Finite Group Databasebeta

← #24913596

Subgroups — #24913596

Order 10624.

598 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581777C211yesmaximal, characteristic, complemented
2C2×#14581767C211yesmaximal, characteristic, complemented
3#19127627C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127618C212yesmaximal, complemented
6#19127611C211yesmaximal, characteristic, complemented
7#19127611C212yesmaximal, complemented
8#19127602C212yesmaximal, complemented
9#19127590C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C22×D332C2211yescharacteristic
14C22×D332C2212yes—
15C22×D332C2211yescharacteristic
16C22×C4×D83C2211yescharacteristic
17#14581777—24no—
18#14581767—24no—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×(C2×D83)⋊C4C2212yes—
29C2×C4×Dic83C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2312yes—
39C23×D83C2311yescharacteristic
40C2×D332—24no—
41C2×D332—22no—
42C2×D332—22no—
43C2×D332—24no—
44C2×D332—42no—
45C2×D332—42no—
46C2×D332—42no—
47C2×D332—42no—
48C2×D332—42no—
49C2×D332—42no—
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2311yescharacteristic
53C2×C4×D83—42no—
54C2×C4×D83D414yescomplemented
55Dic83⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64(C2×D83)⋊C4—42no—
65C4×Dic83—42no—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2312yes—
71C4⋊C4×C83—22no—
72C4⋊C4×C83—22no—
73C4⋊C4×C83—42no—
74C23×C83C2411yescharacteristic
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—21no—
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—21no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—21no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83C2×D412yes—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83C2×D412yes—
104C2×C4×C83#4114yes—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83C2×D412yes—
112D332—44no—
113D332—48no—
114D332—44no—
115D332—48no—
116C4×D83—44no—
117C22×C83C22×D411yescharacteristic
118C22×C83C2×#4112yes—
119C22×C83H(5,𝔽2)11yescharacteristic
120C22×C83H(5,𝔽2)12yes—
121C22×C83H(5,𝔽2)11yescharacteristic
122C2×D83—22no—
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—42no—
126C2×D83—42no—
127C2×D83—41no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—41no—
131C2×D83—42no—
132C2×D83—41no—
133C2×D83—42no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C4×C83—22no—
140C4×C83—24no—
141C4×C83—41no—
142C4×C83—42no—
143C4×C83—41no—
144Dic83—22no—
145Dic83—41no—
146Dic83—42no—
147C2×C83#53211yescharacteristic
148C2×C83#53411yescharacteristic
149C2×C83#53511yescharacteristic
150C2×C83#53812yes—
151C2×C83#54012yes—
152D83—42no—
153D83—82no—
154D83—81no—
155#2661—831nomaximal, Sylow
156C83#266111yescharacteristic, complemented, Sylow
157C2×#123—831no—
158C2×#121—831no—
159#398D8311yescharacteristic, complemented
160#397—831no—
161#394—831no—
162#394—832no—
163#392—831no—
164#392—832no—
165#390—832no—
166#388—832no—
167C4×C22⋊C4—831no—
168C22×D4—832no—
169C22×D4—831no—
170C22×D4—831no—
171C22×D4—832no—
172C23×C4—831no—
173#123—1664no—
174#121—1664no—
175C2×C4⋊C4—831no—
176C2×C4⋊C4C2×D8311yescharacteristic
177C2×C4⋊C4C2×D8311yescharacteristic
178C2×C4⋊C4C2×D8312yes—
179C2×C22⋊C4—832no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—831no—
184C2×C22⋊C4—831no—
185C2×C22⋊C4—831no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4—832no—
188C2×C42—832no—
189#95—831no—
190#95—832no—
191#95C2×D8311yescharacteristic
192#95C2×D8312yes—
193C24—831no—
194C24—831no—
195C24—832no—
196C2×D4—3322no—
197C2×D4—3322no—
198C2×D4—3322no—
199C2×D4—3322no—
200C2×D4—3322no—
201C2×D4—3322no—
202C2×D4—1664no—
203C2×D4—1662no—
204C2×D4—1662no—
205C2×D4—1664no—
206C22×C4—3322no—
207C22×C4—834no—
208C22×C4—831no—
209C22×C4—832no—
210C22×C4—831no—
211C22×C4C22×D8312yes—
212C22×C4C22×D8311yescharacteristic
213C22×C4C22×D8311yescharacteristic
214C22×C4C22×D8311yescharacteristic
215C22×C4C22×D8312yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—1662no—
220C22⋊C4—3322no—
221C22⋊C4—3322no—
222C22⋊C4—3322no—
223C22⋊C4—1664no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C22⋊C4—1664no—
228C22⋊C4—1664no—
229C42—3322no—
230C23—832no—
231C23—1661no—
232C23—1662no—
233C23—1662no—
234C23—1662no—
235C23—1662no—
236C23—1662no—
237C23—1661no—
238C23—1661no—
239C23—1662no—
240C23—1662no—
241C23—1662no—
242C23—1662no—
243C23—1662no—
244C23—1661no—
245C23—1662no—
246C23—1661no—
247C23C23×D8311yescharacteristic
248D4—3324no—
249D4—3328no—
250D4—3324no—
251D4—3328no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—3324no—
264C2×C4—832no—
265C2×C4—1661no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4D4×D8312yes—
272C2×C4#1305059114yes—
273C22—3322no—
274C22—3322no—
275C22—3321no—
276C22—3322no—
277C22—3322no—
278C22—3321no—
279C22—3322no—
280C22—3321no—
281C22—3322no—
282C22—3322no—
283C22—3322no—
284C22—3322no—
285C22—3322no—
286C22—3322no—
287C22—1662no—
288C22—1662no—
289C22—1662no—
290C22C2×D4×D8311yescharacteristic
291C22C2×#1305059112yes—
292C22#1458183711yescharacteristic
293C22#1458183812yes—
294C22#1458183811yescharacteristic
295C4—22no—
296C4—24no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—3321no—
301C4—3322no—
302C4—1662no—
303C2—3322no—
304C2—6642no—
305C2—6641no—
306C2#1912825211yescharacteristic
307C2#1912826211yescharacteristic
308C2#1912826711yescharacteristic
309C2#1912829112yes—
310C2#1912830712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
hypercenter#398D83
hypocenterC83#2661
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14581777C2semidirect
2. C2×#14581767C2semidirect
3. #19127627C2semidirect
4. #19127618C2semidirect
5. #19127618C2semidirect
6. #19127611C2semidirect
7. #19127611C2semidirect
8. #19127602C2semidirect
9. #19127590C2semidirect
10. #19127548C2semidirect
11. #398×C83C2semidirect
54. C2×C4×D83D4semidirect
156. C83#2661semidirect
159. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817672
#191276272
#191276182
#191276112
#191276022
#191275902
#191275482
#398×C832
#266183