Finite Group Databasebeta

← #22594913

Subgroups — #22594913

Order 9088.

456 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
1C2×#14556123C211yesmaximal, characteristic
2C2×#14556108C211yesmaximal, characteristic
3#17985931C211yesmaximal, characteristic, complemented
4#17985931C211yesmaximal, characteristic, complemented
5#17985928C211yesmaximal, characteristic
6#17985926C211yesmaximal, characteristic
7#17985920C211yesmaximal, characteristic, complemented
8#17985903C211yesmaximal, characteristic, complemented
9#17985894C211yesmaximal, characteristic, complemented
10#17985888C211yesmaximal, characteristic
11#17985886C211yesmaximal, characteristic, complemented
12#17985873C211yesmaximal, characteristic
13#17985864C211yesmaximal, characteristic, complemented
14#17985861C211yesmaximal, characteristic
15#400×C71C211yesmaximal, characteristic, complemented
16C22×Dic142C2211yescharacteristic
17C22×C4×D71C2211yescharacteristic
18#14556123—24no—
19#14556108—24no—
20C2×C4⋊Dic71C2211yescharacteristic
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic
23C2×C4⋊Dic71C2211yescharacteristic
24C2×C4⋊Dic71C2211yescharacteristic
25C2×Dic71⋊C4C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×(C2×D71)⋊C4C2211yescharacteristic
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2211yescharacteristic
32C2×(C2×D71)⋊C4C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×C4×Dic71C2211yescharacteristic
36C2×C4×Dic71C2211yescharacteristic
37C71⋊#95C2211yescharacteristic
38C71⋊#95C2211yescharacteristic
39C71⋊#95C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C2×C4⋊C4×C71C2211yescharacteristic
47C2×C4⋊C4×C71C2211yescharacteristic
48#95×C71C2211yescharacteristic
49#95×C71C2211yescharacteristic
50#95×C71C2211yescharacteristic
51#95×C71C2211yescharacteristic
52#95×C71C2211yescharacteristic
53C23×D71C2311yescharacteristic
54C2×Dic142—22no—
55C2×Dic142—22no—
56C2×Dic142—41no—
57C22×Dic71C2311yescharacteristic
58C22×Dic71C2311yescharacteristic
59C22×Dic71C2311yescharacteristic
60C22×Dic71C2311yescharacteristic
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C2×C4×D71—22no—
65C2×C4×D71—22no—
66C2×C4×D71—41no—
67C4⋊Dic71—22no—
68C4⋊Dic71—22no—
69C4⋊Dic71—22no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72Dic71⋊C4—22no—
73Dic71⋊C4—22no—
74Dic71⋊C4—22no—
75Dic71⋊C4—41no—
76(C2×D71)⋊C4—22no—
77(C2×D71)⋊C4—22no—
78(C2×D71)⋊C4—22no—
79(C2×D71)⋊C4—22no—
80(C2×D71)⋊C4—41no—
81(C2×D71)⋊C4D414yes—
82C4×Dic71—22no—
83C4×Dic71—22no—
84C22×C4×C71C2311yescharacteristic
85C22×C4×C71C2311yescharacteristic
86C22×C4×C71C2311yescharacteristic
87C22×C4×C71C2311yescharacteristic
88C22×C4×C71C2311yescharacteristic
89C22×C4×C71C2311yescharacteristic
90C22×C4×C71C2311yescharacteristic
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C23×C71C2411yescharacteristic
94C22×D71—21no—
95C22×D71—21no—
96C22×D71—21no—
97C22×D71—21no—
98C22×D71—21no—
99C22×D71—21no—
100C22×D71C2×D412yes—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71C2×D412yes—
121C2×C4×C71#4112yes—
122C2×Dic71—21no—
123C2×Dic71—21no—
124C2×Dic71—21no—
125C2×Dic71—21no—
126C2×Dic71—21no—
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—21no—
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71C2×D412yes—
142C2×Dic71#4112yes—
143C4×D71—44no—
144Dic142—44no—
145C22×C71C22×D411yescharacteristic
146C22×C71C2×#4111yescharacteristic
147C22×C71C2×#4111yescharacteristic
148C22×C71H(5,𝔽2)11yescharacteristic
149C22×C712−1+411yescharacteristic
150C22×C712−1+411yescharacteristic
151C22×C712−1+411yescharacteristic
152C2×D71—22no—
153C2×D71—22no—
154C2×D71—22no—
155C2×D71—41no—
156C2×D71—41no—
157C2×D71—41no—
158C2×D71—41no—
159C4×C71—22no—
160C4×C71—22no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—41no—
166Dic71—22no—
167Dic71—22no—
168Dic71—41no—
169Dic71—41no—
170Dic71—41no—
171Dic71—41no—
172Dic71—41no—
