Finite Group Databasebeta

← #22602430

Subgroups — #22602430

Order 9088.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986727C213yesmaximal, complemented
2#17986725C211yesmaximal, characteristic, complemented
3#17986717C213yesmaximal, complemented
4#17986690C213yesmaximal, complemented
5#17986688C213yesmaximal, complemented
6Q8×D284C211yesmaximal, characteristic, complemented
7Q8×C22⋊D71C211yesmaximal, characteristic, complemented
8Q8×C22⋊D71C211yesmaximal, characteristic, complemented
9#17986611C213yesmaximal, complemented
10#17986607C213yesmaximal, complemented
11#17986553C213yesmaximal, complemented
12#17986527C213yesmaximal
13#17986475C211yesmaximal, characteristic, complemented
14C4×Q8×D71C211yesmaximal, characteristic, complemented
15#519×C71C211yesmaximal, characteristic, complemented
16C2×Q8×D71C2211yescharacteristic
17C2×Q8×D71C2211yescharacteristic
18#14556128C2213yes—
19#14556127C2213yes—
20#14556126C2213yes—
21#14556126C2213yes—
22#14556123C2213yes—
23#14556122C2216yes—
24#14556121C2213yes—
25#14556121C2216yes—
26#14556120C2216yes—
27#14556120C2213yes—
28#14556119C2213yes—
29#14556118C2216yes—
30#14556118C2216yes—
31#14556118C2213yes—
32#14556114C2213yes—
33#14556113C2213yes—
34#14556110C2213yes—
35#14556108C2213yes—
36#14556108C2213yes—
37#14556107C2213yes—
38#14556107C2213yes—
39#14556106C2213yes—
40#14556104C2213yes—
41#14556104C2213yes—
42#14556101C2211yescharacteristic
43Q8×Dic71C2211yescharacteristic
44Q8×Dic71C2211yescharacteristic
45C4×Dic142C2213yes—
46C4×Dic142C2213yes—
47C4×Dic142C2216yes—
48C4×Dic142C2213yes—
49#14556093C2213yes—
50#14556093C2213yes—
51C4×D284C2213yes—
52C4×C22⋊D71C2213yes—
53C4×C22⋊D71C2213yes—
54#14556086C2213yes—
55#14556085C2213yes—
56#14556085C2216yes—
57C4⋊C4×D71C2213yes—
58C42×D71C2213yes—
59C22×Q8×C71C2211yescharacteristic
60C4×Q8×C71C2213yes—
61C4×Q8×C71C2211yescharacteristic
62C4×Q8×C71C2213yes—
63C4×Q8×C71C2211yescharacteristic
64#117×C71C2213yes—
65#117×C71C2213yes—
66C2×Dic142C2313yes—
67C2×Dic142C2313yes—
68C2×Dic142C2313yes—
69Q8×D71—24no—
70Q8×D71—24no—
71C2×D284C2311yescharacteristic
72C2×C22⋊D71C2311yescharacteristic
73C2×C22⋊D71C2311yescharacteristic
74C2×C4×D71C2313yes—
75C2×C4×D71C2313yes—
76C2×C4×D71C2311yescharacteristic
77C2×C4×D71C2313yes—
78C4⋊Dic71C2313yes—
79C4⋊Dic71C2313yes—
80C4⋊Dic71C2313yes—
81C4⋊Dic71C2316yes—
82C4⋊Dic71C2313yes—
83Dic71⋊C4C2316yes—
84Dic71⋊C4C2313yes—
85Dic71⋊C4C2313yes—
86Dic71⋊C4C2316yes—
87Dic71⋊C4C2316yes—
88Dic71⋊C4C2313yes—
89Dic71⋊C4C2313yes—
90Dic71⋊C4C2311yescharacteristic
91Dic71⋊C4C2316yes—
92Dic71⋊C4C2313yes—
93C22⋊Dic71C2313yes—
94C22⋊Dic71C2313yes—
95(C2×D71)⋊C4C2311yescharacteristic
