Finite Group Databasebeta

← #23777394

Subgroups — #23777394

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#19099289C211yesmaximal, characteristic, complemented
2#19099287C213yesmaximal, complemented
3#19099271C213yesmaximal, complemented
4#19099259C213yesmaximal, complemented
5#19099247C213yesmaximal, complemented
6#19099224C211yesmaximal, characteristic, complemented
7#19099224C211yesmaximal, characteristic, complemented
8#19099186C213yesmaximal, complemented
9#19099174C213yesmaximal, complemented
10#19099099C213yesmaximal, complemented
11#19099093C213yesmaximal, complemented
12#19099045C211yesmaximal, characteristic, complemented
13#19099041C211yesmaximal, characteristic, complemented
14C2×Q8×Dic79C211yesmaximal, characteristic, complemented
15D4×Q8×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17#14579160C2213yes—
18#14579156C2216yes—
19#14579156C2216yes—
20#14579156C2213yes—
21#14579154C2213yes—
22#14579153C2213yes—
23#14579152C2213yes—
24#14579151C2213yes—
25#14579151C2216yes—
26#14579150C2213yes—
27#14579150C2213yes—
28#14579147C2213yes—
29#14579145C2213yes—
30#14579143C2216yes—
31#14579142C2211yescharacteristic
32#14579141C2216yes—
33#14579138C2213yes—
34#14579137C2213yes—
35#14579137C2213yes—
36#14579135C2213yes—
37#14579135C2213yes—
38#14579133C2213yes—
39Q8×Dic79C2213yes—
40Q8×Dic79C2213yes—
41Q8×Dic79C2211yescharacteristic
42Q8×Dic79C2211yescharacteristic
43Q8×Dic79C2213yes—
44Q8×Dic79C2213yes—
45Q8×Dic79C2211yescharacteristic
46C4×Dic158C2213yes—
47C4×D316C2213yes—
48#14579123C2213yes—
49C4×C22⋊D79C2213yes—
50C4×C22⋊D79C2213yes—
51(C22⋊D79)⋊C4C2216yes—
52#14579120C2213yes—
53#14579120C2213yes—
54#14579119C2216yes—
55#14579118C2213yes—
56C2×C4⋊Dic79C2213yes—
57C4⋊C4×D79C2213yes—
58C2×C4×Dic79C2213yes—
59C22×Q8×C79C2211yescharacteristic
60C22×Q8×C79C2211yescharacteristic
61C4⋊Q8×C79C2213yes—
62C22⋊Q8×C79C2213yes—
63C22⋊Q8×C79C2213yes—
64C4×Q8×C79C2211yescharacteristic
65C4×D4×C79C2213yes—
66#11929625—24no—
67C2×Dic158C2313yes—
68C2×D316C2313yes—
69C2×C22⋊D79C2313yes—
70C2×C22⋊D79C2313yes—
71C22×Dic79C2311yescharacteristic
72C22×Dic79C2313yes—
73C2×C4×D79C2313yes—
74C2×C4×D79C2316yes—
75C4⋊Dic79C2313yes—
76C4⋊Dic79C2313yes—
77C4⋊Dic79C2316yes—
78C4⋊Dic79C2313yes—
79C4⋊Dic79C2313yes—
80C4⋊Dic79C2313yes—
81C4⋊Dic79C2313yes—
82Dic79⋊C4C2313yes—
83Dic79⋊C4C2311yescharacteristic
84Dic79⋊C4C2316yes—
85Dic79⋊C4C2313yes—
86Dic79⋊C4C2313yes—
87Dic79⋊C4C2311yescharacteristic
88Dic79⋊C4C2313yes—
89C22⋊Dic79C2313yes—
90C22⋊Dic79C2311yescharacteristic
91C22⋊Dic79C2311yescharacteristic
92C22⋊Dic79C2311yescharacteristic
93(C2×D79)⋊C4C2313yes—
94(C2×D79)⋊C4C2313yes—
95(C2×D79)⋊C4C2313yes—
