Finite Group Databasebeta

← #23777512

Subgroups — #23777512

Order 10112.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099298C213yesmaximal, complemented
2#19099279C213yesmaximal, complemented
3#19099266C213yesmaximal, complemented
4#19099251C213yesmaximal, complemented
5#19099235C211yesmaximal, characteristic, complemented
6#19099225C211yesmaximal, characteristic, complemented
7#19099225C211yesmaximal, characteristic, complemented
8#19099187C213yesmaximal, complemented
9#19099169C213yesmaximal, complemented
10#19099100C213yesmaximal, complemented
11#19099098C213yesmaximal, complemented
12#19099045C211yesmaximal, characteristic, complemented
13#19099039C211yesmaximal, characteristic, complemented
14C4×#11929625C211yesmaximal, characteristic, complemented
15#557×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17C2×#11929625C2211yescharacteristic
18C2×Q8×D79C2211yescharacteristic
19#14579158C2213yes—
20#14579156C2216yes—
21#14579156C2213yes—
22#14579156C2211yescharacteristic
23#14579152C2213yes—
24#14579150C2216yes—
25#14579150C2213yes—
26#14579149C2213yes—
27#14579149C2213yes—
28#14579146C2216yes—
29#14579146C2216yes—
30#14579146C2213yes—
31#14579146C2213yes—
32#14579141C2213yes—
33#14579141C2213yes—
34#14579137C2213yes—
35#14579136C2213yes—
36#14579136C2213yes—
37#14579136C2213yes—
38#14579136C2216yes—
39#14579136C2213yes—
40#14579136C2213yes—
41Q8×Dic79C2211yescharacteristic
42C4×Dic158C2213yes—
43#14579126C2213yes—
44C4×D316C2213yes—
45C4×D316C2213yes—
46C4×D316C2213yes—
47C4×D316C2213yes—
48C4×D316C2213yes—
49#14579123C2213yes—
50#14579123C2213yes—
51#14579123C2216yes—
52#14579119C2213yes—
53#14579119C2216yes—
54#14579119C2213yes—
55#14579118C2213yes—
56#14579118C2213yes—
57C4⋊C4×D79C2213yes—
58C42×D79C2213yes—
59C4⋊Q8×C79C2213yes—
60#125×C79C2213yes—
61#125×C79C2213yes—
62C4×Q8×C79C2211yescharacteristic
63C4×Q8×C79C2211yescharacteristic
64C4×Q8×C79C2211yescharacteristic
65C4×Q8×C79C2213yes—
66#11929625—24no—
67#11929625—24no—
68C2×Dic158C2313yes—
69Q8×D79—24no—
70C2×D316C2313yes—
71C2×D316C2313yes—
72C2×D316C2316yes—
73C2×D316C2313yes—
74C2×D316C2313yes—
75C2×D316C2313yes—
76C2×C4×D79C2313yes—
77C2×C4×D79C2313yes—
78C2×C4×D79C2313yes—
79C2×C4×D79C2313yes—
80C2×C4×D79C2313yes—
81C2×C4×D79C2316yes—
82C4⋊Dic79C2313yes—
83C4⋊Dic79C2313yes—
84C4⋊Dic79C2311yescharacteristic
85C4⋊Dic79C2313yes—
86Dic79⋊C4C2313yes—
87Dic79⋊C4C2311yescharacteristic
88Dic79⋊C4C2313yes—
89Dic79⋊C4C2313yes—
90(C2×D79)⋊C4C2311yescharacteristic
91(C2×D79)⋊C4C2313yes—
92(C2×D79)⋊C4C2313yes—
93(C2×D79)⋊C4C2313yes—
94(C2×D79)⋊C4C2313yes—
95(C2×D79)⋊C4C2316yes—
96(C2×D79)⋊C4C2316yes—
97(C2×D79)⋊C4C2313yes—
98(C2×D79)⋊C4C2316yes—
