Finite Group Databasebeta

← #23766758

Subgroups — #23766758

Order 10112.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×#14579138C212yesmaximal, complemented
3C2×(C22⋊D79)⋊C4C211yesmaximal, characteristic
4#19098484C211yesmaximal, characteristic, complemented
5#19098482C211yesmaximal, characteristic, complemented
6#19098475C211yesmaximal, characteristic, complemented
7#19098467C212yesmaximal, complemented
8#19098439C212yesmaximal
9#19098437C212yesmaximal, complemented
10#19098422C211yesmaximal, characteristic, complemented
11#393×C79C211yesmaximal, characteristic, complemented
12C22×Dic158C2211yescharacteristic
13C22×Dic158C2212yes—
14C2×Q8×D79—28no—
15C2×Q8×D79—24no—
16C22×C22⋊D79C2211yescharacteristic
17C23×Dic79C2211yescharacteristic
18C22×C4×D79C2212yes—
19C22×C4×D79C2211yescharacteristic
20#14579138—28no—
21(C22⋊D79)⋊C4—24no—
22C2×C4⋊Dic79C2212yes—
23C2×Dic79⋊C4C2212yes—
24C2×Dic79⋊C4C2212yes—
25C2×Dic79⋊C4C2211yescharacteristic
26C2×C22⋊Dic79C2212yes—
27C2×(C2×D79)⋊C4C2212yes—
28C2×(C2×D79)⋊C4C2211yescharacteristic
29C2×C4×Dic79C2211yescharacteristic
30C79⋊#95C2211yescharacteristic
31C79⋊#95C2212yes—
32C79⋊#95C2212yes—
33C79⋊#95C2212yes—
34C22×Q8×C79C2211yescharacteristic
35C2×C22⋊C4×C79C2212yes—
36C2×C22⋊C4×C79C2211yescharacteristic
37#95×C79C2211yescharacteristic
38#95×C79C2212yes—
39C23×D79C2311yescharacteristic
40C2×Dic158—24no—
41C2×Dic158—24no—
42C2×Dic158—24no—
43C2×Dic158—22no—
44C2×Dic158D418yescomplemented
45Q8×D79—232no—
46C2×C22⋊D79—22no—
47C2×C22⋊D79—22no—
48C2×C22⋊D79Q814yescomplemented
49C22×Dic79—22no—
50C22×Dic79—24no—
51C22×Dic79C2312yes—
52C22×Dic79C2311yescharacteristic
53C22×Dic79C2312yes—
54C22×Dic79C2311yescharacteristic
55C22×Dic79C2311yescharacteristic
56C2×C4×D79—24no—
57C2×C4×D79—24no—
58C2×C4×D79—24no—
59C2×C4×D79—24no—
60C2×C4×D79D414yes—
61C4⋊Dic79—24no—
62Dic79⋊C4—24no—
63Dic79⋊C4—22no—
64Dic79⋊C4—24no—
65C22⋊Dic79—24no—
66(C2×D79)⋊C4—22no—
67(C2×D79)⋊C4—24no—
68C4×Dic79—22no—
69C24×C79C2311yescharacteristic
70C2×Q8×C79—24no—
71C2×Q8×C79—22no—
72C22×C4×C79C2311yescharacteristic
73C22×C4×C79C2312yes—
74C22×C4×C79C2312yes—
75C22×C4×C79C2311yescharacteristic
76C22⋊C4×C79—22no—
77C22⋊C4×C79—24no—
78C23×C79—22no—
79C23×C79—21no—
80C23×C79—21no—
81C23×C79—22no—
82C23×C79C2×Q812yescomplemented
83C23×C79C2411yescharacteristic
84C22×D79—22no—
85C22×D79—22no—
86C22×D79C2×D412yes—
87C22×D79C2×Q812yes—
88C22×D79#4112yes—
89Q8×C79—28no—
90C2×C4×C79—22no—
91C2×C4×C79—22no—
92C2×C4×C79—22no—
93C2×C4×C79—21no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79C2×D414yes—
99C2×C4×C79C2×D412yes—
100C22⋊D79—28no—
101C2×Dic79—21no—
102C2×Dic79—22no—
103C2×Dic79—22no—
104C2×Dic79—21no—
105C2×Dic79—21no—
106C2×Dic79—22no—
107C2×Dic79—22no—
108C2×Dic79—22no—
109C2×Dic79—22no—
110C2×Dic79—44no—
111C2×Dic79C2×D414yes—
112C2×Dic79C2×D412yes—
113C2×Dic79C2×D414yes—
114C2×Dic79C2×Q812yescomplemented
115C4×D79—216no—
116C4×D79—28no—
117Dic158—28no—
118Dic158—216no—
119C22×C79—22no—
120C22×C79—24no—
121C22×C79—41no—
122C22×C79—41no—
123C22×C79—42no—
124C22×C79C22×D411yescharacteristic
125C22×C79C22×D412yes—
126C22×C79C22×Q811yescharacteristic
127C22×C79C2×#4111yescharacteristic
128C22×C792−1+412yes—
129C2×D79—22no—
130C2×D79—22no—
131C2×D79—24no—
132C2×D79—24no—
133C2×D79C22⋊Q814yescomplemented
134C4×C79—24no—
135C4×C79—22no—
136C4×C79—41no—
137C4×C79—42no—
138Dic79—24no—
139Dic79—22no—
140Dic79—41no—
141Dic79—42no—
142Dic79#12014yes—
143C2×C79—42no—
