Finite Group Databasebeta

← #23777400

Subgroups — #23777400

Order 10112.

1122 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#19099289C211yesmaximal, characteristic, complemented
2#19099263C213yesmaximal, complemented
3#19099261C213yesmaximal, complemented
4#19099235C211yesmaximal, characteristic, complemented
5#19099234C213yesmaximal, complemented
6#19099232C211yesmaximal, characteristic, complemented
7#19099232C211yesmaximal, characteristic, complemented
8#19099214C213yesmaximal, complemented
9#19099168C213yesmaximal, complemented
10#19099164C213yesmaximal, complemented
11#19099110C213yesmaximal, complemented
12#19099091C213yesmaximal, complemented
13#19099045C211yesmaximal, characteristic, complemented
14C4×#11929625C211yesmaximal, characteristic, complemented
15#520×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17C2×#11929625C2211yescharacteristic
18C2×#11929624C2216yes—
19#14579160C2213yes—
20#14579158C2213yes—
21#14579157C2213yes—
22#14579157C2213yes—
23#14579156C2213yes—
24#14579154C2213yes—
25#14579153C2213yes—
26#14579152C2213yes—
27#14579151C2211yescharacteristic
28#14579147C2213yes—
29#14579146C2213yes—
30#14579146C2216yes—
31#14579144C2216yes—
32#14579143C2213yes—
33#14579139C2213yes—
34#14579136C2213yes—
35#14579136C2213yes—
36#14579135C2216yes—
37#14579135C2213yes—
38#14579135C2213yes—
39#14579134C2216yes—
40#14579134C2216yes—
41#14579134C2213yes—
42#14579132C2213yes—
43#14579132C2213yes—
44Q8×Dic79C2211yescharacteristic
45Q8×Dic79C2211yescharacteristic
46C4×Dic158C2213yes—
47C4×D316C2213yes—
48C4×D316C2213yes—
49C4×D316C2213yes—
50#14579123C2213yes—
51#14579123C2213yes—
52C4×C22⋊D79C2213yes—
53C4×C22⋊D79C2213yes—
54C4×C22⋊D79C2216yes—
55#14579119C2213yes—
56#14579118C2213yes—
57C4⋊C4×D79C2213yes—
58C42×D79C2213yes—
59C2×#41×C79C2211yescharacteristic
60C4×Q8×C79C2211yescharacteristic
61C4×Q8×C79C2211yescharacteristic
62C4×D4×C79C2213yes—
63C4×D4×C79C2213yes—
64#117×C79C2213yes—
65C2×C4⋊C4×C79C2213yes—
66#11929625—24no—
67#11929625—24no—
68#11929624—412no—
69C2×Dic158C2313yes—
70C2×D316C2313yes—
71C2×D316C2313yes—
72C2×D316C2316yes—
73C2×D316C2313yes—
74C2×C22⋊D79C2313yes—
75C2×C22⋊D79C2316yes—
76C2×C22⋊D79C2316yes—
77C2×C22⋊D79C2313yes—
78C2×C4×D79C2313yes—
79C2×C4×D79C2313yes—
80C2×C4×D79C2313yes—
81C2×C4×D79C2316yes—
82C2×C4×D79C2313yes—
83C4⋊Dic79C2313yes—
84C4⋊Dic79C2313yes—
85C4⋊Dic79C2311yescharacteristic
86C4⋊Dic79C2313yes—
87Dic79⋊C4C2313yes—
88Dic79⋊C4C2311yescharacteristic
89Dic79⋊C4C2311yescharacteristic
90Dic79⋊C4C2311yescharacteristic
91Dic79⋊C4C2316yes—
92Dic79⋊C4C2311yescharacteristic
93Dic79⋊C4C2313yes—
94C22⋊Dic79C2313yes—
95C22⋊Dic79C2313yes—
96(C2×D79)⋊C4C2313yes—
97(C2×D79)⋊C4C2313yes—
98(C2×D79)⋊C4C2316yes—
99(C2×D79)⋊C4C2313yes—
100(C2×D79)⋊C4C2313yes—
101(C2×D79)⋊C4C2313yes—
