Finite Group Databasebeta

← #22602419

Subgroups — #22602419

Order 9088.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986732C213yesmaximal, complemented
2Q8×Dic142C211yesmaximal, characteristic, complemented
3#17986704C213yesmaximal, complemented
4#17986668C211yesmaximal, characteristic, complemented
5#17986661C211yesmaximal, characteristic, complemented
6#17986656C211yesmaximal, characteristic, complemented
7#17986654C213yesmaximal, complemented
8#17986650C213yesmaximal, complemented
9#17986620C213yesmaximal, complemented
10#17986614C213yesmaximal, complemented
11#17986546C213yesmaximal, complemented
12#17986527C213yesmaximal, complemented
13#17986481C211yesmaximal, characteristic, complemented
14C4×Q8×D71C211yesmaximal, characteristic, complemented
15#520×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765865C2216yes—
18C2×Q8×D71C2211yescharacteristic
19#14556128C2213yes—
20#14556127C2213yes—
21#14556127C2213yes—
22#14556124C2213yes—
23#14556123C2213yes—
24#14556121C2213yes—
25#14556118C2213yes—
26#14556116C2213yes—
27#14556115C2213yes—
28#14556114C2216yes—
29#14556114C2213yes—
30#14556113C2213yes—
31#14556111C2216yes—
32#14556110C2216yes—
33#14556110C2213yes—
34#14556108C2213yes—
35#14556107C2216yes—
36#14556107C2213yes—
37#14556106C2216yes—
38#14556106C2211yescharacteristic
39#14556103C2213yes—
40#14556102C2213yes—
41#14556101C2213yes—
42#14556099C2213yes—
43#14556099C2213yes—
44Q8×Dic71C2211yescharacteristic
45Q8×Dic71C2211yescharacteristic
46C4×Dic142C2213yes—
47C4×Dic142C2213yes—
48#14556093C2213yes—
49C4×D284C2213yes—
50C4×D284C2213yes—
51#14556090C2213yes—
52C4×C22⋊D71C2213yes—
53C4×C22⋊D71C2213yes—
54C4×C22⋊D71C2216yes—
55#14556085C2213yes—
56C4⋊C4×D71C2213yes—
57C4⋊C4×D71C2213yes—
58C42×D71C2213yes—
59C2×#41×C71C2211yescharacteristic
60C4×Q8×C71C2211yescharacteristic
61C4×Q8×C71C2211yescharacteristic
62C4×D4×C71C2213yes—
63C4×D4×C71C2213yes—
64#117×C71C2213yes—
65C2×C4⋊C4×C71C2213yes—
66#11765866—24no—
67#11765865—412no—
68C2×Dic142C2313yes—
69C2×Dic142C2311yescharacteristic
70C2×Dic142C2316yes—
71Q8×D71—24no—
72C2×D284C2313yes—
73C2×D284C2313yes—
74C2×C22⋊D71C2313yes—
75C2×C22⋊D71C2313yes—
76C2×C22⋊D71C2316yes—
77C2×C4×D71C2313yes—
78C2×C4×D71C2313yes—
79C2×C4×D71C2313yes—
80C2×C4×D71C2311yescharacteristic
81C2×C4×D71C2316yes—
82C4⋊Dic71C2313yes—
83C4⋊Dic71C2313yes—
84C4⋊Dic71C2313yes—
85C4⋊Dic71C2313yes—
86Dic71⋊C4C2313yes—
87Dic71⋊C4C2316yes—
88Dic71⋊C4C2311yescharacteristic
89Dic71⋊C4C2311yescharacteristic
90Dic71⋊C4C2313yes—
91Dic71⋊C4C2313yes—
92Dic71⋊C4C2313yes—
93C22⋊Dic71C2313yes—
94C22⋊Dic71C2316yes—
95C22⋊Dic71C2313yes—
96(C2×D71)⋊C4C2313yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2311yescharacteristic
99(C2×D71)⋊C4C2313yes—
100(C2×D71)⋊C4C2316yes—
101(C2×D71)⋊C4C2311yescharacteristic
