Finite Group Databasebeta

← #24913347

Subgroups — #24913347

Order 10624.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581766C211yesmaximal, characteristic, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127616C212yesmaximal, complemented
6#19127610C211yesmaximal, characteristic, complemented
7#19127609C212yesmaximal, complemented
8#19127592C212yesmaximal, complemented
9#19127580C212yesmaximal, complemented
10#19127564C211yesmaximal, characteristic, complemented
11C2×C4×D4×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2212yes—
15#14581766—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C22×D4×C83C2211yescharacteristic
30C23×C4×C83C2211yescharacteristic
31C23×C4×C83C2211yescharacteristic
32C4×D4×C83—24no—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C42×C83C2211yescharacteristic
37C23×D83C2312yes—
38C2×D332—22no—
39C2×D332—42no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C2×C22⋊D83—42no—
45C2×C22⋊D83—42no—
46C22×Dic83C2312yes—
47C22×Dic83C2312yes—
48C22×Dic83C2312yes—
49C4⋊Dic83—22no—
50Dic83⋊C4—42no—
51C22⋊Dic83—42no—
52C22⋊Dic83—42no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C24×C83C2311yescharacteristic
58C24×C83C2311yescharacteristic
59C2×D4×C83—22no—
60C2×D4×C83—22no—
61C2×D4×C83—22no—
62C22×C4×C83—22no—
63C22×C4×C83—42no—
64C22×C4×C83—42no—
65C22×C4×C83D414yescomplemented
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2311yescharacteristic
71C4⋊C4×C83—22no—
72C22⋊C4×C83—22no—
73C22⋊C4×C83—22no—
74C42×C83—22no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—21no—
78C23×C83—21no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83C2×D412yes—
84C23×C83#4112yes—
85C23×C83C2411yescharacteristic
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83—22no—
92C22×D83—22no—
93D4×C83—28no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—44no—
103C2×C4×C83—44no—
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D412yes—
106C2×C4×C83#4112yes—
107C22⋊D83—48no—
108C22⋊D83—48no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83—22no—
116C2×Dic83—22no—
117C2×Dic83—22no—
118D332—44no—
119C22×C83—22no—
120C22×C83—22no—
121C22×C83—22no—
122C22×C83—22no—
123C22×C83—24no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83C22×D411yescharacteristic
130C22×C83C2×#4111yescharacteristic
131C22×C83C2×#4111yescharacteristic
132C22×C83H(5,𝔽2)12yes—
133C22×C83H(5,𝔽2)12yes—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C2×D83—42no—
140C2×D83—42no—
141C4×C83—22no—
142C4×C83—22no—
143C4×C83—22no—
144C4×C83—41no—
145C4×C83—41no—
146Dic83—42no—
147Dic83—42no—
148Dic83—42no—
149C2×C83—42no—
150C2×C83—42no—
151C2×C83#52511yescharacteristic
152C2×C83#53411yescharacteristic
153C2×C83#53511yescharacteristic
154C2×C83#53812yes—
155C2×C83#54012yes—
156D83—82no—
157#2635—831nomaximal, Sylow
158C83#263511yescharacteristic, complemented, Sylow
159C2×#121—831no—
160C2×C4×D4D8311yescharacteristic, complemented
161#400—831no—
162#399—831no—
163#394—831no—
164#394—832no—
165#392—832no—
166#392—831no—
167#388—832no—
168#386—832no—
169#381—831no—
170C22×D4—832no—
171C22×D4—832no—
172C22×D4—831no—
173C22×D4C2×D8311yescharacteristic
174C23×C4C2×D8311yescharacteristic
175C23×C4C2×D8311yescharacteristic
176#121—1664no—
177C4×D4—24no—
178C2×C4⋊C4—832no—
179C2×C4⋊C4—831no—
180C2×C4⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—832no—
185C2×C22⋊C4—832no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4C2×D8311yescharacteristic
188C2×C22⋊C4C2×D8311yescharacteristic
189C2×C42C2×D8311yescharacteristic
190#95—832no—
191#95—831no—
192#95—831no—
193#95—832no—
194#95—832no—
195C24—832no—
196C24C22×D8311yescharacteristic
197C24C22×D8311yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—3322no—
202C2×D4—3322no—
203C2×D4—3322no—
204C2×D4—3322no—
205C2×D4—3322no—
206C2×D4—1664no—
207C2×D4—1664no—
208C2×D4—1662no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—832no—
213C22×C4—832no—
214C22×C4—832no—
215C22×C4C22⋊D8314yescomplemented
216C22×C4C22×D8311yescharacteristic
217C22×C4C22×D8311yescharacteristic
218C22×C4C22×D8311yescharacteristic
219C22×C4C22×D8311yescharacteristic
220C22×C4C22×D8311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—3322no—
223C4⋊C4—1662no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—3322no—
227C22⋊C4—3322no—
228C22⋊C4—3322no—
229C22⋊C4—3322no—
230C22⋊C4—1664no—
231C22⋊C4—1664no—
232C42—22no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—1662no—
242C23—1662no—
243C23—1662no—
244C23—1662no—
245C23—1662no—
246C23—1662no—
247C23—1662no—
248C23C2×C22⋊D8312yes—
249C23#1305059012yes—
250C23C23×D8311yescharacteristic
251D4—28no—
252D4—3324no—
253D4—3328no—
254D4—3328no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4C2×C22⋊D8312yes—
275C2×C4C2×C22⋊D8312yes—
276C2×C4#1305059012yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—3322no—
288C22—3322no—
289C22—3322no—
290C22—3322no—
291C22—3322no—
292C22—3322no—
293C22—3322no—
294C22C22×C22⋊D8311yescharacteristic
295C22C2×#1305059011yescharacteristic
296C22C2×#1305059011yescharacteristic
297C22#1458183712yes—
298C22#1458183812yes—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—3322no—
305C4—3322no—
306C4—3322no—
307C2—42no—
308C2—42no—
309C2—6642no—
310C2#1912821211yescharacteristic
311C2#1912826011yescharacteristic
312C2#1912826511yescharacteristic
313C2#1912828412yes—
314C2#1912830212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4×D4×C83C2
hypercenterC2×C4×D4D83
hypocenterC83#2635
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581766C2semidirect
2. #19127640C2semidirect
3. #19127637C2semidirect
4. #19127618C2semidirect
5. #19127616C2semidirect
6. #19127610C2semidirect
7. #19127609C2semidirect
8. #19127592C2semidirect
9. #19127580C2semidirect
10. #19127564C2semidirect
11. C2×C4×D4×C83C2semidirect
65. C22×C4×C83D4semidirect
158. C83#2635semidirect
160. C2×C4×D4D83semidirect
215. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
#191276402
#191276372
#191276182
#191276162
#191276102
#191276092
#191275922
#191275802
#191275642
C2×C4×D4×C832
#263583