96(C2×D79)⋊C4C2316yes—
97(C2×D79)⋊C4C2316yes—
98(C2×D79)⋊C4C2313yes—
99C4×Dic79C2313yes—
100C4×Dic79C2313yes—
101C4×Dic79C2313yes—
102C4×Dic79C2313yes—
103C4×Dic79C2316yes—
104C4×Dic79C2313yes—
105C2×Q8×C79—24no—
106C2×Q8×C79—24no—
107C2×Q8×C79C2313yes—
108C2×Q8×C79C2313yes—
109C2×Q8×C79C2311yescharacteristic
110C2×D4×C79C2311yescharacteristic
111C22×C4×C79C2313yes—
112C22×C4×C79C2313yes—
113C4⋊C4×C79C2313yes—
114C4⋊C4×C79C2313yes—
115C4⋊C4×C79C2316yes—
116C22⋊C4×C79C2313yes—
117C22⋊C4×C79C2313yes—
118C42×C79C2313yes—
119C23×C79C2411yescharacteristic
120C23×C79C2411yescharacteristic
121C22×D79C2413yes—
122Q8×C79—26no—
123Q8×C79—26no—
124Q8×C79#4114yes—
125D4×C79—22no—
126C2×C4×C79—26no—
127C2×C4×C79—26no—
128C2×C4×C79C2413yes—
129C2×C4×C79C2413yes—
130C2×C4×C79C2411yescharacteristic
131C2×C4×C79C2413yes—
132C2×C4×C79C2413yes—
133C22⋊D79—26no—
134C22⋊D79—43no—
135C2×Dic79—22no—
136C2×Dic79—43no—
137C2×Dic79C2411yescharacteristic
138C2×Dic79C2411yescharacteristic
139C2×Dic79C2413yes—
140C2×Dic79C2411yescharacteristic
141C2×Dic79C2413yes—
142C2×Dic79C2413yes—
143C2×Dic79C2411yescharacteristic
144D316—26no—
145C4×D79—26no—
146C4×D79—46no—
147Dic158—43no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C792−1+412yes—
154C22×C79C2511yescharacteristic
155C2×D79—23no—
156C2×D79—23no—
157C2×D79—23no—
158C4×C79—23no—
159C4×C79—23no—
160C4×C79—21no—
161C4×C79—23no—
162C4×C79C2×#4116yes—
163Dic79—21no—
164Dic79—23no—
165Dic79—21no—
166Dic79—21no—
167Dic79—23no—
168Dic79—21no—
169Dic79—23no—
170C2×C79—22no—
171C2×C79—41no—
172C2×C79C22×#4111yescharacteristic
173C2×C79C2×2−1+411yescharacteristic
174C2×C79#58511yescharacteristic
175D79—43no—
176#3516—791nomaximal, Sylow
177C79#351611yescharacteristic, complemented, Sylow
178#557—791no—
179#556—793no—
180#554—793no—
181#553—793no—
182#552—793no—
183D4×Q8D7911yescharacteristic, complemented
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×Q8C2×D7911yescharacteristic
195C22×Q8C2×D7911yescharacteristic
196C4⋊Q8—793no—
197C4⋊Q8C2×D7913yes—
198#126—796no—
199#126—793no—
200#126—796no—
201#126—793no—
202#125—793no—
203#125—793no—
204#125—796no—
205#125—793no—
206#124—793no—
207#124—793no—
208#124—793no—
209#123—791no—
210#123—793no—
211#123—796no—
212C22⋊Q8—793no—
213C22⋊Q8—793no—
214C22⋊Q8—796no—
215C22⋊Q8—793no—
216C22⋊Q8C2×D7913yes—
217C22⋊Q8C2×D7913yes—
218#121—793no—
219#121—793no—
220#121—793no—
221C4×Q8—791no—
222C4×Q8—793no—
223C4×Q8—791no—
224C4×Q8—793no—
225C4×Q8—793no—
226C4×Q8—791no—
227C4×Q8—793no—
228C4×Q8—793no—
229C4×Q8C2×D7911yescharacteristic
230C4×D4—793no—
231C4×D4—793no—
232C4×D4—793no—
233C4×D4—796no—
234C4×D4—793no—
235C4×D4C2×D7913yes—
