Finite Group Databasebeta

← #24908686

Subgroups — #24908686

Order 10624.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#19128279C211yesmaximal, characteristic
2C2×#14581720C211yesmaximal, characteristic, complemented
3#19127703C211yesmaximal, characteristic, complemented
4#19127702C211yesmaximal, characteristic, complemented
5#19127571C211yesmaximal, characteristic, complemented
6#19127469C211yesmaximal, characteristic, complemented
7C2×#104×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10#14581771—21no—
11#14581768—21no—
12#14581766—21no—
13#14581764—22no—
14#14581763—21no—
15C2×C4⋊Dic83—22no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic
18#14581720—22no—
19#14581717—21no—
20#14581716—21no—
21#14581714—21no—
22#14581713—21no—
23C2×#41×C83C2211yescharacteristic
24C2×M16×C83C2211yescharacteristic
25C2×C42×C83C2211yescharacteristic
26#104×C83—22no—
27#13050590—21no—
28#13050590—21no—
29#13050590—21no—
30#13050590—21no—
31C2×Dic166D411yescharacteristic
32C2×D332D411yescharacteristic
33C2×C22⋊D83—21no—
34C2×C22⋊D83—22no—
35C22×Dic83—21no—
36C22×Dic83D412yes—
37C2×C4×D83—21no—
38#13050574—21no—
39#13050574—21no—
40C2×C83⋊C8—21no—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83—21no—
45C4⋊Dic83—44no—
46Dic83⋊C4—22no—
47C22⋊Dic83—22no—
48(C2×D83)⋊C4—22no—
49#41×C83—21no—
50#41×C83—21no—
51#41×C83—21no—
52#41×C83—21no—
53C2×Q8×C83D411yescharacteristic
54C2×D4×C83—21no—
55C2×D4×C83D411yescharacteristic
56C22×C4×C83—21no—
57C22×C4×C83—21no—
58C22×C4×C83C2311yescharacteristic
59M16×C83—21no—
60M16×C83—21no—
61C2×C8×C83—21no—
62C42×C83—21no—
63C42×C83D412yescomplemented
64C23×C83—21no—
65C23×C83C2×D411yescharacteristic
66C22×D83—21no—
67Q8×C83—21no—
68Q8×C83—21no—
69D4×C83—21no—
70D4×C83—21no—
71D4×C83—41no—
72D4×C83—41no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—41no—
78C2×C4×C83—41no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83#4111yescharacteristic
83C2×C4×C83#4111yescharacteristic
84C2×C4×C83#4111yescharacteristic
85C22⋊D83—41no—
86C22⋊D83—41no—
87C22⋊D83—82no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—42no—
91C2×Dic83—42no—
92C2×Dic83—41no—
93D332—21no—
94D332—21no—
95C4×D83—41no—
96C4×D83—41no—
97Dic166—21no—
98Dic166—21no—
99C8×C83—41no—
100C83⋊C8—41no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83—41no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83#12011yescharacteristic
107C22×C83#12111yescharacteristic
108C22×C83#12411yescharacteristic
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—41no—
112C4×C83—24no—
113C4×C83—41no—
114C4×C83#12111yescharacteristic
115C4×C83#12111yescharacteristic
116C4×C83#12311yescharacteristic
117C4×C83#12311yescharacteristic
118Dic83—41no—
119Dic83—42no—
120Dic83—81no—
121C2×C83—21no—
122C2×C83—21no—
123C2×C83—81no—
124C2×C83#39411yescharacteristic
125C2×C83#45311yescharacteristic
126C2×C83#45611yescharacteristic
127D83—81no—
128#2039—831nomaximal, Sylow
129C83#203911yescharacteristic, complemented, Sylow
130#537—831no—
131C2×#104—831no—
132C2×#104D8311yescharacteristic, complemented
133#417—831no—
134#417—831no—
135#384—831no—
136#343—831no—
137C2×#41—831no—
138C2×#41C2×D8311yescharacteristic
139C2×M16—831no—
140C2×M16C2×D8311yescharacteristic
141C22⋊Q8—1661no—
142C22⋊Q8—1661no—
143#121—1661no—
144#121—1662no—
145#121—1661no—
146C2×C4⋊C4—831no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—1662no—
149C2×C42C2×D8311yescharacteristic
150#104—22no—
151#104—1662no—
152#103—1661no—
153#103—1661no—
154#102—1661no—
155#102—1661no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—1661no—
161#41—1661no—
162#41—1661no—
163#41—1661no—
164C2×Q8—831no—
165C2×Q8D33211yescharacteristic
166C2×D4—21no—
167C2×D4—831no—
168C2×D4—1661no—
169C2×D4—1662no—
170C2×D4D33211yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—832no—
174C22×C4—1661no—
175C22×C4—1661no—
176C22×C4C22×D8311yescharacteristic
177M16—21no—
178M16—21no—
179M16—1661no—
180M16—1661no—
181C2×C8—21no—
182C2×C8—1661no—
183C4⋊C4—3324no—
184C4⋊C4—1662no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4—1661no—
188C4⋊C4—1661no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C42—21no—
192C42C22⋊D8312yescomplemented
193C23—21no—
194C23—1661no—
195C23D4×D8311yescharacteristic
196Q8—21no—
197Q8—21no—
198Q8—1661no—
199Q8—1661no—
200D4—21no—
201D4—21no—
202D4—41no—
203D4—41no—
204D4—3321no—
205D4—3321no—
206D4—6642no—
207D4—1661no—
208D4—1661no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—3321no—
216C2×C4—3321no—
217C2×C4—3322no—
218C2×C4—3322no—
219C2×C4—3321no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4D4×D8311yescharacteristic
223C2×C4C2×C22⋊D8311yescharacteristic
224C2×C4C2×D33211yescharacteristic
225C2×C4#1305058911yescharacteristic
226C2×C4#1305058911yescharacteristic
227C2×C4#1305059011yescharacteristic
228C8—41no—
229C8—3321no—
230C22—21no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—3321no—
236C22—3321no—
237C22—3321no—
238C22#1458176211yescharacteristic
239C22#1458176411yescharacteristic
240C22#1458177811yescharacteristic
241C4—24no—
242C4—41no—
243C4—3322no—
244C4—3321no—
245C4—6641no—
246C4#1458176311yescharacteristic
247C4#1458176611yescharacteristic
248C4#1458177411yescharacteristic
249C4#1458177611yescharacteristic
250C2—21no—
251C2—21no—
252C2—81no—
253C2—6641no—
254C2#1912761711yescharacteristic
255C2#1912787011yescharacteristic
256C2#1912788511yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
FittingC2×#104×C83C2
socleC22×C83#120
prime radical#19128279C2
hypercenterC2×#104D83
hypocenterC83#2039
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#14581720C2semidirect
3. #19127703C2semidirect
4. #19127702C2semidirect
5. #19127571C2semidirect
6. #19127469C2semidirect
7. C2×#104×C83C2semidirect
63. C42×C83D4semidirect
129. C83#2039semidirect
132. C2×#104D83semidirect
192. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191282792
C2×#145817202
#191277032
#191277022
#191275712
#191274692
C2×#104×C832
#203983