Finite Group Databasebeta

← #23766785

Subgroups — #23766785

Order 10112.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14579144C211yesmaximal, characteristic, complemented
2C2×#14579130C211yesmaximal, characteristic
3C2×#14579120C211yesmaximal, characteristic, complemented
4#19098486C212yesmaximal, complemented
5#19098479C212yesmaximal, complemented
6#19098463C212yesmaximal, complemented
7#19098457C212yesmaximal, complemented
8#19098442C211yesmaximal, characteristic, complemented
9#19098436C211yesmaximal, characteristic, complemented
10#19098419C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C79C211yesmaximal, characteristic, complemented
12C22×C22⋊D79C2211yescharacteristic
13C22×C22⋊D79C2212yes—
14#14579144—24no—
15#14579130—24no—
16#14579120—24no—
17C2×C4⋊Dic79C2212yescomplemented
18C2×Dic79⋊C4C2211yescharacteristic, complemented
19C2×Dic79⋊C4C2211yescharacteristic, complemented
20C2×Dic79⋊C4C2212yescomplemented
21C2×C22⋊Dic79C2211yescharacteristic
22C2×C22⋊Dic79C2211yescharacteristic
23C2×C22⋊Dic79C2212yes—
24C2×C22⋊Dic79C2212yes—
25C2×C22⋊Dic79C2212yes—
26C2×C22⋊Dic79C2211yescharacteristic
27C2×(C2×D79)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D79)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D79)⋊C4C2212yescomplemented
30C2×C4×Dic79C2212yescomplemented
31C79⋊#95C2212yescomplemented
32C79⋊#95C2212yescomplemented
33C25×C79C2211yescharacteristic
34C23×C4×C79C2211yescharacteristic
35C23×C4×C79C2211yescharacteristic
36C2×C22⋊C4×C79—24no—
37C2×C22⋊C4×C79C2212yes—
38C2×C22⋊C4×C79C2212yes—
39C23×D79C2311yescharacteristic
40C2×C22⋊D79—24no—
41C2×C22⋊D79—22no—
42C2×C22⋊D79—22no—
43C2×C22⋊D79—24no—
44C2×C22⋊D79—22no—
45C2×C22⋊D79—42no—
46C22×Dic79C2311yescharacteristic
47C22×Dic79C2311yescharacteristic
48C22×Dic79C2312yes—
49C22×Dic79C2311yescharacteristic
50C22×Dic79C2312yes—
51C4⋊Dic79—24no—
52Dic79⋊C4—22no—
53Dic79⋊C4—22no—
54Dic79⋊C4—42no—
55C22⋊Dic79—24no—
56C22⋊Dic79—22no—
57C22⋊Dic79—22no—
58C22⋊Dic79—24no—
59C22⋊Dic79—42no—
60C22⋊Dic79D414yes—
61(C2×D79)⋊C4—22no—
62(C2×D79)⋊C4—22no—
63(C2×D79)⋊C4—42no—
64C4×Dic79—24no—
65C24×C79—22no—
66C24×C79—24no—
67C24×C79—22no—
68C24×C79—22no—
69C24×C79—42no—
70C24×C79C2312yes—
71C24×C79C2311yescharacteristic
72C22×C4×C79—22no—
73C22×C4×C79—22no—
74C22×C4×C79—22no—
75C22×C4×C79—42no—
76C22×C4×C79D414yescomplemented
77C22×C4×C79C2312yes—
78C22×C4×C79C2311yescharacteristic
79C22×C4×C79C2311yescharacteristic
80C22⋊C4×C79—24no—
81C22⋊C4×C79—24no—
82C23×C79—22no—
83C23×C79—22no—
84C23×C79—28no—
85C23×C79—22no—
86C23×C79—22no—
87C23×C79—22no—
88C23×C79—22no—
89C23×C79—22no—
90C23×C79—22no—
91C23×C79—48no—
92C23×C79—48no—
93C23×C79—44no—
94C23×C79—44no—
95C23×C79C2×D412yes—
96C23×C79C2×D412yes—
97C23×C79#4112yes—
98C23×C79#4114yes—
99C23×C79C2411yescharacteristic
100C22×D79—21no—
101C22×D79—21no—
102C22×D79—21no—
103C22×D79—22no—
104C22×D79—22no—
105C2×C4×C79—28no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—44no—
114C2×C4×C79C2×D412yes—
115C2×C4×C79C2×D412yes—
116C22⋊D79—48no—
117C22⋊D79—44no—
118C2×Dic79—22no—
119C2×Dic79—22no—
120C2×Dic79—22no—
121C2×Dic79—21no—
122C2×Dic79—21no—
123C2×Dic79—21no—
124C2×Dic79—21no—
125C2×Dic79—22no—
126C2×Dic79—22no—
127C2×Dic79—21no—
128C2×Dic79—22no—
129C2×Dic79—22no—
130C2×Dic79C2×D414yes—
131C22×C79—22no—
132C22×C79—22no—
133C22×C79—24no—
134C22×C79—24no—
135C22×C79—24no—
136C22×C79—24no—
137C22×C79—24no—
138C22×C79—416no—
139C22×C79—42no—
140C22×C79—42no—
141C22×C79—42no—
142C22×C79—42no—
143C22×C79#12114yes—
144C22×C79C22×D411yescharacteristic
145C22×C79C22×D411yescharacteristic
146C22×C79C2×#4111yescharacteristic
147C22×C79C2×#4112yes—
148C22×C79H(5,𝔽2)12yes—
149C2×D79—41no—
150C2×D79—41no—
151C2×D79—41no—
152C2×D79—42no—
153C2×D79—42no—
154C4×C79—22no—
155C4×C79—22no—
156C4×C79—42no—
157Dic79—24no—
158Dic79—41no—
159Dic79—41no—
160Dic79—41no—
