Finite Group Databasebeta

← #24918166

Subgroups — #24918166

Order 10624.

366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128107C211yesmaximal, characteristic, complemented
3#19128093C211yesmaximal, characteristic, complemented
4#19128069C211yesmaximal, characteristic, complemented
5#19128063C211yesmaximal, characteristic, complemented
6#19127940C212yesmaximal, complemented
7#19127932C212yesmaximal
8#19127806C212yesmaximal, complemented
9#19127799C212yesmaximal, complemented
10C2×#14581703C211yesmaximal, characteristic, complemented
11#533×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050577C2212yes—
14C22×C83⋊C8C2212yes—
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—
27#14581716C2212yes—
28#14581716C2212yes—
29#14581715C2212yes—
30#14581712C2212yes—
31#14581703C2211yescharacteristic
32#14581703C2211yescharacteristic
33#14581703C2212yes—
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—24no—
48#13050577—24no—
49C2×C83⋊C8—24no—
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—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
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×C83C2411yescharacteristic
97C22⋊D83—42no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101D332—22no—
102C4×D83—42no—
103Dic166—22no—
104C83⋊C8—22no—
105C83⋊C8#4114yes—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83#13712yes—
109C22×C83C22×D411yescharacteristic
110C2×D83—41no—
111C2×D83—41no—
112C2×D83—41no—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—21no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—42no—
119C4×C83—42no—
120C4×C83C2×#4112yes—
121Dic83—41no—
122Dic83—41no—
123Dic83—41no—
124C2×C83—22no—
125C2×C83#53211yescharacteristic
126C2×C83C2×#13711yescharacteristic
127C2×C83#57511yescharacteristic
128D83—81no—
129#3181—831nomaximal, Sylow
130C83#318111yescharacteristic, complemented, Sylow
131#533D8311yescharacteristic, complemented
132#529—831no—
133#501—831no—
134#497—831no—
135#491—831no—
136#489—831no—
137#467—832no—
138#465—832no—
139#441—832no—
140#440—832no—
141C2×#97—831no—
142C2×Q16—832no—
143C2×SD16—832no—
144C22×C8—832no—
145C4⋊Q8C2×D8311yescharacteristic
146C4⋊Q8C2×D8311yescharacteristic
147#126—1661no—
148#125—21no—
149#125—831no—
150#124—831no—
151#123—1661no—
152C22⋊Q8—21no—
153C22⋊Q8—831no—
154C22⋊Q8C2×D8312yes—
155#121—831no—
156C4×Q8—21no—
157C4×Q8—831no—
158C4×Q8C2×D8312yes—
159C4×D4—831no—
160C4×D4—1661no—
161#117—21no—
162#117—1661no—
163C2×C42C2×D8311yescharacteristic
164#107—832no—
165#106—832no—
166#103—832no—
167#103—832no—
168#103—832no—
169#102—832no—
170#97—832no—
171#97—831no—
172#97—831no—
173C2×Q8—21no—
174C2×Q8—831no—
175C2×Q8C22×D8312yes—
176C2×D4—831no—
177C2×D4—1661no—
178C22×C4—1661no—
179C22×C4C22⋊D8311yescharacteristic
180C22×C4C22⋊D8311yescharacteristic
181C22×C4C22×D8311yescharacteristic
182Q16—1664no—
183SD16—1664no—
184C2×C8—832no—
185C2×C8—832no—
186C2×C8—1664no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—831no—
193C4⋊C4—831no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4C22×D8312yes—
199C22⋊C4—22no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C42—22no—
206C42—1661no—
207C42C22⋊D8311yescharacteristic
208C42C22⋊D8311yescharacteristic
209C42C22×D8311yescharacteristic
210C42C22×D8311yescharacteristic
211C23—1661no—
212C23C2×C22⋊D8311yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—1662no—
217D4—3322no—
218D4—1662no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—3322no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4—1661no—
228C2×C4C2×C22⋊D8311yescharacteristic
229C2×C4C2×C22⋊D8311yescharacteristic
230C2×C4C2×C22⋊D8311yescharacteristic
231C2×C4C2×C22⋊D8311yescharacteristic
232C2×C4C2×C22⋊D8311yescharacteristic
233C2×C4C23×D8311yescharacteristic
234C8—834no—
235C8—1662no—
236C22—21no—
237C22—21no—
238C22—3321no—
239C22—3321no—
240C22—3321no—
241C22#1458182312yes—
242C22C22×C22⋊D8311yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—3321no—
251C4—3321no—
252C4—3321no—
253C4#41×D8312yes—
254C2—22no—
255C2—6641no—
256C2#1912824711yescharacteristic
257C2C2×#1458182311yescharacteristic
258C2#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#3181
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. #19128107C2semidirect
3. #19128093C2semidirect
4. #19128069C2semidirect
5. #19128063C2semidirect
6. #19127940C2semidirect
8. #19127806C2semidirect
9. #19127799C2semidirect
10. C2×#14581703C2semidirect
11. #533×C83C2semidirect
130. C83#3181semidirect
131. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191281072
#191280932
#191280692
#191280632
#191279402
#191279322
#191278062
#191277992
C2×#145817032
#533×C832
#318183