Finite Group Databasebeta

← #22602326

Subgroups — #22602326

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#17986725C211yesmaximal, characteristic, complemented
2#17986718C213yesmaximal, complemented
3#17986707C213yesmaximal, complemented
4#17986696C213yesmaximal, complemented
5#17986689C213yesmaximal, complemented
6Q8×C22⋊D71C211yesmaximal, characteristic, complemented
7#17986656C211yesmaximal, characteristic, complemented
8#17986616C213yesmaximal, complemented
9#17986609C213yesmaximal, complemented
10#17986531C213yesmaximal, complemented
11#17986528C213yesmaximal, complemented
12#17986482C211yesmaximal, characteristic, complemented
13#17986477C211yesmaximal, characteristic, complemented
14C4×Q8×D71C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic, complemented
16C2×Q8×D71C2211yescharacteristic
17#14556127C2213yes—
18#14556126C2213yes—
19#14556121C2216yes—
20#14556121C2216yes—
21#14556121C2213yes—
22#14556121C2213yes—
23#14556120C2213yes—
24#14556119C2213yes—
25#14556119C2213yes—
26#14556118C2216yes—
27#14556115C2213yes—
28#14556115C2213yes—
29#14556110C2213yes—
30#14556110C2211yescharacteristic
31#14556109C2216yes—
32#14556108C2213yes—
33#14556107C2213yes—
34#14556107C2213yes—
35#14556105C2216yes—
36#14556104C2213yes—
37#14556099C2213yes—
38#14556099C2213yes—
39Q8×Dic71C2213yes—
40Q8×Dic71C2211yescharacteristic
41Q8×Dic71C2211yescharacteristic
42Q8×Dic71C2213yes—
43Q8×Dic71C2211yescharacteristic
44C4×Dic142C2213yes—
45C4×Dic142C2213yes—
46#14556093C2213yes—
47D4×Dic71C2213yes—
48D4×Dic71C2213yes—
49C4×C22⋊D71C2213yes—
50C4×C22⋊D71C2213yes—
51(C22⋊D71)⋊C4C2216yes—
52#14556087C2213yes—
53#14556087C2213yes—
54#14556087C2213yes—
55#14556084C2216yes—
56C2×C4⋊Dic71C2213yes—
57C4⋊C4×D71C2213yes—
58C42×D71C2213yes—
59C2×#41×C71C2211yescharacteristic
60C22×Q8×C71C2211yescharacteristic
61#124×C71C2213yes—
62C22⋊Q8×C71C2213yes—
63#121×C71C2213yes—
64C4×Q8×C71C2211yescharacteristic
65C4×D4×C71C2213yes—
66C2×Dic142C2313yes—
67C2×Dic142C2313yes—
68Q8×D71—24no—
69C2×C22⋊D71C2313yes—
70C2×C22⋊D71C2311yescharacteristic
71C22×Dic71C2316yes—
72C2×C4×D71C2313yes—
73C2×C4×D71C2311yescharacteristic
74C2×C4×D71C2313yes—
75C4⋊Dic71C2313yes—
76C4⋊Dic71C2313yes—
77C4⋊Dic71C2313yes—
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2313yes—
80C4⋊Dic71C2313yes—
81C4⋊Dic71C2313yes—
82C4⋊Dic71C2313yes—
83Dic71⋊C4C2316yes—
84Dic71⋊C4C2313yes—
85Dic71⋊C4C2316yes—
86Dic71⋊C4C2316yes—
87Dic71⋊C4C2313yes—
88Dic71⋊C4C2313yes—
89Dic71⋊C4C2311yescharacteristic
90C22⋊Dic71C2316yes—
91C22⋊Dic71C2313yes—
92C22⋊Dic71C2311yescharacteristic
93C22⋊Dic71C2313yes—
94C22⋊Dic71C2313yes—
95C22⋊Dic71C2313yes—
96C22⋊Dic71C2313yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2311yescharacteristic