161Dic79—42no—
162C2×C79—24no—
163C2×C79—44no—
164C2×C79C2×#12111yescharacteristic
165C2×C79#52511yescharacteristic
166C2×C79#53511yescharacteristic
167C2×C79#54612yes—
168C2×C79#55112yes—
169D79—81no—
170#2411—791nomaximal, Sylow
171C79#241111yescharacteristic, complemented, Sylow
172C2×#123—791no—
173C2×#120—791no—
174C2×#117—791no—
175C22×C22⋊C4D7911yescharacteristic, complemented
176#396—792no—
177#394—792no—
178#390—792no—
179#388—792no—
180#386—791no—
181#385—791no—
182#379—791no—
183C25C2×D7911yescharacteristic
184C22×D4—792no—
185C22×D4—791no—
186C23×C4C2×D7911yescharacteristic
187C23×C4C2×D7911yescharacteristic
188#123—1584no—
189#120—1584no—
190#117—1584no—
191C2×C4⋊C4—792no—
192C2×C4⋊C4—792no—
193C2×C4⋊C4—791no—
194C2×C4⋊C4—791no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—792no—
197C2×C22⋊C4—792no—
198C2×C22⋊C4—791no—
199C2×C22⋊C4—791no—
200C2×C22⋊C4—792no—
201C2×C22⋊C4—791no—
202C2×C22⋊C4—792no—
203C2×C22⋊C4—791no—
204C2×C22⋊C4—791no—
205C2×C22⋊C4C2×D7912yes—
206C2×C22⋊C4C2×D7912yes—
207C2×C42—792no—
208#95—792no—
209#95—792no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—791no—
216C24C22×D7912yes—
217C24C22×D7911yescharacteristic
218C2×D4—3162no—
219C2×D4—1584no—
220C2×D4—1582no—
221C2×D4—1582no—
222C2×D4—1582no—
223C2×D4—1584no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—792no—
229C22×C4—791no—
230C22×C4—791no—
231C22×C4—792no—
232C22×C4—791no—
233C22×C4C22⋊D7914yescomplemented
234C22×C4C22×D7912yes—
235C22×C4C22×D7911yescharacteristic
236C22×C4C22×D7911yescharacteristic
237C4⋊C4—3162no—
238C4⋊C4—1584no—
239C4⋊C4—1582no—
240C4⋊C4—1582no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—3162no—
244C22⋊C4—3162no—
245C22⋊C4—794no—
246C22⋊C4—1584no—
247C22⋊C4—1582no—
248C22⋊C4—1582no—
249C22⋊C4—1584no—
250C22⋊C4—1582no—
251C22⋊C4—1582no—
252C42—1584no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—1581no—
267C23—1582no—
268C23—1582no—
269C23—1581no—
270C23—1581no—
271C23D4×D7912yes—
272C23C2×C22⋊D7912yes—
273C23#1192962312yes—
274C23#1192962414yes—
275C23C23×D7911yescharacteristic
276D4—3164no—
277D4—3168no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—794no—
288C2×C4—1581no—
289C2×C4—1582no—
290C2×C4—1582no—
291C2×C4—1582no—
292C2×C4—1582no—
293C2×C4—1582no—
294C2×C4—1581no—
295C2×C4—1581no—
296C2×C4—1581no—
297C2×C4—1581no—
298C2×C4—1582no—
299C2×C4—1582no—
300C2×C4C2×C22⋊D7912yes—
301C2×C4C2×C22⋊D7912yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—3162no—
315C22—3162no—
316C22—3161no—
317C22—3161no—
318C22—3161no—
319C22#1457913314yes—
320C22C2×D4×D7911yescharacteristic
321C22C22×C22⋊D7911yescharacteristic
322C22C2×#1192962311yescharacteristic
323C22C2×#1192962412yes—
324C22#1457920612yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—3162no—
329C4—3161no—
330C4—3161no—
331C4—3161no—
332C4—1584no—
333C2—24no—
334C2—44no—
335C2—6321no—
336C2C2×#1457913311yescharacteristic
337C2#1909907411yescharacteristic
338C2#1909913011yescharacteristic
339C2#1909920312yes—
340C2#1909923912yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC22×C22⋊C4×C79C2
prime radicalC2×#14579130C2
hypercenterC22×C22⋊C4D79
hypocenterC79#2411
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14579144C2semidirect
3. C2×#14579120C2semidirect
4. #19098486C2semidirect
5. #19098479C2semidirect
6. #19098463C2semidirect
7. #19098457C2semidirect
8. #19098442C2semidirect
9. #19098436C2semidirect
10. #19098419C2semidirect
11. C22×C22⋊C4×C79C2semidirect
17. C2×C4⋊Dic79C22semidirect
18. C2×Dic79⋊C4C22semidirect
19. C2×Dic79⋊C4C22semidirect
20. C2×Dic79⋊C4C22semidirect
27. C2×(C2×D79)⋊C4C22semidirect
28. C2×(C2×D79)⋊C4C22semidirect
29. C2×(C2×D79)⋊C4C22semidirect
30. C2×C4×Dic79C22semidirect
31. C79⋊#95C22semidirect
32. C79⋊#95C22semidirect
76. C22×C4×C79D4semidirect
171. C79#2411semidirect
175. C22×C22⋊C4D79semidirect
233. C22×C4C22⋊D79semidirect