Finite Group Databasebeta

← #22598617

Subgroups — #22598617

Order 9088.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986620C211yesmaximal, characteristic, complemented
2#17986329C211yesmaximal, characteristic, complemented
3#17986317C211yesmaximal, characteristic, complemented
4#17986289C211yesmaximal, characteristic, complemented
5#17986284C211yesmaximal, characteristic, complemented
6#17986221C211yesmaximal, characteristic, complemented
7#17986214C211yesmaximal, characteristic, complemented
8#17986161C211yesmaximal, characteristic, complemented
9#17986144C211yesmaximal, characteristic
10#17986109C211yesmaximal, characteristic, complemented
11#17986104C211yesmaximal, characteristic, complemented
12C4×#11765852C211yesmaximal, characteristic, complemented
13C4×#11765853C211yesmaximal, characteristic, complemented
14#17986060C211yesmaximal, characteristic, complemented
15#520×C71C211yesmaximal, characteristic, complemented
16C2×#11765855C2211yescharacteristic
17C2×#11765853C2211yescharacteristic
18C2×#11765852C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20#14556128C2211yescharacteristic
21#14556123—21no—
22#14556116C2211yescharacteristic
23#14556114—21no—
24#14556110—21no—
25#14556107C2211yescharacteristic
26#14556106—21no—
27#14556102C2211yescharacteristic
28C4×Dic142C2211yescharacteristic
29C4×D284C2211yescharacteristic
30#14556059C2211yescharacteristic
31#14556059C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35C4⋊(C71⋊C8)C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44C22⋊(C71⋊C8)C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46#14556039C2211yescharacteristic
47C4×C71⋊C8C2211yescharacteristic
48C2×#41×C71C2211yescharacteristic
49C4×Q8×C71C2211yescharacteristic
50C4×Q8×C71C2211yescharacteristic
51C4×D4×C71—21no—
52C4×D4×C71—21no—
53C4×D4×C71C2211yescharacteristic
54C4×D4×C71C2211yescharacteristic
55#117×C71—21no—
56#117×C71C2211yescharacteristic
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71C2211yescharacteristic
59C2×Dic142—21no—
60C2×Dic142C2311yescharacteristic
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C4×D71—21no—
64#11765855—22no—
65#11765853—22no—
66#11765852—22no—
67#11765850—22no—
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C4⋊Dic71—21no—
73C4⋊Dic71C2311yescharacteristic
74C4⋊Dic71C2311yescharacteristic
75Dic71⋊C4—21no—
76Dic71⋊C4—21no—
77C22⋊Dic71—21no—
78(C2×D71)⋊C4—21no—
79(C2×D71)⋊C4—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82C4×Dic71—21no—
83#41×C71—22no—
84#41×C71—22no—
85C2×Q8×C71C2311yescharacteristic
86C2×D4×C71—21no—
87C2×D4×C71C2311yescharacteristic
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71C2311yescharacteristic
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71D411yescharacteristic
97C4⋊C4×C71D411yescharacteristic
98C4⋊C4×C71C2311yescharacteristic
99C4⋊C4×C71C2311yescharacteristic
100C22⋊C4×C71—21no—
101C22⋊C4×C71—21no—
102C22⋊C4×C71D411yescharacteristic
103C22⋊C4×C71D411yescharacteristic
104C42×C71—21no—
105C42×C71—21no—
106C42×C71C2311yescharacteristic
107C42×C71C2311yescharacteristic
108C23×C71—21no—
109C23×C71C2×D411yescharacteristic
110C22×D71—21no—
111Q8×C71—21no—
112Q8×C71#4112yes—
113D4×C71—24no—
114D4×C71—21no—
115D4×C71#4112yes—
116C2×C4×C71—24no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—81no—
123C2×C4×C71—81no—
124C2×C4×C71—81no—
125C2×C4×C71C2×D411yescharacteristic
126C2×C4×C71C2×D411yescharacteristic
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—81no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135D284—21no—
136D284—21no—
137C4×D71—81no—
138Dic142—21no—
139Dic142—21no—
140Dic142—81no—
141C71⋊C8—21no—
142C71⋊C8—21no—
143C71⋊C8—21no—
144C71⋊C8—21no—
145C22×C71—21no—
146C22×C71—22no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71C22×D411yescharacteristic
152C2×D71—41no—
153C2×D71—41no—
154C2×D71—41no—
155C4×C71—21no—
156C4×C71—22no—
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71C2×#4111yescharacteristic
