Finite Group Databasebeta

← #23774213

Subgroups — #23774213

Order 10112.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099073C211yesmaximal, characteristic, complemented
2C2×#14579136C211yesmaximal, characteristic, complemented
3#19098893C211yesmaximal, characteristic, complemented
4#19098890C212yesmaximal, complemented
5#19098815C212yesmaximal, complemented
6#19098810C212yesmaximal, complemented
7#19098716C211yesmaximal, characteristic, complemented
8#19098709C212yesmaximal, complemented
9#19098696C211yesmaximal, characteristic, complemented
10#19098525C211yesmaximal, characteristic, complemented
11#480×C79C211yesmaximal, characteristic, complemented
12C2×#11929623C2211yescharacteristic, complemented
13C22×D316—21no—
14C22×D316C2211yescharacteristic
15C22×C4×D79C2211yescharacteristic
16C2×#11929613C2212yescomplemented
17C2×#11929609C2212yescomplemented
18C2×#11929607C2212yescomplemented
19#14579143—21no—
20#14579141C2211yescharacteristic, complemented
21#14579140C2211yescharacteristic
22#14579136—21no—
23#14579136—21no—
24#14579136C2212yes—
25#14579136C2212yes—
26#14579132C2211yescharacteristic, complemented
27#14579129—21no—
28D4×Dic79—21no—
29(C22⋊D79)⋊C4—21no—
30#14579119C2211yescharacteristic, complemented
31C2×(C2×D79)⋊C4—21no—
32#14579092C2212yescomplemented
33#14579084C2212yescomplemented
34#14579083C2212yescomplemented
35#14579081C2212yescomplemented
36C22⋊(C79⋊C8)C2211yescharacteristic
37(C2×D79)⋊C8C2212yescomplemented
38#121×C79C2211yescharacteristic
39C2×C4⋊C4×C79C2211yescharacteristic
40#107×C79C2212yescomplemented
41#102×C79C2212yescomplemented
42C22⋊C8×C79C2211yescharacteristic
43C23×D79—21no—
44C23×D79D411yescharacteristic
45#11929623—42no—
46C2×Dic158C2311yescharacteristic
47C2×D316—22no—
48C2×D316—22no—
49C2×D316—21no—
50C2×D316—42no—
51C2×D316—44no—
52C2×D316C2312yes—
53C2×C22⋊D79—21no—
54C2×C22⋊D79—21no—
55C22×Dic79—21no—
56C22×Dic79D411yescharacteristic
57C2×C4×D79—42no—
58C2×C4×D79D411yescharacteristic
59C2×C4×D79D411yescharacteristic
60C2×C4×D79C2311yescharacteristic
61C2×C4×D79C2311yescharacteristic
62#11929613—42no—
63#11929609—42no—
64#11929607—42no—
65C2×C79⋊C8C2312yes—
66C4⋊Dic79—21no—
67C4⋊Dic79C2311yescharacteristic
68Dic79⋊C4—21no—
69C22⋊Dic79—21no—
70C22⋊Dic79—21no—
71(C2×D79)⋊C4—22no—
72(C2×D79)⋊C4—22no—
73(C2×D79)⋊C4—21no—
74(C2×D79)⋊C4—21no—
75C4×Dic79—21no—
76C2×D4×C79—21no—
77C2×D4×C79C2311yescharacteristic
78C22×C4×C79—21no—
79C22×C4×C79C2311yescharacteristic
80C2×C8×C79C2312yes—
81C4⋊C4×C79—21no—
82C4⋊C4×C79C2312yes—
83C4⋊C4×C79C2311yescharacteristic
84C22⋊C4×C79—21no—
85C23×C79—21no—
86C23×C79C2×D411yescharacteristic
87C22×D79—22no—
88C22×D79—22no—
89C22×D79—41no—
90C22×D79—41no—
91C22×D79—42no—
92C22×D79—42no—
93C22×D79—42no—
94C22×D79C2×D411yescharacteristic
95C22×D79C2×D411yescharacteristic
96D4×C79—22no—
97D4×C79—42no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—42no—
102C2×C4×C79C2×D411yescharacteristic
103C2×C4×C79C2411yescharacteristic
104C22⋊D79—42no—
105C22⋊D79—42no—
106C2×Dic79—21no—
107C2×Dic79—21no—
108C2×Dic79—41no—
109C2×Dic79—42no—
110C2×Dic79C2×D411yescharacteristic
111C2×Dic79C2×D411yescharacteristic
112D316—22no—
113D316—22no—
114D316—42no—
115D316—44no—
116D316—44no—
117C4×D79—22no—
118C4×D79—22no—
119C4×D79—41no—
120C4×D79—41no—
121Dic158—22no—
122C8×C79—22no—
123C79⋊C8—22no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—41no—
127C22×C79—41no—
128C22×C79—41no—
129C22×C79AGL(1,ℤ/8)12yes—
130C22×C79C22×D411yescharacteristic
131C2×D79—21no—
132C2×D79—21no—
133C2×D79—21no—
134C2×D79—21no—
135C2×D79—21no—
136C2×D79—21no—
137C2×D79—44no—
138C2×D79—42no—
139C2×D79—42no—
140C2×D79—42no—
141C2×D79—84no—
142C4×C79—21no—
143C4×C79—42no—
144C4×C79—41no—
145C4×C79H(5,𝔽2)12yes—
146Dic79—21no—
147Dic79—21no—
148Dic79—41no—
149Dic79—41no—
150C2×C79—22no—
151C2×C79—81no—
152C2×C79#53411yescharacteristic
