Finite Group Databasebeta

← #23774205

Subgroups — #23774205

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
1C22⋊Q8×D79C211yesmaximal, characteristic, complemented
2C2×#14579136C211yesmaximal, characteristic
3#19098898C211yesmaximal, characteristic, complemented
4#19098878C212yesmaximal, complemented
5#19098812C212yesmaximal, complemented
6#19098810C212yesmaximal, complemented
7#19098716C211yesmaximal, characteristic, complemented
8#19098713C211yesmaximal, characteristic, complemented
9#19098703C212yesmaximal, complemented
10#19098525C211yesmaximal, characteristic, complemented
11#484×C79C211yesmaximal, characteristic, complemented
12C2×Q8×D79C2211yescharacteristic, complemented
13C22×D316—21no—
14C22×D316C2211yescharacteristic
15C22×C4×D79C2211yescharacteristic
16C2×#11929613C2212yescomplemented
17C2×#11929611C2212yescomplemented
18C2×#11929607C2212yescomplemented
19#14579141C2211yescharacteristic, complemented
20#14579140C2211yescharacteristic
21#14579139—21no—
22#14579138—21no—
23#14579137C2211yescharacteristic, complemented
24#14579136—21no—
25#14579136—21no—
26#14579136C2212yes—
27#14579136C2212yes—
28C4⋊C4×D79—21no—
29C4⋊C4×D79C2211yescharacteristic, complemented
30C2×(C2×D79)⋊C4—21no—
31C22⋊C4×D79—21no—
32#14579094C2212yescomplemented
33#14579084C2212yescomplemented
34#14579083C2212yescomplemented
35#14579081C2212yescomplemented
36C22⋊(C79⋊C8)C2211yescharacteristic
37(C2×D79)⋊C8C2212yescomplemented
38C22⋊Q8×C79C2211yescharacteristic
39C2×C4⋊C4×C79C2211yescharacteristic
40#107×C79C2212yescomplemented
41#103×C79C2212yescomplemented
42C22⋊C8×C79C2211yescharacteristic
43C23×D79—21no—
44C23×D79D411yescharacteristic
45C2×Dic158—21no—
46C2×Dic158C2311yescharacteristic
47Q8×D79—42no—
48C2×D316—22no—
49C2×D316—22no—
50C2×D316—21no—
51C2×D316—42no—
52C2×D316—44no—
53C2×D316C2312yes—
54C22×Dic79D411yescharacteristic
55C2×C4×D79—21no—
56C2×C4×D79—21no—
57C2×C4×D79—42no—
58C2×C4×D79D411yescharacteristic
59C2×C4×D79D411yescharacteristic
60C2×C4×D79C2311yescharacteristic
61C2×C4×D79C2311yescharacteristic
62#11929613—42no—
63#11929611—42no—
64#11929607—42no—
65C2×C79⋊C8C2312yes—
66C4⋊Dic79—21no—
67C4⋊Dic79C2311yescharacteristic
68Dic79⋊C4—21no—
69Dic79⋊C4—21no—
70Dic79⋊C4—21no—
71C22⋊Dic79—21no—
72(C2×D79)⋊C4—22no—
73(C2×D79)⋊C4—22no—
74(C2×D79)⋊C4—21no—
75(C2×D79)⋊C4—21no—
76C2×Q8×C79C2311yescharacteristic
77C22×C4×C79—21no—
78C22×C4×C79C2311yescharacteristic
79C2×C8×C79C2312yes—
80C4⋊C4×C79—21no—
81C4⋊C4×C79—21no—
82C4⋊C4×C79C2312yes—
83C4⋊C4×C79C2311yescharacteristic
84C22⋊C4×C79—21no—
85C23×C79C2×D411yescharacteristic
86C22×D79—22no—
87C22×D79—22no—
88C22×D79—22no—
89C22×D79—41no—
90C22×D79—42no—
91C22×D79—42no—
92C22×D79—42no—
93C22×D79C2×D411yescharacteristic
94C22×D79C2×D411yescharacteristic
95Q8×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—42no—
101C2×C4×C79C2×D411yescharacteristic
102C2×C4×C79C2411yescharacteristic
103C2×Dic79—21no—
104C2×Dic79—21no—
105C2×Dic79—41no—
106C2×Dic79C2×D411yescharacteristic
107C2×Dic79C2×D411yescharacteristic
108D316—22no—
109D316—22no—
110D316—42no—
111D316—44no—
112D316—44no—
113C4×D79—22no—
114C4×D79—22no—
115C4×D79—41no—
116C4×D79—41no—
117C4×D79—42no—
118C4×D79—42no—
119Dic158—22no—
120Dic158—42no—
121C8×C79—22no—
122C79⋊C8—22no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79#13712yes—
126C22×C79C22×D411yescharacteristic
127C2×D79—21no—
128C2×D79—21no—
129C2×D79—21no—
130C2×D79—21no—
131C2×D79—21no—
132C2×D79—44no—
133C2×D79—42no—
134C2×D79—42no—
135C2×D79—42no—
136C2×D79—84no—
137C2×D79AGL(1,ℤ/8)12yes—
138C4×C79—21no—
139C4×C79—41no—
140C4×C79—42no—
141C4×C79—41no—
142C4×C79H(5,𝔽2)12yes—
143Dic79—21no—
144Dic79—21no—
145Dic79—41no—
146Dic79—41no—
147C2×C79—22no—
148C2×C79#53411yescharacteristic
