Finite Group Databasebeta

← #22600608

Subgroups — #22600608

Order 9088.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986689C211yesmaximal, characteristic, complemented
2#17986353C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986320C211yesmaximal, characteristic, complemented
5#17986306C211yesmaximal, characteristic, complemented
6#17986255C211yesmaximal, characteristic, complemented
7#17986243C211yesmaximal, characteristic, complemented
8#17986218C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986155C211yesmaximal, characteristic, complemented
11C4×#11765855C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986008C211yesmaximal, characteristic, complemented
14#17986000C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic
16C2×#11765855C2211yescharacteristic
17C2×#11765855C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765849C2211yescharacteristic
20C2×#11765849C2211yescharacteristic
21#14556121—21no—
22#14556120C2211yescharacteristic
23#14556118—21no—
24#14556109—21no—
25#14556107C2211yescharacteristic
26#14556107C2211yescharacteristic
27C4×Dic142C2211yescharacteristic
28#14556087C2211yescharacteristic
29#14556084—21no—
30C2×C4⋊Dic71C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556061C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556052C2211yescharacteristic
39#14556052C2211yescharacteristic
40#14556052C2211yescharacteristic
41#14556051C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556049C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47C4×C71⋊C8C2211yescharacteristic
48C2×#41×C71C2211yescharacteristic
49C22×Q8×C71C2211yescharacteristic
50#124×C71—21no—
51#124×C71C2211yescharacteristic
52C22⋊Q8×C71—21no—
53C22⋊Q8×C71C2211yescharacteristic
54#121×C71—21no—
55#121×C71C2211yescharacteristic
56C4×Q8×C71C2211yescharacteristic
57C4×D4×C71—21no—
58C4×D4×C71C2211yescharacteristic
59C2×Dic142C2311yescharacteristic
60C22×Dic71—21no—
61#11765855—22no—
62#11765855—41no—
63#11765853—41no—
64#11765849—41no—
65#11765849—41no—
66C2×C71⋊C8C2311yescharacteristic
67C2×C71⋊C8C2311yescharacteristic
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C4⋊Dic71—21no—
71C4⋊Dic71C2311yescharacteristic
72C4⋊Dic71C2311yescharacteristic
73C4⋊Dic71C2311yescharacteristic
74Dic71⋊C4—21no—
75Dic71⋊C4—21no—
76Dic71⋊C4—21no—
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80C22⋊Dic71—21no—
81C22⋊Dic71—21no—
82C4×Dic71—21no—
83C4×Dic71—21no—
84#41×C71—22no—
85#41×C71—41no—
86C2×Q8×C71—21no—
87C2×Q8×C71—22no—
88C2×Q8×C71—41no—
89C2×Q8×C71C2311yescharacteristic
90C2×Q8×C71C2311yescharacteristic
91C2×D4×C71—21no—
92C2×D4×C71—21no—
93C2×D4×C71D411yescharacteristic
94C2×D4×C71C2311yescharacteristic
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71C2311yescharacteristic
98C22×C4×C71C2311yescharacteristic
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114Q8×C71—21no—
115Q8×C71—21no—
116Q8×C71—21no—
117Q8×C71—42no—
118Q8×C71#4112yes—
119D4×C71—21no—
120D4×C71—21no—
121D4×C71—41no—
122D4×C71—42no—
123D4×C71—42no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2411yescharacteristic
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—42no—
143Dic142—21no—
144Dic142—21no—
145C71⋊C8—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C71⋊C8—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71C22×D411yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71C2×#4111yescharacteristic
167C4×C71H(5,𝔽2)11yescharacteristic
168Dic71—41no—
169Dic71—41no—
170Dic71—41no—
171Dic71—41no—
172C2×C71—41no—
173C2×C71—41no—
174C2×C71—81no—
175C2×C71#54611yescharacteristic
176C2×C71#57511yescharacteristic
