Finite Group Databasebeta

← #22598525

Subgroups — #22598525

Order 9088.

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#17986622C211yesmaximal, characteristic, complemented
2#17986474C211yesmaximal, characteristic, complemented
3#17986337C211yesmaximal, characteristic, complemented
4#17986324C211yesmaximal, characteristic, complemented
5#17986287C211yesmaximal, characteristic, complemented
6#17986277C211yesmaximal, characteristic, complemented
7#17986216C211yesmaximal, characteristic, complemented
8#17986216C211yesmaximal, characteristic, complemented
9#17986133C211yesmaximal, characteristic
10#17986108C211yesmaximal, characteristic, complemented
11#17986100C211yesmaximal, characteristic, complemented
12SD16×Dic71C211yesmaximal, characteristic, complemented
13#17986079C211yesmaximal, characteristic, complemented
14#17986057C211yesmaximal, characteristic, complemented
15#450×C71C211yesmaximal, characteristic, complemented
16C2×#11765850C2211yescharacteristic
17C2×#11765850C2211yescharacteristic
18#14556127C2211yescharacteristic, complemented
19#14556123—21no—
20#14556117C2211yescharacteristic, complemented
21#14556114—21no—
22#14556110—21no—
23#14556108—21no—
24#14556102C2211yescharacteristic
25Q8×Dic71C2211yescharacteristic, complemented
26D4×Dic71—21no—
27D4×Dic71—21no—
28D4×Dic71C2211yescharacteristic
29D4×Dic71C2211yescharacteristic
30D4×Dic71C2211yescharacteristic
31D4×Dic71C2211yescharacteristic
32#14556090C2211yescharacteristic, complemented
33#14556087C2211yescharacteristic
34#14556087C2211yescharacteristic
35C2×C22⋊Dic71—21no—
36C2×C22⋊Dic71—21no—
37#14556062C2211yescharacteristic, complemented
38#14556061C2211yescharacteristic, complemented
39#14556060C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic, complemented
47#14556049C2211yescharacteristic
48#14556049C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic
50#14556038C2211yescharacteristic, complemented
51C8×Dic71C2211yescharacteristic, complemented
52C22×D4×C71C2211yescharacteristic
53C2×SD16×C71C2211yescharacteristic
54C2×SD16×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic
58C22⋊C8×C71C2211yescharacteristic
59C2×Dic142—21no—
60C2×D284C2311yescharacteristic
61C2×C22⋊D71—21no—
62C22×Dic71—21no—
63C22×Dic71—21no—
64C22×Dic71—21no—
65C22×Dic71—21no—
66C2×C4×D71—21no—
67#11765850—22no—
68#11765850—22no—
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C4⋊Dic71—21no—
72C4⋊Dic71D411yescharacteristic
73C4⋊Dic71D411yescharacteristic
74C4⋊Dic71C2311yescharacteristic
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77Dic71⋊C4—21no—
78C22⋊Dic71—21no—
79C22⋊Dic71—21no—
80C22⋊Dic71—21no—
81C22⋊Dic71—21no—
82C22⋊Dic71—21no—
83C22⋊Dic71D411yescharacteristic
84C22⋊Dic71D411yescharacteristic
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88(C2×D71)⋊C4—21no—
89C4×Dic71—21no—
90C4×Dic71—21no—
91C4×Dic71C2311yescharacteristic
92C4×Dic71C2311yescharacteristic
93C24×C71—21no—
94C2×Q8×C71C2311yescharacteristic
95C2×D4×C71—22no—
96C2×D4×C71—22no—
97C2×D4×C71—21no—
98C2×D4×C71C2311yescharacteristic
99C2×D4×C71C2311yescharacteristic
100C22×C4×C71C2311yescharacteristic
101SD16×C71—22no—
102SD16×C71—22no—
103C2×C8×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71C2311yescharacteristic
107C22⋊C4×C71—21no—
108C23×C71—24no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—81no—
112C23×C71—81no—
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115Q8×C71—21no—
116Q8×C71—21no—
117D4×C71—24no—
118D4×C71—21no—
119D4×C71—21no—
120D4×C71#4112yes—
121D4×C71#4112yes—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71C2×D411yescharacteristic
127C2×C4×C71C2411yescharacteristic
128C22⋊D71—81no—
129C2×Dic71—21no—
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—81no—
133C2×Dic71—81no—
134C2×Dic71—81no—
135C2×Dic71—81no—
136C2×Dic71C2×D411yescharacteristic
137C2×Dic71C2×D411yescharacteristic
138C2×Dic71C2×D411yescharacteristic
139C2×Dic71C2×D411yescharacteristic
140D284—21no—
141D284—21no—
142C4×D71—81no—
143Dic142—81no—
144C8×C71—21no—
145C8×C71—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C22×C71—21no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—84no—
158C22×C71C22×D411yescharacteristic
159C2×D71—41no—
160C2×D71—41no—
161C2×D71—41no—
162C4×C71—21no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71C2×#4111yescharacteristic
166C4×C71C2×#4111yescharacteristic
167Dic71—21no—
168Dic71—21no—