173C2×C71#53611yescharacteristic
174C2×C71#54111yescharacteristic
175C2×C71#54211yescharacteristic
176C2×C71#54611yescharacteristic
177C2×C71#54711yescharacteristic
178C2×C71#55211yescharacteristic
179C2×C71#56311yescharacteristic
180D71—42no—
181#2698—711nomaximal, Sylow
182C71#269811yescharacteristic, complemented, Sylow
183C2×#126—711no—
184C2×C22⋊Q8—711no—
185#400D7111yescharacteristic, complemented
186#398—711no—
187#398—711no—
188#397—711no—
189#397—711no—
190#395—711no—
191#391—711no—
192#389—711no—
193#388—711no—
194#387—711no—
195#385—711no—
196#382—711no—
197#381—711no—
198C22×Q8—711no—
199C23×C4—711no—
200#126—1424no—
201C22⋊Q8—1424no—
202C2×C4⋊C4—711no—
203C2×C4⋊C4—711no—
204C2×C4⋊C4—711no—
205C2×C4⋊C4—711no—
206C2×C4⋊C4—711no—
207C2×C4⋊C4—711no—
208C2×C4⋊C4—711no—
209C2×C4⋊C4—711no—
210C2×C4⋊C4—711no—
211C2×C4⋊C4C2×D7111yescharacteristic
212C2×C4⋊C4C2×D7111yescharacteristic
213C2×C22⋊C4—711no—
214C2×C22⋊C4—711no—
215C2×C22⋊C4—711no—
216C2×C22⋊C4—711no—
217C2×C22⋊C4—711no—
218C2×C22⋊C4—711no—
219C2×C42—711no—
220C2×C42—711no—
221#95—711no—
222#95—711no—
223#95—711no—
224#95—711no—
225#95—711no—
226#95—711no—
227#95—711no—
228#95—711no—
229#95—711no—
230#95C2×D7111yescharacteristic
231#95C2×D7111yescharacteristic
232#95C2×D7111yescharacteristic
233#95C2×D7111yescharacteristic
234#95C2×D7111yescharacteristic
235C24—711no—
236C2×Q8—2841no—
237C2×Q8—1422no—
238C2×Q8—1422no—
239C22×C4—2841no—
240C22×C4—711no—
241C22×C4—711no—
242C22×C4—711no—
243C22×C4—711no—
244C22×C4—711no—
245C22×C4—711no—
246C22×C4—711no—
247C22×C4—1422no—
248C22×C4—1422no—
249C22×C4C22×D7111yescharacteristic
250C22×C4C22×D7111yescharacteristic
251C22×C4C22×D7111yescharacteristic
252C22×C4C22×D7111yescharacteristic
253C22×C4C22×D7111yescharacteristic
254C22×C4C22×D7111yescharacteristic
255C22×C4C22×D7111yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—2841no—
259C4⋊C4—2841no—
260C4⋊C4—2841no—
261C4⋊C4—1422no—
262C4⋊C4—1422no—
263C4⋊C4—1422no—
264C4⋊C4—1422no—
265C4⋊C4—1422no—
266C4⋊C4—1422no—
267C22⋊C4—2841no—
268C22⋊C4—714no—
269C22⋊C4—1422no—
270C22⋊C4—1422no—
271C22⋊C4—1422no—
272C22⋊C4—1422no—
273C42—1422no—
274C42—1422no—
275C23—712no—
276C23—1421no—
277C23—1421no—
278C23—1421no—
279C23—1421no—
280C23—1421no—
281C23—1421no—
282C23C23×D7111yescharacteristic
283Q8—2844no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2844no—
304C2×C4—712no—
305C2×C4—712no—
306C2×C4—1421no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1421no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4—1421no—
313C2×C4—1421no—
314C2×C4—1421no—
315C2×C4—1421no—
316C2×C4—1421no—
317C2×C4—1421no—
318C2×C4—1421no—
319C2×C4—1421no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4—1421no—
324C2×C4—1421no—
325C2×C4D4×D7112yes—
326C2×C4#1176586612yes—
327C22—2841no—
328C22—2841no—
329C22—2841no—
330C22—2841no—
331C22—1422no—
332C22—1422no—
333C22—1422no—
334C22C2×D4×D7111yescharacteristic
335C22#41×D7111yescharacteristic
336C22C2×#1176586611yescharacteristic
337C22#1455617411yescharacteristic
338C22#1455617511yescharacteristic
339C22#1455617511yescharacteristic
340C22#1455617511yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2841no—
349C4—2841no—
350C4—2841no—
351C4—2841no—
352C4—2841no—
353C4—1422no—
354C4—1422no—
355C2—2842no—
356C2#1798656911yescharacteristic
357C2#1798661211yescharacteristic
358C2#1798661511yescharacteristic
359C2#1798664011yescharacteristic
360C2#1798664711yescharacteristic
361C2#1798668711yescharacteristic
362C2#1798675111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2698
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17985931C2semidirect
4. #17985931C2semidirect
7. #17985920C2semidirect
8. #17985903C2semidirect
9. #17985894C2semidirect
11. #17985886C2semidirect
13. #17985864C2semidirect
15. #400×C71C2semidirect
182. C71#2698semidirect
185. #400D71semidirect