102(C2×D79)⋊C4C2313yes—
103C4×Dic79C2311yescharacteristic
104C4×Dic79C2313yes—
105C4×Dic79C2313yes—
106C4×Dic79C2313yes—
107#41×C79—24no—
108C2×Q8×C79C2311yescharacteristic
109C2×D4×C79C2313yes—
110C22×C4×C79C2313yes—
111C22×C4×C79C2316yes—
112C4⋊C4×C79C2313yes—
113C4⋊C4×C79C2313yes—
114C4⋊C4×C79C2313yes—
115C4⋊C4×C79C2311yescharacteristic
116C22⋊C4×C79C2313yes—
117C22⋊C4×C79C2313yes—
118C42×C79C2313yes—
119C42×C79C2313yes—
120C23×C79C2413yes—
121C22×D79C2413yes—
122C22×D79C2413yes—
123Q8×C79#4114yescomplemented
124D4×C79—26no—
125C2×C4×C79—26no—
126C2×C4×C79—46no—
127C2×C4×C79C2411yescharacteristic
128C2×C4×C79C2413yes—
129C2×C4×C79C2411yescharacteristic
130C2×C4×C79C2413yes—
131C2×C4×C79C2411yescharacteristic
132C2×C4×C79C2413yes—
133C22⋊D79—26no—
134C22⋊D79—43no—
135C22⋊D79—46no—
136C22⋊D79—46no—
137C2×Dic79C2411yescharacteristic
138C2×Dic79C2411yescharacteristic
139C2×Dic79C2413yes—
140C2×Dic79C2413yes—
141C2×Dic79C2411yescharacteristic
142C2×Dic79C2411yescharacteristic
143D316—26no—
144D316—26no—
145D316—43no—
146D316—46no—
147C4×D79—26no—
148C4×D79—26no—
149C4×D79—26no—
150C4×D79—43no—
151C4×D79—46no—
152Dic158—26no—
153C22×C79—23no—
154C22×C79—23no—
155C22×C79—23no—
156C22×C79C2511yescharacteristic
157C2×D79—23no—
158C2×D79—23no—
159C2×D79—23no—
160C2×D79—23no—
161C2×D79—23no—
162C2×D79—23no—
163C4×C79—21no—
164C4×C79—23no—
165C4×C79—23no—
166C4×C79—21no—
167C4×C79—21no—
168C4×C79C2×#4116yes—
169Dic79—21no—
170Dic79—21no—
171Dic79—23no—
172Dic79—21no—
173Dic79—23no—
174Dic79H(5,𝔽2)12yes—
175C2×C79—43no—
176C2×C79C22×#4111yescharacteristic
177C2×C79C2×H(5,𝔽2)11yescharacteristic
178C2×C79#58511yescharacteristic
179D79—43no—
180D79—43no—
181#3517—791nomaximal, Sylow
182C79#351711yescharacteristic, complemented, Sylow
183#557—791no—
184#553—793no—
185#553—793no—
186#550—791no—
187#550—791no—
188#550—791no—
189#550—793no—
190#547—793no—
191#540—793no—
192#540—793no—
193#532—793no—
194#529—793no—
195#520—791no—
196#520D7911yescharacteristic, complemented
197C4×#41—791no—
198C2×#41—791no—
199C2×#41—791no—
200C2×#41—796no—
201C2×#41C2×D7911yescharacteristic
202C4⋊Q8—793no—
203#127—793no—
204#127—793no—
205#127—793no—
206#126—793no—
207#126—793no—
208#125—791no—
209#125—793no—
210#125—793no—
211#124—793no—
212#123—796no—
213#123—793no—
214#123—796no—
215#123—793no—
216C22⋊Q8—793no—
217#121—793no—
218#121—793no—
219#121—793no—
220#121—796no—
221#121—793no—
222#121—793no—
223#121—793no—
224#121—793no—
225#121—796no—
226#121—796no—
227C4×Q8—791no—
228C4×Q8—791no—
229C4×Q8—793no—
230C4×Q8C2×D7911yescharacteristic
231C4×Q8C2×D7911yescharacteristic
232C4×D4—793no—
233C4×D4—793no—
234C4×D4—796no—
235C4×D4—793no—
236C4×D4—793no—
237C4×D4—793no—
238C4×D4—793no—
239C4×D4—793no—
240C4×D4C2×D7913yes—
241C4×D4C2×D7913yes—
