Finite Group Databasebeta

← #24918066

Subgroups — #24918066

Order 10624.

364 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 119 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#19128256C211yesmaximal, characteristic, complemented
2#19128230C211yesmaximal, characteristic
3#19128110C211yesmaximal, characteristic, complemented
4#19128069C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19127956C212yesmaximal, complemented
7#19127944C212yesmaximal
8Q16×Dic83C212yesmaximal, complemented
9SD16×Dic83C212yesmaximal, complemented
10#19127653C211yesmaximal, characteristic, complemented
11#497×C83C211yesmaximal, characteristic, complemented
12C2×#13050573C2212yes—
13#14581792C2211yescharacteristic
14#14581785—21no—
15#14581784C2211yescharacteristic
16#14581783—21no—
17#14581774—21no—
18#14581772—21no—
19#14581768C2212yes—
20#14581768C2211yescharacteristic
21#14581763C2211yescharacteristic
22Q8×Dic83C2211yescharacteristic
23Q8×Dic83C2212yes—
24#14581757—21no—
25D4×Dic83C2211yescharacteristic
26(C22⋊D83)⋊C4—21no—
27#14581750—21no—
28#14581748—21no—
29C42×D83C2211yescharacteristic
30#14581726C2212yes—
31#14581724C2212yes—
32#14581716C2212yes—
33#14581716C2212yes—
34#14581716C2212yes—
35#14581713C2212yes—
36#14581703C2211yescharacteristic
37C8×Dic83C2212yes—
38C2×Q16×C83C2212yes—
39C2×SD16×C83C2212yes—
40C4⋊Q8×C83C2211yescharacteristic
41#124×C83C2211yescharacteristic
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×C22⋊D83—21no—
45C22×Dic83—21no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49#13050573—24no—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2311yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2312yes—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—21no—
65C4×Dic83—21no—
66C4×Dic83—22no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83C2311yescharacteristic
71C2×Q8×C83C2312yes—
72C2×D4×C83C2311yescharacteristic
73Q16×C83—24no—
74SD16×C83—24no—
75C2×C8×C83C2312yes—
76C4⋊C4×C83—21no—
77C4⋊C4×C83—21no—
78C22⋊C4×C83—21no—
79C22⋊C4×C83—21no—
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85Q8×C83—22no—
86D4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—42no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—42no—
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C4×D83—22no—
101C4×D83—22no—
102C4×D83—41no—
103Dic166—42no—
104C8×C83#4114yes—
105C83⋊C8—22no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83C22×D411yescharacteristic
110C2×D83—21no—
111C2×D83—21no—
112C2×D83—21no—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—41no—
116C4×C83—42no—
117C4×C83C2×#4112yes—
118Dic83—21no—
119Dic83—21no—
120Dic83—41no—
121Dic83—42no—
122Dic83—41no—
123Dic83#13712yes—
124C2×C83—81no—
125C2×C83#53211yescharacteristic
126C2×C83C2×#13711yescharacteristic
127C2×C83#57511yescharacteristic
128D83—41no—
129#3172—831nomaximal, Sylow
130C83#317211yescharacteristic, complemented, Sylow
131#533—831no—
132#529—831no—
133#501—831no—
134#497D8311yescharacteristic, complemented
135#491—831no—
136#489—831no—
137#470—832no—
138#468—832no—
139C4×Q16—832no—
140C4×SD16—832no—
141C4×M16—831no—
142C2×Q16C2×D8312yes—
143C2×SD16C2×D8312yes—
144C2×M16—832no—
145C4⋊Q8—831no—
146C4⋊Q8C2×D8311yescharacteristic
147#126—1661no—
148#125—831no—
149#125—1661no—
150#124C2×D8311yescharacteristic
151#123—1661no—
152C22⋊Q8—831no—
153C22⋊Q8—832no—
154C22⋊Q8—1661no—
155#121—831no—
156C4×Q8—831no—
157C4×Q8—832no—
158C4×Q8—1661no—
159C4×D4—831no—
160C4×D4—1661no—
161#117—1661no—
162#117—1661no—
163C2×C42—831no—
164#107—832no—
165#106—832no—
166#103—832no—
167#103—832no—
168#103—832no—
169#102—832no—
170#97—831no—
171#97C2×D8311yescharacteristic
172C4×C8—832no—
173C2×Q8—1661no—
174C2×Q8C22×D8311yescharacteristic
175C2×Q8C22×D8312yes—
176C2×D4—1661no—
177C2×D4C22×D8311yescharacteristic
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—1661no—
182Q16—24no—
183SD16—24no—
184M16—1664no—
185C2×C8—832no—
186C2×C8C22×D8312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—832no—
190C4⋊C4—831no—
191C4⋊C4—831no—
192C4⋊C4—1661no—
193C4⋊C4—1662no—
194C4⋊C4—1661no—
195C4⋊C4—1662no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—1662no—
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C42—831no—
206C42—831no—
207C42—831no—
208C42—1661no—
209C42—1662no—
210C42C22×D8311yescharacteristic
211C23—21no—
212C23—831no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—3322no—
217D4—22no—
218D4—3322no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—3321no—
222C2×C4—3322no—
223C2×C4—831no—
224C2×C4—831no—
225C2×C4—831no—
226C2×C4—1662no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1661no—
230C2×C4—1662no—
231C2×C4D4×D8311yescharacteristic
232C2×C4D4×D8311yescharacteristic
233C2×C4C23×D8311yescharacteristic
234C8—1662no—
235C8#1305058914yes—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—1661no—
240C22—1661no—
241C22—1661no—
242C22C2×D4×D8311yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—42no—
247C4—3321no—
248C4—3321no—
249C4—3322no—
250C4—832no—
251C4—1661no—
252C4—1661no—
253C4C2×#1305058912yes—
254C2—81no—
255C2—3321no—
256C2#1912824911yescharacteristic
257C2#137×D8311yescharacteristic
258C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
prime radical#14581763C22
hypercenter#497D83
hypocenterC83#3172
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128110C2semidirect
4. #19128069C2semidirect
5. #19128060C2semidirect
6. #19127956C2semidirect
8. Q16×Dic83C2semidirect
9. SD16×Dic83C2semidirect
10. #19127653C2semidirect
11. #497×C83C2semidirect
130. C83#3172semidirect
134. #497D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282302
#191281102
#191280692
#191280602
#191279562
#191279442
Q16×Dic832
SD16×Dic832
#191276532
#497×C832
#317283