Finite Group Databasebeta

← #24908489

Subgroups — #24908489

Order 10624.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#19128265C211yesmaximal, characteristic
2C2×#14581781C211yesmaximal, characteristic, complemented
3C2×#14581719C211yesmaximal, characteristic, complemented
4#19127710C211yesmaximal, characteristic, complemented
5C2×#14581708C211yesmaximal, characteristic, complemented
6#19127462C211yesmaximal, characteristic, complemented
7C2×#137×C83C211yesmaximal, characteristic, complemented
8C22×D332C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10#14581788—21no—
11#14581781—22no—
12#14581781—21no—
13#14581781—22no—
14#14581766—22no—
15#14581760—22no—
16#14581751C2211yescharacteristic
17C2×(C2×D83)⋊C4—22no—
18C2×C4×Dic83C2211yescharacteristic
19#14581719—22no—
20#14581716—22no—
21#14581708—22no—
22C2×#41×C83C2211yescharacteristic
23C22×Q8×C83C2211yescharacteristic
24#137×C83—22no—
25#137×C83—22no—
26C2×Q16×C83—21no—
27C2×SD16×C83—21no—
28C2×M16×C83C2211yescharacteristic
29C23×D83D412yes—
30C2×D332—21no—
31C2×D332—21no—
32C2×D332—21no—
33C2×D332—21no—
34C2×D332—21no—
35C2×D332—21no—
36C2×C22⋊D83—24no—
37C22×Dic83—21no—
38#13050574—21no—
39#13050574—21no—
40C2×C83⋊C8—21no—
41C4⋊Dic83D412yes—
42C22⋊Dic83—21no—
43(C2×D83)⋊C4—24no—
44(C2×D83)⋊C4—28no—
45C4×Dic83—21no—
46C4×Dic83—21no—
47C4×Dic83D412yescomplemented
48#41×C83—21no—
49#41×C83—21no—
50#41×C83—21no—
51#41×C83—21no—
52C2×Q8×C83—21no—
53C2×Q8×C83—21no—
54C2×Q8×C83—21no—
55C2×Q8×C83—21no—
56C2×Q8×C83—21no—
57C2×Q8×C83D411yescharacteristic
58C2×Q8×C83D412yescomplemented
59C2×D4×C83—21no—
60C2×D4×C83D411yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83C2311yescharacteristic
64Q16×C83—24no—
65SD16×C83—24no—
66M16×C83—21no—
67M16×C83D412yescomplemented
68C2×C8×C83—21no—
69C23×C83—21no—
70C23×C83C2×D411yescharacteristic
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—44no—
75Q8×C83—22no—
76Q8×C83—21no—
77Q8×C83—22no—
78Q8×C83—21no—
79Q8×C83—41no—
80Q8×C83—41no—
81D4×C83—21no—
82D4×C83—21no—
83D4×C83—41no—
84D4×C83—41no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—41no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83#4111yescharacteristic
97C22⋊D83—84no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—42no—
101D332—41no—
102D332—41no—
103D332—41no—
104D332—41no—
105C8×C83—22no—
106C83⋊C8—41no—
107C22×C83—22no—
108C22×C83—41no—
109C22×C83—42no—
110C22×C83#12011yescharacteristic
111C22×C83#12011yescharacteristic
112C22×C83#12111yescharacteristic
113C2×D83—44no—
114C2×D83—42no—
115C2×D83—44no—
116C2×D83—82no—
117C4×C83—41no—
118C4×C83—42no—
119C4×C83#12011yescharacteristic
120C4×C83#12111yescharacteristic
121C4×C83#12111yescharacteristic
122C4×C83#12711yescharacteristic
123Dic83—42no—
124Dic83—42no—
125C2×C83—21no—
126C2×C83—21no—
127C2×C83—81no—
128C2×C83#39211yescharacteristic
129C2×C83#45512yes—
130D83—82no—
131#2004—831nomaximal, Sylow
132C83#200411yescharacteristic, complemented, Sylow
133C2×#137D8311yescharacteristic, complemented
134#535—831no—
135C2×#124—831no—
136C2×#104—831no—
137#419—831no—
138C2×#100—831no—
139#339—831no—
140C2×#41C2×D8311yescharacteristic
141C22×Q8C2×D8311yescharacteristic
142C22×D4—831no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—831no—
148C2×M16C2×D8311yescharacteristic
149#127—1661no—
150#124—1662no—
151#124—1661no—
152#124—1662no—
153#121—1662no—
154#120—1662no—
155#117—831no—
156C2×C22⋊C4—1662no—
157C2×C42—831no—
158#104—1662no—
159#103—1662no—
160#100—1662no—
161C24—832no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D8311yescharacteristic
172C2×Q8C22⋊D8312yescomplemented
173C2×D4—21no—
174C2×D4—1664no—
175C2×D4—1661no—
176C2×D4—1661no—
177C2×D4—1661no—
178C2×D4—1661no—
179C2×D4—1661no—
180C2×D4—1661no—
181C2×D4C22⋊D8311yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—1661no—
185C22×C4C22×D8311yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—1661no—
190M16—1661no—
191M16C22⋊D8312yescomplemented
192C2×C8—21no—
193C2×C8—1661no—
194C4⋊C4—832no—
195C22⋊C4—1668no—
196C22⋊C4—1664no—
197C22⋊C4—1661no—
198C42—832no—
199C42—1661no—
200C42—1661no—
201C23—21no—
202C23—3324no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23D4×D8311yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—3321no—
218D4—3321no—
219D4—3321no—
220D4—3321no—
221D4—6644no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3322no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4D4×D8311yescharacteristic
232C2×C4D4×D8311yescharacteristic
233C2×C4C2×C22⋊D8311yescharacteristic
234C2×C4C2×C22⋊D8311yescharacteristic
235C2×C4C2×C22⋊D8311yescharacteristic
236C2×C4#1305058911yescharacteristic
237C8—22no—
238C8—3321no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—3324no—
243C22—3322no—
244C22—3324no—
245C22—6642no—
246C22#1458176011yescharacteristic
247C22#1458176011yescharacteristic
248C22#1458176311yescharacteristic
249C4—41no—
250C4—42no—
251C4—3322no—
252C4—3322no—
253C4#1458176111yescharacteristic
254C4#1458176411yescharacteristic
255C4#1458176411yescharacteristic
256C4#1458178811yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—6642no—
261C2#1912760911yescharacteristic
262C2#1912787712yes—
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
FittingC2×#137×C83C2
socleC22×C83#120
prime radical#19128265C2
hypercenterC2×#137D83
hypocenterC83#2004
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#14581781C2semidirect
3. C2×#14581719C2semidirect
4. #19127710C2semidirect
5. C2×#14581708C2semidirect
6. #19127462C2semidirect
7. C2×#137×C83C2semidirect
47. C4×Dic83D4semidirect
58. C2×Q8×C83D4semidirect
67. M16×C83D4semidirect
132. C83#2004semidirect
133. C2×#137D83semidirect
172. C2×Q8C22⋊D83semidirect
191. M16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
C2×#145817812
C2×#145817192
#191277102
C2×#145817082
#191274622
C2×#137×C832
#200483