236#117—793no—
237#117—793no—
238#117—796no—
239#117—793no—
240C2×C4⋊C4—793no—
241C2×C4⋊C4—793no—
242C2×C42—793no—
243#41—1584no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—793no—
247C2×Q8C22×D7913yes—
248C2×Q8C22×D7913yes—
249C2×Q8C22×D7911yescharacteristic
250C2×D4—793no—
251C2×D4—793no—
252C2×D4—793no—
253C2×D4C22×D7911yescharacteristic
254C22×C4—793no—
255C22×C4—796no—
256C22×C4—791no—
257C22×C4—793no—
258C22×C4C22×D7913yes—
259C22×C4C22×D7913yes—
260C4⋊C4—793no—
261C4⋊C4—791no—
262C4⋊C4—793no—
263C4⋊C4—793no—
264C4⋊C4—796no—
265C4⋊C4—791no—
266C4⋊C4—793no—
267C4⋊C4—793no—
268C4⋊C4—793no—
269C4⋊C4—796no—
270C4⋊C4—793no—
271C4⋊C4—793no—
272C4⋊C4—793no—
273C4⋊C4—793no—
274C4⋊C4C22×D7913yes—
275C4⋊C4C22×D7916yes—
276C4⋊C4C22×D7913yes—
277C22⋊C4—793no—
278C22⋊C4—791no—
279C22⋊C4—791no—
280C22⋊C4—791no—
281C22⋊C4—793no—
282C22⋊C4—793no—
283C22⋊C4—793no—
284C22⋊C4—796no—
285C22⋊C4—796no—
286C22⋊C4—793no—
287C22⋊C4C22×D7913yes—
288C22⋊C4C22×D7913yes—
289C42—793no—
290C42—793no—
291C42—793no—
292C42—793no—
293C42—796no—
294C42—793no—
295C42C22×D7913yes—
296C23—793no—
297C23C23×D7911yescharacteristic
298C23C23×D7911yescharacteristic
299Q8—26no—
300Q8—26no—
301Q8—3163no—
302Q8#1192962314yes—
303D4—22no—
304D4—3163no—
305D4—1586no—
306D4—1586no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—3166no—
310C2×C4—3163no—
311C2×C4—791no—
312C2×C4—791no—
313C2×C4—793no—
314C2×C4—791no—
315C2×C4—793no—
316C2×C4—793no—
317C2×C4—791no—
318C2×C4—1582no—
319C2×C4—1586no—
320C2×C4C23×D7913yes—
321C2×C4C23×D7913yes—
322C2×C4C23×D7911yescharacteristic
323C2×C4C23×D7913yes—
324C2×C4C23×D7913yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—1583no—
331C22—1583no—
332C22—1583no—
333C22#1457920912yes—
334C22C24×D7911yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—23no—
339C4—1581no—
340C4—1581no—
341C4—1583no—
342C4—1583no—
343C4—1583no—
344C4—1581no—
345C4—1581no—
346C4C2×#1192962316yes—
347C2—22no—
348C2—41no—
349C2—3163no—
350C2C22×#1192962311yescharacteristic
351C2C2×#1457920911yescharacteristic
352C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
FittingD4×Q8×C79C2
hypercenterD4×Q8D79
hypocenterC79#3516
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #19099289C2semidirect
2. #19099287C2semidirect
3. #19099271C2semidirect
4. #19099259C2semidirect
5. #19099247C2semidirect
6. #19099224C2semidirect
7. #19099224C2semidirect
8. #19099186C2semidirect
9. #19099174C2semidirect
10. #19099099C2semidirect
11. #19099093C2semidirect
12. #19099045C2semidirect
13. #19099041C2semidirect
14. C2×Q8×Dic79C2semidirect
15. D4×Q8×C79C2semidirect
177. C79#3516semidirect
183. D4×Q8D79semidirect