Finite Group Databasebeta

← #24918836

Subgroups — #24918836

Order 10624.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128258C211yesmaximal, characteristic, complemented
2#19128005C212yesmaximal, complemented
3#19127996C211yesmaximal, characteristic, complemented
4#19127991C211yesmaximal, characteristic, complemented
5#19127957C212yesmaximal, complemented
6#19127945C212yesmaximal
7#19127920C212yesmaximal, complemented
8#19127911C211yesmaximal, characteristic, complemented
9#19127906C211yesmaximal, characteristic, complemented
10#19127725C211yesmaximal, characteristic, complemented
11#529×C83C211yesmaximal, characteristic
12C2×#13050580C2212yes—
13C2×#13050578C2212yes—
14C2×#13050574C2212yes—
15#14581792C2211yescharacteristic
16#14581792C2211yescharacteristic
17#14581784—21no—
18#14581771—21no—
19#14581771C2212yes—
20C4×Dic166—21no—
21C4×Dic166C2212yes—
22#14581751—21no—
23#14581725C2212yes—
24#14581723C2212yes—
25C4⋊(C83⋊C8)C2211yescharacteristic
26C4⋊(C83⋊C8)C2211yescharacteristic
27C4⋊(C83⋊C8)C2212yes—
28#14581716C2212yes—
29#14581715C2212yes—
30#14581715C2212yes—
31#14581713C2212yes—
32#126×C83—21no—
33#125×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#123×C83—21no—
36C22⋊Q8×C83C2211yescharacteristic
37#121×C83C2211yescharacteristic
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83—21no—
40C4×D4×C83C2211yescharacteristic
41#117×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166C2312yes—
45#13050580—42no—
46#13050578—42no—
47#13050574—42no—
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83C2312yes—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56C22⋊Dic83—22no—
57C4×Dic83—22no—
58C2×Q8×C83C2311yescharacteristic
59C2×D4×C83—21no—
60C2×D4×C83C2311yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83D411yescharacteristic
63C22×C4×C83D411yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—21no—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C22⋊C4×C83—21no—
74C22⋊C4×C83—21no—
75C22⋊C4×C83—21no—
76C42×C83—21no—
77C42×C83D411yescharacteristic
78C42×C83D411yescharacteristic
79C42×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C23×C83C2×D411yescharacteristic
83Q8×C83—21no—
84Q8×C83—21no—
85D4×C83—21no—
86D4×C83—21no—
87D4×C83—42no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2411yescharacteristic
102C2×Dic83—22no—
103C2×Dic83—22no—
104Dic166—22no—
105Dic166—22no—
106Dic166—42no—
107C83⋊C8—22no—
108C83⋊C8—22no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83—21no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83C22×D411yescharacteristic
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—21no—
120C4×C83—41no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83#13712yes—
124C4×C83H(5,𝔽2)11yescharacteristic
125C4×C832−1+411yescharacteristic
126Dic83—42no—
127Dic83—42no—
128C2×C83—41no—
129C2×C83—81no—
130C2×C83#53711yescharacteristic
131C2×C83C2×#13711yescharacteristic
132C2×C83#57511yescharacteristic
133#3252—831nomaximal, Sylow
134C83#325211yescharacteristic, complemented, Sylow
135#533—831no—
136#529D8311yescharacteristic
137#479—832no—
138#477—831no—
139#476—831no—
140#470—832no—
141#468—832no—
142#464—832no—
143#462—831no—
144#461—831no—
145#423—831no—
146C2×Q16—832no—
147C2×SD16—832no—
148C2×M16—832no—
149C4⋊Q8—831no—
150C4⋊Q8—831no—
151#126—21no—
152#125—1661no—
153#125C2×D8311yescharacteristic
154#124C2×D8311yescharacteristic
155#123—21no—
156C22⋊Q8—832no—
157C22⋊Q8—1661no—
158C22⋊Q8C2×D8311yescharacteristic
159#121C2×D8311yescharacteristic
160C4×Q8—832no—
161C4×Q8—1661no—
162C4×Q8C2×D8311yescharacteristic
163C4×D4—21no—
164C4×D4C2×D8311yescharacteristic
165#117—21no—
166#117—1661no—
167C2×C42C2×D8311yescharacteristic
168#107—832no—
169#106—832no—
170C4⋊C8—831no—
171C4⋊C8—831no—
172C4⋊C8—832no—
173#103—832no—
174#103—832no—
175#103—832no—
176#102—832no—
177C2×Q8—832no—
178C2×Q8—1661no—
179C2×Q8C22×D8311yescharacteristic
180C2×D4—21no—
181C2×D4C22×D8311yescharacteristic
182C22×C4—21no—
183C22×C4C22⋊D8311yescharacteristic
184C22×C4C22⋊D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186Q16—3322no—
187SD16—3322no—
188M16—3322no—
189C2×C8—832no—
190C2×C8—832no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—832no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1662no—
199C4⋊C4—1662no—
200C4⋊C4—1661no—
201C4⋊C4C22×D8311yescharacteristic
202C4⋊C4C22×D8311yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1662no—
209C42—21no—
210C42—1662no—
211C42C22⋊D8311yescharacteristic
212C42C22⋊D8311yescharacteristic
213C42C22×D8311yescharacteristic
214C42C22×D8311yescharacteristic
215C23—21no—
216C23C2×C22⋊D8311yescharacteristic
217Q8—21no—
218Q8—21no—
219Q8—3322no—
220Q8—1662no—
221Q8—1662no—
222D4—21no—
223D4—21no—
224D4—42no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4C2×C22⋊D8311yescharacteristic
236C2×C4C2×C22⋊D8311yescharacteristic
237C2×C4C2×C22⋊D8311yescharacteristic
238C2×C4C2×C22⋊D8311yescharacteristic
239C2×C4C2×C22⋊D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—1662no—
242C8—1662no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22C22×C22⋊D8311yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—3322no—
258C4—3322no—
259C4#1458182212yes—
260C4#1458183811yescharacteristic
261C4#1458183911yescharacteristic
262C2—41no—
263C2—81no—
264C2#1912827911yescharacteristic
265C2C2×#1458182211yescharacteristic
266C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#529×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#529D83
hypocenterC83#3252
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
2. #19128005C2semidirect
3. #19127996C2semidirect
4. #19127991C2semidirect
5. #19127957C2semidirect
7. #19127920C2semidirect
8. #19127911C2semidirect
9. #19127906C2semidirect
10. #19127725C2semidirect
134. C83#3252semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191280052
#191279962
#191279912
#191279572
#191279452
#191279202
#191279112
#191279062
#191277252
#529×C832
#325283