Finite Group Databasebeta

← #22601930

Subgroups — #22601930

Order 9088.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986731C213yesmaximal, complemented
2#17986725C211yesmaximal, characteristic, complemented
3#17986720C213yesmaximal, complemented
4#17986701C213yesmaximal, complemented
5#17986695C213yesmaximal, complemented
6#17986668C211yesmaximal, characteristic, complemented
7#17986660C211yesmaximal, characteristic, complemented
8#17986544C213yesmaximal, complemented
9#17986535C213yesmaximal, complemented
10#17986528C213yesmaximal, complemented
11#41×Dic71C211yesmaximal, characteristic, complemented
12C4×#11765866C211yesmaximal, characteristic, complemented
13C4×#11765864C213yesmaximal, complemented
14C2×Q8×Dic71C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765864C2213yes—
18#14556127C2213yes—
19#14556124C2213yes—
20#14556123C2216yes—
21#14556121C2216yes—
22#14556120C2213yes—
23#14556119C2213yes—
24#14556117C2213yes—
25#14556114C2216yes—
26#14556112C2216yes—
27#14556110C2216yes—
28#14556104C2213yes—
29#14556102C2213yes—
30#14556102C2213yes—
31#14556100C2216yes—
32#14556099C2213yes—
33Q8×Dic71C2211yescharacteristic
34Q8×Dic71C2211yescharacteristic
35Q8×Dic71C2213yes—
36Q8×Dic71C2211yescharacteristic
37Q8×Dic71C2213yes—
38C4×Dic142C2213yes—
39#14556093C2213yes—
40C4×D284C2213yes—
41D4×Dic71C2213yes—
42D4×Dic71C2213yes—
43D4×Dic71C2213yes—
44#14556090C2213yes—
45C4×C22⋊D71C2213yes—
46C4×C22⋊D71C2213yes—
47(C22⋊D71)⋊C4C2213yes—
48(C22⋊D71)⋊C4C2213yes—
49(C22⋊D71)⋊C4C2216yes—
50#14556087C2213yes—
51#14556086C2213yes—
52#14556086C2213yes—
53#14556084C2211yescharacteristic
54#14556084C2213yes—
55C2×C4⋊Dic71C2213yes—
56C2×C4×Dic71C2213yes—
57C2×C4×Dic71C2213yes—
58C42×D71C2213yes—
59C2×#41×C71C2211yescharacteristic
60C22×Q8×C71C2211yescharacteristic
61#124×C71C2213yes—
62C22⋊Q8×C71C2213yes—
63#121×C71C2213yes—
64C4×Q8×C71C2211yescharacteristic
65C4×D4×C71C2213yes—
66#11765866—24no—
67#11765864—212no—
68C2×Dic142C2313yes—
69C2×D284C2313yes—
70C2×C22⋊D71C2313yes—
71C2×C22⋊D71C2313yes—
72C2×C22⋊D71C2316yes—
73C22×Dic71C2313yes—
74C22×Dic71C2313yes—
75C22×Dic71C2313yes—
76C22×Dic71C2311yescharacteristic
77C2×C4×D71C2313yes—
78C2×C4×D71C2313yes—
79C2×C4×D71C2313yes—
80C4⋊Dic71C2313yes—
81C4⋊Dic71C2313yes—
82C4⋊Dic71C2316yes—
83C4⋊Dic71C2313yes—
84C4⋊Dic71C2313yes—
85Dic71⋊C4C2311yescharacteristic
86Dic71⋊C4C2313yes—
87Dic71⋊C4C2311yescharacteristic
88Dic71⋊C4C2313yes—
89Dic71⋊C4C2316yes—
90C22⋊Dic71C2311yescharacteristic
91C22⋊Dic71C2313yes—
92C22⋊Dic71C2316yes—
93(C2×D71)⋊C4C2313yes—
94(C2×D71)⋊C4C2313yes—
95(C2×D71)⋊C4C2316yes—
96(C2×D71)⋊C4C2316yes—
97C4×Dic71C2311yescharacteristic
98C4×Dic71C2313yes—
99C4×Dic71C2313yes—
100C4×Dic71C2311yescharacteristic
101C4×Dic71C2313yes—
102C4×Dic71C2313yes—
103C4×Dic71C2313yes—