153C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C79#57511yescharacteristic
155D79—41no—
156D79—41no—
157D79—82no—
158#3194—791nomaximal, Sylow
159C79#319411yescharacteristic, complemented, Sylow
160#525—791no—
161C2×#121—791no—
162#483—791no—
163#482—792no—
164#480D7911yescharacteristic, complemented
165#469—792no—
166#468—792no—
167#450—791no—
168#449—792no—
169#447—791no—
170#407—791no—
171C2×#41—791no—
172C22×D4—791no—
173C22×D4—1581no—
174C23×C4—791no—
175C2×SD16—792no—
176C2×D8—792no—
177C2×M16—792no—
178#123—1581no—
179C22⋊Q8—791no—
180C22⋊Q8—791no—
181#121—791no—
182#121—792no—
183#121—792no—
184#121—1581no—
185#121—1581no—
186#121C2×D7911yescharacteristic
187#120—1581no—
188C4×D4—1581no—
189C4×D4—1581no—
190#117—791no—
191C2×C4⋊C4C2×D7911yescharacteristic
192C2×C22⋊C4—1581no—
193#107C2×D7912yescomplemented
194#106—792no—
195#103—792no—
196#102—792no—
197#102—792no—
198#102C2×D7912yescomplemented
199C22⋊C8—791no—
200C22⋊C8—792no—
201C22⋊C8C2×D7911yescharacteristic
202C24—791no—
203C24—1581no—
204#41—3162no—
205C2×Q8—791no—
206C2×D4—21no—
207C2×D4—3162no—
208C2×D4—3164no—
209C2×D4—792no—
210C2×D4—1582no—
211C2×D4—1582no—
212C2×D4—1581no—
213C2×D4—1581no—
214C2×D4—1581no—
215C2×D4C22×D7911yescharacteristic
216C22×C4—21no—
217C22×C4—3162no—
218C22×C4—791no—
219C22×C4—791no—
220C22×C4—791no—
221C22×C4—791no—
222C22×C4—791no—
223C22×C4—1581no—
224C22×C4C22×D7911yescharacteristic
225SD16—3162no—
226D8—3162no—
227M16—3162no—
228C2×C8—792no—
229C2×C8C22×D7912yes—
230C4⋊C4—21no—
231C4⋊C4—791no—
232C4⋊C4—1581no—
233C4⋊C4—1581no—
234C4⋊C4C22×D7911yescharacteristic
235C4⋊C4C22×D7912yes—
236C22⋊C4—21no—
237C22⋊C4—1582no—
238C22⋊C4—1582no—
239C22⋊C4—1581no—
240C22⋊C4—1581no—
241C22⋊C4—1581no—
242C22⋊C4—1581no—
243C42—1581no—
244C23—21no—
245C23—3161no—
246C23—3161no—
247C23—3162no—
248C23—3162no—
249C23—3162no—
250C23—791no—
251C23—791no—
252C23—1582no—
253C23—1582no—
254C23D4×D7911yescharacteristic
255Q8—1582no—
256D4—22no—
257D4—42no—
258D4—3162no—
259D4—3164no—
260D4—3164no—
261D4—3162no—
262D4—3162no—
263D4—1582no—
264D4—1582no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—3161no—
270C2×C4—3161no—
271C2×C4—3162no—
272C2×C4—3161no—
273C2×C4—791no—
274C2×C4—791no—
275C2×C4—1581no—
276C2×C4—1581no—
277C2×C4—1582no—
278C2×C4—1582no—
279C2×C4D4×D7911yescharacteristic
280C2×C4C23×D7911yescharacteristic
281C8—22no—
282C8—1582no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—3164no—
289C22—3162no—
290C22—3162no—
291C22—3162no—
292C22—6324no—
293C22—1581no—
294C22—1581no—
295C22—1581no—
296C22—1581no—
297C22—1581no—
298C22—1581no—
299C22#1457918512yes—
300C22C2×D4×D7911yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—3161no—
305C4—3161no—
306C4—1581no—
307C4—1581no—
308C4#1457920712yes—
309C2—22no—
310C2—81no—
311C2—3161no—
312C2—3161no—
313C2—6322no—
314C2#1909912411yescharacteristic
315C2C2×#1457918511yescharacteristic
316C2#1909936211yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#480×C79C2
socleC22×C79C22×D4
hypercenter#480D79
hypocenterC79#3194
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #19099073C2semidirect
2. C2×#14579136C2semidirect
3. #19098893C2semidirect
4. #19098890C2semidirect
5. #19098815C2semidirect
6. #19098810C2semidirect
7. #19098716C2semidirect
8. #19098709C2semidirect
9. #19098696C2semidirect
10. #19098525C2semidirect
11. #480×C79C2semidirect
12. C2×#11929623C22semidirect
16. C2×#11929613C22semidirect
17. C2×#11929609C22semidirect
18. C2×#11929607C22semidirect
20. #14579141C22semidirect
26. #14579132C22semidirect
30. #14579119C22semidirect
32. #14579092C22semidirect
33. #14579084C22semidirect
34. #14579083C22semidirect
35. #14579081C22semidirect
37. (C2×D79)⋊C8C22semidirect
40. #107×C79C22semidirect
41. #102×C79C22semidirect
159. C79#3194semidirect
164. #480D79semidirect
193. #107C2×D79semidirect
198. #102C2×D79semidirect