Finite Group Databasebeta

← #23774416

Subgroups — #23774416

Order 10112.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#14579157C211yesmaximal, characteristic, complemented
2#19099094C211yesmaximal, characteristic, complemented
3#19098951C212yesmaximal, complemented
4#19098926C212yesmaximal, complemented
5#19098762C212yesmaximal, complemented
6#19098756C212yesmaximal, complemented
7#19098716C211yesmaximal, characteristic, complemented
8#19098714C211yesmaximal, characteristic, complemented
9#19098696C211yesmaximal, characteristic, complemented
10#19098617C211yesmaximal, characteristic, complemented
11#447×C79C211yesmaximal, characteristic, complemented
12C22×D316C2211yescharacteristic
13C22×C22⋊D79—21no—
14C22×C22⋊D79—21no—
15C2×#11929613C2212yescomplemented
16C2×#11929612C2212yescomplemented
17C2×#11929609C2212yescomplemented
18#14579157—21no—
19#14579157—21no—
20#14579157C2212yes—
21#14579157C2212yes—
22#14579154—21no—
23#14579152C2211yescharacteristic, complemented
24#14579148C2211yescharacteristic, complemented
25#14579142—21no—
26#14579140C2211yescharacteristic
27#14579126C2211yescharacteristic, complemented
28D4×Dic79—21no—
29D4×Dic79C2211yescharacteristic, complemented
30#14579117—21no—
31C2×C4×Dic79C2211yescharacteristic
32Dic79⋊C8C2212yescomplemented
33#14579083C2212yescomplemented
34#14579082C2212yescomplemented
35#14579081C2212yescomplemented
36C22⋊(C79⋊C8)C2211yescharacteristic
37#14579071C2212yescomplemented
38C22×D4×C79C2211yescharacteristic
39C2×D8×C79C2212yescomplemented
40#121×C79C2211yescharacteristic
41#102×C79C2212yescomplemented
42C22⋊C8×C79C2211yescharacteristic
43C23×D79—21no—
44C2×Dic158C2311yescharacteristic
45C2×D316—21no—
46C2×D316—42no—
47C2×D316C2312yes—
48C2×C22⋊D79—22no—
49C2×C22⋊D79—22no—
50C2×C22⋊D79—22no—
51C2×C22⋊D79—22no—
52C2×C22⋊D79—44no—
53C2×C22⋊D79—44no—
54C22×Dic79—21no—
55C22×Dic79D411yescharacteristic
56C22×Dic79D411yescharacteristic
57#11929613—42no—
58#11929612—42no—
59#11929609—42no—
60C2×C79⋊C8C2312yes—
61C4⋊Dic79—21no—
62C4⋊Dic79C2311yescharacteristic
63Dic79⋊C4—21no—
64Dic79⋊C4—21no—
65Dic79⋊C4—21no—
66C22⋊Dic79—21no—
67C22⋊Dic79—21no—
68C22⋊Dic79—21no—
69C22⋊Dic79—21no—
70C4×Dic79—21no—
71C4×Dic79D411yescharacteristic
72C4×Dic79D411yescharacteristic
73C4×Dic79C2311yescharacteristic
74C4×Dic79C2311yescharacteristic
75C24×C79—21no—
76C2×D4×C79—21no—
77C2×D4×C79—21no—
78C2×D4×C79—42no—
79C2×D4×C79C2312yes—
80C2×D4×C79C2311yescharacteristic
81C22×C4×C79C2311yescharacteristic
82D8×C79—42no—
83C2×C8×C79C2312yes—
84C4⋊C4×C79C2311yescharacteristic
85C22⋊C4×C79—21no—
86C23×C79—21no—
87C23×C79—22no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79—42no—
91C23×C79C2×D411yescharacteristic
92C22×D79—22no—
93C22×D79—42no—
94C22×D79—42no—
95C22×D79—42no—
96D4×C79—22no—
97D4×C79—22no—
98D4×C79—22no—
99D4×C79—42no—
100D4×C79—42no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79C2×D411yescharacteristic
104C2×C4×C79C2411yescharacteristic
105C22⋊D79—44no—
106C22⋊D79—44no—
107C22⋊D79—44no—
108C22⋊D79—44no—
109C2×Dic79—22no—
110C2×Dic79—22no—
111C2×Dic79—21no—
112C2×Dic79—21no—
113C2×Dic79—42no—
114C2×Dic79C2×D411yescharacteristic
115C2×Dic79C2×D411yescharacteristic
116C2×Dic79C2×D411yescharacteristic
117C2×Dic79C2×D411yescharacteristic
118D316—22no—
119D316—22no—
120D316—42no—
121Dic158—22no—
122C8×C79—22no—
123C79⋊C8—22no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—42no—
127C22×C79—41no—
128C22×C79—42no—
129C22×C79—42no—
130C22×C79—41no—
131C22×C79—41no—
132C22×C79—84no—
133C22×C79AGL(1,ℤ/8)12yes—
134C22×C79C22×D411yescharacteristic
135C2×D79—42no—
136C2×D79—42no—
137C2×D79—42no—
138C2×D79—84no—
139C4×C79—21no—
140C4×C79—41no—
141C4×C79H(5,𝔽2)12yes—
142Dic79—21no—
143Dic79—21no—
144Dic79—21no—
145Dic79—41no—
146Dic79—41no—
147Dic79AGL(1,ℤ/8)12yes—
