Finite Group Databasebeta

← #23777397

Subgroups — #23777397

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#19099260C213yesmaximal, complemented
3#19099259C213yesmaximal, complemented
4#19099233C213yesmaximal, complemented
5#19099232C211yesmaximal, characteristic, complemented
6#19099232C211yesmaximal, characteristic, complemented
7#19099213C213yesmaximal, complemented
8#19099167C213yesmaximal, complemented
9#19099166C213yesmaximal, complemented
10#19099108C213yesmaximal, complemented
11#19099093C213yesmaximal, complemented
12#19099046C211yesmaximal, characteristic, complemented
13#19099045C211yesmaximal, characteristic, complemented
14#41×Dic79C211yesmaximal, characteristic, complemented
15#550×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17C2×#11929623C2216yes—
18#14579160C2213yes—
19#14579157C2213yes—
20#14579157C2213yes—
21#14579156C2213yes—
22#14579154C2213yes—
23#14579153C2213yes—
24#14579152C2213yes—
25#14579151C2211yescharacteristic
26#14579147C2213yes—
27#14579145C2213yes—
28#14579143C2216yes—
29#14579142C2213yes—
30#14579142C2216yes—
31#14579138C2213yes—
32#14579135C2213yes—
33#14579135C2213yes—
34#14579133C2216yes—
35#14579133C2216yes—
36#14579133C2213yes—
37#14579132C2216yes—
38#14579132C2213yes—
39#14579132C2213yes—
40Q8×Dic79C2211yescharacteristic
41Q8×Dic79C2211yescharacteristic
42Q8×Dic79C2211yescharacteristic
43C4×Dic158C2213yes—
44C4×D316C2213yes—
45D4×Dic79C2213yes—
46D4×Dic79C2213yes—
47D4×Dic79C2213yes—
48D4×Dic79C2213yes—
49#14579123C2213yes—
50C4×C22⋊D79C2213yes—
51C4×C22⋊D79C2213yes—
52(C22⋊D79)⋊C4C2216yes—
53#14579120C2213yes—
54#14579120C2213yes—
55#14579118C2213yes—
56C2×C4⋊Dic79C2213yes—
57C4⋊C4×D79C2213yes—
58C2×C4×Dic79C2213yes—
59C2×#41×C79C2211yescharacteristic
60C2×#41×C79C2211yescharacteristic
61#127×C79C2213yes—
62#121×C79C2213yes—
63#121×C79C2213yes—
64C4×Q8×C79C2211yescharacteristic
65C4×D4×C79C2213yes—
66#11929625—24no—
67#11929623—412no—
68C2×Dic158C2313yes—
69C2×D316C2313yes—
70C2×C22⋊D79C2313yes—
71C2×C22⋊D79C2316yes—
72C2×C22⋊D79C2316yes—
73C2×C22⋊D79C2313yes—
74C22×Dic79C2313yes—
75C22×Dic79C2316yes—
76C22×Dic79C2313yes—
77C2×C4×D79C2313yes—
78C2×C4×D79C2316yes—
79C4⋊Dic79C2313yes—
80C4⋊Dic79C2313yes—
81C4⋊Dic79C2313yes—
82C4⋊Dic79C2311yescharacteristic
83C4⋊Dic79C2313yes—
84C4⋊Dic79C2313yes—
85Dic79⋊C4C2313yes—
86Dic79⋊C4C2311yescharacteristic
87Dic79⋊C4C2316yes—
88Dic79⋊C4C2311yescharacteristic
89Dic79⋊C4C2311yescharacteristic
90Dic79⋊C4C2311yescharacteristic
91Dic79⋊C4C2313yes—
92C22⋊Dic79C2313yes—
93C22⋊Dic79C2316yes—
94C22⋊Dic79C2313yes—
95C22⋊Dic79C2313yes—
96C22⋊Dic79C2313yes—
97(C2×D79)⋊C4C2313yes—
98(C2×D79)⋊C4C2313yes—
99(C2×D79)⋊C4C2313yes—
100(C2×D79)⋊C4C2313yes—
101C4×Dic79C2311yescharacteristic
102C4×Dic79C2313yes—
103C4×Dic79C2313yes—
104C4×Dic79C2313yes—
105C4×Dic79C2313yes—
106#41×C79—24no—
107#41×C79—24no—
108C2×Q8×C79C2311yescharacteristic
109C2×D4×C79C2313yes—
110C2×D4×C79C2313yes—
111C2×D4×C79C2316yes—
112C2×D4×C79C2313yes—
113C22×C4×C79C2313yes—
114C22×C4×C79C2313yes—
115C4⋊C4×C79C2311yescharacteristic
116C4⋊C4×C79C2313yes—
117C22⋊C4×C79C2313yes—
118C22⋊C4×C79C2313yes—
119C42×C79C2313yes—
120C23×C79C2413yes—
121C23×C79C2413yes—
122C22×D79C2413yes—
123Q8×C79#4114yescomplemented
124D4×C79—26no—
125D4×C79—26no—
126D4×C79—46no—
127D4×C79—43no—
128C2×C4×C79—26no—
129C2×C4×C79—26no—
130C2×C4×C79C2411yescharacteristic
131C2×C4×C79C2411yescharacteristic
132C2×C4×C79C2411yescharacteristic
133C2×C4×C79C2413yes—
134C2×C4×C79C2413yes—
135C22⋊D79—26no—
136C22⋊D79—43no—
137C22⋊D79—46no—
138C22⋊D79—46no—
139C2×Dic79—26no—
140C2×Dic79—43no—
141C2×Dic79—46no—
142C2×Dic79C2411yescharacteristic
143C2×Dic79C2411yescharacteristic
144C2×Dic79C2413yes—
145C2×Dic79C2413yes—
146C2×Dic79C2413yes—
147C2×Dic79C2411yescharacteristic
148C2×Dic79C2411yescharacteristic
149D316—26no—
150C4×D79—26no—
151C4×D79—46no—
152Dic158—26no—
153C22×C79—23no—
154C22×C79—23no—
