Finite Group Databasebeta

← #23774204

Subgroups — #23774204

Order 10112.

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

normal subgroups only

This group has 462 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×#14579137C211yesmaximal, characteristic, complemented
2#121×D79C211yesmaximal, characteristic
3#19098897C212yesmaximal, complemented
4#19098877C211yesmaximal, characteristic, complemented
5#19098874C211yesmaximal, characteristic, complemented
6#19098812C212yesmaximal, complemented
7#19098806C212yesmaximal, complemented
8#19098712C212yesmaximal, complemented
9#19098702C211yesmaximal, characteristic, complemented
10#19098525C211yesmaximal, characteristic, complemented
11#448×C79C211yesmaximal, characteristic, complemented
12C2×D4×D79—21no—
13C2×D4×D79C2211yescharacteristic
14C22×C4×D79C2211yescharacteristic
15C2×#11929611C2212yescomplemented
16C2×#11929607C2212yescomplemented
17#14579137—21no—
18#14579137—21no—
19#14579137C2212yes—
20#14579137C2212yescomplemented
21#14579136C2211yescharacteristic
22#14579135C2211yescharacteristic
23#14579134—21no—
24#14579133—21no—
25#14579132C2211yescharacteristic
26C2×C4⋊Dic79C2211yescharacteristic, complemented
27C2×Dic79⋊C4—21no—
28C4⋊C4×D79C2211yescharacteristic
29C2×(C2×D79)⋊C4—21no—
30C22⋊C4×D79—21no—
31#14579095C2212yescomplemented
32#14579094C2212yescomplemented
33#14579085C2212yescomplemented
34#14579083C2212yescomplemented
35#14579082C2212yescomplemented
36C22⋊(C79⋊C8)C2211yescharacteristic, complemented
37(C2×D79)⋊C8C2212yes—
38C22×Q8×C79C2211yescharacteristic, complemented
39C2×SD16×C79C2212yescomplemented
40#121×C79C2211yescharacteristic
41#103×C79C2212yescomplemented
42C22⋊C8×C79C2211yescharacteristic, complemented
43C23×D79—21no—
44C23×D79D411yescharacteristic
45C2×D316—21no—
46C2×D316C2311yescharacteristic
47C2×C22⋊D79—21no—
48C2×C22⋊D79—21no—
49C2×C22⋊D79—21no—
50D4×D79—42no—
51D4×D79—44no—
52C22×Dic79—21no—
53C22×Dic79D411yescharacteristic
54C2×C4×D79—21no—
55C2×C4×D79—42no—
56C2×C4×D79D411yescharacteristic
57C2×C4×D79D411yescharacteristic
58C2×C4×D79C2311yescharacteristic
59C2×C4×D79C2311yescharacteristic
60#11929611—42no—
61#11929607—42no—
62C2×C79⋊C8C2312yes—
63C4⋊Dic79—21no—
64C4⋊Dic79C2312yes—
65C4⋊Dic79C2311yescharacteristic
66Dic79⋊C4—21no—
67Dic79⋊C4—22no—
68Dic79⋊C4—22no—
69C22⋊Dic79—21no—
70(C2×D79)⋊C4—21no—
71(C2×D79)⋊C4—22no—
72(C2×D79)⋊C4—22no—
73(C2×D79)⋊C4—21no—
74C2×Q8×C79—21no—
75C2×Q8×C79—42no—
76C2×Q8×C79C2312yes—
77C2×D4×C79—21no—
78C2×D4×C79C2311yescharacteristic
79C22×C4×C79—21no—
80C22×C4×C79C2311yescharacteristic
81SD16×C79—42no—
82C2×C8×C79C2312yes—
83C4⋊C4×C79C2311yescharacteristic
84C22⋊C4×C79—21no—
85C23×C79—21no—
86C23×C79C2×D411yescharacteristic
87C22×D79—22no—
88C22×D79—22no—
89C22×D79—21no—
90C22×D79—41no—
91C22×D79—42no—
92C22×D79—42no—
93C22×D79—42no—
94C22×D79C2×D411yescharacteristic
95C22×D79C2×D411yescharacteristic
96Q8×C79—22no—
97Q8×C79—22no—
98Q8×C79—42no—
99D4×C79—22no—
100D4×C79—42no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—21no—
104C2×C4×C79—42no—
105C2×C4×C79C2×D411yescharacteristic
106C2×C4×C79C2411yescharacteristic
107C22⋊D79—42no—
108C22⋊D79—42no—
109C22⋊D79—42no—
110C2×Dic79—22no—
111C2×Dic79—21no—
112C2×Dic79—41no—
113C2×Dic79—42no—
114C2×Dic79C2×D411yescharacteristic
115C2×Dic79C2×D411yescharacteristic
116D316—22no—
117D316—42no—
118C4×D79—22no—
119C4×D79—22no—
120C4×D79—41no—
121C4×D79—41no—
122C4×D79—42no—
123C8×C79—22no—
124C79⋊C8—22no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—41no—
128C22×C79—41no—
129C22×C79—41no—
130C22×C79AGL(1,ℤ/8)12yes—
131C22×C79C22×D411yescharacteristic
132C2×D79—21no—
133C2×D79—21no—
134C2×D79—21no—
135C2×D79—21no—
136C2×D79—21no—
137C2×D79—44no—
138C2×D79—41no—
139C2×D79—41no—
140C2×D79—41no—
141C2×D79—84no—
142C2×D79#13712yes—
143C4×C79—21no—
144C4×C79—42no—
145C4×C79—41no—
146C4×C79H(5,𝔽2)12yes—
147Dic79—21no—
148Dic79—21no—
149Dic79—41no—
150Dic79—42no—
151C2×C79—22no—
152C2×C79—81no—
153C2×C79#53411yescharacteristic