102(C2×D71)⋊C4C2313yes—
103C4×Dic71C2313yes—
104C4×Dic71C2313yes—
105C4×Dic71C2313yes—
106C4×Dic71C2313yes—
107#41×C71—24no—
108C2×Q8×C71C2311yescharacteristic
109C2×D4×C71C2313yes—
110C22×C4×C71C2313yes—
111C22×C4×C71C2316yes—
112C4⋊C4×C71C2313yes—
113C4⋊C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C4⋊C4×C71C2311yescharacteristic
116C22⋊C4×C71C2313yes—
117C22⋊C4×C71C2313yes—
118C42×C71C2313yes—
119C42×C71C2313yes—
120C23×C71C2413yes—
121C22×D71C2413yes—
122C22×D71C2411yescharacteristic
123Q8×C71#4114yescomplemented
124D4×C71—26no—
125C2×C4×C71—26no—
126C2×C4×C71—46no—
127C2×C4×C71C2411yescharacteristic
128C2×C4×C71C2411yescharacteristic
129C2×C4×C71C2413yes—
130C2×C4×C71C2413yes—
131C2×C4×C71C2411yescharacteristic
132C2×C4×C71C2413yes—
133C22⋊D71—26no—
134C22⋊D71—26no—
135C22⋊D71—46no—
136C2×Dic71C2411yescharacteristic
137C2×Dic71C2413yes—
138C2×Dic71C2413yes—
139C2×Dic71C2413yes—
140C2×Dic71C2411yescharacteristic
141C2×Dic71C2411yescharacteristic
142D284—26no—
143D284—26no—
144C4×D71—26no—
145C4×D71—26no—
146C4×D71—26no—
147C4×D71—22no—
148C4×D71—46no—
149Dic142—26no—
150Dic142—22no—
151Dic142—46no—
152C22×C71—23no—
153C22×C71—23no—
154C22×C71—23no—
155C22×C71C2511yescharacteristic
156C2×D71—21no—
157C2×D71—21no—
158C2×D71—23no—
159C2×D71—23no—
160C2×D71—23no—
161C2×D71H(5,𝔽2)12yes—
162C4×C71—21no—
163C4×C71—23no—
164C4×C71—23no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71C2×#4116yes—
168Dic71—23no—
169Dic71—21no—
170Dic71—23no—
171Dic71—23no—
172Dic71—21no—
173Dic712−1+412yes—
174C2×C71—43no—
175C2×C71C22×#4111yescharacteristic
176C2×C71C2×H(5,𝔽2)11yescharacteristic
177C2×C71C2×2−1+411yescharacteristic
178D71—22no—
179D71—43no—
180#3522—711nomaximal, Sylow
181C71#352211yescharacteristic, complemented, Sylow
182#559—713no—
183Q82—711no—
184#554—713no—
185#550—711no—
186#548—711no—
187#548—711no—
188#547—713no—
189#547—713no—
190#542—713no—
191#542—713no—
192#532—713no—
193#529—713no—
194#520—711no—
195#520D7111yescharacteristic, complemented
196C2×C4×Q8—711no—
197C2×#41—711no—
198C2×#41—716no—
199C2×#41C2×D7111yescharacteristic
200C22×Q8—711no—
201C4⋊Q8—713no—
202C4⋊Q8—713no—
203C4⋊Q8—713no—
204#127—713no—
205#126—713no—
206#126—713no—
207#125—713no—
208#124—713no—
209#124—713no—
210#124—716no—
211#124—713no—
212#123—716no—
213#123—713no—
214#123—713no—
215#123—716no—
216C22⋊Q8—716no—
217C22⋊Q8—713no—
218C22⋊Q8—711no—
219C22⋊Q8—716no—
220C22⋊Q8—713no—
221#121—713no—
222#121—713no—
223#121—713no—
224#121—713no—
225#121—713no—
226C4×Q8—711no—
227C4×Q8—711no—
228C4×Q8—713no—
229C4×Q8—713no—
230C4×Q8—713no—
231C4×Q8C2×D7111yescharacteristic
232C4×Q8C2×D7111yescharacteristic
233C4×D4—713no—
234C4×D4—713no—
235C4×D4—716no—
236C4×D4—713no—
237C4×D4—713no—
238C4×D4—713no—
239C4×D4C2×D7113yes—