99C4×Dic71C2313yes—
100C4×Dic71C2313yes—
101C4×Dic71C2313yes—
102C4×Dic71C2316yes—
103C4×Dic71C2313yes—
104C4×Dic71C2313yes—
105#41×C71—24no—
106C2×Q8×C71—24no—
107C2×Q8×C71C2313yes—
108C2×Q8×C71C2311yescharacteristic
109C2×D4×C71C2313yes—
110C2×D4×C71C2313yes—
111C22×C4×C71C2313yes—
112C22×C4×C71C2313yes—
113C4⋊C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C22⋊C4×C71C2311yescharacteristic
116C22⋊C4×C71C2313yes—
117C22⋊C4×C71C2316yes—
118C42×C71C2313yes—
119C23×C71C2413yes—
120C23×C71C2411yescharacteristic
121C22×D71C2411yescharacteristic
122Q8×C71—26no—
123Q8×C71#4114yes—
124D4×C71—26no—
125D4×C71—43no—
126C2×C4×C71—26no—
127C2×C4×C71—26no—
128C2×C4×C71C2411yescharacteristic
129C2×C4×C71C2413yes—
130C2×C4×C71C2413yes—
131C2×C4×C71C2411yescharacteristic
132C2×C4×C71C2413yes—
133C22⋊D71—26no—
134C22⋊D71—22no—
135C2×Dic71—46no—
136C2×Dic71C2411yescharacteristic
137C2×Dic71C2413yes—
138C2×Dic71C2411yescharacteristic
139C2×Dic71C2413yes—
140C2×Dic71C2411yescharacteristic
141C2×Dic71C2413yes—
142C2×Dic71C2413yes—
143C4×D71—22no—
144C4×D71—26no—
145C4×D71—43no—
146Dic142—26no—
147Dic142—43no—
148C22×C71—21no—
149C22×C71—23no—
150C22×C71—23no—
151C22×C71—21no—
152C22×C71—23no—
153C22×C71—21no—
154C22×C71C2511yescharacteristic
155C2×D71—21no—
156C2×D71—21no—
157C2×D712−1+412yes—
158C4×C71—21no—
159C4×C71—23no—
160C4×C71—21no—
161C4×C71—23no—
162C4×C71C2×#4116yes—
163Dic71—23no—
164Dic71—21no—
165Dic71—23no—
166Dic71—21no—
167Dic71—21no—
168Dic71—23no—
169Dic71—23no—
170C2×C71—41no—
171C2×C71—43no—
172C2×C71C22×#4111yescharacteristic
173C2×C71C2×2−1+411yescharacteristic
174C2×C71#58511yescharacteristic
175D71—22no—
176#3516—711nomaximal, Sylow
177C71#351611yescharacteristic, complemented, Sylow
178#557—711no—
179#556—713no—
180#554—713no—
181#553—713no—
182#552—713no—
183D4×Q8—711no—
184#548—711no—
185#548D7111yescharacteristic, complemented
186#542—713no—
187#541—713no—
188#529—713no—
189#529—713no—
190#520—711no—
191#519—711no—
192C2×C4×Q8—711no—
193C2×#41C2×D7111yescharacteristic
194C22×Q8—711no—
195C22×Q8C2×D7111yescharacteristic
196C4⋊Q8—713no—
197C4⋊Q8—713no—
198#126—716no—
199#126—713no—
200#126—713no—
201#126—716no—
202#125—713no—
203#125—716no—
204#125—713no—
205#125—713no—
206#124—713no—
207#124—713no—
208#124C2×D7113yes—
209#123—716no—
210#123—711no—
211#123—713no—
212C22⋊Q8—713no—
213C22⋊Q8—713no—
214C22⋊Q8—713no—
215C22⋊Q8—713no—
216C22⋊Q8—716no—
217C22⋊Q8C2×D7113yes—
218#121—713no—
219#121—713no—
220#121C2×D7113yes—
221C4×Q8—711no—
222C4×Q8—713no—
223C4×Q8—711no—
224C4×Q8—713no—
225C4×Q8—711no—
226C4×Q8—713no—
227C4×Q8—713no—
228C4×Q8—713no—
229C4×Q8C2×D7111yescharacteristic
230C4×D4—713no—
231C4×D4—713no—
