Finite Group Databasebeta

← #24913321

Subgroups — #24913321

Order 10624.

628 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14581781C211yesmaximal, characteristic, complemented
2C2×(C22⋊D83)⋊C4C211yesmaximal, characteristic, complemented
3#19127630C212yesmaximal, complemented
4#19127612C211yesmaximal, characteristic, complemented
5#19127611C211yesmaximal, characteristic, complemented
6#19127610C212yesmaximal, complemented
7#19127610C211yesmaximal, characteristic, complemented
8#19127579C212yesmaximal, complemented
9#19127578C212yesmaximal, complemented
10#19127568C211yesmaximal, characteristic, complemented
11#393×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C22×D332C2211yescharacteristic
14C22×C22⋊D83C2211yescharacteristic
15C22×C22⋊D83C2212yes—
16C23×Dic83C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18#14581781—24no—
19(C22⋊D83)⋊C4—24no—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×Q8×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2312yes—
38C23×D83C2311yescharacteristic
39C2×D332—24no—
40C2×D332—22no—
41C2×D332—42no—
42C2×D332—42no—
43C2×D332—42no—
44C2×C22⋊D83—22no—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—24no—
47C2×C22⋊D83—22no—
48C2×C22⋊D83—42no—
49C2×C22⋊D83—42no—
50C22×Dic83—22no—
51C22×Dic83—42no—
52C22×Dic83C2311yescharacteristic
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C2×C4×D83—22no—
57C2×C4×D83—42no—
58Dic83⋊C4—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64(C2×D83)⋊C4—42no—
65C4×Dic83D414yescomplemented
66C24×C83C2311yescharacteristic
67C2×Q8×C83—22no—
68C2×Q8×C83—42no—
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2312yes—
71C22×C4×C83C2312yes—
72C22×C4×C83C2311yescharacteristic
73C22⋊C4×C83—22no—
74C22⋊C4×C83—42no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83#4112yes—
80C23×C83C2411yescharacteristic
81C22×D83—21no—
82C22×D83—21no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83—22no—
92C22×D83#4112yes—
93Q8×C83—44no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83C2×D412yes—
104C22⋊D83—28no—
105C22⋊D83—48no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—44no—
115C2×Dic83C2×D412yes—
116C2×Dic83C2×D412yes—
117D332—44no—
118D332—48no—
119C4×D83—44no—
120C22×C83—22no—
121C22×C83—24no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—42no—
125C22×C83C22×D411yescharacteristic
126C22×C83C2×#4111yescharacteristic
127C22×C83C2×#4111yescharacteristic
128C22×C83H(5,𝔽2)12yes—
129C22×C83H(5,𝔽2)12yes—
130C2×D83—22no—
131C2×D83—24no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C2×D83—42no—
140C2×D83—42no—
141C2×D83—42no—
142C4×C83—22no—
143C4×C83—41no—
144C4×C83—42no—
145C4×C83—42no—
146Dic83—22no—
147Dic83—22no—
148Dic83—41no—
149Dic83—42no—
150C2×C83—42no—
151C2×C83#52511yescharacteristic
152C2×C83#53511yescharacteristic
153C2×C83#53511yescharacteristic
154C2×C83#53812yes—
155C2×C83#53812yes—
156D83—42no—
157D83—82no—
158#2631—831nomaximal, Sylow
159C83#263111yescharacteristic, complemented, Sylow
160C2×#124—831no—
161C2×C4×D4—831no—
162#397—832no—
163#393—831no—
164#393D8311yescharacteristic, complemented
165#392—831no—
166#392—832no—
167#392—831no—
168#386—832no—
169#386—832no—
170#383—831no—
171C22×Q8C2×D8311yescharacteristic
172C22×D4—832no—
173C22×D4—831no—
174C22×D4—831no—
175C22×D4—832no—
176C23×C4—831no—
177C23×C4—831no—
178#124—1664no—
179C4×D4—1664no—
180C2×C4⋊C4—831no—
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—831no—
185C2×C22⋊C4—832no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4C2×D8311yescharacteristic
188C2×C22⋊C4C2×D8312yes—
189C2×C42—831no—
190#95—832no—
191#95—831no—
192#95—832no—
193#95—832no—
194#95C2×D8311yescharacteristic
195#95C2×D8312yes—
196C24—831no—
197C24—832no—
198C24C22×D8311yescharacteristic
199C2×Q8—22no—
200C2×Q8—42no—
201C2×D4—3322no—
202C2×D4—3322no—
203C2×D4—3322no—
204C2×D4—3322no—
205C2×D4—3322no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1664no—
209C2×D4—1662no—
210C2×D4—1662no—
211C2×D4—1662no—
212C22×C4—3322no—
213C22×C4—3322no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—832no—
217C22×C4—831no—
218C22×C4—1662no—
219C22×C4—1662no—
220C22×C4C22×D8312yes—
221C22×C4C22×D8312yes—
222C22×C4C22×D8311yescharacteristic
223C22×C4C22×D8311yescharacteristic
224C4⋊C4—1662no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—3322no—
228C22⋊C4—3322no—
229C22⋊C4—3322no—
230C22⋊C4—1664no—
231C22⋊C4—1664no—
232C22⋊C4—1662no—
233C42—834no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—832no—
239C23—1662no—
240C23—1662no—
241C23—1662no—
242C23—1662no—
243C23—1662no—
244C23—1662no—
245C23—1662no—
246C23—1661no—
247C23—1661no—
248C23—1662no—
249C23—1662no—
250C23#1305059112yes—
251C23C23×D8311yescharacteristic
252Q8—44no—
253D4—3324no—
254D4—3328no—
255D4—3328no—
256D4—1668no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3324no—
267C2×C4—3324no—
268C2×C4—832no—
269C2×C4—832no—
270C2×C4—1661no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1662no—
275C2×C4—1661no—
276C2×C4—1661no—
277C2×C4—1662no—
278C2×C4D4×D8312yes—
279C22—22no—
280C22—24no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—3321no—
285C22—3321no—
286C22—3322no—
287C22—3322no—
288C22—3322no—
289C22—3322no—
290C22—3322no—
291C22—3322no—
292C22—3322no—
293C22—3322no—
294C22—1662no—
295C22—1664no—
296C22C2×D4×D8311yescharacteristic
297C22#41×D8311yescharacteristic
298C22C2×#1305059111yescharacteristic
299C22#1458183712yes—
300C22#1458183812yes—
301C4—22no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—3321no—
306C4—3322no—
307C4—1662no—
308C4—1662no—
309C2—42no—
310C2—3322no—
311C2—6642no—
312C2#1912821411yescharacteristic
313C2#1912826611yescharacteristic
314C2#1912826911yescharacteristic
315C2#1912828512yes—
316C2#1912829112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
hypercenter#393D83
hypocenterC83#2631
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14581781C2semidirect
2. C2×(C22⋊D83)⋊C4C2semidirect
3. #19127630C2semidirect
4. #19127612C2semidirect
5. #19127611C2semidirect
6. #19127610C2semidirect
7. #19127610C2semidirect
8. #19127579C2semidirect
9. #19127578C2semidirect
10. #19127568C2semidirect
11. #393×C83C2semidirect
65. C4×Dic83D4semidirect
159. C83#2631semidirect
164. #393D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×(C22⋊D83)⋊C42
#191276302
#191276122
#191276112
#191276102
#191275792
#191275782
#191275682
#393×C832
#263183