149C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C79C2×#13711yescharacteristic
151D79—22no—
152D79—41no—
153D79—82no—
154#3193—791nomaximal, Sylow
155C79#319311yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—791no—
157C2×#121—791no—
158#484—791no—
159#484D7911yescharacteristic, complemented
160#480—792no—
161#468—792no—
162#468—792no—
163#450—791no—
164#450—791no—
165#448—792no—
166#407—791no—
167C22×Q8—791no—
168C22×D4—791no—
169C22×D4—1581no—
170C23×C4—791no—
171C2×SD16—792no—
172C2×SD16—792no—
173C2×M16—792no—
174C22⋊Q8—791no—
175C22⋊Q8—791no—
176C22⋊Q8—791no—
177C22⋊Q8—1581no—
178C22⋊Q8—1581no—
179C22⋊Q8C2×D7911yescharacteristic
180#121—792no—
181#121—792no—
182#121—1581no—
183#121—1581no—
184C2×C4⋊C4—791no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4C2×D7911yescharacteristic
187C2×C22⋊C4—1581no—
188C2×C22⋊C4—1581no—
189#107—792no—
190#107C2×D7912yescomplemented
191#103—792no—
192#103C2×D7912yescomplemented
193#102—792no—
194#102—792no—
195C22⋊C8—791no—
196C22⋊C8—792no—
197C22⋊C8C2×D7911yescharacteristic
198C24—791no—
199C24—1581no—
200C2×Q8—3162no—
201C2×Q8—791no—
202C2×Q8—1581no—
203C2×Q8C22×D7911yescharacteristic
204C2×D4—3162no—
205C2×D4—3164no—
206C2×D4—792no—
207C2×D4—1582no—
208C2×D4—1582no—
209C2×D4—1581no—
210C22×C4—21no—
211C22×C4—3162no—
212C22×C4—791no—
213C22×C4—791no—
214C22×C4—791no—
215C22×C4—791no—
216C22×C4—791no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C22×D7911yescharacteristic
220SD16—3162no—
221SD16—3162no—
222M16—3162no—
223C2×C8—792no—
224C2×C8C22×D7912yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—791no—
228C4⋊C4—1581no—
229C4⋊C4—1581no—
230C4⋊C4—1581no—
231C4⋊C4—1581no—
232C4⋊C4C22×D7911yescharacteristic
233C4⋊C4C22×D7912yes—
234C22⋊C4—21no—
235C22⋊C4—1582no—
236C22⋊C4—1582no—
237C22⋊C4—1581no—
238C22⋊C4—1581no—
239C22⋊C4—1581no—
240C23—3161no—
241C23—3162no—
242C23—3162no—
243C23—3162no—
244C23—791no—
245C23—791no—
246C23—1582no—
247C23—1582no—
248C23—1582no—
249C23D4×D7911yescharacteristic
250Q8—22no—
251Q8—3162no—
252Q8—1582no—
253D4—3162no—
254D4—3164no—
255D4—3164no—
256D4—1582no—
257D4—1582no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3161no—
268C2×C4—791no—
269C2×C4—791no—
270C2×C4—1581no—
271C2×C4—1581no—
272C2×C4—1582no—
273C2×C4—1582no—
274C2×C4D4×D7911yescharacteristic
275C2×C4C23×D7911yescharacteristic
276C8—22no—
277C8—1582no—
278C22—21no—
279C22—21no—
280C22—3164no—
281C22—3162no—
282C22—3162no—
283C22—3162no—
284C22—792no—
285C22—6324no—
286C22—1581no—
287C22—1581no—
288C22—1581no—
289C22—1581no—
290C22—1581no—
291C22#1457919312yes—
292C22C2×D4×D7911yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3161no—
298C4—3161no—
299C4—1581no—
300C4—1581no—
301C4#1457920712yes—
302C2—22no—
303C2—3161no—
304C2—6322no—
305C2—1582no—
306C2#1909912411yescharacteristic
307C2AGL(1,ℤ/8)×D7911yescharacteristic
308C2C2×#1457919311yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#484×C79C2
socleC22×C79C22×D4
prime radicalC2×#14579136C2
hypercenter#484D79
hypocenterC79#3193
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D79C2semidirect
3. #19098898C2semidirect
4. #19098878C2semidirect
5. #19098812C2semidirect
6. #19098810C2semidirect
7. #19098716C2semidirect
8. #19098713C2semidirect
9. #19098703C2semidirect
10. #19098525C2semidirect
11. #484×C79C2semidirect
12. C2×Q8×D79C22semidirect
16. C2×#11929613C22semidirect
17. C2×#11929611C22semidirect
18. C2×#11929607C22semidirect
19. #14579141C22semidirect
23. #14579137C22semidirect
29. C4⋊C4×D79C22semidirect
32. #14579094C22semidirect
33. #14579084C22semidirect
34. #14579083C22semidirect
35. #14579081C22semidirect
37. (C2×D79)⋊C8C22semidirect
40. #107×C79C22semidirect
41. #103×C79C22semidirect
155. C79#3193semidirect
159. #484D79semidirect
190. #107C2×D79semidirect
192. #103C2×D79semidirect