Finite Group Databasebeta

← #28346646

Subgroups — #28346646

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#20293250C211yesmaximal, characteristic, complemented
2#20293102C211yesmaximal, characteristic, complemented
3#20292965C211yesmaximal, characteristic, complemented
4#20292952C211yesmaximal, characteristic, complemented
5#20292915C211yesmaximal, characteristic, complemented
6#20292905C211yesmaximal, characteristic, complemented
7#20292844C211yesmaximal, characteristic, complemented
8#20292844C211yesmaximal, characteristic, complemented
9#20292761C211yesmaximal, characteristic
10#20292736C211yesmaximal, characteristic, complemented
11#20292728C211yesmaximal, characteristic, complemented
12SD16×Dic107C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14#20292685C211yesmaximal, characteristic, complemented
15#450×C107C211yesmaximal, characteristic, complemented
16C2×#13133803C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18#16812108C2211yescharacteristic, complemented
19#16812104—21no—
20#16812098C2211yescharacteristic, complemented
21#16812095—21no—
22#16812091—21no—
23#16812089—21no—
24#16812083C2211yescharacteristic
25Q8×Dic107C2211yescharacteristic, complemented
26D4×Dic107—21no—
27D4×Dic107—21no—
28D4×Dic107C2211yescharacteristic
29D4×Dic107C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32#16812071C2211yescharacteristic, complemented
33#16812068C2211yescharacteristic
34#16812068C2211yescharacteristic
35C2×C22⋊Dic107—21no—
36C2×C22⋊Dic107—21no—
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C22×Dic107—21no—
63C22×Dic107—21no—
64C22×Dic107—21no—
65C22×Dic107—21no—
66C2×C4×D107—21no—
67#13133803—22no—
68#13133803—22no—
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C4⋊Dic107—21no—
72C4⋊Dic107D411yescharacteristic
73C4⋊Dic107D411yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78C22⋊Dic107—21no—
79C22⋊Dic107—21no—
80C22⋊Dic107—21no—
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107D411yescharacteristic
84C22⋊Dic107D411yescharacteristic
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93C24×C107—21no—
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107—22no—
96C2×D4×C107—22no—
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C2×D4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101SD16×C107—22no—
102SD16×C107—22no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C23×C107—24no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107—81no—
112C23×C107—81no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—24no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107#4112yes—
121D4×C107#4112yes—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—81no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—81no—
133C2×Dic107—81no—
134C2×Dic107—81no—
135C2×Dic107—81no—
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140D428—21no—
141D428—21no—
142C4×D107—81no—
143Dic214—81no—
144C8×C107—21no—
145C8×C107—21no—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C22×C107—21no—
149C22×C107—22no—
150C22×C107—22no—
151C22×C107—21no—
152C22×C107—21no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107—84no—
158C22×C107C22×D411yescharacteristic
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C4×C107—21no—
163C4×C107—41no—
164C4×C107—41no—
165C4×C107C2×#4111yescharacteristic
166C4×C107C2×#4111yescharacteristic
167Dic107—21no—
168Dic107—21no—
169Dic107—21no—
170Dic107—21no—
171Dic107—41no—
172Dic107—41no—
173Dic107—41no—
174C2×C107—41no—
175C2×C107—42no—
176C2×C107—42no—
177C2×C107#52511yescharacteristic
178C2×C107#57611yescharacteristic
179C2×C107#57711yescharacteristic
180D107—81no—
181#3118—1071nomaximal, Sylow
182C107#311811yescharacteristic, complemented, Sylow
183#542—1071no—
184#518—1071no—
185#485—1071no—
186#482—1071no—
187#476—1071no—
188#474—1071no—
189#463—1071no—
190#463—1071no—
191#450D10711yescharacteristic, complemented
192#447—1071no—
193#442—1071no—
194#440—1071no—
195C4×SD16—1071no—
196C4×D8—1071no—
197#433—1071no—
198C22×D4C2×D10711yescharacteristic
199C2×SD16C2×D10711yescharacteristic
200C2×SD16C2×D10711yescharacteristic
201C2×D8—1071no—
202C2×D8—1071no—
203C4⋊Q8—1071no—
204#126—2141no—
205#124—1071no—
206#124—2141no—
207#123—2141no—
208C22⋊Q8—2141no—
209C22⋊Q8C2×D10711yescharacteristic
210#121—1071no—
211C4×Q8—1071no—
212C4×D4—1071no—
213C4×D4—1071no—
214C4×D4—1071no—
215C4×D4—1071no—
216C4×D4—1071no—
217C4×D4—2141no—
218C4×D4—2141no—
219#117—1071no—
220#117—1071no—
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#102C2×D10711yescharacteristic
236#102C2×D10711yescharacteristic
237C22⋊C8—1071no—
238C22⋊C8C2×D10711yescharacteristic
239#97—1071no—
240C4×C8—1071no—
241C24—21no—
242C2×Q8—2141no—
243C2×Q8C22×D10711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—1071no—
248C2×D4—2141no—
249C2×D4C22×D10711yescharacteristic
250C2×D4C22×D10711yescharacteristic
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4C22×D10711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—2142no—
260D8—2142no—
261C2×C8—1071no—
262C2×C8—1071no—
263C2×C8C22×D10711yescharacteristic
264C2×C8C22×D10711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4C22×D10711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—1071no—
276C22⋊C4—1071no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—1071no—
287C42—1071no—
288C42—2141no—
289C42—2141no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—2141no—
296C23D4×D10711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—8561no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—8561no—
304D4—2141no—
305D4—2141no—
306D4#1313381712yes—
307D4#1313381712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—8561no—
313C2×C4—8561no—
314C2×C4—8561no—
315C2×C4—8561no—
316C2×C4—8561no—
317C2×C4—1071no—
318C2×C4—1071no—
319C2×C4—1071no—
320C2×C4—1071no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4D4×D10711yescharacteristic
325C2×C4C23×D10711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2141no—
329C8—2141no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—4281no—
341C22—4281no—
342C22—4281no—
343C22C2×D4×D10711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—4281no—
348C4—4281no—
349C4—4281no—
350C4—2141no—
351C4—2141no—
352C4—2141no—
353C4—2141no—
354C4C2×#1313381711yescharacteristic
355C4C2×#1313381711yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—8561no—
360C2#2029313311yescharacteristic
361C2#2029343111yescharacteristic
362C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radical#20292761C2
hypercenter#450D107
hypocenterC107#3118
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20293250C2semidirect
2. #20293102C2semidirect
3. #20292965C2semidirect
4. #20292952C2semidirect
5. #20292915C2semidirect
6. #20292905C2semidirect
7. #20292844C2semidirect
8. #20292844C2semidirect
10. #20292736C2semidirect
11. #20292728C2semidirect
12. SD16×Dic107C2semidirect
13. #20292707C2semidirect
14. #20292685C2semidirect
15. #450×C107C2semidirect
18. #16812108C22semidirect
20. #16812098C22semidirect
25. Q8×Dic107C22semidirect
32. #16812071C22semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812033C22semidirect
44. #16812033C22semidirect
45. #16812031C22semidirect
46. #16812031C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
182. C107#3118semidirect
191. #450D107semidirect

Maximal subgroups

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

subgroupindex
#202932502
#202931022
#202929652
#202929522
#202929152
#202929052
#202928442
#202927612
#202927362
#202927282
SD16×Dic1072
#202927072
#202926852
#450×C1072
#3118107