Finite Group Databasebeta

← #23772778

Subgroups — #23772778

Order 10112.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#19099399C211yesmaximal, characteristic
2C2×#14579192C212yesmaximal, complemented
3#19098745C214yesmaximal, complemented
4#19098740C214yesmaximal, complemented
5#19098581C212yesmaximal, complemented
6#19098547C211yesmaximal, characteristic, complemented
7#536×C79C211yesmaximal, characteristic, complemented
8#14579209C2212yes—
9#14579208—22no—
10#14579207—22no—
11#14579206—22no—
12#14579206C2212yes—
13C2×#11929624—22no—
14C2×#11929624C2212yes—
15#41×D79—24no—
16#14579192—24no—
17#14579192C2218yescomplemented
18C2×#11929614—22no—
19C2×#11929611—22no—
20C2×#11929608C2212yescomplemented
21#14579089C2218yescomplemented
22#14579080C2212yescomplemented
23#14579078C2212yescomplemented
24C2×#41×C79C2211yescharacteristic
25C22×Q8×C79C2211yescharacteristic, complemented
26C4⋊Q8×C79C2212yescomplemented
27#124×C79C2212yescomplemented
28#123×C79—22no—
29C22⋊Q8×C79—22no—
30#117×C79C2211yescharacteristic, complemented
31#11929625—24no—
32#11929624—24no—
33#11929624—24no—
34#11929624—22no—
35#11929624—24no—
36#11929624—22no—
37#11929624D414yes—
38#11929624C2314yes—
39#11929623—28no—
40#11929623—24no—
41C2×Dic158—22no—
42C2×Dic158D412yes—
43Q8×D79—24no—
44C2×D316—22no—
45C2×D316D412yes—
46C2×C22⋊D79—24no—
47C2×C22⋊D79—22no—
48D4×D79—28no—
49D4×D79—24no—
50C2×C4×D79—24no—
51C2×C4×D79—22no—
52#11929614—28no—
53#11929611—28no—
54#11929608—22no—
55#11929608C2314yes—
56C2×C79⋊C8—22no—
57#41×C79—22no—
58#41×C79—22no—
59C2×Q8×C79—21no—
60C2×Q8×C79—22no—
61C2×Q8×C79—22no—
62C2×Q8×C79C2312yes—
63C2×Q8×C79C2311yescharacteristic
64C2×D4×C79D412yes—
65C2×D4×C79C2311yescharacteristic
66C22×C4×C79—21no—
67C22×C4×C79D412yes—
68C22×C4×C79C2311yescharacteristic
69C4⋊C4×C79—21no—
70C4⋊C4×C79—24no—
71C22⋊C4×C79—21no—
72C22⋊C4×C79—24no—
73C42×C79C2312yes—
74C23×C79C2×D412yes—
75C23×C79C2×D411yescharacteristic
76C22×D79—24no—
77C22×D79—22no—
78Q8×C79—22no—
79Q8×C79—24no—
80Q8×C79—41no—
81Q8×C79—41no—
82D4×C79—24no—
83D4×C79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—42no—
89C2×C4×C79C2×D412yes—
90C2×C4×C79C2×D412yes—
91C2×C4×C79C2×D412yes—
92C2×C4×C79C2×D411yescharacteristic
93C2×C4×C79C2411yescharacteristic
94C22⋊D79—28no—
95C22⋊D79—24no—
96C22⋊D79—24no—
97C2×Dic79—24no—
98C2×Dic79—22no—
99D316—24no—
100D316—22no—
101D316C2×D414yes—
102C4×D79—28no—
103C4×D79—24no—
104C4×D79—24no—
105Dic158—24no—
106Dic158—22no—
107Dic158C2×D414yes—
108C79⋊C8—24no—
109C22×C79—22no—
110C22×C79—41no—
111C22×C79—42no—
112C22×C79#12012yes—
113C22×C79C22×D411yescharacteristic
114C2×D79—24no—
115C2×D79—82no—
116C2×D79—84no—
117C4×C79—22no—
118C4×C79—42no—
119C4×C79—42no—
120C4×C79#12011yescharacteristic
121C4×C79#12011yescharacteristic
122C4×C79C22×D412yes—
123Dic79—24no—
124C2×C79—21no—
125C2×C79—22no—
126C2×C79—42no—
127C2×C79C2×#12011yescharacteristic
128D79—44no—
129#3022—791nomaximal, Sylow
130C79#302211yescharacteristic, complemented, Sylow
131#585—791no—
132C2×#137—792no—
