Finite Group Databasebeta

← #24919762

Subgroups — #24919762

Order 10624.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128356C211yesmaximal, characteristic, complemented
2#19128053C211yesmaximal, characteristic, complemented
3#19128036C212yesmaximal, complemented
4#19127978C211yesmaximal, characteristic, complemented
5#19127945C212yesmaximal, complemented
6#19127906C211yesmaximal, characteristic, complemented
7#19127844C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127700C212yesmaximal, complemented
11#547×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yes—
13C2×#13050578C2212yes—
14C2×#13050575C2211yescharacteristic
15C2×#13050574C2212yes—
16#14581792C2211yescharacteristic
17#14581774—22no—
18#14581771C2212yes—
19#14581763—21no—
20C4×D332C2211yescharacteristic
21C4⋊C4×D83—21no—
22#14581725C2212yes—
23#14581725C2211yescharacteristic
24C4⋊(C83⋊C8)C2211yescharacteristic
25#14581715C2212yes—
26#14581715C2212yes—
27#14581715C2212yes—
28#14581713C2211yescharacteristic
29#14581712C2212yes—
30#14581712C2211yescharacteristic
31C22⋊(C83⋊C8)C2212yes—
32C4×C83⋊C8C2211yescharacteristic
33C2×#41×C83—21no—
34C4⋊Q8×C83C2211yescharacteristic
35#123×C83—22no—
36C22⋊Q8×C83—21no—
37#121×C83C2212yes—
38C4×D4×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2212yes—
41#13050590—24no—
42#13050590—42no—
43C2×Dic166C2312yes—
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—22no—
46C2×C4×D83—21no—
47C2×C4×D83—22no—
48#13050578—42no—
49#13050575—22no—
50#13050574—42no—
51C2×C83⋊C8C2311yescharacteristic
52C2×C83⋊C8C2312yes—
53C2×C83⋊C8C2311yescharacteristic
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—22no—
58C22⋊Dic83—22no—
59(C2×D83)⋊C4—21no—
60#41×C83—44no—
61C2×Q8×C83—21no—
62C2×D4×C83—22no—
63C2×D4×C83D411yescharacteristic
64C2×D4×C83C2311yescharacteristic
65C22×C4×C83—22no—
66C22×C4×C83—21no—
67C22×C4×C83C2312yes—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83D411yescharacteristic
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2312yes—
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83C2×D412yes—
79C22×D83—21no—
80Q8×C83—42no—
81D4×C83—22no—
82D4×C83—21no—
83D4×C83—21no—
84D4×C83—44no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—44no—
90C2×C4×C83—42no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D412yes—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—44no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98D332—21no—
99D332#4112yescomplemented
100C4×D83—44no—
101C4×D83—42no—
102Dic166—22no—
103Dic166—22no—
104C83⋊C8—21no—
105C83⋊C8—21no—
106C83⋊C8—22no—
107C22×C83—22no—
108C22×C83—22no—
109C22×C83—22no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83—41no—
113C22×C83C22×D411yescharacteristic
114C2×D83—22no—
115C2×D83—41no—
116C2×D83—41no—
117C4×C83—22no—
118C4×C83—21no—
119C4×C83—41no—
120C4×C83—42no—
121C4×C83AGL(1,ℤ/8)12yes—
122C4×C83C2×#4111yescharacteristic
123C4×C83H(5,𝔽2)11yescharacteristic
124Dic83—42no—
125Dic83—41no—
126C2×C83—42no—
127C2×C83—81no—
128C2×C83#54611yescharacteristic
129C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
130C2×C83#57811yescharacteristic
131D83—42no—
132#3330—831nomaximal, Sylow
133C83#333011yescharacteristic, complemented, Sylow
134#547—831no—
135#547D8311yescharacteristic, complemented
136#487—831no—
137#484—832no—
138#474—831no—
139#468—832no—
140#461—831no—
141#449—832no—
142C4×D8—831no—
143#436—831no—
144#417—832no—
145C2×#41—21no—
146C2×#41—832no—
147C2×SD16—832no—
148C2×D8—831no—
149C2×M16—832no—
150C4⋊Q8—831no—
151C4⋊Q8C2×D8311yescharacteristic
152#123—22no—
153#123—1662no—
154C22⋊Q8—21no—
155C22⋊Q8—832no—
156#121—1661no—
157#121C2×D8312yes—
158C4×D4—831no—
159C4×D4C2×D8311yescharacteristic
160C4×D4C2×D8311yescharacteristic
161C2×C4⋊C4—1661no—
162C2×C4⋊C4C2×D8312yes—
163#107—831no—
164#107—832no—
165C4⋊C8—831no—
166#103—832no—
167#103—832no—
168#103—832no—
169#102—831no—
170#102—831no—
171#102—832no—
172C22⋊C8—832no—
173C4×C8—831no—
174#41—44no—
175#41—3322no—
176#41—1664no—
177C2×Q8—21no—
178C2×Q8—832no—
179C2×D4—22no—
180C2×D4—831no—
181C2×D4—1662no—
182C2×D4C22⋊D8311yescharacteristic
183C2×D4C22×D8311yescharacteristic
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—1661no—
187C22×C4—1662no—
188C22×C4C22×D8312yes—
189SD16—3322no—
190D8—1662no—
191M16—3322no—
192C2×C8—831no—
193C2×C8—831no—
194C2×C8—832no—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—831no—
198C4⋊C4—1661no—
199C4⋊C4—1662no—
200C4⋊C4—1661no—
201C4⋊C4C22⋊D8311yescharacteristic
202C4⋊C4C22×D8311yescharacteristic
203C4⋊C4C22×D8312yes—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—1662no—
207C22⋊C4—1661no—
208C22⋊C4C22⋊D8312yes—
209C42C22×D8311yescharacteristic
210C23—21no—
211C23—1661no—
212C23C2×C22⋊D8312yes—
213Q8—42no—
214Q8—1662no—
215Q8—1662no—
216D4—22no—
217D4—21no—
218D4—21no—
219D4—44no—
220D4—3324no—
221D4—832no—
222D4—1661no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—3324no—
230C2×C4—3322no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4C2×C22⋊D8311yescharacteristic
234C2×C4C2×C22⋊D8312yes—
235C2×C4C2×C22⋊D8311yescharacteristic
236C2×C4C23×D8311yescharacteristic
237C8—1661no—
238C8—1661no—
239C8—1662no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—3321no—
247C22—3321no—
248C22—1662no—
249C22C22×C22⋊D8311yescharacteristic
250C4—22no—
251C4—21no—
252C4—41no—
253C4—42no—
254C4—3322no—
255C4—3321no—
256C4#1458181812yes—
257C4#41×D8311yescharacteristic
258C4#1458183811yescharacteristic
259C2—42no—
260C2—81no—
261C2—3322no—
262C2#1912833911yescharacteristic
263C2C2×#1458181811yescharacteristic
264C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#547×C83C2
socleC22×C83C22×D4
hypercenter#547D83
hypocenterC83#3330
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #19128356C2semidirect
2. #19128053C2semidirect
3. #19128036C2semidirect
4. #19127978C2semidirect
5. #19127945C2semidirect
6. #19127906C2semidirect
7. #19127844C2semidirect
8. C4×#13050575C2semidirect
9. #19127779C2semidirect
10. #19127700C2semidirect
11. #547×C83C2semidirect
99. D332#41semidirect
133. C83#3330semidirect
135. #547D83semidirect

Maximal subgroups

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

subgroupindex
#191283562
#191280532
#191280362
#191279782
#191279452
#191279062
#191278442
C4×#130505752
#191277792
#191277002
#547×C832
#333083