Finite Group Databasebeta

← #24918830

Subgroups — #24918830

Order 10624.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C211yesmaximal, characteristic
2#19128008C212yesmaximal, complemented
3#19127993C211yesmaximal, characteristic, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127957C212yesmaximal, complemented
6#19127949C212yesmaximal
7#19127923C212yesmaximal, complemented
8#19127908C211yesmaximal, characteristic, complemented
9#19127903C211yesmaximal, characteristic, complemented
10#19127725C211yesmaximal, characteristic, complemented
11#533×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050577C2212yes—
14C2×#13050574C2212yes—
15#14581786—21no—
16#14581784C2211yescharacteristic
17#14581780C2211yescharacteristic
18#14581775—21no—
19#14581771C2211yescharacteristic
20#14581766C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22C4×D332C2211yescharacteristic
23C4×C22⋊D83—21no—
24#14581749—21no—
25#14581725C2212yes—
26#14581723C2212yes—
27C4⋊(C83⋊C8)C2211yescharacteristic
28C4⋊(C83⋊C8)C2211yescharacteristic
29C4⋊(C83⋊C8)C2212yes—
30#14581716C2212yes—
31#14581716C2212yes—
32#14581715C2212yes—
33#14581712C2212yes—
34C4⋊Q8×C83C2211yescharacteristic
35C4⋊Q8×C83C2211yescharacteristic
36#125×C83—21no—
37C22⋊Q8×C83—21no—
38C22⋊Q8×C83C2212yes—
39C4×Q8×C83—21no—
40C4×Q8×C83C2212yes—
41#117×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050580—42no—
48#13050577—42no—
49#13050574—42no—
50C2×C83⋊C8C2312yes—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58C22⋊Dic83—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63C4×Dic83—21no—
64C2×Q8×C83—21no—
65C2×Q8×C83C2312yes—
66C22×C4×C83D411yescharacteristic
67C22×C4×C83D411yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—21no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—22no—
73C4⋊C4×C83—22no—
74C4⋊C4×C83C2312yes—
75C22⋊C4×C83—22no—
76C42×C83—22no—
77C42×C83D411yescharacteristic
78C42×C83D411yescharacteristic
79C42×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C23×C83C2×D411yescharacteristic
82C22×D83—21no—
83Q8×C83—22no—
84Q8×C83—22no—
85Q8×C83—42no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—41no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—42no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102D332—21no—
103D332—21no—
104C4×D83—42no—
105Dic166—21no—
106Dic166—21no—
107C83⋊C8—22no—
108C83⋊C8—22no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83—21no—
112C22×C83C22×D411yescharacteristic
113C2×D83—41no—
114C2×D83—41no—
115C2×D83—41no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—21no—
120C4×C83—42no—
121C4×C83—42no—
122C4×C83#13712yes—
123C4×C83H(5,𝔽2)11yescharacteristic
124C4×C832−1+411yescharacteristic
125Dic83—41no—
126Dic83—41no—
127Dic83—41no—
128C2×C83—41no—
129C2×C83#53711yescharacteristic
130C2×C83C2×#13711yescharacteristic
131C2×C83#57511yescharacteristic
132D83—81no—
133#3252—831nomaximal, Sylow
134C83#325211yescharacteristic, complemented, Sylow
135#533D8311yescharacteristic, complemented
136#529—831no—
137#479—832no—
138#477—831no—
139#476—831no—
140#470—832no—
141#468—832no—
142#464—832no—
143#462—831no—
144#461—831no—
145#423—831no—
146C2×Q16—832no—
147C2×SD16—832no—
148C2×M16—832no—
149C4⋊Q8C2×D8311yescharacteristic
150C4⋊Q8C2×D8311yescharacteristic
151#126—1661no—
152#125—21no—
153#125—831no—
154#124—831no—
155#123—1661no—
156C22⋊Q8—21no—
157C22⋊Q8—831no—
158C22⋊Q8C2×D8312yes—
159#121—831no—
160C4×Q8—21no—
161C4×Q8—831no—
162C4×Q8C2×D8312yes—
163C4×D4—831no—
164C4×D4—1661no—
165#117—21no—
166#117—1661no—
167C2×C42C2×D8311yescharacteristic
168#107—832no—
169#106—832no—
170C4⋊C8—831no—
171C4⋊C8—831no—
172C4⋊C8—832no—
173#103—832no—
174#103—832no—
175#103—832no—
176#102—832no—
177C2×Q8—21no—
178C2×Q8—831no—
179C2×Q8C22×D8312yes—
180C2×D4—831no—
181C2×D4—1661no—
182C22×C4—1661no—
183C22×C4C22⋊D8311yescharacteristic
184C22×C4C22⋊D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186Q16—3322no—
187SD16—3322no—
188M16—3322no—
189C2×C8—832no—
190C2×C8—832no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—831no—
197C4⋊C4—831no—
198C4⋊C4—1661no—
199C4⋊C4—1661no—
200C4⋊C4—1661no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8312yes—
203C22⋊C4—22no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C22⋊C4—1661no—
207C22⋊C4—1661no—
208C22⋊C4—1661no—
209C42—22no—
210C42—1661no—
211C42C22⋊D8311yescharacteristic
212C42C22⋊D8311yescharacteristic
213C42C22×D8311yescharacteristic
214C42C22×D8311yescharacteristic
215C23—1661no—
216C23C2×C22⋊D8311yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—1661no—
221Q8—1661no—
222D4—3322no—
223D4—1661no—
224D4—1661no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—3322no—
232C2×C4—1661no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4C2×C22⋊D8311yescharacteristic
236C2×C4C2×C22⋊D8311yescharacteristic
237C2×C4C2×C22⋊D8311yescharacteristic
238C2×C4C2×C22⋊D8311yescharacteristic
239C2×C4C2×C22⋊D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—1662no—
242C8—1662no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—3321no—
247C22—3321no—
248C22—3321no—
249C22C22×C22⋊D8311yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—3321no—
257C4—3321no—
258C4—3321no—
259C4#1458182312yes—
260C4#1458183811yescharacteristic
261C4#1458183911yescharacteristic
262C2—41no—
263C2—6641no—
264C2#1912827911yescharacteristic
265C2C2×#1458182311yescharacteristic
266C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#533D83
hypocenterC83#3252
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #19128008C2semidirect
3. #19127993C2semidirect
4. #19127988C2semidirect
5. #19127957C2semidirect
7. #19127923C2semidirect
8. #19127908C2semidirect
9. #19127903C2semidirect
10. #19127725C2semidirect
11. #533×C83C2semidirect
134. C83#3252semidirect
135. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280082
#191279932
#191279882
#191279572
#191279492
#191279232
#191279082
#191279032
#191277252
#533×C832
#325283