Finite Group Databasebeta

← #24918171

Subgroups — #24918171

Order 10624.

366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128236C211yesmaximal, characteristic
3#19128111C211yesmaximal, characteristic, complemented
4#19128093C211yesmaximal, characteristic, complemented
5#19128061C211yesmaximal, characteristic, complemented
6#19127941C212yesmaximal, complemented
7#19127933C212yesmaximal
8#19127805C212yesmaximal, complemented
9#19127804C212yesmaximal, complemented
10#97×D83C211yesmaximal, characteristic, complemented
11#491×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050577C2212yes—
14C2×C8×D83C2212yes—
15#14581792C2211yescharacteristic
16#14581787—21no—
17#14581783—21no—
18#14581780C2211yescharacteristic
19#14581777—21no—
20#14581772—21no—
21#14581772C2211yescharacteristic
22#14581768C2212yes—
23#14581767C2211yescharacteristic
24Q8×Dic83C2212yes—
25#14581757—21no—
26#14581757C2211yescharacteristic
27#14581754—21no—
28#14581754C2211yescharacteristic
29#14581750—21no—
30#14581750—21no—
31C42×D83C2211yescharacteristic
32#14581725C2212yes—
33#14581723C2212yes—
34#14581717C2212yes—
35#14581714C2212yes—
36#14581703C2211yescharacteristic
37#14581702C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#103×C83C2212yes—
41#103×C83C2212yes—
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C2×D332—21no—
46C2×D332C2311yescharacteristic
47C2×C4×D83—21no—
48C2×C4×D83D411yescharacteristic
49C2×C4×D83D411yescharacteristic
50C2×C4×D83C2311yescharacteristic
51#13050580—24no—
52#13050577—24no—
53C2×C83⋊C8C2312yes—
54C8×D83—24no—
55C4⋊Dic83—21no—
56C4⋊Dic83—22no—
57C4⋊Dic83—21no—
58C4⋊Dic83C2312yes—
59Dic83⋊C4—22no—
60Dic83⋊C4—21no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—21no—
68C4×Dic83—22no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×Q8×C83C2312yes—
73C2×C8×C83C2312yes—
74C4⋊C4×C83—21no—
75C4⋊C4×C83—21no—
76C4⋊C4×C83—21no—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C4⋊C4×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C22×D83—21no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83C2×D411yescharacteristic
94C2×Dic83C2×D411yescharacteristic
95C2×Dic83C2×D411yescharacteristic
96D332—22no—
97D332—42no—
98C4×D83—22no—
99C4×D83—22no—
100C4×D83—41no—
101C4×D83—42no—
102Dic166—22no—
103Dic166—42no—
104C8×C83—22no—
105C83⋊C8#4114yes—
106C22×C83C22×D411yescharacteristic
107C2×D83—21no—
108C2×D83—21no—
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—41no—
112C2×D83#13712yes—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—42no—
116C4×C83—41no—
117C4×C83—41no—
118C4×C83C2×#4112yes—
119Dic83—21no—
120Dic83—21no—
121Dic83—21no—
122Dic83—41no—
123Dic83—42no—
124C2×C83#53211yescharacteristic
125C2×C83C2×#13711yescharacteristic
126C2×C83#57511yescharacteristic
127D83—22no—
128D83—81no—
129#3181—831nomaximal, Sylow
130C83#318111yescharacteristic, complemented, Sylow
131#533—831no—
132#529—831no—
133#501—831no—
134#497—831no—
135#491D8311yescharacteristic, complemented
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⋊Q8—831no—
146C4⋊Q8C2×D8311yescharacteristic
147#126—1661no—
148#125—1661no—
149#125C2×D8311yescharacteristic
150#124—831no—
151#123—1661no—
152C22⋊Q8—832no—
153C22⋊Q8—831no—
154C22⋊Q8—1661no—
155#121—831no—
156C4×Q8—832no—
157C4×Q8—831no—
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#103C2×D8312yes—
168#103C2×D8312yes—
169#102—832no—
170#97—832no—
171#97—831no—
172#97C2×D8311yescharacteristic
173C2×Q8—831no—
174C2×Q8—1661no—
175C2×Q8C22×D8312yes—
176C2×D4—831no—
177C2×D4—1661no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—1661no—
182Q16—1664no—
183SD16—1664no—
184C2×C8—832no—
185C2×C8—1664no—
186C2×C8C22×D8312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—832no—
192C4⋊C4—1661no—
193C4⋊C4—1662no—
194C4⋊C4—1661no—
195C4⋊C4—1662no—
196C4⋊C4—1661no—
197C4⋊C4C22×D8311yescharacteristic
198C4⋊C4C22×D8311yescharacteristic
199C22⋊C4—1661no—
200C22⋊C4—1661no—
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—831no—
212C23—1661no—
213Q8—22no—
214Q8—22no—
215Q8—3322no—
216Q8—1662no—
217D4—3322no—
218D4—1662no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—3321no—
223C2×C4—3322no—
224C2×C4—831no—
225C2×C4—831no—
226C2×C4—831no—
227C2×C4—1662no—
228C2×C4—1661no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4D4×D8311yescharacteristic
232C2×C4D4×D8311yescharacteristic
233C2×C4C23×D8311yescharacteristic
234C8—22no—
235C8—834no—
236C22—3321no—
237C22—3321no—
238C22—3321no—
239C22—832no—
240C22—1661no—
241C22—1661no—
242C22C2×D4×D8311yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—3322no—
249C4—3321no—
250C4—1661no—
251C4—1661no—
252C4—1661no—
253C4#41×D8312yes—
254C2—6641no—
255C2—1662no—
256C2#1912825011yescharacteristic
257C2#137×D8311yescharacteristic
258C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#491×C83C2
socleC22×C83C22×D4
prime radical#14581767C22
hypercenter#491D83
hypocenterC83#3181
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128111C2semidirect
4. #19128093C2semidirect
5. #19128061C2semidirect
6. #19127941C2semidirect
8. #19127805C2semidirect
9. #19127804C2semidirect
10. #97×D83C2semidirect
11. #491×C83C2semidirect
130. C83#3181semidirect
135. #491D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282362
#191281112
#191280932
#191280612
#191279412
#191279332
#191278052
#191278042
#97×D832
#491×C832
#318183