242#117—793no—
243#117—793no—
244#117C2×D7913yes—
245C2×C4⋊C4—793no—
246C2×C4⋊C4C2×D7913yes—
247C2×C42—793no—
248#41—24no—
249#41—31612no—
250#41—1584no—
251#41—1584no—
252C2×Q8—793no—
253C2×Q8C22×D7911yescharacteristic
254C2×D4—796no—
255C2×D4—793no—
256C2×D4—796no—
257C2×D4—793no—
258C2×D4—793no—
259C2×D4—796no—
260C2×D4—793no—
261C2×D4—793no—
262C2×D4C22×D7913yes—
263C22×C4—793no—
264C22×C4—796no—
265C22×C4—793no—
266C22×C4—793no—
267C22×C4—793no—
268C22×C4C22×D7913yes—
269C22×C4C22×D7916yes—
270C4⋊C4—793no—
271C4⋊C4—791no—
272C4⋊C4—791no—
273C4⋊C4—791no—
274C4⋊C4—796no—
275C4⋊C4—791no—
276C4⋊C4—793no—
277C4⋊C4—793no—
278C4⋊C4—793no—
279C4⋊C4—791no—
280C4⋊C4—793no—
281C4⋊C4C22×D7913yes—
282C4⋊C4C22×D7913yes—
283C4⋊C4C22×D7913yes—
284C4⋊C4C22×D7911yescharacteristic
285C22⋊C4—793no—
286C22⋊C4—793no—
287C22⋊C4—793no—
288C22⋊C4—796no—
289C22⋊C4—793no—
290C22⋊C4—793no—
291C22⋊C4—793no—
292C22⋊C4—793no—
293C22⋊C4—793no—
294C22⋊C4C22×D7913yes—
295C22⋊C4C22×D7913yes—
296C42—791no—
297C42—793no—
298C42—793no—
299C42—793no—
300C42C22×D7913yes—
301C42C22×D7913yes—
302C23—793no—
303C23—793no—
304C23C23×D7913yes—
305Q8—1586no—
306Q8#1192962414yescomplemented
307D4—26no—
308D4—3166no—
309D4—3163no—
310D4—3166no—
311D4—3163no—
312D4—3166no—
313D4—1586no—
314D4—1586no—
315D4—1586no—
316C2×C4—26no—
317C2×C4—46no—
318C2×C4—3166no—
319C2×C4—3163no—
320C2×C4—791no—
321C2×C4—791no—
322C2×C4—791no—
323C2×C4—793no—
324C2×C4—793no—
325C2×C4—791no—
326C2×C4—1586no—
327C2×C4—1586no—
328C2×C4—1586no—
329C2×C4C23×D7911yescharacteristic
330C2×C4C23×D7913yes—
331C2×C4C23×D7911yescharacteristic
332C2×C4C23×D7913yes—
333C2×C4C23×D7911yescharacteristic
334C2×C4C23×D7913yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1583no—
339C22—1583no—
340C22—1583no—
341C22—1583no—
342C22—1583no—
343C22—1583no—
344C22C24×D7911yescharacteristic
345C4—23no—
346C4—23no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—792no—
351C4—1583no—
352C4—1583no—
353C4—1581no—
354C4—1581no—
355C4—1581no—
356C4C2×#1192962416yes—
357C2—43no—
358C2—3163no—
359C2—3163no—
360C2C22×#1192962411yescharacteristic
361C2H(5,𝔽2)×D7911yescharacteristic
362C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#520×C79C2
hypercenter#520D79
hypocenterC79#3517
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #19099289C2semidirect
2. #19099263C2semidirect
3. #19099261C2semidirect
4. #19099235C2semidirect
5. #19099234C2semidirect
6. #19099232C2semidirect
7. #19099232C2semidirect
8. #19099214C2semidirect
9. #19099168C2semidirect
10. #19099164C2semidirect
11. #19099110C2semidirect
12. #19099091C2semidirect
13. #19099045C2semidirect
14. C4×#11929625C2semidirect
15. #520×C79C2semidirect
123. Q8×C79#41semidirect
182. C79#3517semidirect
196. #520D79semidirect
306. Q8#11929624semidirect