Finite Group Databasebeta

← #22602336

Subgroups — #22602336

Order 9088.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986722C213yesmaximal, complemented
2#17986710C213yesmaximal, complemented
3#17986702C213yesmaximal, complemented
4#17986687C213yesmaximal, complemented
5Q8×D284C211yesmaximal, characteristic, complemented
6#17986661C211yesmaximal, characteristic, complemented
7#17986661C211yesmaximal, characteristic, complemented
8#17986623C213yesmaximal, complemented
9#17986612C213yesmaximal, complemented
10#17986536C213yesmaximal, complemented
11#17986534C213yesmaximal, complemented
12#17986481C211yesmaximal, characteristic, complemented
13#17986475C211yesmaximal, characteristic, complemented
14C4×Q8×D71C211yesmaximal, characteristic, complemented
15#557×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×Q8×D71C2211yescharacteristic
18C2×Q8×D71C2211yescharacteristic
19#14556128C2213yes—
20#14556123C2216yes—
21#14556123C2216yes—
22#14556123C2213yes—
23#14556123C2213yes—
24#14556119C2213yes—
25#14556119C2216yes—
26#14556117C2213yes—
27#14556116C2213yes—
28#14556116C2213yes—
29#14556113C2216yes—
30#14556113C2213yes—
31#14556113C2211yescharacteristic
32#14556108C2213yes—
33#14556108C2213yes—
34#14556108C2213yes—
35#14556108C2213yes—
36#14556104C2216yes—
37#14556104C2213yes—
38#14556103C2213yes—
39#14556103C2213yes—
40#14556103C2213yes—
41Q8×Dic71C2211yescharacteristic
42C4×Dic142C2213yes—
43C4×Dic142C2213yes—
44#14556093C2213yes—
45#14556093C2213yes—
46C4×D284C2213yes—
47C4×D284C2213yes—
48C4×D284C2213yes—
49C4×D284C2213yes—
50#14556090C2213yes—
51#14556090C2216yes—
52#14556086C2213yes—
53#14556086C2216yes—
54#14556085C2213yes—
55#14556085C2213yes—
56C4⋊C4×D71C2213yes—
57C4⋊C4×D71C2213yes—
58C42×D71C2213yes—
59C4⋊Q8×C71C2213yes—
60#125×C71C2213yes—
61#125×C71C2213yes—
62C4×Q8×C71C2211yescharacteristic
63C4×Q8×C71C2211yescharacteristic
64C4×Q8×C71C2211yescharacteristic
65C4×Q8×C71C2213yes—
66#11765866—24no—
67C2×Dic142C2313yes—
68C2×Dic142C2313yes—
69Q8×D71—24no—
70Q8×D71—24no—
71C2×D284C2313yes—
72C2×D284C2313yes—
73C2×D284C2311yescharacteristic
74C2×D284C2313yes—
75C2×C4×D71C2313yes—
76C2×C4×D71C2313yes—
77C2×C4×D71C2313yes—
78C2×C4×D71C2311yescharacteristic
79C2×C4×D71C2313yes—
80C2×C4×D71C2316yes—
81C4⋊Dic71C2313yes—
82C4⋊Dic71C2316yes—
83C4⋊Dic71C2313yes—
84C4⋊Dic71C2313yes—
85C4⋊Dic71C2313yes—
86Dic71⋊C4C2313yes—
87Dic71⋊C4C2316yes—
88Dic71⋊C4C2311yescharacteristic
89Dic71⋊C4C2313yes—
90Dic71⋊C4C2313yes—
91(C2×D71)⋊C4C2311yescharacteristic
92(C2×D71)⋊C4C2313yes—
93(C2×D71)⋊C4C2313yes—
94(C2×D71)⋊C4C2313yes—
95(C2×D71)⋊C4C2313yes—
96(C2×D71)⋊C4C2316yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2316yes—