177C2×C71#57811yescharacteristic
178#3327—711nomaximal, Sylow
179C71#332711yescharacteristic, complemented, Sylow
180#552—711no—
181#548D7111yescharacteristic
182#488—711no—
183#484—711no—
184#482—711no—
185#479—711no—
186#470—711no—
187#468—711no—
188#464—711no—
189#452—711no—
190#451—711no—
191C4×Q16—711no—
192#436—711no—
193#419—711no—
194#417—711no—
195C2×#41C2×D7111yescharacteristic
196C22×Q8C2×D7111yescharacteristic
197C2×Q16—711no—
198C2×Q16—711no—
199C2×SD16—711no—
200C2×M16—711no—
201C2×M16—711no—
202#126—1421no—
203#125—711no—
204#125—1421no—
205#124—21no—
206#124C2×D7111yescharacteristic
207#123—1421no—
208C22⋊Q8—21no—
209C22⋊Q8—711no—
210C22⋊Q8—711no—
211C22⋊Q8C2×D7111yescharacteristic
212#121—21no—
213#121C2×D7111yescharacteristic
214C4×Q8—711no—
215C4×Q8C2×D7111yescharacteristic
216C4×D4—21no—
217C4×D4C2×D7111yescharacteristic
218#117—711no—
219#117—1421no—
220C2×C4⋊C4—711no—
221#107—711no—
222#107—711no—
223#106—711no—
224C4⋊C8—711no—
225#103—711no—
226#103—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231#103—711no—
232#102—711no—
233#102—711no—
234#102—711no—
235C22⋊C8—711no—
236C22⋊C8—711no—
237C4×C8—711no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—711no—
244C2×Q8C22×D7111yescharacteristic
245C2×Q8C22×D7111yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D7111yescharacteristic
249C2×D4C22×D7111yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—1421no—
253C22×C4C22×D7111yescharacteristic
254C22×C4C22×D7111yescharacteristic
255Q16—2841no—
256Q16—1422no—
257SD16—2841no—
258M16—2841no—
259M16—2841no—
260C2×C8—711no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8—711no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—711no—
267C4⋊C4—711no—
268C4⋊C4—711no—
269C4⋊C4—1421no—
270C4⋊C4—1421no—
271C4⋊C4—1421no—
272C4⋊C4—1421no—
273C4⋊C4—1421no—
274C4⋊C4—1421no—
275C4⋊C4C22⋊D7111yescharacteristic
276C4⋊C4C22×D7111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1421no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4C22⋊D7111yescharacteristic
285C22⋊C4C22⋊D7111yescharacteristic
286C42—21no—
287C42—1421no—
288C42—1421no—
289C42C22×D7111yescharacteristic
290C23—21no—
291C23C2×C22⋊D7111yescharacteristic
292C23C2×C22⋊D7111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—1421no—
298Q8—1421no—
299Q8#1176586412yes—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—2842no—
315C2×C4—1421no—
316C2×C4—1421no—
317C2×C4—1421no—
318C2×C4—1421no—
319C2×C4C2×C22⋊D7111yescharacteristic
320C2×C4C2×C22⋊D7111yescharacteristic
321C2×C4C2×C22⋊D7111yescharacteristic
322C2×C4C2×C22⋊D7111yescharacteristic
323C2×C4C23×D7111yescharacteristic
324C8—1421no—
325C8—1421no—
326C8—1421no—
327C8—1421no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊D7111yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—2841no—
346C4—2841no—
347C4—2841no—
348C4—2841no—
349C4C2×#1176586411yescharacteristic
350C4#1455617311yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2#1798664311yescharacteristic
355C2#1798679711yescharacteristic
356C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#548×C71C2
socleC22×C71C22×D4
hypercenter#548D71
hypocenterC71#3327
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986689C2semidirect
2. #17986353C2semidirect
3. #17986332C2semidirect
4. #17986320C2semidirect
5. #17986306C2semidirect
6. #17986255C2semidirect
7. #17986243C2semidirect
8. #17986218C2semidirect
9. #17986161C2semidirect
10. #17986155C2semidirect
11. C4×#11765855C2semidirect
12. #17986077C2semidirect
13. #17986008C2semidirect
14. #17986000C2semidirect
179. C71#3327semidirect