Finite Group Databasebeta

← #24910844

Subgroups — #24910844

Order 10624.

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×#14581773C211yesmaximal, characteristic, complemented
2C2×#14581748C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic83C211yesmaximal, characteristic, complemented
4#19127624C212yesmaximal, complemented
5#19127615C212yesmaximal, complemented
6#19127604C212yesmaximal, complemented
7#19127589C212yesmaximal, complemented
8#19127581C211yesmaximal, characteristic, complemented
9#19127576C211yesmaximal, characteristic, complemented
10#19127557C211yesmaximal, characteristic, complemented
11C2×#120×C83C211yesmaximal, characteristic
12C23×Dic83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14#14581773—24no—
15#14581748—24no—
16C2×C4⋊Dic83C2212yescomplemented
17C2×Dic83⋊C4C2211yescharacteristic, complemented
18C2×Dic83⋊C4C2212yescomplemented
19C2×Dic83⋊C4C2211yescharacteristic, complemented
20C2×C22⋊Dic83—24no—
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C22⋊Dic83C2212yes—
27C2×C22⋊Dic83C2211yescharacteristic, complemented
28C2×C22⋊Dic83C2211yescharacteristic, complemented
29C2×C22⋊Dic83C2212yescomplemented
30C2×C4×Dic83C2212yescomplemented
31C83⋊#95C2212yescomplemented
32C83⋊#95C2212yescomplemented
33C25×C83C2211yescharacteristic
34C22×D4×C83C2211yescharacteristic
35C22×D4×C83C2212yes—
36#120×C83—24no—
37C2×C22⋊C4×C83C2212yes—
38C2×C22⋊C4×C83C2211yescharacteristic
39C22×Dic83—22no—
40C22×Dic83—22no—
41C22×Dic83—22no—
42C22×Dic83—42no—
43C22×Dic83D414yescomplemented
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2311yescharacteristic
50C4⋊Dic83—42no—
51Dic83⋊C4—24no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—22no—
56C22⋊Dic83—24no—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59C22⋊Dic83—24no—
60C22⋊Dic83—24no—
61C22⋊Dic83—42no—
62C22⋊Dic83—42no—
63C4×Dic83—24no—
64C24×C83—22no—
65C24×C83—24no—
66C24×C83—22no—
67C24×C83—22no—
68C24×C83—42no—
69C24×C83C2311yescharacteristic
70C24×C83C2312yes—
71C24×C83C2311yescharacteristic
72C2×D4×C83—22no—
73C2×D4×C83—22no—
74C2×D4×C83—24no—
75C2×D4×C83—22no—
76C2×D4×C83—24no—
77C2×D4×C83—42no—
78C22×C4×C83C2311yescharacteristic
79C22×C4×C83C2312yes—
80C22⋊C4×C83—24no—
81C22⋊C4×C83D414yes—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—28no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—22no—
90C23×C83—22no—
91C23×C83—21no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—22no—
95C23×C83—22no—
96C23×C83—48no—
97C23×C83—48no—
98C23×C83—44no—
99C23×C83—44no—
100C23×C83C2×D412yes—
101C23×C83C2×D412yes—
102C23×C83#4114yes—
103C23×C83#4112yes—
104C23×C83C2411yescharacteristic
105D4×C83—48no—
106D4×C83—44no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83C2×D414yes—
113C2×Dic83—28no—
114C2×Dic83—21no—
115C2×Dic83—21no—
116C2×Dic83—21no—
117C2×Dic83—21no—
118C2×Dic83—22no—
119C2×Dic83—21no—
120C2×Dic83—21no—
121C2×Dic83—22no—
122C2×Dic83—22no—
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—44no—
129C2×Dic83C2×D412yes—
130C2×Dic83C2×D412yes—
131C22×C83—22no—
132C22×C83—22no—
133C22×C83—24no—
134C22×C83—24no—
135C22×C83—24no—
136C22×C83—24no—
137C22×C83—24no—
138C22×C83—416no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83—41no—
144C22×C83—41no—
145C22×C83—41no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83#12114yes—
149C22×C83C22×D411yescharacteristic
150C22×C83C22×D411yescharacteristic
151C22×C83C2×#4111yescharacteristic
152C22×C83C2×#4112yes—
153C22×C83H(5,𝔽2)12yes—
154C4×C83—24no—
155C4×C83—41no—
156Dic83—22no—
157Dic83—22no—
158Dic83—41no—
159Dic83—41no—
160Dic83—42no—
161Dic83—42no—
162C2×C83—24no—
163C2×C83—44no—
164C2×C83—81no—
165C2×C83C2×#12111yescharacteristic
