Finite Group Databasebeta

← #23772675

Subgroups — #23772675

Order 10112.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14579207C211yesmaximal, characteristic
2C2×#14579192C211yesmaximal, characteristic, complemented
3C22×#11929611C211yesmaximal, characteristic, complemented
4#19098723C214yesmaximal, complemented
5#19098713C214yesmaximal, complemented
6#19098567C211yesmaximal, characteristic, complemented
7#19098557C211yesmaximal, characteristic, complemented
8#19098530C211yesmaximal, characteristic, complemented
9#536×C79C211yesmaximal, characteristic, complemented
10#14579207—24no—
11#14579207—24no—
12C2×#11929625—21no—
13C2×#11929624C2212yes—
14C2×#11929624C2211yescharacteristic
15C22×D316C2212yes—
16C22×D316C2211yescharacteristic
17C2×D4×D79—22no—
18C2×D4×D79—21no—
19#14579192—24no—
20C2×#11929614C2212yescomplemented
21C2×#11929611—24no—
22C2×#11929611C2212yescomplemented
23C2×#11929611C2214yescomplemented
24C2×#11929608C2211yescharacteristic, complemented
25C22×C79⋊C8C2211yescharacteristic, complemented
26#14579084C2212yescomplemented
27#14579081C2212yescomplemented
28#14579081C2214yescomplemented
29C22⋊(C79⋊C8)C2214yescomplemented
30C2×#41×C79C2211yescharacteristic
31C22×Q8×C79C2211yescharacteristic, complemented
32C4⋊Q8×C79—21no—
33#124×C79—21no—
34#123×C79—22no—
35C22⋊Q8×C79C2214yescomplemented
36#117×C79C2211yescharacteristic, complemented
37C23×D79—22no—
38C23×D79—21no—
39#11929625—28no—
40#11929624—24no—
41#11929624—24no—
42#11929624—22no—
43#11929624D414yes—
44C2×Dic158C2311yescharacteristic
45C2×D316—24no—
46C2×D316—24no—
47C2×D316—22no—
48C2×D316—21no—
49C2×D316—22no—
50C2×D316D414yes—
51C2×D316C2311yescharacteristic
52C2×D316C2312yes—
53C2×C22⋊D79—22no—
54C2×C22⋊D79—21no—
55D4×D79—28no—
56D4×D79—216no—
57C2×C4×D79—22no—
58C2×C4×D79—21no—
59#11929614—24no—
60#11929611—28no—
61#11929611—24no—
62#11929608—22no—
63C2×C79⋊C8—22no—
64C2×C79⋊C8C2312yes—
65C2×C79⋊C8C2312yes—
66#41×C79—22no—
67#41×C79—22no—
68C2×Q8×C79—21no—
69C2×Q8×C79—22no—
70C2×Q8×C79—22no—
71C2×Q8×C79D411yescharacteristic
72C2×Q8×C79C2312yes—
73C2×D4×C79D412yes—
74C2×D4×C79D411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79C2312yes—
77C22×C4×C79C2311yescharacteristic
78C4⋊C4×C79—24no—
79C4⋊C4×C79C2312yes—
80C22⋊C4×C79—21no—
81C22⋊C4×C79—24no—
82C42×C79—21no—
83C23×C79C2×D412yes—
84C23×C79C2×D411yescharacteristic
85C22×D79—22no—
86C22×D79—21no—
87C22×D79—28no—
88C22×D79—24no—
89C22×D79—82no—
90C22×D79—84no—
91Q8×C79—22no—
92Q8×C79—22no—
93Q8×C79—22no—
94Q8×C79—42no—
95D4×C79—24no—
96D4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—42no—
102C2×C4×C79C2×D412yes—
103C2×C4×C79C2×D412yes—
104C2×C4×C79C2×D412yes—
105C2×C4×C79C2×D411yescharacteristic
106C2×C4×C79C2411yescharacteristic
107C22⋊D79—28no—
108C22⋊D79—24no—
109C2×Dic79—21no—
110D316—28no—
111D316—24no—
112D316—21no—
113D316—22no—
114D316C2×D412yes—
115D316C2×D414yes—
116C4×D79—28no—
117C4×D79—24no—
118Dic158—21no—
119Dic158C2×D412yes—
120C79⋊C8—22no—
121C79⋊C8—22no—
122C22×C79—22no—
123C22×C79—22no—
124C22×C79—24no—
125C22×C79#12012yes—
126C22×C79C22×D411yescharacteristic
127C2×D79—24no—
128C2×D79—22no—
129C2×D79—44no—
130C2×D79—42no—
131C2×D79—84no—
132C2×D79—88no—
133C4×C79—22no—
134C4×C79—42no—
135C4×C79—42no—
136C4×C79#12012yes—
137C4×C79C22×D411yescharacteristic
138C4×C79C22×D411yescharacteristic
139Dic79—22no—
140C2×C79—22no—
141C2×C79—42no—
142C2×C79C2×#12011yescharacteristic
143C2×C79#57712yes—
144D79—44no—
145D79—42no—
146#3009—791nomaximal, Sylow