164C4×C71C2×#4111yescharacteristic
165Dic71—41no—
166Dic71—41no—
167Dic71—41no—
168C2×C71—41no—
169C2×C71—42no—
170C2×C71#52511yescharacteristic
171C2×C71#57711yescharacteristic
172C2×C71#57711yescharacteristic
173D71—81no—
174#3125—711nomaximal, Sylow
175C71#312511yescharacteristic, complemented, Sylow
176#542—711no—
177#520D7111yescharacteristic, complemented
178#483—711no—
179#481—711no—
180#476—711no—
181#475—711no—
182#464—711no—
183#463—711no—
184#452—711no—
185#449—711no—
186#442—711no—
187#441—711no—
188C4×SD16—711no—
189C4×SD16—711no—
190#433—711no—
191C2×#41C2×D7111yescharacteristic
192C2×Q16—711no—
193C2×SD16—711no—
194C2×SD16—711no—
195C2×D8—711no—
196C4⋊Q8—711no—
197#126—1421no—
198#124—711no—
199#124—1421no—
200#123—1421no—
201C22⋊Q8—711no—
202C22⋊Q8—1421no—
203#121—711no—
204C4×Q8—711no—
205C4×Q8C2×D7111yescharacteristic
206C4×Q8C2×D7111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—711no—
210C4×D4C2×D7111yescharacteristic
211C4×D4C2×D7111yescharacteristic
212#117—21no—
213#117C2×D7111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D7111yescharacteristic
216#106—711no—
217#106—711no—
218#106—711no—
219#106—711no—
220C4⋊C8—711no—
221C4⋊C8—711no—
222#103—711no—
223#103—711no—
224#103—711no—
225#103—711no—
226#102—711no—
227#102—711no—
228#102—711no—
229#102—711no—
230C22⋊C8—711no—
231C22⋊C8—711no—
232#97—711no—
233C4×C8—711no—
234#41—22no—
235#41—22no—
236C2×Q8—711no—
237C2×Q8—1421no—
238C2×Q8C22×D7111yescharacteristic
239C2×D4—21no—
240C2×D4—711no—
241C2×D4—1421no—
242C2×D4C22×D7111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1421no—
248C22×C4C22×D7111yescharacteristic
249Q16—1422no—
250SD16—1422no—
251SD16—1422no—
252D8—1422no—
253C2×C8—711no—
254C2×C8—711no—
255C2×C8—711no—
256C2×C8—711no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—711no—
261C4⋊C4—711no—
262C4⋊C4—1421no—
263C4⋊C4—1421no—
264C4⋊C4—1421no—
265C4⋊C4C22⋊D7111yescharacteristic
266C4⋊C4C22⋊D7111yescharacteristic
267C4⋊C4C22×D7111yescharacteristic
268C4⋊C4C22×D7111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1421no—
272C22⋊C4—1421no—
273C22⋊C4—1421no—
274C22⋊C4—1421no—
275C22⋊C4—1421no—
276C22⋊C4C22⋊D7111yescharacteristic
277C22⋊C4C22⋊D7111yescharacteristic
278C42—21no—
279C42—21no—
280C42—1421no—
281C42C22×D7111yescharacteristic
282C42C22×D7111yescharacteristic
283C23—21no—
284C23—1421no—
285C23C2×C22⋊D7111yescharacteristic
286Q8—21no—
287Q8—5681no—
288Q8—1421no—
289Q8—1421no—
290Q8#1176586512yes—
291D4—24no—
292D4—21no—
293D4—5681no—
294D4—1421no—
295D4—1421no—
296D4#1176586512yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—5681no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1421no—
310C2×C4C2×C22⋊D7111yescharacteristic
311C2×C4C2×C22⋊D7111yescharacteristic
312C2×C4C2×C22⋊D7111yescharacteristic
313C2×C4C2×C22⋊D7111yescharacteristic
314C2×C4C2×C22⋊D7111yescharacteristic
315C2×C4C23×D7111yescharacteristic
316C8—1421no—
317C8—1421no—
318C8—1421no—
319C8—1421no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—2841no—
327C22—2841no—
328C22—2841no—
329C22C22×C22⋊D7111yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—2841no—
339C4—2841no—
340C4—2841no—
341C4C2×#1176586511yescharacteristic
342C4C2×#1176586511yescharacteristic
343C2—41no—
344C2—42no—
345C2—5681no—
346C2#1798651011yescharacteristic
347C2#1798680711yescharacteristic
348C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#520×C71C2
socleC22×C71C22×D4
hypercenter#520D71
hypocenterC71#3125
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #17986620C2semidirect
2. #17986329C2semidirect
3. #17986317C2semidirect
4. #17986289C2semidirect
5. #17986284C2semidirect
6. #17986221C2semidirect
7. #17986214C2semidirect
8. #17986161C2semidirect
10. #17986109C2semidirect
11. #17986104C2semidirect
12. C4×#11765852C2semidirect
13. C4×#11765853C2semidirect
14. #17986060C2semidirect
15. #520×C71C2semidirect
175. C71#3125semidirect
177. #520D71semidirect