148C2×C79—22no—
149C2×C79—81no—
150C2×C79—82no—
151C2×C79#53511yescharacteristic
152C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
154D79—82no—
155#3217—791nomaximal, Sylow
156C79#321711yescharacteristic, complemented, Sylow
157C2×#127—791no—
158#529—791no—
159#496—792no—
160#490—792no—
161#460—792no—
162#459—792no—
163#450—791no—
164#450—791no—
165#447—791no—
166#447D7911yescharacteristic, complemented
167#432—791no—
168C22×D4—791no—
169C22×D4—1581no—
170C22×D4—1581no—
171C22×D4C2×D7911yescharacteristic
172C2×SD16—792no—
173C2×SD16—792no—
174C2×D8—792no—
175C2×D8C2×D7912yescomplemented
176#127—792no—
177#127—792no—
178#127—1581no—
179#127—1581no—
180#126—1581no—
181#125—791no—
182#124—791no—
183#123—1581no—
184C22⋊Q8—791no—
185#121C2×D7911yescharacteristic
186C4×Q8—791no—
187C4×D4—791no—
188C4×D4—1581no—
189#117—1581no—
190C2×C42—791no—
191C4⋊C8—792no—
192#102—792no—
193#102—792no—
194#102—792no—
195#102C2×D7912yescomplemented
196C22⋊C8—791no—
197C22⋊C8C2×D7911yescharacteristic
198#97—792no—
199C24—21no—
200C24—1581no—
201C2×Q8—791no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—42no—
205C2×D4—3162no—
206C2×D4—3164no—
207C2×D4—3164no—
208C2×D4—792no—
209C2×D4—1582no—
210C2×D4—1582no—
211C2×D4—1582no—
212C2×D4—1582no—
213C2×D4—1581no—
214C2×D4C22×D7912yes—
215C2×D4C22×D7911yescharacteristic
216C22×C4—791no—
217C22×C4—791no—
218C22×C4—1581no—
219C22×C4C22×D7911yescharacteristic
220SD16—3162no—
221SD16—3162no—
222D8—42no—
223D8—3162no—
224C2×C8—792no—
225C2×C8C22×D7912yes—
226C4⋊C4—791no—
227C4⋊C4—1581no—
228C4⋊C4—1581no—
229C4⋊C4—1581no—
230C4⋊C4—1581no—
231C4⋊C4C22×D7911yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—1581no—
234C22⋊C4—1581no—
235C22⋊C4—1581no—
236C22⋊C4—1581no—
237C42—791no—
238C42—791no—
239C42—791no—
240C42—791no—
241C42—1581no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—3162no—
248C23—3162no—
249C23—3162no—
250C23—1582no—
251C23D4×D7911yescharacteristic
252Q8—1582no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3162no—
259D4—3164no—
260D4—3164no—
261D4—3164no—
262D4—3164no—
263D4—1582no—
264D4—1582no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3162no—
268C2×C4—791no—
269C2×C4—791no—
270C2×C4—791no—
271C2×C4—791no—
272C2×C4—1581no—
273C2×C4—1581no—
274C2×C4—1582no—
275C2×C4—1582no—
276C2×C4D4×D7911yescharacteristic
277C2×C4C23×D7911yescharacteristic
278C8—22no—
279C8—1582no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—84no—
289C22—3162no—
290C22—3162no—
291C22—3162no—
292C22—6324no—
293C22#1457918512yes—
294C22C2×D4×D7911yescharacteristic
295C4—21no—
296C4—41no—
297C4—3161no—
298C4—3161no—
299C4—792no—
300C4—1581no—
301C4—1581no—
302C4—1581no—
303C4#1457920612yes—
304C2—22no—
305C2—82no—
306C2—81no—
307C2—6322no—
308C2#1909912811yescharacteristic
309C2AGL(1,ℤ/8)×D7911yescharacteristic
310C2C2×#1457918511yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#447×C79C2
socleC22×C79C22×D4
hypercenter#447D79
hypocenterC79#3217
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#14579157C2semidirect
2. #19099094C2semidirect
3. #19098951C2semidirect
4. #19098926C2semidirect
5. #19098762C2semidirect
6. #19098756C2semidirect
7. #19098716C2semidirect
8. #19098714C2semidirect
9. #19098696C2semidirect
10. #19098617C2semidirect
11. #447×C79C2semidirect
15. C2×#11929613C22semidirect
16. C2×#11929612C22semidirect
17. C2×#11929609C22semidirect
23. #14579152C22semidirect
24. #14579148C22semidirect
27. #14579126C22semidirect
29. D4×Dic79C22semidirect
32. Dic79⋊C8C22semidirect
33. #14579083C22semidirect
34. #14579082C22semidirect
35. #14579081C22semidirect
37. #14579071C22semidirect
39. C2×D8×C79C22semidirect
41. #102×C79C22semidirect
156. C79#3217semidirect
166. #447D79semidirect
175. C2×D8C2×D79semidirect
195. #102C2×D79semidirect