Finite Group Databasebeta

← #24913616

Subgroups — #24913616

Order 10624.

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

normal subgroups only

This group has 498 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×#14581774C211yesmaximal, characteristic
2C2×#14581772C211yesmaximal, characteristic, complemented
3#19127639C211yesmaximal, characteristic, complemented
4#19127627C212yesmaximal, complemented
5#19127622C211yesmaximal, characteristic, complemented
6#19127617C212yesmaximal
7#19127613C211yesmaximal, characteristic, complemented
8#19127600C212yesmaximal, complemented
9#19127594C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#399×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581774—24no—
16#14581772—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×C4×Dic83C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—42no—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C2×C4×D83—42no—
48C2×C4×D83D414yes—
49C4⋊Dic83—24no—
50C4⋊Dic83—22no—
51C4⋊Dic83—42no—
52Dic83⋊C4—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—42no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—24no—
58C4×Dic83—42no—
59C24×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C4⋊C4×C83—22no—
65C4⋊C4×C83—24no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—24no—
68C23×C83—21no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—21no—
72C23×C83—21no—
73C23×C83C2411yescharacteristic
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—21no—
77C22×D83—21no—
78C22×D83C2×D412yes—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83#4114yes—
89C22⋊D83—44no—
90C2×Dic83—22no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83C2×D412yes—
103C4×D83—44no—
104Dic166—44no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)12yes—
113C22×C832−1+411yescharacteristic
114C22×C832−1+411yescharacteristic
115C2×D83—22no—
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—24no—
122C4×C83—41no—
123C4×C83—42no—
124Dic83—22no—
125Dic83—42no—
126Dic83—41no—
127Dic83—41no—
128Dic83—42no—
129C2×C83—81no—
130C2×C83#53211yescharacteristic
131C2×C83#53411yescharacteristic
132C2×C83#53611yescharacteristic
133C2×C83#53912yes—
134C2×C83#54212yes—
135D83—42no—
136#2663—831nomaximal, Sylow
137C83#266311yescharacteristic, complemented, Sylow
138C2×#123—831no—
139C2×C22⋊Q8—831no—
140#400—831no—
141#399D8311yescharacteristic, complemented
142#397—832no—
143#395—831no—
144#394—832no—
145#393—831no—
146#389—832no—
147#388—832no—
148C4×C22⋊C4—831no—
149C22×Q8—831no—
150C22×D4—831no—
151C23×C4—831no—
152#123—1664no—
153C22⋊Q8—1664no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8312yes—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—831no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4C2×D8312yes—
167C2×C42—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95C2×D8311yescharacteristic
174C24—831no—
175C24C22×D8311yescharacteristic
176C2×Q8—3322no—
177C2×Q8—1662no—
178C2×D4—3322no—
179C2×D4—1662no—
180C22×C4—3322no—
181C22×C4—834no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—831no—
185C22×C4—832no—
186C22×C4—831no—
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8312yes—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—3322no—
194C4⋊C4—1664no—
195C4⋊C4—1662no—
196C4⋊C4—1662no—
197C22⋊C4—22no—
198C22⋊C4—24no—
199C22⋊C4—3322no—
200C22⋊C4—1664no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1664no—
204C42—3322no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—832no—
211C23—1662no—
212C23—1662no—
213C23—1661no—
214C23—1661no—
215C23C23×D8311yescharacteristic
216Q8—3324no—
217D4—3324no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—3324no—
227C2×C4—832no—
228C2×C4—1661no—
229C2×C4—1662no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4—1662no—
240C2×C4D4×D8312yes—
241C2×C4#1305058914yes—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—3322no—
248C22—3322no—
249C22—1662no—
250C22—1662no—
251C22—1662no—
252C22C2×D4×D8311yescharacteristic
253C22C2×#1305058912yes—
254C22#1458183812yes—
255C22#1458183911yescharacteristic
256C22#1458184011yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—3322no—
262C4—3321no—
263C4—3322no—
264C4—3321no—
265C4—1662no—
266C2—81no—
267C2—3322no—
268C2#1912824911yescharacteristic
269C2#1912826211yescharacteristic
270C2#1912827411yescharacteristic
271C2#1912829512yes—
272C2#1912832412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#399D83
hypocenterC83#2663
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#14581772C2semidirect
3. #19127639C2semidirect
4. #19127627C2semidirect
5. #19127622C2semidirect
7. #19127613C2semidirect
8. #19127600C2semidirect
9. #19127594C2semidirect
10. #19127548C2semidirect
11. #399×C83C2semidirect
137. C83#2663semidirect
141. #399D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817742
C2×#145817722
#191276392
#191276272
#191276222
#191276172
#191276132
#191276002
#191275942
#191275482
#399×C832
#266383