99(C2×D79)⋊C4C2311yescharacteristic
100(C2×D79)⋊C4C2311yescharacteristic
101(C2×D79)⋊C4C2313yes—
102(C2×D79)⋊C4C2313yes—
103C4×Dic79C2311yescharacteristic
104C4×Dic79C2313yes—
105C4×Dic79C2313yes—
106C4×Dic79C2316yes—
107C2×Q8×C79C2313yes—
108C2×Q8×C79C2311yescharacteristic
109C4⋊C4×C79C2311yescharacteristic
110C4⋊C4×C79C2313yes—
111C4⋊C4×C79C2313yes—
112C4⋊C4×C79C2313yes—
113C4⋊C4×C79C2313yes—
114C4⋊C4×C79C2313yes—
115C4⋊C4×C79C2316yes—
116C42×C79C2313yes—
117C42×C79C2313yes—
118C42×C79C2313yes—
119C22×D79C2411yescharacteristic
120C22×D79C2413yes—
121C22×D79C2413yes—
122Q8×C79—43no—
123Q8×C79#4114yes—
124C2×C4×C79C2413yes—
125C2×C4×C79C2413yes—
126C2×C4×C79C2411yescharacteristic
127C2×C4×C79C2411yescharacteristic
128C2×C4×C79C2413yes—
129C2×C4×C79C2411yescharacteristic
130C2×C4×C79C2413yes—
131C2×Dic79C2411yescharacteristic
132C2×Dic79C2413yes—
133C2×Dic79C2411yescharacteristic
134C2×Dic79C2413yes—
135C2×Dic79C2411yescharacteristic
136D316—26no—
137D316—26no—
138D316—43no—
139D316—46no—
140D316—43no—
141D316—43no—
142C4×D79—26no—
143C4×D79—26no—
144C4×D79—26no—
145C4×D79—43no—
146C4×D79—43no—
147C4×D79—46no—
148Dic158—26no—
149C22×C79C2511yescharacteristic
150C2×D79—21no—
151C2×D79—23no—
152C2×D79—23no—
153C2×D79—23no—
154C2×D79—23no—
155C2×D79—23no—
156C2×D79—21no—
157C2×D79—23no—
158C2×D79—21no—
159C4×C79—21no—
160C4×C79—23no—
161C4×C79—23no—
162C4×C79—21no—
163C4×C79—23no—
164C4×C79—21no—
165C4×C79C2×#4116yes—
166Dic79—23no—
167Dic79—21no—
168Dic79—21no—
169Dic79—21no—
170Dic79—23no—
171C2×C79C22×#4111yescharacteristic
172C2×C79#58511yescharacteristic
173C2×C79#58511yescharacteristic
174D79—41no—
175D79—43no—
176D79—43no—
177#3524—791nomaximal, Sylow
178C79#352411yescharacteristic, complemented, Sylow
179#559—793no—
180#557D7911yescharacteristic, complemented
181#555—793no—
182#553—793no—
183#552—793no—
184#550—791no—
185#548—791no—
186#548—791no—
187#542—793no—
188#540—793no—
189#529—793no—
190#529—793no—
191#520—791no—
192#519—791no—
193C4×#41—791no—
194C2×#41—791no—
195C2×#41—791no—
196C22×Q8—791no—
197C4⋊Q8C2×D7913yes—
198#127—793no—
199#126—791no—
200#126—793no—
201#126—796no—
202#125—793no—
203#125C2×D7913yes—
204#125C2×D7913yes—
205#124—793no—
206#124—796no—
207#124—793no—
208#124—793no—
209#123—796no—
210#123—793no—
211#123—793no—
212#123—796no—
213C22⋊Q8—793no—
214C22⋊Q8—793no—
215C22⋊Q8—793no—
216#121—793no—
217#121—793no—
218#121—793no—
219#121—793no—
220#121—796no—
221#121—793no—
222C4×Q8—791no—
223C4×Q8—793no—
224C4×Q8—793no—
225C4×Q8C2×D7911yescharacteristic
226C4×Q8C2×D7911yescharacteristic
227C4×Q8C2×D7911yescharacteristic
228C4×Q8C2×D7913yes—
229C4×D4—793no—
230C4×D4—793no—
231C4×D4—796no—
232C4×D4—793no—
