Finite Group Databasebeta

← #23777396

Subgroups — #23777396

Order 10112.

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#19099286C213yesmaximal, complemented
2#19099274C213yesmaximal, complemented
3#19099266C213yesmaximal, complemented
4#19099251C213yesmaximal, complemented
5Q8×D316C211yesmaximal, characteristic, complemented
6#19099225C211yesmaximal, characteristic, complemented
7#19099225C211yesmaximal, characteristic, complemented
8#19099187C213yesmaximal, complemented
9#19099176C213yesmaximal, complemented
10#19099100C213yesmaximal, complemented
11#19099098C213yesmaximal, complemented
12#19099045C211yesmaximal, characteristic, complemented
13#19099039C211yesmaximal, characteristic, complemented
14C4×Q8×D79C211yesmaximal, characteristic, complemented
15#557×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17C2×Q8×D79C2211yescharacteristic
18C2×Q8×D79C2211yescharacteristic
19#14579161C2213yes—
20#14579156C2216yes—
21#14579156C2216yes—
22#14579156C2213yes—
23#14579156C2213yes—
24#14579152C2213yes—
25#14579152C2216yes—
26#14579150C2213yes—
27#14579149C2213yes—
28#14579149C2213yes—
29#14579146C2216yes—
30#14579146C2213yes—
31#14579146C2211yescharacteristic
32#14579141C2213yes—
33#14579141C2213yes—
34#14579141C2213yes—
35#14579141C2213yes—
36#14579137C2216yes—
37#14579137C2213yes—
38#14579136C2213yes—
39#14579136C2213yes—
40#14579136C2213yes—
41Q8×Dic79C2211yescharacteristic
42C4×Dic158C2213yes—
43C4×Dic158C2213yes—
44#14579126C2213yes—
45#14579126C2213yes—
46C4×D316C2213yes—
47C4×D316C2213yes—
48C4×D316C2213yes—
49C4×D316C2213yes—
50#14579123C2213yes—
51#14579123C2216yes—
52#14579119C2213yes—
53#14579119C2216yes—
54#14579118C2213yes—
55#14579118C2213yes—
56C4⋊C4×D79C2213yes—
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—
67C2×Dic158C2313yes—
68C2×Dic158C2313yes—
69Q8×D79—24no—
70Q8×D79—24no—
71C2×D316C2313yes—
72C2×D316C2313yes—
73C2×D316C2311yescharacteristic
74C2×D316C2313yes—
75C2×C4×D79C2313yes—
76C2×C4×D79C2313yes—
77C2×C4×D79C2313yes—
78C2×C4×D79C2311yescharacteristic
79C2×C4×D79C2313yes—
80C2×C4×D79C2316yes—
81C4⋊Dic79C2313yes—
82C4⋊Dic79C2316yes—
83C4⋊Dic79C2313yes—
84C4⋊Dic79C2313yes—
85C4⋊Dic79C2313yes—
86Dic79⋊C4C2313yes—
87Dic79⋊C4C2316yes—
88Dic79⋊C4C2311yescharacteristic
89Dic79⋊C4C2313yes—
90Dic79⋊C4C2313yes—
91(C2×D79)⋊C4C2311yescharacteristic
92(C2×D79)⋊C4C2313yes—
93(C2×D79)⋊C4C2313yes—
94(C2×D79)⋊C4C2313yes—
95(C2×D79)⋊C4C2313yes—
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×Dic79C2313yes—
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×D79C2411yescharacteristic
122Q8×C79—43no—
123Q8×C79#4114yes—
124C2×C4×C79C2411yescharacteristic
125C2×C4×C79C2413yes—
126C2×C4×C79C2413yes—
127C2×C4×C79C2411yescharacteristic
128C2×C4×C79C2411yescharacteristic
129C2×C4×C79C2413yes—
130C2×C4×C79C2413yes—
131C2×Dic79C2411yescharacteristic
132C2×Dic79C2413yes—
133C2×Dic79C2413yes—
134C2×Dic79C2413yes—
135C2×Dic79C2411yescharacteristic
136D316—26no—
137D316—22no—
138D316—26no—
139D316—43no—
140C4×D79—22no—
141C4×D79—26no—
142C4×D79—26no—
143C4×D79—26no—
144C4×D79—46no—
145C4×D79—43no—
146Dic158—26no—
147Dic158—26no—
148C22×C79C2511yescharacteristic
149C2×D79—21no—
150C2×D79—21no—
151C2×D79—21no—
152C2×D79—23no—
153C2×D79—21no—
154C2×D79—23no—
155C2×D79—23no—
156C2×D79—21no—
157C2×D792−1+412yes—
158C4×C79—21no—
159C4×C79—23no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—23no—
163C4×C79—23no—
164C4×C79C2×#4116yes—
165Dic79—23no—
166Dic79—23no—
167Dic79—21no—
168Dic79—21no—
169Dic79—23no—
170C2×C79C22×#4111yescharacteristic
171C2×C79C2×2−1+411yescharacteristic
172C2×C79#58511yescharacteristic
173D79—22no—
174D79—41no—
175D79—43no—
176#3516—791nomaximal, Sylow
