Finite Group Databasebeta

← #24920111

Subgroups — #24920111

Order 10624.

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

normal subgroups only

This group has 350 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#19128421C211yesmaximal, characteristic, complemented
2#19128412C211yesmaximal, characteristic, complemented
3#19128054C212yesmaximal, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19128011C212yesmaximal, complemented
6#19127915C212yesmaximal
7#19127907C211yesmaximal, characteristic, complemented
8#19127825C211yesmaximal, characteristic, complemented
9#19127788C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#476×C83C211yesmaximal, characteristic, complemented
12C2×#13050579C2212yes—
13C2×#13050575C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581785—22no—
16#14581783C2212yes—
17#14581779C2212yes—
18#14581772—21no—
19#14581771—21no—
20#14581768—22no—
21C4×Dic166C2211yescharacteristic
22#14581757C2212yes—
23C4×D332C2211yescharacteristic
24#14581754C2212yes—
25#14581751—21no—
26C4⋊C4×D83—21no—
27#14581725C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581717C2212yes—
30#14581715C2212yes—
31#14581715C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4×D4×C83C2211yescharacteristic
38#106×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#102×C83C2212yes—
41C2×Dic166D411yescharacteristic
42C2×Dic166C2312yes—
43C2×D332C2311yescharacteristic
44C2×C4×D83—21no—
45C2×C4×D83—22no—
46#13050579—24no—
47#13050575—22no—
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58C22⋊Dic83—22no—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83—21no—
65C2×D4×C83C2311yescharacteristic
66C22×C4×C83—21no—
67C2×C8×C83C2312yes—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2312yes—
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C42×C83C2311yescharacteristic
73C23×C83—21no—
74C22×D83—21no—
75Q8×C83—81no—
76D4×C83—21no—
77D4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—42no—
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2411yescharacteristic
84C2×Dic83—22no—
85C2×Dic83—21no—
86C2×Dic83C2×D412yes—
87C2×Dic83C2×D412yes—
88D332—21no—
89D332#4112yes—
90C4×D83—42no—
91C4×D83—82no—
92Dic166—22no—
93Dic166—22no—
94Dic166—22no—
95C8×C83—22no—
96C83⋊C8—21no—
97C83⋊C8—21no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83C22×D411yescharacteristic
102C2×D83—22no—
103C2×D83—41no—
104C2×D83—41no—
105C4×C83—21no—
106C4×C83—41no—
107C4×C83—42no—
108C4×C83#13512yes—
109C4×C83C2×#4111yescharacteristic
110C4×C832−1+411yescharacteristic
111Dic83—22no—
112Dic83—22no—
113Dic83—42no—
114Dic83—41no—
115C2×C83—81no—
116C2×C83#54811yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57711yescharacteristic
119D83—42no—
120#3360—831nomaximal, Sylow
121C83#336011yescharacteristic, complemented, Sylow
122#556—831no—
123#554—831no—
124#488—832no—
125#487—831no—
126#479—832no—
127#476D8311yescharacteristic, complemented
128#463—832no—
129#461—831no—
130C8×Q8—831no—
131C4×SD16—832no—
132C4×D8—831no—
133C2×SD16—832no—
134C2×D8—831no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#126—1662no—
138#125—832no—
139#124—832no—
140C22⋊Q8—1661no—
141C22⋊Q8—1662no—
142C22⋊Q8—1661no—
143C4×Q8—831no—
144C4×Q8—832no—
145C4×D4—832no—
146C4×D4—831no—
147C4×D4C2×D8311yescharacteristic
148#117—1661no—
149C2×C4⋊C4—1661no—
150#107—831no—
151#106C2×D8312yes—
152C4⋊C8—832no—
153C4⋊C8C2×D8311yescharacteristic
154#103—832no—
155#103—832no—
156#103—832no—
157#102—831no—
158#102—831no—
159#102C2×D8312yes—
160C4×C8—832no—
161C4×C8—831no—
162C2×Q8—21no—
163C2×Q8—831no—
164C2×Q8—832no—
165C2×D4—831no—
166C2×D4C22×D8311yescharacteristic
167C22×C4—21no—
168C22×C4—1662no—
169C22×C4—1661no—
170SD16—1664no—
171D8—1662no—
172C2×C8—831no—
173C2×C8—831no—
174C2×C8C22×D8312yes—
175C4⋊C4—21no—
176C4⋊C4—831no—
177C4⋊C4—832no—
178C4⋊C4—831no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4C22×D8311yescharacteristic
185C4⋊C4C22×D8312yes—
186C22⋊C4—21no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C22⋊C4—1662no—
190C22⋊C4—1661no—
191C42—832no—
192C42—1662no—
193C42C22×D8311yescharacteristic
194C23—21no—
195C23—1661no—
196Q8—81no—
197Q8—1662no—
198Q8—1662no—
199Q8—1662no—
200D4—21no—
201D4—21no—
202D4—832no—
203D4—1661no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—3322no—
208C2×C4—832no—
209C2×C4—832no—
210C2×C4—6642no—
211C2×C4—1662no—
212C2×C4—1661no—
213C2×C4D4×D8311yescharacteristic
214C2×C4D4×D8311yescharacteristic
215C2×C4C23×D8311yescharacteristic
216C8—22no—
217C8—1661no—
218C8—1661no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3321no—
223C22—3321no—
224C22—1662no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—3322no—
230C4—3321no—
231C4—1662no—
232C4—1662no—
233C4#1458181312yes—
234C4#41×D8311yescharacteristic
235C4#1458184011yescharacteristic
236C2—81no—
237C2—3322no—
238C2#1912836111yescharacteristic
239C2C2×#1458181311yescharacteristic
240C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#476×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#476D83
hypocenterC83#3360
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
2. #19128412C2semidirect
3. #19128054C2semidirect
4. #19128053C2semidirect
5. #19128011C2semidirect
7. #19127907C2semidirect
8. #19127825C2semidirect
9. #19127788C2semidirect
11. #476×C83C2semidirect
121. C83#3360semidirect
127. #476D83semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191284122
#191280542
#191280532
#191280112
#191279152
#191279072
#191278252
#191277882
C4×#130505752
#476×C832
#336083