240C4×D4C2×D7113yes—
241#117—713no—
242#117C2×D7113yes—
243C2×C4⋊C4—713no—
244C2×C4⋊C4—713no—
245C2×C4⋊C4C2×D7113yes—
246C2×C42—713no—
247#41—24no—
248#41—28412no—
249#41—1424no—
250C2×Q8—716no—
251C2×Q8—711no—
252C2×Q8—713no—
253C2×Q8—1424no—
254C2×Q8C22×D7111yescharacteristic
255C2×D4—716no—
256C2×D4—713no—
257C2×D4—713no—
258C2×D4—713no—
259C2×D4—713no—
260C2×D4C22×D7113yes—
261C22×C4—716no—
262C22×C4—711no—
263C22×C4—713no—
264C22×C4—713no—
265C22×C4—713no—
266C22×C4C22×D7113yes—
267C22×C4C22×D7116yes—
268C4⋊C4—713no—
269C4⋊C4—716no—
270C4⋊C4—711no—
271C4⋊C4—711no—
272C4⋊C4—713no—
273C4⋊C4—713no—
274C4⋊C4—713no—
275C4⋊C4—713no—
276C4⋊C4—713no—
277C4⋊C4—713no—
278C4⋊C4—713no—
279C4⋊C4C22×D7113yes—
280C4⋊C4C22×D7113yes—
281C4⋊C4C22×D7113yes—
282C4⋊C4C22×D7111yescharacteristic
283C22⋊C4—713no—
284C22⋊C4—716no—
285C22⋊C4—713no—
286C22⋊C4—713no—
287C22⋊C4—713no—
288C22⋊C4—711no—
289C22⋊C4—711no—
290C22⋊C4—713no—
291C22⋊C4—716no—
292C22⋊C4—713no—
293C22⋊C4C22×D7113yes—
294C22⋊C4C22×D7113yes—
295C42—713no—
296C42—713no—
297C42—713no—
298C42—713no—
299C42C22×D7113yes—
300C42C22×D7113yes—
301C23—711no—
302C23—713no—
303C23C23×D7113yes—
304Q8—2846no—
305Q8—1422no—
306Q8—1426no—
307Q8#1176586514yescomplemented
308D4—26no—
309D4—2846no—
310D4—1426no—
311D4—1426no—
312D4—1426no—
313D4—1426no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—2846no—
317C2×C4—711no—
318C2×C4—711no—
319C2×C4—713no—
320C2×C4—713no—
321C2×C4—713no—
322C2×C4—711no—
323C2×C4—1422no—
324C2×C4—1426no—
325C2×C4—1426no—
326C2×C4—1426no—
327C2×C4C23×D7111yescharacteristic
328C2×C4C23×D7113yes—
329C2×C4C23×D7111yescharacteristic
330C2×C4C23×D7113yes—
331C2×C4C23×D7111yescharacteristic
332C2×C4C23×D7113yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—712no—
337C22—1423no—
338C22—1423no—
339C22—1423no—
340C22—1421no—
341C22—1421no—
342C22C24×D7111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—712no—
349C4—1421no—
350C4—1423no—
351C4—1423no—
352C4—1423no—
353C4—1421no—
354C4C2×#1176586516yes—
355C2—43no—
356C2—2843no—
357C2—1422no—
358C2C22×#1176586511yescharacteristic
359C2H(5,𝔽2)×D7111yescharacteristic
360C22−1+4×D7111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#520×C71C2
hypercenter#520D71
hypocenterC71#3522
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #17986732C2semidirect
2. Q8×Dic142C2semidirect
3. #17986704C2semidirect
4. #17986668C2semidirect
5. #17986661C2semidirect
6. #17986656C2semidirect
7. #17986654C2semidirect
8. #17986650C2semidirect
9. #17986620C2semidirect
10. #17986614C2semidirect
11. #17986546C2semidirect
12. #17986527C2semidirect
13. #17986481C2semidirect
14. C4×Q8×D71C2semidirect
15. #520×C71C2semidirect
123. Q8×C71#41semidirect
181. C71#3522semidirect
195. #520D71semidirect
307. Q8#11765865semidirect