154C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C79C2×#13711yescharacteristic
156D79—22no—
157D79—41no—
158D79—81no—
159#3193—791nomaximal, Sylow
160C79#319311yescharacteristic, complemented, Sylow
161C2×C22⋊Q8—791no—
162C2×#121—791no—
163#484—792no—
164#480—791no—
165#480—791no—
166#468—792no—
167#468—792no—
168#450—792no—
169#448—791no—
170#448D7911yescharacteristic, complemented
171#407—791no—
172C22×Q8C2×D7911yescharacteristic, complemented
173C22×D4—791no—
174C22×D4—1581no—
175C23×C4—791no—
176C2×SD16—792no—
177C2×SD16C2×D7912yescomplemented
178C2×M16—792no—
179C22⋊Q8—792no—
180C22⋊Q8—792no—
181C22⋊Q8—1581no—
182C22⋊Q8—1581no—
183#121—791no—
184#121—791no—
185#121—791no—
186#121—1581no—
187#121—1581no—
188#121C2×D7911yescharacteristic
189C2×C4⋊C4—791no—
190C2×C4⋊C4—791no—
191C2×C4⋊C4—1581no—
192C2×C22⋊C4—1581no—
193C2×C22⋊C4—1581no—
194#107—792no—
195#107—792no—
196#103—792no—
197#103C2×D7912yescomplemented
198#102—792no—
199#102—792no—
200C22⋊C8—791no—
201C22⋊C8—792no—
202C22⋊C8C2×D7911yescharacteristic, complemented
203C24—791no—
204C24—1581no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D7912yes—
208C2×D4—21no—
209C2×D4—3164no—
210C2×D4—3162no—
211C2×D4—791no—
212C2×D4—1581no—
213C2×D4—1581no—
214C2×D4—1581no—
215C2×D4—1581no—
216C2×D4C22×D7911yescharacteristic
217C22×C4—21no—
218C22×C4—3162no—
219C22×C4—791no—
220C22×C4—791no—
221C22×C4—791no—
222C22×C4—791no—
223C22×C4—791no—
224C22×C4—1581no—
225C22×C4—1581no—
226C22×C4C22×D7911yescharacteristic
227SD16—42no—
228SD16—3162no—
229M16—3162no—
230C2×C8—792no—
231C2×C8C22×D7912yes—
232C4⋊C4—792no—
233C4⋊C4—791no—
234C4⋊C4—1581no—
235C4⋊C4—1582no—
236C4⋊C4—1582no—
237C4⋊C4—1581no—
238C4⋊C4C22×D7911yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1581no—
241C22⋊C4—1582no—
242C22⋊C4—1582no—
243C22⋊C4—1581no—
244C22⋊C4—1581no—
245C23—21no—
246C23—3161no—
247C23—3162no—
248C23—3162no—
249C23—3162no—
250C23—791no—
251C23—791no—
252C23—1582no—
253C23—1581no—
254C23—1582no—
255C23D4×D7911yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—3162no—
262D4—3162no—
263D4—3162no—
264D4—3162no—
265D4—1582no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3161no—
271C2×C4—3161no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—3161no—
275C2×C4—791no—
276C2×C4—791no—
277C2×C4—1582no—
278C2×C4—1581no—
279C2×C4—1582no—
280C2×C4—1582no—
281C2×C4D4×D7911yescharacteristic
282C2×C4C23×D7911yescharacteristic
283C8—22no—
284C8—1582no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—3164no—
291C22—3161no—
292C22—3161no—
293C22—3161no—
294C22—792no—
295C22—6324no—
296C22—1581no—
297C22—1581no—
298C22—1581no—
299C22—1581no—
300C22—1581no—
301C22#1457918812yes—
302C22C2×D4×D7911yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—3161no—
307C4—3162no—
308C4—1581no—
309C4—1581no—
310C4#1457920612yes—
311C2—22no—
312C2—81no—
313C2—3161no—
314C2—6321no—
315C2—1582no—
316C2#1909912311yescharacteristic
317C2C2×#1457918811yescharacteristic
318C2#137×D7911yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#448×C79C2
socleC22×C79C22×D4
prime radical#121×D79C2
hypercenter#448D79
hypocenterC79#3193
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#14579137C2semidirect
3. #19098897C2semidirect
4. #19098877C2semidirect
5. #19098874C2semidirect
6. #19098812C2semidirect
7. #19098806C2semidirect
8. #19098712C2semidirect
9. #19098702C2semidirect
10. #19098525C2semidirect
11. #448×C79C2semidirect
15. C2×#11929611C22semidirect
16. C2×#11929607C22semidirect
20. #14579137C22semidirect
26. C2×C4⋊Dic79C22semidirect
31. #14579095C22semidirect
32. #14579094C22semidirect
33. #14579085C22semidirect
34. #14579083C22semidirect
35. #14579082C22semidirect
36. C22⋊(C79⋊C8)C22semidirect
38. C22×Q8×C79C22semidirect
39. C2×SD16×C79C22semidirect
41. #103×C79C22semidirect
42. C22⋊C8×C79C22semidirect
160. C79#3193semidirect
170. #448D79semidirect
172. C22×Q8C2×D79semidirect
177. C2×SD16C2×D79semidirect
197. #103C2×D79semidirect
202. C22⋊C8C2×D79semidirect