99(C2×D71)⋊C4C2311yescharacteristic
100(C2×D71)⋊C4C2311yescharacteristic
101(C2×D71)⋊C4C2313yes—
102(C2×D71)⋊C4C2313yes—
103C4×Dic71C2313yes—
104C4×Dic71C2313yes—
105C4×Dic71C2313yes—
106C4×Dic71C2316yes—
107C2×Q8×C71C2313yes—
108C2×Q8×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C4⋊C4×C71C2313yes—
111C4⋊C4×C71C2313yes—
112C4⋊C4×C71C2313yes—
113C4⋊C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C4⋊C4×C71C2316yes—
116C42×C71C2313yes—
117C42×C71C2313yes—
118C42×C71C2313yes—
119C22×D71C2411yescharacteristic
120C22×D71C2413yes—
121C22×D71C2411yescharacteristic
122Q8×C71—43no—
123Q8×C71#4114yes—
124C2×C4×C71C2411yescharacteristic
125C2×C4×C71C2411yescharacteristic
126C2×C4×C71C2411yescharacteristic
127C2×C4×C71C2413yes—
128C2×C4×C71C2413yes—
129C2×C4×C71C2413yes—
130C2×C4×C71C2413yes—
131C2×Dic71C2411yescharacteristic
132C2×Dic71C2413yes—
133C2×Dic71C2413yes—
134C2×Dic71C2413yes—
135C2×Dic71C2411yescharacteristic
136D284—26no—
137D284—22no—
138D284—26no—
139D284—43no—
140C4×D71—22no—
141C4×D71—26no—
142C4×D71—26no—
143C4×D71—26no—
144C4×D71—46no—
145C4×D71—43no—
146Dic142—26no—
147Dic142—26no—
148C22×C71C2511yescharacteristic
149C2×D71—21no—
150C2×D71—21no—
151C2×D71—21no—
152C2×D71—23no—
153C2×D71—21no—
154C2×D71—23no—
155C2×D71—23no—
156C2×D71—21no—
157C2×D712−1+412yes—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—23no—
161C4×C71—23no—
162C4×C71—23no—
163C4×C71—21no—
164C4×C71C2×#4116yes—
165Dic71—23no—
166Dic71—23no—
167Dic71—21no—
168Dic71—21no—
169Dic71—23no—
170C2×C71C22×#4111yescharacteristic
171C2×C71C2×2−1+411yescharacteristic
172C2×C71#58511yescharacteristic
173D71—22no—
174D71—41no—
175D71—43no—
176#3516—711nomaximal, Sylow
177C71#351611yescharacteristic, complemented, Sylow
178#557D7111yescharacteristic, complemented
179#556—713no—
180#554—713no—
181#553—713no—
182#552—713no—
183D4×Q8—711no—
184#548—711no—
185#548—711no—
186#542—713no—
187#541—713no—
188#529—713no—
189#529—713no—
190#520—711no—
191#519—711no—
192C2×C4×Q8—711no—
193C2×#41—711no—
194C22×Q8—711no—
195C22×Q8—711no—
196C4⋊Q8—713no—
197C4⋊Q8C2×D7113yes—
198#126—716no—
199#126—713no—
200#126—713no—
201#126—716no—
202#125—713no—
203#125—716no—
204#125C2×D7113yes—
205#125C2×D7113yes—
206#124—713no—
207#124—713no—
208#124—713no—
209#123—711no—
210#123—713no—
211#123—716no—
212C22⋊Q8—713no—
213C22⋊Q8—716no—
214C22⋊Q8—713no—
215C22⋊Q8—713no—
216C22⋊Q8—713no—
217C22⋊Q8—713no—
218#121—713no—
219#121—713no—
220#121—713no—
221C4×Q8—711no—
222C4×Q8—713no—
223C4×Q8—713no—
224C4×Q8—713no—
225C4×Q8—713no—
226C4×Q8C2×D7111yescharacteristic
227C4×Q8C2×D7111yescharacteristic
228C4×Q8C2×D7111yescharacteristic
229C4×Q8C2×D7113yes—
230C4×D4—713no—
231C4×D4—716no—