96(C2×D71)⋊C4C2313yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2313yes—
99(C2×D71)⋊C4C2313yes—
100(C2×D71)⋊C4C2313yes—
101C4×Dic71C2313yes—
102C4×Dic71C2313yes—
103C4×Dic71C2313yes—
104C4×Dic71C2316yes—
105C4×Dic71C2313yes—
106C2×Q8×C71—24no—
107C2×Q8×C71C2313yes—
108C2×Q8×C71C2311yescharacteristic
109C22×C4×C71C2313yes—
110C4⋊C4×C71C2313yes—
111C4⋊C4×C71C2313yes—
112C4⋊C4×C71C2313yes—
113C4⋊C4×C71C2313yes—
114C22⋊C4×C71C2311yescharacteristic
115C22⋊C4×C71C2313yes—
116C42×C71C2313yes—
117C42×C71C2316yes—
118C42×C71C2313yes—
119C23×C71C2411yescharacteristic
120C22×D71C2411yescharacteristic
121C22×D71C2411yescharacteristic
122Q8×C71—26no—
123Q8×C71#4114yes—
124C2×C4×C71—26no—
125C2×C4×C71C2411yescharacteristic
126C2×C4×C71C2413yes—
127C2×C4×C71C2413yes—
128C2×C4×C71C2411yescharacteristic
129C2×C4×C71C2413yes—
130C2×C4×C71C2413yes—
131C22⋊D71—22no—
132C22⋊D71—22no—
133C2×Dic71C2413yes—
134C2×Dic71C2411yescharacteristic
135C2×Dic71C2413yes—
136C2×Dic71C2413yes—
137C2×Dic71C2413yes—
138C2×Dic71C2411yescharacteristic
139D284—22no—
140C4×D71—26no—
141C4×D71—26no—
142C4×D71—26no—
143C4×D71—22no—
144Dic142—26no—
145Dic142—26no—
146Dic142—26no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71C2511yescharacteristic
151C2×D71—21no—
152C2×D71—21no—
153C2×D71—21no—
154C2×D71—21no—
155C2×D71—21no—
156C2×D712−1+412yes—
157C4×C71—23no—
158C4×C71—23no—
159C4×C71—23no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71C2×#4116yes—
163Dic71—23no—
164Dic71—23no—
165Dic71—23no—
166Dic71—23no—
167Dic71—21no—
168Dic712−1+412yes—
169C2×C71—41no—
170C2×C71C22×#4111yescharacteristic
171C2×C71C2×2−1+411yescharacteristic
172C2×C71C2×2−1+411yescharacteristic
173D71—22no—
174D71—41no—
175#3523—711nomaximal, Sylow
176C71#352311yescharacteristic, complemented, Sylow
177#557—711no—
178#557—713no—
179#556—713no—
180#552—713no—
181#552—713no—
182D4×Q8—711no—
183D4×Q8—711no—
184D4×Q8—711no—
185#541—713no—
186#541—713no—
187#533—713no—
188#529—713no—
189#519—711no—
190#519D7111yescharacteristic, complemented
191C2×C4×Q8—711no—
192C22×Q8—711no—
193C22×Q8—711no—
194C22×Q8C2×D7111yescharacteristic
195C4⋊Q8—713no—
196C4⋊Q8—713no—
197C4⋊Q8—713no—
198C4⋊Q8—713no—
199#126—713no—
200#126—713no—
201#126—716no—
202#126—716no—
203#125—713no—
204#125—716no—
205#125—716no—
206#125—713no—
207#125—716no—
208#125—713no—
209#124—713no—
210#123—713no—
211#123—713no—
212C22⋊Q8—713no—
213C22⋊Q8—713no—
214C22⋊Q8—713no—
215C22⋊Q8—713no—
216C22⋊Q8—713no—
217C22⋊Q8—713no—
218C22⋊Q8—713no—
219#121—711no—
220C4×Q8—711no—
221C4×Q8—711no—
222C4×Q8—713no—
223C4×Q8—713no—
224C4×Q8—713no—
225C4×Q8—713no—
226C4×Q8—713no—
227C4×Q8—716no—
228C4×Q8C2×D7111yescharacteristic
229C4×Q8C2×D7113yes—
