Finite Group Databasebeta

← #23768289

Subgroups — #23768289

Order 10112.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14579129C211yesmaximal, characteristic
2C2×D4×Dic79C211yesmaximal, characteristic, complemented
3C22×(C2×D79)⋊C4C211yesmaximal, characteristic, complemented
4#19098485C212yesmaximal, complemented
5#19098478C212yesmaximal, complemented
6#19098453C212yesmaximal, complemented
7#19098440C212yesmaximal, complemented
8#19098426C211yesmaximal, characteristic, complemented
9#19098421C211yesmaximal, characteristic, complemented
10#19098418C211yesmaximal, characteristic, complemented
11C2×#123×C79C211yesmaximal, characteristic, complemented
12C24×D79C2211yescharacteristic
13C22×C22⋊D79C2212yes—
14C23×Dic79C2211yescharacteristic
15C23×Dic79C2211yescharacteristic, complemented
16#14579129—24no—
17D4×Dic79—24no—
18C2×C4⋊Dic79C2211yescharacteristic, complemented
19C2×Dic79⋊C4C2212yescomplemented
20C2×C22⋊Dic79C2211yescharacteristic
21C2×C22⋊Dic79C2211yescharacteristic
22C2×C22⋊Dic79C2211yescharacteristic, complemented
23C2×(C2×D79)⋊C4—24no—
24C2×(C2×D79)⋊C4C2212yes—
25C2×(C2×D79)⋊C4C2212yes—
26C2×(C2×D79)⋊C4C2212yes—
27C2×(C2×D79)⋊C4C2212yes—
28C2×(C2×D79)⋊C4C2212yes—
29C2×C4×Dic79C2211yescharacteristic, complemented
30C79⋊#95C2212yescomplemented
31C79⋊#95C2212yescomplemented
32C79⋊#95C2212yescomplemented
33C22×D4×C79C2211yescharacteristic
34C23×C4×C79C2211yescharacteristic
35#123×C79—24no—
36C2×C4⋊C4×C79C2212yescomplemented
37C2×C22⋊C4×C79C2211yescharacteristic
38C2×C22⋊C4×C79C2212yescomplemented
39C23×D79—22no—
40C23×D79—24no—
41C23×D79—22no—
42C23×D79—22no—
43C23×D79—42no—
44C23×D79C2312yes—
45C2×C22⋊D79—24no—
46C2×C22⋊D79—24no—
47C2×C22⋊D79—42no—
48C22×Dic79—22no—
49C22×Dic79—42no—
50C22×Dic79—42no—
51C22×Dic79D414yescomplemented
52C22×Dic79C2312yes—
53C22×Dic79C2311yescharacteristic
54C22×Dic79C2311yescharacteristic
55C22×Dic79C2311yescharacteristic
56C22×Dic79C2311yescharacteristic
57C4⋊Dic79—22no—
58Dic79⋊C4—42no—
59C22⋊Dic79—22no—
60C22⋊Dic79—22no—
61C22⋊Dic79—22no—
62(C2×D79)⋊C4—24no—
63(C2×D79)⋊C4—24no—
64(C2×D79)⋊C4—24no—
65(C2×D79)⋊C4—24no—
66(C2×D79)⋊C4—42no—
67C4×Dic79—22no—
68C24×C79C2311yescharacteristic
69C24×C79C2311yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—22no—
72C2×D4×C79D414yes—
73C22×C4×C79—22no—
74C22×C4×C79—22no—
75C22×C4×C79—22no—
76C22×C4×C79C2312yes—
77C22×C4×C79C2312yes—
78C22×C4×C79C2311yescharacteristic
79C4⋊C4×C79—24no—
80C22⋊C4×C79—22no—
81C22⋊C4×C79—24no—
82C23×C79—22no—
83C23×C79—22no—
84C23×C79—21no—
85C23×C79—22no—
86C23×C79—21no—
87C23×C79—22no—
88C23×C79C2×D412yes—
89C23×C79C2×D412yes—
90C23×C79C2×D412yes—
91C23×C79#4112yes—
92C23×C79C2411yescharacteristic
93C22×D79—28no—
94C22×D79—22no—
95C22×D79—22no—
96C22×D79—22no—
97C22×D79—22no—
98C22×D79—22no—
99C22×D79—22no—
100C22×D79—48no—
101C22×D79—48no—
102C22×D79—44no—
103C22×D79—44no—
104C22×D79#4114yes—
105D4×C79—28no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—44no—
115C2×C4×C79C2×D412yes—
116C22⋊D79—48no—
117C2×Dic79—22no—
118C2×Dic79—22no—
119C2×Dic79—21no—
120C2×Dic79—22no—
121C2×Dic79—21no—
122C2×Dic79—22no—
123C2×Dic79—22no—
124C2×Dic79—22no—
125C2×Dic79—22no—
126C2×Dic79—44no—
127C2×Dic79—44no—
128C2×Dic79C2×D412yes—
129C2×Dic79C2×D412yes—
130C22×C79—22no—
131C22×C79—22no—
132C22×C79—24no—
133C22×C79—24no—
134C22×C79—22no—
135C22×C79—24no—
136C22×C79—41no—
137C22×C79—41no—
138C22×C79—42no—
139C22×C79#12114yes—
140C22×C79C22×D411yescharacteristic
141C22×C79C22×D411yescharacteristic
142C22×C79C2×#4112yes—
143C22×C79C2×#4111yescharacteristic
144C22×C79H(5,𝔽2)12yes—
145C2×D79—24no—
146C2×D79—24no—
147C2×D79—24no—
148C2×D79—416no—
149C2×D79—42no—
150C2×D79—42no—
151C2×D79—42no—
152C2×D79—42no—
153C4×C79—22no—
154C4×C79—42no—
155C4×C79—42no—
156Dic79—22no—
157Dic79—22no—
158Dic79—41no—
159Dic79—41no—
160Dic79—42no—
161C2×C79—24no—
162C2×C79—42no—
163C2×C79C2×#12111yescharacteristic
164C2×C79#52511yescharacteristic
165C2×C79#53411yescharacteristic