133#536D7911yescharacteristic, complemented
134#456—794no—
135#455—794no—
136#421—792no—
137#413—791no—
1382−1+4—792no—
1392−1+4—1582no—
140H(5,𝔽2)—792no—
141H(5,𝔽2)—1582no—
142H(5,𝔽2)—1582no—
143C2×#41—792no—
144C2×#41—1584no—
145C2×#41—1582no—
146C2×#41C2×D7911yescharacteristic
147C22×Q8C2×D7911yescharacteristic, complemented
148#137—798no—
149#137—1584no—
150C2×Q16—1582no—
151C2×SD16—1582no—
152C2×M16—792no—
153C4⋊Q8C2×D7912yescomplemented
154#124C2×D7912yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D7911yescharacteristic, complemented
158#104—798no—
159#101—792no—
160#100—792no—
161#41—22no—
162#41—22no—
163#41—794no—
164#41—794no—
165#41—1584no—
166#41—1584no—
167#41—1584no—
168#41—1588no—
169#41—1584no—
170#41—1582no—
171#41—1582no—
172#41—1584no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—792no—
177C2×Q8—1582no—
178C2×Q8—1584no—
179C2×Q8C22×D7912yes—
180C2×Q8C22×D7911yescharacteristic
181C2×D4—792no—
182C2×D4—1582no—
183C2×D4—1584no—
184C2×D4—1588no—
185C2×D4—1582no—
186C2×D4—1584no—
187C2×D4C22⋊D7912yes—
188C2×D4C22×D7911yescharacteristic
189C22×C4—21no—
190C22×C4—1584no—
191C22×C4—1582no—
192C22×C4C22⋊D7912yes—
193C22×C4C22×D7911yescharacteristic
194Q16—1588no—
195SD16—1588no—
196M16—794no—
197M16—1582no—
198C2×C8—1582no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D7912yes—
204C23—1584no—
205C23—1582no—
206C23C2×C22⋊D7912yes—
207C23C2×C22⋊D7911yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—794no—
213Q8—1584no—
214Q8—1582no—
215D4—24no—
216D4—22no—
217D4—794no—
218D4—1584no—
219D4—1588no—
220D4—1584no—
221D4—1584no—
222D4—1582no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1588no—
229C2×C4—1584no—
230C2×C4—1582no—
231C2×C4—1584no—
232C2×C4—1584no—
233C2×C4D4×D7912yes—
234C2×C4D4×D7912yes—
235C2×C4C2×C22⋊D7912yes—
236C2×C4C2×C22⋊D7911yescharacteristic
237C2×C4C23×D7911yescharacteristic
238C8—1584no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—6322no—
243C22—6324no—
244C22—1584no—
245C22#1457912912yes—
246C22C22×C22⋊D7911yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—1584no—
251C4#1457912911yescharacteristic
252C4#1457912911yescharacteristic
253C4C2×D4×D7912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—3164no—
258C2C2×#1457912911yescharacteristic
subgroupG/H
centerC2C2×#14579129
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#536×C79C2
socleC2×C79C2×#120
prime radical#19099399C2
hypercenter#536D79
hypocenterC79#3022
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#14579192C2semidirect
3. #19098745C2semidirect
4. #19098740C2semidirect
5. #19098581C2semidirect
6. #19098547C2semidirect
7. #536×C79C2semidirect
17. #14579192C22semidirect
20. C2×#11929608C22semidirect
21. #14579089C22semidirect
22. #14579080C22semidirect
23. #14579078C22semidirect
25. C22×Q8×C79C22semidirect
26. C4⋊Q8×C79C22semidirect
27. #124×C79C22semidirect
30. #117×C79C22semidirect
130. C79#3022semidirect
133. #536D79semidirect
147. C22×Q8C2×D79semidirect
153. C4⋊Q8C2×D79semidirect
154. #124C2×D79semidirect
157. #117C2×D79semidirect