230C4×Q8C2×D7111yescharacteristic
231C4×Q8C2×D7113yes—
232C4×D4—713no—
233C4×D4—713no—
234C4×D4—713no—
235#117—713no—
236#117—713no—
237#117—716no—
238#117C2×D7113yes—
239#117C2×D7113yes—
240C2×C4⋊C4—713no—
241C2×C42—713no—
242C2×Q8—24no—
243C2×Q8—713no—
244C2×Q8—713no—
245C2×Q8—713no—
246C2×Q8—1424no—
247C2×Q8—1424no—
248C2×Q8C22×D7113yes—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—711no—
251C2×D4—711no—
252C2×D4—711no—
253C22×C4—711no—
254C22×C4—713no—
255C22×C4—713no—
256C22×C4—713no—
257C22×C4C22×D7113yes—
258C4⋊C4—716no—
259C4⋊C4—713no—
260C4⋊C4—713no—
261C4⋊C4—716no—
262C4⋊C4—716no—
263C4⋊C4—713no—
264C4⋊C4—713no—
265C4⋊C4—711no—
266C4⋊C4—716no—
267C4⋊C4—713no—
268C4⋊C4—713no—
269C4⋊C4—713no—
270C4⋊C4—713no—
271C4⋊C4—716no—
272C4⋊C4—713no—
273C4⋊C4C22×D7113yes—
274C4⋊C4C22×D7113yes—
275C4⋊C4C22×D7113yes—
276C4⋊C4C22×D7113yes—
277C22⋊C4—713no—
278C22⋊C4—713no—
279C22⋊C4—711no—
280C22⋊C4—713no—
281C22⋊C4—713no—
282C22⋊C4—713no—
283C22⋊C4—713no—
284C22⋊C4—713no—
285C22⋊C4C22×D7111yescharacteristic
286C22⋊C4C22×D7113yes—
287C42—713no—
288C42—713no—
289C42—713no—
290C42—716no—
291C42—713no—
292C42C22×D7113yes—
293C42C22×D7116yes—
294C42C22×D7113yes—
295C23—711no—
296C23—711no—
297C23C23×D7111yescharacteristic
298Q8—26no—
299Q8—1426no—
300Q8—1426no—
301Q8—1426no—
302Q8#1176586514yes—
303D4—1422no—
304D4—1422no—
305D4—1422no—
306C2×C4—26no—
307C2×C4—711no—
308C2×C4—713no—
309C2×C4—711no—
310C2×C4—713no—
311C2×C4—713no—
312C2×C4—713no—
313C2×C4—1422no—
314C2×C4—1426no—
315C2×C4—1426no—
316C2×C4—1426no—
317C2×C4C23×D7113yes—
318C2×C4C23×D7111yescharacteristic
319C2×C4C23×D7113yes—
320C2×C4C23×D7111yescharacteristic
321C2×C4C23×D7113yes—
322C2×C4C23×D7113yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—712no—
327C22—1421no—
328C22—1421no—
329C22—1421no—
330C22—1421no—
331C22—1421no—
332C22C24×D7111yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—712no—
339C4—1423no—
340C4—1423no—
341C4—1423no—
342C4—1421no—
343C4—1423no—
344C4C2×#1176586516yes—
345C2—41no—
346C2—2841no—
347C2—1422no—
348C2C22×#1176586511yescharacteristic
349C22−1+4×D7111yescharacteristic
350C22−1+4×D7111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#519×C71C2
hypercenter#519D71
hypocenterC71#3523
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #17986727C2semidirect
2. #17986725C2semidirect
3. #17986717C2semidirect
4. #17986690C2semidirect
5. #17986688C2semidirect
6. Q8×D284C2semidirect
7. Q8×C22⋊D71C2semidirect
8. Q8×C22⋊D71C2semidirect
9. #17986611C2semidirect
10. #17986607C2semidirect
11. #17986553C2semidirect
13. #17986475C2semidirect
14. C4×Q8×D71C2semidirect
15. #519×C71C2semidirect
176. C71#3523semidirect
190. #519D71semidirect