144C2×C79C2×#12011yescharacteristic
145C2×C79C2×C22⋊Q811yescharacteristic
146C2×C79#53611yescharacteristic
147C2×C79#54812yes—
148C2×C79D4×Q812yes—
149D79—24no—
150#2409—791nomaximal, Sylow
151C79#240911yescharacteristic, complemented, Sylow
152C23×Q8—791no—
153C2×C22⋊Q8—792no—
154C2×C4×D4—791no—
155#395—791no—
156#395—791no—
157#393—791no—
158#393D7911yescharacteristic, complemented
159#391—792no—
160#386—792no—
161#385—792no—
162#380—791no—
163C22×Q8—792no—
164C22×Q8—791no—
165C22×Q8—1584no—
166C22×Q8—1588no—
167C22×Q8C2×D7911yescharacteristic
168C22×D4—791no—
169C23×C4—791no—
170C23×C4—791no—
171C23×C4—792no—
172C22⋊Q8—1588no—
173C4×D4—1584no—
174C2×C4⋊C4—791no—
175C2×C4⋊C4—792no—
176C2×C4⋊C4—792no—
177C2×C4⋊C4—792no—
178C2×C22⋊C4—792no—
179C2×C22⋊C4—791no—
180C2×C22⋊C4—792no—
181C2×C22⋊C4C2×D7911yescharacteristic
182C2×C22⋊C4C2×D7912yes—
183C2×C42—791no—
184#95—791no—
185#95—792no—
186#95—792no—
187#95—792no—
188#95C2×D7911yescharacteristic
189#95C2×D7912yes—
190C24—791no—
191C24C22×D7911yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—798no—
195C2×Q8—1584no—
196C2×Q8—1584no—
197C2×Q8—1584no—
198C2×Q8—15832no—
199C2×Q8—1582no—
200C2×D4—794no—
201C2×D4—1582no—
202C2×D4—1582no—
203C22×C4—791no—
204C22×C4—791no—
205C22×C4—792no—
206C22×C4—794no—
207C22×C4—792no—
208C22×C4—791no—
209C22×C4—1584no—
210C22×C4—1584no—
211C22×C4—1584no—
212C22×C4—1582no—
213C22×C4—1584no—
214C22×C4—1584no—
215C22×C4C22×D7911yescharacteristic
216C22×C4C22×D7912yes—
217C22×C4C22×D7911yescharacteristic
218C22×C4C22×D7912yes—
219C4⋊C4—1584no—
220C4⋊C4—1584no—
221C4⋊C4—1582no—
222C4⋊C4—1584no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1584no—
226C22⋊C4—1582no—
227C22⋊C4—1584no—
228C42—1582no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—792no—
234C23—792no—
235C23—792no—
236C23—1582no—
237C23—1582no—
238C23Q8×D7912yescomplemented
239C23C23×D7911yescharacteristic
240Q8—28no—
241Q8—15816no—
242Q8—1588no—
243D4—1588no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—3164no—
253C2×C4—792no—
254C2×C4—794no—
255C2×C4—792no—
256C2×C4—794no—
257C2×C4—1588no—
258C2×C4—15816no—
259C2×C4—1582no—
260C2×C4—1582no—
261C2×C4—1582no—
262C2×C4—1581no—
263C2×C4—1582no—
264C2×C4—1581no—
265C2×C4—1581no—
266C2×C4—1582no—
267C2×C4—1582no—
268C2×C4D4×D7912yes—
269C2×C4D4×D7914yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—794no—
276C22—1584no—
277C22—1584no—
278C22—1582no—
279C22—1582no—
280C22C2×D4×D7912yes—
281C22C2×D4×D7911yescharacteristic
282C22C2×Q8×D7911yescharacteristic
283C22#41×D7911yescharacteristic
284C22#1457920812yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—3162no—
290C4—3161no—
291C4—794no—
292C4—1584no—
293C4—1582no—
294C2—42no—
295C2—1584no—
296C2#120×D7911yescharacteristic
297C2C22⋊Q8×D7911yescharacteristic
298C2#1909913311yescharacteristic
299C2#1909922112yes—
300C2#1909922612yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
Fitting#393×C79C2
prime radicalC22×C22⋊D79C22
hypercenter#393D79
hypocenterC79#2409
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D79C2semidirect
2. C2×#14579138C2semidirect
4. #19098484C2semidirect
5. #19098482C2semidirect
6. #19098475C2semidirect
7. #19098467C2semidirect
9. #19098437C2semidirect
10. #19098422C2semidirect
11. #393×C79C2semidirect
44. C2×Dic158D4semidirect
48. C2×C22⋊D79Q8semidirect
82. C23×C79C2×Q8semidirect
114. C2×Dic79C2×Q8semidirect
133. C2×D79C22⋊Q8semidirect
151. C79#2409semidirect
158. #393D79semidirect
238. C23Q8×D79semidirect