21no—
88C2×D4×C107C2311yescharacteristic
89C22×C4×C107—21no—
90C22×C4×C107C2311yescharacteristic
91C4⋊C4×C107—21no—
92C4⋊C4×C107—21no—
93C4⋊C4×C107—21no—
94C4⋊C4×C107D411yescharacteristic
95C4⋊C4×C107D411yescharacteristic
96C4⋊C4×C107C2311yescharacteristic
97C4⋊C4×C107C2311yescharacteristic
98C22⋊C4×C107—21no—
99C22⋊C4×C107—21no—
100C22⋊C4×C107—21no—
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C22⋊C4×C107D411yescharacteristic
104C22⋊C4×C107D411yescharacteristic
105C42×C107—21no—
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111C22×D107—21no—
112C22×D107—24no—
113C22×D107—81no—
114C22×D107—81no—
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—81no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—81no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—81no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2411yescharacteristic
133C2×Dic107—21no—
134C2×Dic107—21no—
135D428—24no—
136D428—21no—
137D428—21no—
138D428#4112yes—
139D428#4112yes—
140C4×D107—81no—
141C4×D107—81no—
142C4×D107—81no—
143C4×D107—81no—
144C107⋊C8—21no—
145C107⋊C8—21no—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C22×C107—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107C22×D411yescharacteristic
155C2×D107—22no—
156C2×D107—22no—
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—84no—
162C4×C107—21no—
163C4×C107—21no—
164C4×C107—21no—
165C4×C107—21no—
166C4×C107—21no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107C2×#4111yescharacteristic
171C4×C107C2×#4111yescharacteristic
172Dic107—41no—
173Dic107—41no—
174C2×C107—41no—
175C2×C107—81no—
176C2×C107#52511yescharacteristic
177C2×C107#57611yescharacteristic
178C2×C107#57711yescharacteristic
179D107—42no—
180D107—42no—
181#3118—1071nomaximal, Sylow
182C107#311811yescharacteristic, complemented, Sylow
183#542D10711yescharacteristic, complemented
184#518—1071no—
185#485—1071no—
186#482—1071no—
187#476—1071no—
188#474—1071no—
189#463—1071no—
190#463—1071no—
191#450—1071no—
192#447—1071no—
193#442—1071no—
194#440—1071no—
195C4×SD16—1071no—
196C4×D8—1071no—
197#433—1071no—
198C22×D4—1071no—
199C2×SD16—1071no—
200C2×SD16—1071no—
201C2×D8—1071no—
202C2×D8—1071no—
203C4⋊Q8C2×D10711yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D10711yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D10711yescharacteristic
210#121C2×D10711yescharacteristic
211C4×Q8C2×D10711yescharacteristic, complemented
212C4×D4—1071no—
213C4×D4—1071no—
214C4×D4—1071no—
215C4×D4—1071no—
216C4×D4—2141no—
217C4×D4—2141no—
218C4×D4C2×D10711yescharacteristic, complemented
219#117—1071no—
220#117C2×D10711yescharacteristic
221C2×C22⋊C4—2141no—
222C2×C22⋊C4—2141no—
223#107—1071no—
224#107—1071no—
225#106—1071no—
226#106—1071no—
227C4⋊C8—1071no—
228C4⋊C8—1071no—
229#103—1071no—
230#103—1071no—
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235#102—1071no—
236#102—1071no—
237C22⋊C8—1071no—
238C22⋊C8—1071no—
239#97—1071no—
240C4×C8—1071no—
241C24—2141no—
242C2×Q8—21no—
243C2×Q8C22×D10711yescharacteristic
244C2×D4—21no—
245C2×D4—1071no—
246C2×D4—1071no—
247C2×D4—2142no—
248C2×D4—2142no—
249C2×D4—2141no—
250C2×D4C22×D10711yescharacteristic
251C22×C4—21no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4C22×D10711yescharacteristic
257SD16—2142no—
258SD16—2142no—
259D8—2142no—
260D8—2142no—
261C2×C8—1071no—
262C2×C8—1071no—
263C2×C8—1071no—
264C2×C8—1071no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4C22⋊D10711yescharacteristic
271C4⋊C4C22⋊D10711yescharacteristic
272C4⋊C4C22×D10711yescharacteristic
273C4⋊C4C22×D10711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4C22⋊D10711yescharacteristic
285C22⋊C4C22⋊D10711yescharacteristic
286C42—21no—
287C42—2141no—
288C42C22×D10711yescharacteristic
289C42C22×D10711yescharacteristic
290C23—21no—
291C23—8561no—
292C23—8561no—
293C23—2144no—
294C23—2141no—
295C23—2141no—
296C23C2×C22⋊D10711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—1072no—
304D4—1072no—
305D4—2144no—
306D4—2141no—
307D4—2141no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—8561no—
315C2×C4—8561no—
316C2×C4—8561no—
317C2×C4—8561no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4C2×C22⋊D10711yescharacteristic
321C2×C4C2×C22⋊D10711yescharacteristic
322C2×C4C2×C22⋊D10711yescharacteristic
323C2×C4C2×C22⋊D10711yescharacteristic
324C2×C4C2×C22⋊D10711yescharacteristic
325C2×C4C23×D10711yescharacteristic
326C8—2141no—
327C8—2141no—
328C8—2141no—
329C8—2141no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—8564no—
337C22—4281no—
338C22—4281no—
339C22—4281no—
340C22—4281no—
341C22—2142no—
342C22—2142no—
343C22C22×C22⋊D10711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—4281no—
353C4—4281no—
354C4#41×D10711yescharacteristic
355C4#41×D10711yescharacteristic
356C2—41no—
357C2—81no—
358C2—4282no—
359C2—4282no—
360C2#2029313411yescharacteristic
361C2#2029343011yescharacteristic
362C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#542×C107C2
socleC22×C107C22×D4
prime radical#20292760C2
hypercenter#542D107
hypocenterC107#3118
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20293101C2semidirect
2. #20292965C2semidirect
3. #20292948C2semidirect
4. #20292914C2semidirect
5. #20292904C2semidirect
6. #20292839C2semidirect
7. #20292839C2semidirect
8. #20292777C2semidirect
10. #20292737C2semidirect
11. #20292725C2semidirect
12. C4×#13133805C2semidirect
13. C4×#13133803C2semidirect
14. #20292688C2semidirect
15. #542×C107C2semidirect
30. #16812042C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812040C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. C4⋊(C107⋊C8)C22semidirect
36. #16812033C22semidirect
37. #16812033C22semidirect
38. #16812030C22semidirect
39. #16812030C22semidirect
46. #16812020C22semidirect
47. C4×C107⋊C8C22semidirect
48. C4⋊Q8×C107C22semidirect
51. #124×C107C22semidirect
56. C4×Q8×C107C22semidirect
57. C4×D4×C107C22semidirect
182. C107#3118semidirect
183. #542D107semidirect
203. C4⋊Q8C2×D107semidirect
206. #124C2×D107semidirect
211. C4×Q8C2×D107semidirect
218. C4×D4C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931012
#202929652
#202929482
#202929142
#202929042
#202928392
#202927772
#202927602
#202927372
#202927252
C4×#131338052
C4×#131338032
#202926882
#542×C1072
#3118107