166C2×C83#52511yescharacteristic
167C2×C83#53511yescharacteristic
168C2×C83#54612yes—
169C2×C83#55112yes—
170#2411—831nomaximal, Sylow
171C83#241111yescharacteristic, complemented, Sylow
172C2×#123—831no—
173C2×#120D8311yescharacteristic
174C2×#117—831no—
175C22×C22⋊C4—831no—
176#396—832no—
177#394—832no—
178#390—832no—
179#388—832no—
180#386—831no—
181#385—831no—
182#379—831no—
183C25C2×D8311yescharacteristic
184C22×D4C2×D8311yescharacteristic
185C22×D4C2×D8312yes—
186C23×C4—831no—
187C23×C4—831no—
188#123—1664no—
189#120—24no—
190#117—1664no—
191C2×C4⋊C4—832no—
192C2×C4⋊C4—832no—
193C2×C4⋊C4—831no—
194C2×C4⋊C4—831no—
195C2×C22⋊C4—832no—
196C2×C22⋊C4—832no—
197C2×C22⋊C4—832no—
198C2×C22⋊C4—832no—
199C2×C22⋊C4—831no—
200C2×C22⋊C4—831no—
201C2×C22⋊C4—831no—
202C2×C22⋊C4—831no—
203C2×C22⋊C4—832no—
204C2×C22⋊C4—1664no—
205C2×C22⋊C4C2×D8312yes—
206C2×C22⋊C4C2×D8311yescharacteristic
207C2×C42—832no—
208#95—832no—
209#95—832no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D8311yescharacteristic
216C24C22×D8312yes—
217C24C22×D8311yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—3322no—
225C22×C4—834no—
226C22×C4—831no—
227C22×C4—831no—
228C22×C4—831no—
229C22×C4—832no—
230C22×C4—832no—
231C22×C4—831no—
232C22×C4—1662no—
233C22×C4—1662no—
234C22×C4—1662no—
235C22×C4C22×D8311yescharacteristic
236C22×C4C22×D8312yes—
237C4⋊C4—3322no—
238C4⋊C4—1662no—
239C4⋊C4—1662no—
240C4⋊C4—1664no—
241C22⋊C4—24no—
242C22⋊C4—3322no—
243C22⋊C4—3322no—
244C22⋊C4—1664no—
245C22⋊C4—1664no—
246C22⋊C4—1662no—
247C22⋊C4—1662no—
248C22⋊C4—1662no—
249C22⋊C4—1662no—
250C22⋊C4—1664no—
251C22⋊C4C22⋊D8314yes—
252C42—1664no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D8312yes—
272C23C2×C22⋊D8312yes—
273C23#1305058914yes—
274C23#1305058912yes—
275C23C23×D8311yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—3324no—
284C2×C4—832no—
285C2×C4—832no—
286C2×C4—1661no—
287C2×C4—1661no—
288C2×C4—1662no—
289C2×C4—1662no—
290C2×C4—1662no—
291C2×C4—1662no—
292C2×C4—1662no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4—1662no—
296C2×C4—1668no—
297C2×C4—1661no—
298C2×C4—1661no—
299C2×C4—1661no—
300C2×C4—1661no—
301C2×C4C2×C22⋊D8314yes—
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—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22#1458176314yes—
320C22C2×D4×D8311yescharacteristic
321C22C22×C22⋊D8311yescharacteristic
322C22C2×#1305058912yes—
323C22C2×#1305058911yescharacteristic
324C22#1458183712yes—
325C4—24no—
326C4—41no—
327C4—3322no—
328C4—3321no—
329C4—3322no—
330C4—3321no—
331C4—1662no—
332C4—1662no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#1458176311yescharacteristic
337C2#1912820711yescharacteristic
338C2#1912826611yescharacteristic
339C2#1912834512yes—
340C2#1912838212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#120×C83C2
hypercenterC2×#120D83
hypocenterC83#2411
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14581773C2semidirect
2. C2×#14581748C2semidirect
3. C22×C22⋊Dic83C2semidirect
4. #19127624C2semidirect
5. #19127615C2semidirect
6. #19127604C2semidirect
7. #19127589C2semidirect
8. #19127581C2semidirect
9. #19127576C2semidirect
10. #19127557C2semidirect
16. C2×C4⋊Dic83C22semidirect
17. C2×Dic83⋊C4C22semidirect
18. C2×Dic83⋊C4C22semidirect
19. C2×Dic83⋊C4C22semidirect
27. C2×C22⋊Dic83C22semidirect
28. C2×C22⋊Dic83C22semidirect
29. C2×C22⋊Dic83C22semidirect
30. C2×C4×Dic83C22semidirect
31. C83⋊#95C22semidirect
32. C83⋊#95C22semidirect
43. C22×Dic83D4semidirect
171. C83#2411semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#145817732
C2×#145817482
C22×C22⋊Dic832
#191276242
#191276152
#191276042
#191275892
#191275812
#191275762
#191275572
C2×#120×C832
#241183