232C4×D4—713no—
233C4×D4—713no—
234C4×D4—716no—
235C4×D4C2×D7113yes—
236#117—713no—
237#117—713no—
238#117—713no—
239#117—716no—
240C2×C4⋊C4—713no—
241C2×C4⋊C4—713no—
242C2×C42—713no—
243#41—24no—
244C2×Q8—24no—
245C2×Q8—713no—
246C2×Q8—713no—
247C2×Q8—1424no—
248C2×Q8C22×D7113yes—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—711no—
251C2×D4—713no—
252C2×D4C22×D7113yes—
253C2×D4C22×D7113yes—
254C22×C4—716no—
255C22×C4—713no—
256C22×C4—711no—
257C22×C4—713no—
258C22×C4C22×D7113yes—
259C22×C4C22×D7113yes—
260C4⋊C4—716no—
261C4⋊C4—713no—
262C4⋊C4—716no—
263C4⋊C4—713no—
264C4⋊C4—716no—
265C4⋊C4—713no—
266C4⋊C4—711no—
267C4⋊C4—713no—
268C4⋊C4—713no—
269C4⋊C4—713no—
270C4⋊C4—711no—
271C4⋊C4—713no—
272C4⋊C4—713no—
273C4⋊C4—713no—
274C4⋊C4—713no—
275C4⋊C4C22×D7113yes—
276C4⋊C4C22×D7113yes—
277C22⋊C4—713no—
278C22⋊C4—711no—
279C22⋊C4—713no—
280C22⋊C4—713no—
281C22⋊C4—716no—
282C22⋊C4—713no—
283C22⋊C4—713no—
284C22⋊C4—711no—
285C22⋊C4—713no—
286C22⋊C4C22×D7113yes—
287C22⋊C4C22×D7116yes—
288C22⋊C4C22×D7111yescharacteristic
289C42—713no—
290C42—713no—
291C42—713no—
292C42—716no—
293C42—713no—
294C42—713no—
295C42C22×D7113yes—
296C23—711no—
297C23C23×D7111yescharacteristic
298C23C23×D7113yes—
299Q8—26no—
300Q8—2843no—
301Q8—1426no—
302Q8#1176586414yes—
303D4—26no—
304D4—43no—
305D4—1422no—
306D4—1426no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—2843no—
310C2×C4—2846no—
311C2×C4—711no—
312C2×C4—713no—
313C2×C4—711no—
314C2×C4—713no—
315C2×C4—711no—
316C2×C4—713no—
317C2×C4—713no—
318C2×C4—1422no—
319C2×C4—1426no—
320C2×C4C23×D7111yescharacteristic
321C2×C4C23×D7113yes—
322C2×C4C23×D7113yes—
323C2×C4C23×D7111yescharacteristic
324C2×C4C23×D7113yes—
325C22—21no—
326C22—23no—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—712no—
332C22—1421no—
333C22—1421no—
334C22C24×D7111yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—1423no—
340C4—1421no—
341C4—1423no—
342C4—1421no—
343C4—1423no—
344C4—1423no—
345C4—1421no—
346C4C2×#1176586416yes—
347C2—41no—
348C2—43no—
349C2—1422no—
350C2C22×#1176586411yescharacteristic
351C22−1+4×D7111yescharacteristic
352C2#1798683511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#548×C71C2
hypercenter#548D71
hypocenterC71#3516
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986725C2semidirect
2. #17986718C2semidirect
3. #17986707C2semidirect
4. #17986696C2semidirect
5. #17986689C2semidirect
6. Q8×C22⋊D71C2semidirect
7. #17986656C2semidirect
8. #17986616C2semidirect
9. #17986609C2semidirect
10. #17986531C2semidirect
11. #17986528C2semidirect
12. #17986482C2semidirect
13. #17986477C2semidirect
14. C4×Q8×D71C2semidirect
15. #548×C71C2semidirect
177. C71#3516semidirect
185. #548D71semidirect