155C22×C79—23no—
156C22×C79—23no—
157C22×C79—23no—
158C22×C79—23no—
159C22×C79C2511yescharacteristic
160C2×D79—23no—
161C2×D79—23no—
162C2×D79—23no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—23no—
166C4×C79—21no—
167C4×C79C2×#4116yes—
168Dic79—21no—
169Dic79—23no—
170Dic79—23no—
171Dic79—21no—
172Dic79—23no—
173Dic79—21no—
174Dic79H(5,𝔽2)12yes—
175C2×C79—43no—
176C2×C79—43no—
177C2×C79C22×#4111yescharacteristic
178C2×C79C2×H(5,𝔽2)11yescharacteristic
179C2×C79#58511yescharacteristic
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—793no—
189#550D7911yescharacteristic, complemented
190#547—793no—
191#540—793no—
192#540—793no—
193#532—793no—
194#529—793no—
195#520—791no—
196#520—791no—
197C4×#41—791no—
198C2×#41—791no—
199C2×#41—796no—
200C2×#41C2×D7911yescharacteristic
201C2×#41C2×D7911yescharacteristic
202C4⋊Q8—793no—
203#127—793no—
204#127—793no—
205#127C2×D7913yes—
206#126—793no—
207#126—793no—
208#125—791no—
209#125—793no—
210#125—793no—
211#124—793no—
212#123—793no—
213#123—796no—
214#123—793no—
215#123—796no—
216C22⋊Q8—793no—
217#121—793no—
218#121—796no—
219#121—793no—
220#121—796no—
221#121—793no—
222#121—796no—
223#121—793no—
224#121—793no—
225#121C2×D7913yes—
226#121C2×D7913yes—
227C4×Q8—791no—
228C4×Q8—791no—
229C4×Q8—791no—
230C4×Q8—793no—
231C4×Q8C2×D7911yescharacteristic
232C4×D4—793no—
233C4×D4—793no—
234C4×D4—793no—
235C4×D4—793no—
236C4×D4—793no—
237C4×D4—793no—
238C4×D4—793no—
239C4×D4—796no—
240C4×D4—793no—
241C4×D4C2×D7913yes—
242#117—793no—
243#117—793no—
244#117—793no—
245C2×C4⋊C4—793no—
246C2×C4⋊C4—793no—
247C2×C42—793no—
248#41—24no—
249#41—24no—
250#41—31612no—
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×D4C22×D7913yes—
260C2×D4C22×D7913yes—
261C2×D4C22×D7913yes—
262C2×D4C22×D7916yes—
263C22×C4—793no—
264C22×C4—796no—
265C22×C4—796no—
266C22×C4—793no—
267C22×C4—793no—
268C22×C4C22×D7913yes—
269C22×C4C22×D7913yes—
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—793no—
280C4⋊C4—791no—
281C4⋊C4—793no—
282C4⋊C4—793no—
283C4⋊C4C22×D7913yes—
284C4⋊C4C22×D7911yescharacteristic
285C22⋊C4—793no—
286C22⋊C4—796no—
287C22⋊C4—793no—
288C22⋊C4—793no—
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—
300C42—793no—
301C42C22×D7913yes—
302C23—793no—
303C23C23×D7913yes—
304C23C23×D7913yes—
305Q8—1586no—
306Q8#1192962314yescomplemented
307D4—26no—
308D4—26no—
309D4—43no—
310D4—46no—
311D4—3166no—
312D4—3163no—
313D4—3166no—
314D4—1586no—
315D4—1586no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—3166no—
319C2×C4—3166no—
320C2×C4—3163no—
321C2×C4—791no—
322C2×C4—791no—
323C2×C4—791no—
324C2×C4—793no—
325C2×C4—793no—
326C2×C4—793no—
327C2×C4—791no—
328C2×C4—1586no—
329C2×C4—1586no—
330C2×C4C23×D7911yescharacteristic
331C2×C4C23×D7911yescharacteristic
332C2×C4C23×D7913yes—
333C2×C4C23×D7911yescharacteristic
334C2×C4C23×D7913yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—23no—
341C22—1583no—
342C22—1583no—
343C22—1583no—
344C22C24×D7911yescharacteristic
345C4—23no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—792no—
350C4—1583no—
351C4—1583no—
352C4—1581no—
353C4—1583no—
354C4—1581no—
355C4—1581no—
356C4C2×#1192962316yes—
357C2—43no—
358C2—43no—
359C2—3163no—
360C2C22×#1192962311yescharacteristic
361C2H(5,𝔽2)×D7911yescharacteristic
362C2#1909939911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#550×C79C2
hypercenter#550D79
hypocenterC79#3517
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #19099289C2semidirect
2. #19099260C2semidirect
3. #19099259C2semidirect
4. #19099233C2semidirect
5. #19099232C2semidirect
6. #19099232C2semidirect
7. #19099213C2semidirect
8. #19099167C2semidirect
9. #19099166C2semidirect
10. #19099108C2semidirect
11. #19099093C2semidirect
12. #19099046C2semidirect
13. #19099045C2semidirect
14. #41×Dic79C2semidirect
15. #550×C79C2semidirect
123. Q8×C79#41semidirect
182. C79#3517semidirect
189. #550D79semidirect
306. Q8#11929623semidirect