147C79#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—791no—
149C2×#137—791no—
150C22×SD16—791no—
151#536D7911yescharacteristic, complemented
152#452—794no—
153#450—794no—
154#418—791no—
155#416—791no—
156#408—791no—
157H(5,𝔽2)—1584no—
158H(5,𝔽2)—1584no—
159C2×#41—791no—
160C2×#41—792no—
161C2×#41—1581no—
162C2×#41C2×D7911yescharacteristic
163C22×Q8C2×D7911yescharacteristic, complemented
164C22×D4—791no—
165C22×D4—792no—
166C22×D4—1582no—
167C22×D4—1581no—
168#137—1584no—
169C2×Q16—792no—
170C2×SD16—794no—
171C2×SD16—792no—
172C2×SD16—1584no—
173C2×M16—791no—
174C22×C8—791no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D7914yescomplemented
179#117C2×D7911yescharacteristic, complemented
180#103—792no—
181#102—792no—
182#102—794no—
183C22⋊C8—794no—
184C24—1582no—
185C24—1581no—
186#41—22no—
187#41—22no—
188#41—794no—
189#41—1584no—
190#41—1584no—
191#41—1588no—
192#41—1582no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—791no—
197C2×Q8C22⋊D7911yescharacteristic
198C2×Q8C22×D7912yes—
199C2×D4—794no—
200C2×D4—792no—
201C2×D4—791no—
202C2×D4—1584no—
203C2×D4—1584no—
204C2×D4—15816no—
205C2×D4—1588no—
206C2×D4—1582no—
207C2×D4—1581no—
208C2×D4—1582no—
209C2×D4—1582no—
210C2×D4—1581no—
211C2×D4C22⋊D7912yes—
212C2×D4C22⋊D7911yescharacteristic
213C22×C4—21no—
214C22×C4—1582no—
215C22×C4—1581no—
216C22×C4C22×D7912yes—
217C22×C4C22×D7911yescharacteristic
218Q16—1584no—
219SD16—1588no—
220SD16—1584no—
221M16—1582no—
222C2×C8—792no—
223C2×C8—792no—
224C2×C8—1582no—
225C4⋊C4—24no—
226C4⋊C4C22×D7912yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—6322no—
231C23—6324no—
232C23—1588no—
233C23—1582no—
234C23—1581no—
235C23—1584no—
236C23C2×C22⋊D7912yes—
237C23C2×C22⋊D7911yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—792no—
243Q8—1581no—
244D4—24no—
245D4—22no—
246D4—794no—
247D4—792no—
248D4—1588no—
249D4—1588no—
250D4—1584no—
251D4—1584no—
252D4—1581no—
253D4—1582no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—1588no—
260C2×C4—1581no—
261C2×C4—1584no—
262C2×C4D4×D7912yes—
263C2×C4D4×D7912yes—
264C2×C4C2×C22⋊D7912yes—
265C2×C4C2×C22⋊D7911yescharacteristic
266C2×C4C23×D7911yescharacteristic
267C8—1582no—
268C8—1582no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—3164no—
273C22—3162no—
274C22—6324no—
275C22—6328no—
276C22—1584no—
277C22—1582no—
278C22#1457912912yes—
279C22C22×C22⋊D7911yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—1582no—
284C4#1457912912yes—
285C4C2×D4×D7911yescharacteristic
286C4C2×D4×D7911yescharacteristic
287C2—22no—
288C2—42no—
289C2—3164no—
290C2—3162no—
291C2C2×#1457912911yescharacteristic
292C2#1909937112yes—
subgroupG/H
centerC22C22×C22⋊D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#536×C79C2
socleC22×C79C22×D4
prime radicalC2×#14579207C2
hypercenter#536D79
hypocenterC79#3009
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#14579192C2semidirect
3. C22×#11929611C2semidirect
4. #19098723C2semidirect
5. #19098713C2semidirect
6. #19098567C2semidirect
7. #19098557C2semidirect
8. #19098530C2semidirect
9. #536×C79C2semidirect
20. C2×#11929614C22semidirect
22. C2×#11929611C22semidirect
23. C2×#11929611C22semidirect
24. C2×#11929608C22semidirect
25. C22×C79⋊C8C22semidirect
26. #14579084C22semidirect
27. #14579081C22semidirect
28. #14579081C22semidirect
29. C22⋊(C79⋊C8)C22semidirect
31. C22×Q8×C79C22semidirect
35. C22⋊Q8×C79C22semidirect
36. #117×C79C22semidirect
147. C79#3009semidirect
151. #536D79semidirect
163. C22×Q8C2×D79semidirect
178. C22⋊Q8C2×D79semidirect
179. #117C2×D79semidirect