Finite Group Databasebeta

← #24917428

Subgroups — #24917428

Order 10624.

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

normal subgroups only

This group has 386 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#19128275C211yesmaximal, characteristic
2#19128182C211yesmaximal, characteristic
3#19128041C212yesmaximal, complemented
4#19128040C212yesmaximal, complemented
5#19128035C212yesmaximal, complemented
6#19128030C212yesmaximal, complemented
7#19127741C211yesmaximal, characteristic, complemented
8#19127739C211yesmaximal, characteristic, complemented
9#19127710C211yesmaximal, characteristic, complemented
10#19127667C211yesmaximal, characteristic
11C2×#103×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14#14581790—21no—
15#14581781—21no—
16#14581775—22no—
17#14581768C2212yes—
18#14581768C2212yes—
19#14581757—21no—
20#14581754—21no—
21C4×C22⋊D83—22no—
22#14581751C2211yescharacteristic
23#14581751C2211yescharacteristic
24#14581750C2212yes—
25C4⋊C4×D83C2212yes—
26#14581726C2211yescharacteristic
27#14581726C2211yescharacteristic
28#14581725C2212yes—
29#14581724C2211yescharacteristic
30#14581724C2211yescharacteristic
31#14581723C2212yes—
32#14581716C2212yes—
33#14581716C2212yes—
34(C2×D83)⋊C8C2212yes—
35(C2×D83)⋊C8C2212yes—
36C22×Q8×C83C2211yescharacteristic
37C22×C8×C83C2211yescharacteristic
38C2×C4⋊C4×C83C2211yescharacteristic
39#103×C83C2212yes—
40#103×C83C2212yes—
41#13050590—22no—
42#13050590—22no—
43C2×Dic166D411yescharacteristic
44C2×D332D411yescharacteristic
45C2×C22⋊D83D412yes—
46C2×C4×D83—22no—
47C2×C4×D83C2312yes—
48#13050574—22no—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83C2311yescharacteristic
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57C22⋊Dic83—21no—
58C22⋊Dic83—21no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—21no—
63C4×Dic83—21no—
64C4×Dic83—22no—
65C2×Q8×C83—21no—
66C2×Q8×C83—22no—
67C2×Q8×C83—22no—
68C2×Q8×C83C2312yes—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83C2311yescharacteristic
72C2×C8×C83—22no—
73C2×C8×C83C2311yescharacteristic
74C2×C8×C83C2311yescharacteristic
75C4⋊C4×C83—21no—
76C4⋊C4×C83C2312yes—
77C23×C83C2×D411yescharacteristic
78C22×D83C2×D412yes—
79Q8×C83—22no—
80Q8×C83—22no—
81Q8×C83—42no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—42no—
85C2×C4×C83—81no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83#4112yes—
88C2×C4×C83#4112yes—
89C2×C4×C83C2411yescharacteristic
90C22⋊D83—24no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83C2×D412yes—
96D332—22no—
97C4×D83—22no—
98C4×D83—22no—
99C4×D83—82no—
100Dic166—22no—
101C8×C83—21no—
102C8×C83—21no—
103C83⋊C8—22no—
104C22×C83—21no—
105C22×C83—21no—
106C22×C83#12312yes—
107C22×C83C22×D411yescharacteristic
108C2×D83—22no—
109C2×D83—22no—
110C2×D83—22no—
111C4×C83—42no—
112C4×C83—42no—
113C4×C83#12312yes—
114C4×C83C2×#4111yescharacteristic
115C4×C83C2×#4111yescharacteristic
116Dic83—22no—
117Dic83—41no—
118Dic83—41no—
119Dic83—41no—
120Dic83—41no—
121C2×C83—22no—
122C2×C83C2×#12311yescharacteristic
123C2×C83#57711yescharacteristic
124C2×C83#57811yescharacteristic
125D83—42no—
126#3097—831nomaximal, Sylow
127C83#309711yescharacteristic, complemented, Sylow
128#536—831no—
129#520—831no—
130#485—832no—
131#485—832no—
132#484—832no—
133#483—832no—
134#428—831no—
135#427—831no—
136#419—831no—
137C2×#103D8311yescharacteristic, complemented
138#408—831no—
139C2×#41—831no—
140C22×Q8C2×D8311yescharacteristic
141C2×M16—831no—
142C22×C8C2×D8311yescharacteristic
143C4⋊Q8—1661no—
144#124—1661no—
145#123—1662no—
146C22⋊Q8—832no—
147C22⋊Q8—832no—
148C4×Q8—1661no—
149C4×D4—1661no—
150C4×D4—1662no—
151#117—831no—
152#117—831no—
153#117—832no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4C2×D8311yescharacteristic
156#107—831no—
157#107—831no—
158#107—832no—
159#106—831no—
160#106—831no—
161#106—832no—
162#103—832no—
163#103—832no—
164#103C2×D8312yes—
165#103C2×D8312yes—
166C22⋊C8—832no—
167C22⋊C8—832no—
168#41—1662no—
169#41—1662no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—831no—
174C2×Q8C22×D8312yes—
175C2×D4—832no—
176C2×D4—831no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—832no—
180C22×C4—1662no—
181C22×C4C22×D8311yescharacteristic
182M16—1662no—
183C2×C8—22no—
184C2×C8—832no—
185C2×C8C22×D8311yescharacteristic
186C2×C8C22×D8311yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—831no—
189C4⋊C4—831no—
190C4⋊C4—831no—
191C4⋊C4—831no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C4⋊C4C22×D8312yes—
196C22⋊C4—1662no—
197C22⋊C4—1662no—
198C22⋊C4—1662no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C42—1662no—
202C42—1661no—
203C42—1661no—
204C23—832no—
205C23D4×D8311yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—1662no—
210D4—1664no—
211D4—1662no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—832no—
217C2×C4—6642no—
218C2×C4—1661no—
219C2×C4—1661no—
220C2×C4—1661no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4D4×D8311yescharacteristic
225C2×C4#1305058912yes—
226C2×C4#1305059012yes—
227C2×C4C23×D8311yescharacteristic
228C8—21no—
229C8—21no—
230C8—1662no—
231C22—21no—
232C22—21no—
233C22—1662no—
234C22—1662no—
235C22—1662no—
236C22#1458177412yes—
237C22C2×D4×D8311yescharacteristic
238C4—42no—
239C4—42no—
240C4—3321no—
241C4—3321no—
242C4—3321no—
243C4—3321no—
244C4—1662no—
245C4#1458177412yes—
246C4C2×#1305058911yescharacteristic
247C4C2×#1305059011yescharacteristic
248C2—22no—
249C2—3322no—
250C2C2×#1458177411yescharacteristic
251C2#1912850711yescharacteristic
252C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#103×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenterC2×#103D83
hypocenterC83#3097
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #19128041C2semidirect
4. #19128040C2semidirect
5. #19128035C2semidirect
6. #19128030C2semidirect
7. #19127741C2semidirect
8. #19127739C2semidirect
9. #19127710C2semidirect
11. C2×#103×C83C2semidirect
127. C83#3097semidirect
137. C2×#103D83semidirect

Maximal subgroups

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

subgroupindex
#191282752
#191281822
#191280412
#191280402
#191280352
#191280302
#191277412
#191277392
#191277102
#191276672
C2×#103×C832
#309783