104C4×Dic71C2313yes—
105C4×Dic71C2313yes—
106#41×C71—24no—
107C2×Q8×C71—24no—
108C2×Q8×C71C2313yes—
109C2×Q8×C71C2311yescharacteristic
110C2×D4×C71C2313yes—
111C2×D4×C71C2313yes—
112C22×C4×C71C2313yes—
113C22×C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C4⋊C4×C71C2313yes—
116C22⋊C4×C71C2311yescharacteristic
117C22⋊C4×C71C2313yes—
118C22⋊C4×C71C2316yes—
119C42×C71C2313yes—
120C23×C71C2413yes—
121C23×C71C2411yescharacteristic
122C22×D71C2413yes—
123Q8×C71—26no—
124Q8×C71#4114yescomplemented
125D4×C71—26no—
126D4×C71—26no—
127C2×C4×C71—26no—
128C2×C4×C71—26no—
129C2×C4×C71C2411yescharacteristic
130C2×C4×C71C2413yes—
131C2×C4×C71C2413yes—
132C2×C4×C71C2411yescharacteristic
133C2×C4×C71C2413yes—
134C22⋊D71—26no—
135C22⋊D71—26no—
136C22⋊D71—46no—
137C2×Dic71—26no—
138C2×Dic71—43no—
139C2×Dic71—43no—
140C2×Dic71#4114yescomplemented
141C2×Dic71C2411yescharacteristic
142C2×Dic71C2413yes—
143C2×Dic71C2411yescharacteristic
144C2×Dic71C2413yes—
145C2×Dic71C2413yes—
146C2×Dic71C2411yescharacteristic
147C2×Dic71C2411yescharacteristic
148D284—26no—
149C4×D71—26no—
150C4×D71—26no—
151C4×D71—43no—
152Dic142—26no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—23no—
156C22×C71—23no—
157C22×C71—23no—
158C22×C71C2×#4112yes—
159C22×C71C2511yescharacteristic
160C2×D71—23no—
161C2×D71—23no—
162C2×D71—23no—
163C4×C71—23no—
164C4×C71—21no—
165C4×C71—23no—
166C4×C71—21no—
167C4×C71C2×#4116yes—
168Dic71—23no—
169Dic71—21no—
170Dic71—23no—
171Dic71—21no—
172Dic71—23no—
173Dic71C2×#4112yes—
174Dic71C2×#4112yes—
175C2×C71—22no—
176C2×C71—43no—
177C2×C71C22×#4111yescharacteristic
178C2×C71C22×#4111yescharacteristic
179C2×C71#58511yescharacteristic
180D71—43no—
181#3480—711nomaximal, Sylow
182C71#348011yescharacteristic, complemented, Sylow
183#559—713no—
184#557—711no—
185#556—713no—
186#553—713no—
187#553—713no—
188#550—711no—
189#548—711no—
190#548D7111yescharacteristic, complemented
191#532—713no—
192#529—713no—
193#529—713no—
194C4×#41—711no—
195C4×#41—711no—
196C4×#41—713no—
197C2×C4×Q8—711no—
198C2×#41—711no—
199C2×#41—713no—
200C2×#41C2×D7111yescharacteristic
201C22×Q8C2×D7111yescharacteristic
202C4⋊Q8—713no—
203#127—713no—
204#126—716no—
205#126—716no—
206#125—713no—
207#125—713no—
208#124—713no—
209#124—716no—
210#124C2×D7113yes—
211#123—716no—
212#123—716no—
213C22⋊Q8—713no—
214C22⋊Q8C2×D7113yes—
215#121—713no—
216#121—716no—
217#121—713no—
218#121—713no—
219#121C2×D7113yes—
220C4×Q8—711no—
221C4×Q8—713no—
222C4×Q8—711no—
223C4×Q8—713no—
224C4×Q8—711no—
225C4×Q8—713no—
226C4×Q8—713no—
227C4×Q8C2×D7111yescharacteristic
228C4×D4—713no—
229C4×D4—713no—
230C4×D4—713no—
231C4×D4—713no—
232C4×D4—713no—
233C4×D4—713no—
234C4×D4—713no—
235C4×D4—713no—
236C4×D4—716no—
237C4×D4—713no—
238C4×D4C2×D7113yes—
239#117—713no—
240#117—713no—
