Finite Group Databasebeta

← #24918832

Subgroups — #24918832

Order 10624.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128233C211yesmaximal, characteristic
3#19128010C212yesmaximal, complemented
4#19127997C211yesmaximal, characteristic, complemented
5#19127991C211yesmaximal, characteristic, complemented
6#19127958C212yesmaximal, complemented
7#19127944C212yesmaximal
8#19127921C212yesmaximal, complemented
9#19127908C211yesmaximal, characteristic, complemented
10#19127723C211yesmaximal, characteristic, complemented
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050573C2212yes—
14#14581792C2211yescharacteristic
15#14581785—21no—
16#14581783—21no—
17#14581783C2211yescharacteristic
18#14581779C2211yescharacteristic
19#14581774—21no—
20#14581772—21no—
21#14581772C2211yescharacteristic
22#14581768C2212yes—
23#14581763C2211yescharacteristic
24Q8×Dic83C2212yes—
25C4×Dic166C2211yescharacteristic
26#14581757—21no—
27C4×C22⋊D83—21no—
28#14581751—21no—
29#14581750—21no—
30C42×D83C2211yescharacteristic
31#14581726C2212yes—
32#14581723C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581717C2212yes—
36#14581716C2212yes—
37#14581713C2212yes—
38C2×SD16×C83C2212yes—
39C4⋊Q8×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42#103×C83C2212yes—
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49#13050580—42no—
50#13050573—42no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—21no—
66C4×Dic83—21no—
67C4×Dic83—22no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C2×Q8×C83C2312yes—
72C2×D4×C83C2311yescharacteristic
73C22×C4×C83—21no—
74SD16×C83—42no—
75C2×C8×C83C2312yes—
76C4⋊C4×C83—21no—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C22⋊C4×C83—21no—
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85D4×C83—21no—
86D4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—42no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—42no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C4×D83—22no—
101C4×D83—21no—
102C4×D83—21no—
103C4×D83—41no—
104Dic166—21no—
105Dic166—21no—
106Dic166—42no—
107C8×C83—22no—
108C83⋊C8—22no—
109C22×C83—41no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83C22×D411yescharacteristic
113C2×D83—21no—
114C2×D83—21no—
115C2×D83—21no—
116C4×C83—21no—
117C4×C83—42no—
118C4×C83—41no—
119C4×C83#13712yes—
120C4×C83H(5,𝔽2)11yescharacteristic
121C4×C832−1+411yescharacteristic
122Dic83—21no—
123Dic83—21no—
124Dic83—21no—
125Dic83—41no—
126Dic83—42no—
127Dic83—41no—
128C2×C83—81no—
129C2×C83#53711yescharacteristic
130C2×C83C2×#13711yescharacteristic
131C2×C83#57511yescharacteristic
132D83—41no—
133#3252—831nomaximal, Sylow
134C83#325211yescharacteristic, complemented, Sylow
135#533—831no—
136#529—831no—
137#479—832no—
138#477—831no—
139#476—831no—
140#470—832no—
141#468—832no—
142#464—832no—
143#462—831no—
144#461D8311yescharacteristic, complemented
145#423—831no—
146C2×Q16—832no—
147C2×SD16C2×D8312yes—
148C2×M16—832no—
149C4⋊Q8—831no—
150C4⋊Q8C2×D8311yescharacteristic
151#126—1661no—
152#125—831no—
153#125—1661no—
154#124—831no—
155#123—1661no—
156C22⋊Q8—832no—
157C22⋊Q8—831no—
158C22⋊Q8—1661no—
159#121—831no—
160C4×Q8—832no—
161C4×Q8—831no—
162C4×Q8—1661no—
163C4×D4—1661no—
164C4×D4C2×D8311yescharacteristic
165#117—1661no—
166#117—1661no—
167C2×C42—831no—
168#107—832no—
169#106—832no—
170C4⋊C8—832no—
171C4⋊C8—831no—
172C4⋊C8C2×D8311yescharacteristic
173#103—832no—
174#103—832no—
175#103C2×D8312yes—
176#102—832no—
177C2×Q8—831no—
178C2×Q8—1661no—
179C2×Q8C22×D8312yes—
180C2×D4—1661no—
181C2×D4C22×D8311yescharacteristic
182C22×C4—21no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—831no—
186Q16—3322no—
187SD16—42no—
188M16—3322no—
189C2×C8—832no—
190C2×C8C22×D8312yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—832no—
194C4⋊C4—831no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1662no—
199C4⋊C4—1662no—
200C4⋊C4—1661no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8311yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—1662no—
205C22⋊C4—1661no—
206C22⋊C4—1661no—
207C22⋊C4—1661no—
208C22⋊C4—1661no—
209C42—831no—
210C42—831no—
211C42—831no—
212C42—1662no—
213C42—1661no—
214C42C22×D8311yescharacteristic
215C23—21no—
216C23—831no—
217Q8—22no—
218Q8—22no—
219Q8—3322no—
220Q8—1661no—
221Q8—1661no—
222D4—21no—
223D4—21no—
224D4—3322no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—3321no—
229C2×C4—831no—
230C2×C4—831no—
231C2×C4—831no—
232C2×C4—1661no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4D4×D8311yescharacteristic
239C2×C4D4×D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—22no—
242C8—1662no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—1661no—
247C22—1661no—
248C22—1661no—
249C22C2×D4×D8311yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—3321no—
254C4—3322no—
255C4—3321no—
256C4—1661no—
257C4—1661no—
258C4—1661no—
259C4#1458182112yes—
260C4#1458183711yescharacteristic
261C4#1458183911yescharacteristic
262C2—81no—
263C2—3321no—
264C2#1912828011yescharacteristic
265C2C2×#1458182111yescharacteristic
266C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radical#14581763C22
hypercenter#461D83
hypocenterC83#3252
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128010C2semidirect
4. #19127997C2semidirect
5. #19127991C2semidirect
6. #19127958C2semidirect
8. #19127921C2semidirect
9. #19127908C2semidirect
10. #19127723C2semidirect
11. #461×C83C2semidirect
134. C83#3252semidirect
144. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282332
#191280102
#191279972
#191279912
#191279582
#191279442
#191279212
#191279082
#191277232
#461×C832
#325283