169Dic71—21no—
170Dic71—21no—
171Dic71—41no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—41no—
175C2×C71—42no—
176C2×C71—42no—
177C2×C71#52511yescharacteristic
178C2×C71#57611yescharacteristic
179C2×C71#57711yescharacteristic
180D71—81no—
181#3118—711nomaximal, Sylow
182C71#311811yescharacteristic, complemented, Sylow
183#542—711no—
184#518—711no—
185#485—711no—
186#482—711no—
187#476—711no—
188#474—711no—
189#463—711no—
190#463—711no—
191#450D7111yescharacteristic, complemented
192#447—711no—
193#442—711no—
194#440—711no—
195C4×SD16—711no—
196C4×D8—711no—
197#433—711no—
198C22×D4C2×D7111yescharacteristic
199C2×SD16C2×D7111yescharacteristic
200C2×SD16C2×D7111yescharacteristic
201C2×D8—711no—
202C2×D8—711no—
203C4⋊Q8—711no—
204#126—1421no—
205#124—711no—
206#124—1421no—
207#123—1421no—
208C22⋊Q8—1421no—
209C22⋊Q8C2×D7111yescharacteristic
210#121—711no—
211C4×Q8—711no—
212C4×D4—711no—
213C4×D4—711no—
214C4×D4—711no—
215C4×D4—711no—
216C4×D4—711no—
217C4×D4—1421no—
218C4×D4—1421no—
219#117—711no—
220#117—711no—
221C2×C22⋊C4—1421no—
222C2×C22⋊C4—1421no—
223#107—711no—
224#107—711no—
225#106—711no—
226#106—711no—
227C4⋊C8—711no—
228C4⋊C8—711no—
229#103—711no—
230#103—711no—
231#102—711no—
232#102—711no—
233#102—711no—
234#102—711no—
235#102C2×D7111yescharacteristic
236#102C2×D7111yescharacteristic
237C22⋊C8—711no—
238C22⋊C8C2×D7111yescharacteristic
239#97—711no—
240C4×C8—711no—
241C24—21no—
242C2×Q8—1421no—
243C2×Q8C22×D7111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—711no—
248C2×D4—1421no—
249C2×D4C22×D7111yescharacteristic
250C2×D4C22×D7111yescharacteristic
251C22×C4—1421no—
252C22×C4—1421no—
253C22×C4—1421no—
254C22×C4—1421no—
255C22×C4—1421no—
256C22×C4C22×D7111yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—1422no—
260D8—1422no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8C22×D7111yescharacteristic
264C2×C8C22×D7111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—711no—
267C4⋊C4—711no—
268C4⋊C4—711no—
269C4⋊C4—711no—
270C4⋊C4—711no—
271C4⋊C4—1421no—
272C4⋊C4—1421no—
273C4⋊C4C22×D7111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—711no—
276C22⋊C4—711no—
277C22⋊C4—1421no—
278C22⋊C4—1421no—
279C22⋊C4—1421no—
280C22⋊C4—1421no—
281C22⋊C4—1421no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4—1421no—
285C22⋊C4—1421no—
286C42—711no—
287C42—711no—
288C42—1421no—
289C42—1421no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1421no—
296C23D4×D7111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—5681no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—5681no—
304D4—1421no—
305D4—1421no—
306D4#1176586412yes—
307D4#1176586412yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—5681no—
313C2×C4—5681no—
314C2×C4—5681no—
315C2×C4—5681no—
316C2×C4—5681no—
317C2×C4—711no—
318C2×C4—711no—
319C2×C4—711no—
320C2×C4—711no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4—1421no—
324C2×C4D4×D7111yescharacteristic
325C2×C4C23×D7111yescharacteristic
326C8—21no—
327C8—21no—
328C8—1421no—
329C8—1421no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—2841no—
341C22—2841no—
342C22—2841no—
343C22C2×D4×D7111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—2841no—
348C4—2841no—
349C4—2841no—
350C4—1421no—
351C4—1421no—
352C4—1421no—
353C4—1421no—
354C4C2×#1176586411yescharacteristic
355C4C2×#1176586411yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—5681no—
360C2#1798650511yescharacteristic
361C2#1798680311yescharacteristic
362C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3118
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986622C2semidirect
2. #17986474C2semidirect
3. #17986337C2semidirect
4. #17986324C2semidirect
5. #17986287C2semidirect
6. #17986277C2semidirect
7. #17986216C2semidirect
8. #17986216C2semidirect
10. #17986108C2semidirect
11. #17986100C2semidirect
12. SD16×Dic71C2semidirect
13. #17986079C2semidirect
14. #17986057C2semidirect
15. #450×C71C2semidirect
18. #14556127C22semidirect
20. #14556117C22semidirect
25. Q8×Dic71C22semidirect
32. #14556090C22semidirect
37. #14556062C22semidirect
38. #14556061C22semidirect
39. #14556060C22semidirect
40. #14556059C22semidirect
41. Dic71⋊C8C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556052C22semidirect
44. #14556052C22semidirect
45. #14556050C22semidirect
46. #14556050C22semidirect
50. #14556038C22semidirect
51. C8×Dic71C22semidirect
182. C71#3118semidirect
191. #450D71semidirect