241#117—713no—
242#117—711no—
243#117—713no—
244C2×C4⋊C4—713no—
245C2×C42—713no—
246C2×C42—713no—
247C2×C42—713no—
248#41—24no—
249#41—14212no—
250#41—1424no—
251C2×Q8—24no—
252C2×Q8—713no—
253C2×Q8C22×D7113yes—
254C2×Q8C22×D7111yescharacteristic
255C2×D4—716no—
256C2×D4—713no—
257C2×D4—713no—
258C2×D4—713no—
259C2×D4C22×D7113yes—
260C2×D4C22×D7113yes—
261C22×C4—713no—
262C22×C4—713no—
263C22×C4—713no—
264C22×C4—711no—
265C22×C4—713no—
266C22×C4—713no—
267C22×C4—713no—
268C22×C4C22×D7113yes—
269C22×C4C22×D7113yes—
270C4⋊C4—711no—
271C4⋊C4—711no—
272C4⋊C4—713no—
273C4⋊C4—713no—
274C4⋊C4—716no—
275C4⋊C4—713no—
276C4⋊C4—713no—
277C4⋊C4—716no—
278C4⋊C4—713no—
279C4⋊C4—713no—
280C4⋊C4C22×D7113yes—
281C4⋊C4C22×D7113yes—
282C22⋊C4—711no—
283C22⋊C4—713no—
284C22⋊C4—716no—
285C22⋊C4—713no—
286C22⋊C4—716no—
287C22⋊C4—713no—
288C22⋊C4—716no—
289C22⋊C4C22×D7111yescharacteristic
290C22⋊C4C22×D7113yes—
291C22⋊C4C22×D7116yes—
292C42—711no—
293C42—711no—
294C42—713no—
295C42—713no—
296C42—713no—
297C42—713no—
298C42—713no—
299C42—713no—
300C42—713no—
301C42C22×D7113yes—
302C23—713no—
303C23C23×D7113yes—
304C23C23×D7111yescharacteristic
305Q8—26no—
306Q8—1426no—
307Q8#1176586414yescomplemented
308D4—26no—
309D4—26no—
310D4—2846no—
311D4—1426no—
312D4—1426no—
313D4—1426no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2843no—
317C2×C4—2843no—
318C2×C4—2843no—
319C2×C4—714no—
320C2×C4—711no—
321C2×C4—711no—
322C2×C4—711no—
323C2×C4—711no—
324C2×C4—713no—
325C2×C4—713no—
326C2×C4—713no—
327C2×C4—1426no—
328C2×C4—1426no—
329C2×C4—1426no—
330C2×C4C23×D7111yescharacteristic
331C2×C4C23×D7113yes—
332C2×C4C23×D7111yescharacteristic
333C2×C4C23×D7113yes—
334C2×C4C23×D7113yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—1423no—
341C22—1423no—
342C22—1423no—
343C22#41×D7112yes—
344C22C24×D7111yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—712no—
350C4—712no—
351C4—1423no—
352C4—1421no—
353C4—1423no—
354C4—1421no—
355C4—1423no—
356C4C2×#1176586416yes—
357C2—22no—
358C2—43no—
359C2—2843no—
360C2C2×#41×D7111yescharacteristic
361C2C22×#1176586411yescharacteristic
362C2#1798683611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#548×C71C2
hypercenter#548D71
hypocenterC71#3480
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986731C2semidirect
2. #17986725C2semidirect
3. #17986720C2semidirect
4. #17986701C2semidirect
5. #17986695C2semidirect
6. #17986668C2semidirect
7. #17986660C2semidirect
8. #17986544C2semidirect
9. #17986535C2semidirect
10. #17986528C2semidirect
11. #41×Dic71C2semidirect
12. C4×#11765866C2semidirect
13. C4×#11765864C2semidirect
14. C2×Q8×Dic71C2semidirect
15. #548×C71C2semidirect
124. Q8×C71#41semidirect
140. C2×Dic71#41semidirect
182. C71#3480semidirect
190. #548D71semidirect
307. Q8#11765864semidirect