166C2×C79#54612yes—
167C2×C79#55112yes—
168D79—44no—
169#2534—791nomaximal, Sylow
170C79#253411yescharacteristic, complemented, Sylow
171C2×#123D7911yescharacteristic, complemented
172C2×#120—791no—
173C2×C4×D4—791no—
174C22×C22⋊C4—791no—
175#396—792no—
176#394—792no—
177#388—792no—
178#386—792no—
179#381—791no—
180#380—791no—
181#379—791no—
182C25—791no—
183C22×D4—792no—
184C22×D4C2×D7911yescharacteristic
185C23×C4—791no—
186C23×C4—791no—
187C23×C4C2×D7911yescharacteristic
188#123—24no—
189#120—1584no—
190C4×D4—1584no—
191C2×C4⋊C4—792no—
192C2×C4⋊C4—791no—
193C2×C4⋊C4C2×D7912yescomplemented
194C2×C22⋊C4—792no—
195C2×C22⋊C4—791no—
196C2×C22⋊C4—791no—
197C2×C22⋊C4—791no—
198C2×C22⋊C4—792no—
199C2×C22⋊C4—792no—
200C2×C22⋊C4—792no—
201C2×C22⋊C4—792no—
202C2×C22⋊C4—1584no—
203C2×C22⋊C4C2×D7911yescharacteristic
204C2×C22⋊C4C2×D7912yescomplemented
205C2×C42—791no—
206#95—792no—
207#95—792no—
208#95—792no—
209C24—3162no—
210C24—792no—
211C24—1584no—
212C24—1582no—
213C24—1582no—
214C24—1582no—
215C24C22×D7911yescharacteristic
216C24C22×D7911yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—3162no—
220C2×D4—1584no—
221C2×D4—1584no—
222C2×D4C22⋊D7914yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—3162no—
227C22×C4—3162no—
228C22×C4—794no—
229C22×C4—791no—
230C22×C4—791no—
231C22×C4—792no—
232C22×C4—791no—
233C22×C4—791no—
234C22×C4—1582no—
235C22×C4C22×D7912yes—
236C22×C4C22×D7912yes—
237C22×C4C22×D7911yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—3162no—
240C4⋊C4—1582no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—3162no—
244C22⋊C4—1584no—
245C22⋊C4—1584no—
246C22⋊C4—1584no—
247C22⋊C4—1582no—
248C22⋊C4—1582no—
249C22⋊C4—1582no—
250C22⋊C4—1584no—
251C42—1582no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—22no—
257C23—21no—
258C23—3164no—
259C23—3168no—
260C23—3168no—
261C23—3164no—
262C23—794no—
263C23—1588no—
264C23—1582no—
265C23—1582no—
266C23—1582no—
267C23—1582no—
268C23—1582no—
269C23—1582no—
270C23D4×D7912yes—
271C23C2×C22⋊D7912yes—
272C23C2×C22⋊D7912yes—
273C23#1192962312yes—
274C23C23×D7911yescharacteristic
275D4—28no—
276D4—3168no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—3164no—
287C2×C4—3164no—
288C2×C4—792no—
289C2×C4—792no—
290C2×C4—1582no—
291C2×C4—1582no—
292C2×C4—1582no—
293C2×C4—1582no—
294C2×C4—1581no—
295C2×C4—1581no—
296C2×C4—1582no—
297C2×C4—1582no—
298C2×C4—1582no—
299C2×C4C2×C22⋊D7912yes—
300C22—24no—
301C22—24no—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—31616no—
310C22—3162no—
311C22—3162no—
312C22—3162no—
313C22—3162no—
314C22—1584no—
315C22—1584no—
316C22—1584no—
317C22#1457913314yes—
318C22C2×D4×D7911yescharacteristic
319C22C22×C22⋊D7911yescharacteristic
320C22#41×D7912yes—
321C22C2×#1192962311yescharacteristic
322C22#1457920712yes—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—3162no—
327C4—3161no—
328C4—3161no—
329C4—1582no—
330C4—1582no—
331C2—24no—
332C2—42no—
333C2—3164no—
334C2C2×#1457913311yescharacteristic
335C2#1909907111yescharacteristic
336C2#1909912411yescharacteristic
337C2#1909920112yes—
338C2#1909923612yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC2×#123×C79C2
prime radicalC2×#14579129C2
hypercenterC2×#123D79
hypocenterC79#2534
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic79C2semidirect
3. C22×(C2×D79)⋊C4C2semidirect
4. #19098485C2semidirect
5. #19098478C2semidirect
6. #19098453C2semidirect
7. #19098440C2semidirect
8. #19098426C2semidirect
9. #19098421C2semidirect
10. #19098418C2semidirect
11. C2×#123×C79C2semidirect
15. C23×Dic79C22semidirect
18. C2×C4⋊Dic79C22semidirect
19. C2×Dic79⋊C4C22semidirect
22. C2×C22⋊Dic79C22semidirect
29. C2×C4×Dic79C22semidirect
30. C79⋊#95C22semidirect
31. C79⋊#95C22semidirect
32. C79⋊#95C22semidirect
36. C2×C4⋊C4×C79C22semidirect
38. C2×C22⋊C4×C79C22semidirect
51. C22×Dic79D4semidirect
170. C79#2534semidirect
171. C2×#123D79semidirect
193. C2×C4⋊C4C2×D79semidirect
204. C2×C22⋊C4C2×D79semidirect