233C4×D4—793no—
234C4×D4—793no—
235C4×D4—793no—
236C4×D4—793no—
237#117—793no—
238#117—793no—
239#117—796no—
240#117—793no—
241#117—793no—
242C2×C4⋊C4—793no—
243C2×C42—793no—
244#41—1584no—
245#41—1584no—
246C2×Q8—793no—
247C2×Q8—1584no—
248C2×Q8C22×D7913yes—
249C2×Q8C22×D7911yescharacteristic
250C2×D4—793no—
251C2×D4—793no—
252C2×D4—796no—
253C2×D4—793no—
254C2×D4—793no—
255C2×D4—793no—
256C22×C4—793no—
257C22×C4—793no—
258C22×C4—796no—
259C22×C4—793no—
260C22×C4—793no—
261C22×C4—793no—
262C4⋊C4—793no—
263C4⋊C4—791no—
264C4⋊C4—793no—
265C4⋊C4—793no—
266C4⋊C4—793no—
267C4⋊C4—793no—
268C4⋊C4—791no—
269C4⋊C4—793no—
270C4⋊C4C22×D7913yes—
271C4⋊C4C22×D7913yes—
272C4⋊C4C22×D7913yes—
273C4⋊C4C22×D7916yes—
274C4⋊C4C22×D7913yes—
275C4⋊C4C22×D7911yescharacteristic
276C4⋊C4C22×D7913yes—
277C22⋊C4—791no—
278C22⋊C4—793no—
279C22⋊C4—793no—
280C22⋊C4—793no—
281C22⋊C4—796no—
282C22⋊C4—793no—
283C22⋊C4—796no—
284C22⋊C4—791no—
285C22⋊C4—793no—
286C22⋊C4—791no—
287C22⋊C4—793no—
288C22⋊C4—796no—
289C22⋊C4—793no—
290C42—791no—
291C42—793no—
292C42—793no—
293C42—796no—
294C42C22×D7913yes—
295C42C22×D7913yes—
296C42C22×D7913yes—
297C23—793no—
298C23—793no—
299C23—791no—
300Q8—43no—
301Q8—1586no—
302Q8#1192962514yes—
303D4—3163no—
304D4—3166no—
305D4—3163no—
306D4—3163no—
307D4—1586no—
308D4—1586no—
309C2×C4—3163no—
310C2×C4—3163no—
311C2×C4—3166no—
312C2×C4—791no—
313C2×C4—793no—
314C2×C4—791no—
315C2×C4—793no—
316C2×C4—791no—
317C2×C4—1586no—
318C2×C4—1586no—
319C2×C4—1586no—
320C2×C4C23×D7913yes—
321C2×C4C23×D7911yescharacteristic
322C2×C4C23×D7913yes—
323C2×C4C23×D7911yescharacteristic
324C2×C4C23×D7911yescharacteristic
325C2×C4C23×D7913yes—
326C2×C4C23×D7913yes—
327C22—1583no—
328C22—1583no—
329C22—1581no—
330C22—1583no—
331C22—1583no—
332C22—1581no—
333C22—1581no—
334C22—1583no—
335C22—1583no—
336C22C24×D7911yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—1581no—
344C4—1583no—
345C4—1581no—
346C4—1581no—
347C4—1583no—
348C4C2×#1192962516yes—
349C2—3163no—
350C2—3163no—
351C2—3161no—
352C2C22×#1192962511yescharacteristic
353C2#1909939911yescharacteristic
354C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#557×C79C2
hypercenter#557D79
hypocenterC79#3524
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #19099298C2semidirect
2. #19099279C2semidirect
3. #19099266C2semidirect
4. #19099251C2semidirect
5. #19099235C2semidirect
6. #19099225C2semidirect
7. #19099225C2semidirect
8. #19099187C2semidirect
9. #19099169C2semidirect
10. #19099100C2semidirect
11. #19099098C2semidirect
12. #19099045C2semidirect
13. #19099039C2semidirect
14. C4×#11929625C2semidirect
15. #557×C79C2semidirect
178. C79#3524semidirect
180. #557D79semidirect