177C79#351611yescharacteristic, complemented, Sylow
178#557D7911yescharacteristic, complemented
179#556—793no—
180#554—793no—
181#553—793no—
182#552—793no—
183D4×Q8—791no—
184#548—791no—
185#548—791no—
186#542—793no—
187#541—793no—
188#529—793no—
189#529—793no—
190#520—791no—
191#519—791no—
192C2×C4×Q8—791no—
193C2×#41—791no—
194C22×Q8—791no—
195C22×Q8—791no—
196C4⋊Q8—793no—
197C4⋊Q8C2×D7913yes—
198#126—796no—
199#126—793no—
200#126—793no—
201#126—796no—
202#125—793no—
203#125—796no—
204#125C2×D7913yes—
205#125C2×D7913yes—
206#124—793no—
207#124—793no—
208#124—793no—
209#123—791no—
210#123—793no—
211#123—796no—
212C22⋊Q8—793no—
213C22⋊Q8—796no—
214C22⋊Q8—793no—
215C22⋊Q8—793no—
216C22⋊Q8—793no—
217C22⋊Q8—793no—
218#121—793no—
219#121—793no—
220#121—793no—
221C4×Q8—791no—
222C4×Q8—793no—
223C4×Q8—793no—
224C4×Q8—793no—
225C4×Q8—793no—
226C4×Q8C2×D7911yescharacteristic
227C4×Q8C2×D7911yescharacteristic
228C4×Q8C2×D7911yescharacteristic
229C4×Q8C2×D7913yes—
230C4×D4—793no—
231C4×D4—796no—
232C4×D4—793no—
233C4×D4—793no—
234C4×D4—793no—
235C4×D4—793no—
236#117—793no—
237#117—796no—
238#117—793no—
239#117—793no—
240C2×C4⋊C4—793no—
241C2×C4⋊C4—793no—
242C2×C42—793no—
243#41—1584no—
244C2×Q8—793no—
245C2×Q8—793no—
246C2×Q8—1584no—
247C2×Q8—1584no—
248C2×Q8C22×D7913yes—
249C2×Q8C22×D7911yescharacteristic
250C2×D4—793no—
251C2×D4—793no—
252C2×D4—791no—
253C2×D4—793no—
254C22×C4—793no—
255C22×C4—796no—
256C22×C4—791no—
257C22×C4—793no—
258C22×C4—793no—
259C22×C4—793no—
260C4⋊C4—793no—
261C4⋊C4—796no—
262C4⋊C4—791no—
263C4⋊C4—793no—
264C4⋊C4—793no—
265C4⋊C4—793no—
266C4⋊C4—796no—
267C4⋊C4—793no—
268C4⋊C4—793no—
269C4⋊C4—793no—
270C4⋊C4C22×D7913yes—
271C4⋊C4C22×D7913yes—
272C4⋊C4C22×D7913yes—
273C4⋊C4C22×D7913yes—
274C4⋊C4C22×D7916yes—
275C4⋊C4C22×D7911yescharacteristic
276C4⋊C4C22×D7913yes—
277C22⋊C4—793no—
278C22⋊C4—791no—
279C22⋊C4—793no—
280C22⋊C4—793no—
281C22⋊C4—793no—
282C22⋊C4—796no—
283C22⋊C4—793no—
284C22⋊C4—791no—
285C22⋊C4—791no—
286C22⋊C4—796no—
287C22⋊C4—793no—
288C22⋊C4—793no—
289C42—793no—
290C42—793no—
291C42—793no—
292C42—796no—
293C42C22×D7913yes—
294C42C22×D7913yes—
295C42C22×D7913yes—
296C23—791no—
297C23—793no—
298C23—791no—
299Q8—43no—
300Q8—1586no—
301Q8—1586no—
302Q8#1192962514yes—
303D4—3163no—
304D4—1586no—
305D4—1582no—
306D4—1586no—
307C2×C4—3166no—
308C2×C4—3163no—
309C2×C4—791no—
310C2×C4—793no—
311C2×C4—793no—
312C2×C4—793no—
313C2×C4—791no—
314C2×C4—1582no—
315C2×C4—1586no—
316C2×C4—1586no—
317C2×C4—1586no—
318C2×C4C23×D7911yescharacteristic
319C2×C4C23×D7913yes—
320C2×C4C23×D7913yes—
321C2×C4C23×D7911yescharacteristic
322C2×C4C23×D7911yescharacteristic
323C2×C4C23×D7913yes—
324C2×C4C23×D7913yes—
325C22—792no—
326C22—1583no—
327C22—1581no—
328C22—1583no—
329C22—1581no—
330C22—1581no—
331C22—1583no—
332C22—1581no—
333C22—1581no—
334C22C24×D7911yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—1581no—
342C4—1583no—
343C4—1581no—
344C4—1583no—
345C4—1583no—
346C4C2×#1192962516yes—
347C2—3163no—
348C2—3161no—
349C2—1582no—
350C2C22×#1192962511yescharacteristic
351C22−1+4×D7911yescharacteristic
352C2#1909939911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#557×C79C2
hypercenter#557D79
hypocenterC79#3516
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #19099286C2semidirect
2. #19099274C2semidirect
3. #19099266C2semidirect
4. #19099251C2semidirect
5. Q8×D316C2semidirect
6. #19099225C2semidirect
7. #19099225C2semidirect
8. #19099187C2semidirect
9. #19099176C2semidirect
10. #19099100C2semidirect
11. #19099098C2semidirect
12. #19099045C2semidirect
13. #19099039C2semidirect
14. C4×Q8×D79C2semidirect
15. #557×C79C2semidirect
177. C79#3516semidirect
178. #557D79semidirect