232C4×D4—713no—
233C4×D4—713no—
234C4×D4—713no—
235C4×D4—713no—
236#117—713no—
237#117—716no—
238#117—713no—
239#117—713no—
240C2×C4⋊C4—713no—
241C2×C4⋊C4—713no—
242C2×C42—713no—
243#41—1424no—
244C2×Q8—713no—
245C2×Q8—713no—
246C2×Q8—1424no—
247C2×Q8—1424no—
248C2×Q8C22×D7113yes—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—713no—
251C2×D4—713no—
252C2×D4—711no—
253C2×D4—713no—
254C22×C4—713no—
255C22×C4—716no—
256C22×C4—711no—
257C22×C4—713no—
258C22×C4—713no—
259C22×C4—713no—
260C4⋊C4—713no—
261C4⋊C4—716no—
262C4⋊C4—711no—
263C4⋊C4—713no—
264C4⋊C4—713no—
265C4⋊C4—713no—
266C4⋊C4—716no—
267C4⋊C4—713no—
268C4⋊C4—713no—
269C4⋊C4—713no—
270C4⋊C4C22×D7113yes—
271C4⋊C4C22×D7113yes—
272C4⋊C4C22×D7113yes—
273C4⋊C4C22×D7113yes—
274C4⋊C4C22×D7116yes—
275C4⋊C4C22×D7111yescharacteristic
276C4⋊C4C22×D7113yes—
277C22⋊C4—713no—
278C22⋊C4—711no—
279C22⋊C4—713no—
280C22⋊C4—713no—
281C22⋊C4—713no—
282C22⋊C4—716no—
283C22⋊C4—713no—
284C22⋊C4—711no—
285C22⋊C4—711no—
286C22⋊C4—716no—
287C22⋊C4—713no—
288C22⋊C4—713no—
289C42—713no—
290C42—713no—
291C42—713no—
292C42—716no—
293C42C22×D7113yes—
294C42C22×D7113yes—
295C42C22×D7113yes—
296C23—711no—
297C23—711no—
298C23—713no—
299Q8—43no—
300Q8—1426no—
301Q8—1426no—
302Q8#1176586614yes—
303D4—2843no—
304D4—1426no—
305D4—1422no—
306D4—1426no—
307C2×C4—2846no—
308C2×C4—2843no—
309C2×C4—711no—
310C2×C4—713no—
311C2×C4—713no—
312C2×C4—713no—
313C2×C4—711no—
314C2×C4—1422no—
315C2×C4—1426no—
316C2×C4—1426no—
317C2×C4—1426no—
318C2×C4C23×D7111yescharacteristic
319C2×C4C23×D7113yes—
320C2×C4C23×D7113yes—
321C2×C4C23×D7111yescharacteristic
322C2×C4C23×D7111yescharacteristic
323C2×C4C23×D7113yes—
324C2×C4C23×D7113yes—
325C22—712no—
326C22—1421no—
327C22—1423no—
328C22—1421no—
329C22—1423no—
330C22—1423no—
331C22—1421no—
332C22—1421no—
333C22—1421no—
334C22C24×D7111yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—1421no—
342C4—1423no—
343C4—1423no—
344C4—1421no—
345C4—1423no—
346C4C2×#1176586616yes—
347C2—2841no—
348C2—2843no—
349C2—1422no—
350C2C22×#1176586611yescharacteristic
351C22−1+4×D7111yescharacteristic
352C2#1798683511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#557×C71C2
hypercenter#557D71
hypocenterC71#3516
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #17986722C2semidirect
2. #17986710C2semidirect
3. #17986702C2semidirect
4. #17986687C2semidirect
5. Q8×D284C2semidirect
6. #17986661C2semidirect
7. #17986661C2semidirect
8. #17986623C2semidirect
9. #17986612C2semidirect
10. #17986536C2semidirect
11. #17986534C2semidirect
12. #17986481C2semidirect
13. #17986475C2semidirect
14. C4×Q8×D71C2semidirect
15. #557×C71C2semidirect
177. C71#3516semidirect
178. #557D71semidirect