Finite Group Databasebeta

← #24907688

Subgroups — #24907688

Order 10624.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#19128182C211yesmaximal, characteristic
2#19127741C211yesmaximal, characteristic, complemented
3C2×#14581720C211yesmaximal, characteristic
4#19127700C211yesmaximal, characteristic
5#19127470C211yesmaximal, characteristic, complemented
6#19127461C211yesmaximal, characteristic, complemented
7#384×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10C2×#13050574C2211yescharacteristic
11#14581757—21no—
12#14581754—21no—
13C4×C22⋊D83—22no—
14#14581751C2211yescharacteristic
15#14581750—21no—
16C4⋊C4×D83—21no—
17#14581725—21no—
18#14581723—21no—
19#14581720C414yescomplemented
20#14581715—21no—
21#14581712—21no—
22C2×C4⋊C4×C83—22no—
23C2×C4⋊C4×C83C2211yescharacteristic
24C2×C4⋊C4×C83C2211yescharacteristic
25C2×C42×C83C2211yescharacteristic
26#13050590—21no—
27#13050590—21no—
28#13050590C2×C412yes—
29#13050590Q812yes—
30C2×Dic166D411yescharacteristic
31C2×D332D411yescharacteristic
32C2×C22⋊D83—21no—
33C2×C4×D83—21no—
34C2×C4×D83—22no—
35#13050574—21no—
36#13050574—21no—
37#13050574—21no—
38#13050574C2×C412yescomplemented
39C2×C83⋊C8—21no—
40C2×C83⋊C8—21no—
41C4⋊Dic83D411yescharacteristic
42C4⋊Dic83D411yescharacteristic
43Dic83⋊C4—22no—
44C22⋊Dic83—21no—
45(C2×D83)⋊C4—22no—
46C4×Dic83—22no—
47C4×Dic83—21no—
48C22×C4×C83—21no—
49C22×C4×C83—21no—
50C22×C4×C83D412yes—
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C4⋊C4×C83—21no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83—44no—
57C42×C83—21no—
58C42×C83C2×C412yescomplemented
59C23×C83C2×D411yescharacteristic
60C22×D83—21no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—42no—
66C2×C4×C83—42no—
67C2×C4×C83—41no—
68C2×C4×C83C22×C411yescharacteristic
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×Q811yescharacteristic
72C2×C4×C83#4111yescharacteristic
73C2×C4×C83#4111yescharacteristic
74C22⋊D83—22no—
75C22⋊D83—22no—
76C2×Dic83—21no—
77C2×Dic83—21no—
78C2×Dic83—21no—
79D332—21no—
80D332C4⋊C412yescomplemented
81C4×D83—22no—
82C4×D83—22no—
83C4×D83—82no—
84Dic166—21no—
85Dic166C4⋊C412yescomplemented
86C83⋊C8—22no—
87C83⋊C8—41no—
88C22×C83—21no—
89C22×C83—21no—
90C22×C83C4×D411yescharacteristic
91C22×C83#12011yescharacteristic
92C22×C83C22⋊Q811yescharacteristic
93C2×D83—22no—
94C2×D83—41no—
95C2×D83—41no—
96C4×C83—24no—
97C4×C83—42no—
98C4×C83—81no—
99C4×C83C2×C4⋊C411yescharacteristic
100C4×C83C4×D411yescharacteristic
101C4×C83C22⋊Q811yescharacteristic
102C4×C83#12311yescharacteristic
103Dic83—22no—
104Dic83—41no—
105Dic83—41no—
106C2×C83—21no—
107C2×C83—21no—
108C2×C83#38511yescharacteristic
109C2×C83#45311yescharacteristic
110C2×C83#45611yescharacteristic
111D83—42no—
112#1859—831nomaximal, Sylow
113C83#185911yescharacteristic, complemented, Sylow
114#520—831no—
115#428—831no—
116C2×#104—831no—
117#417—831no—
118#384D8311yescharacteristic, complemented
119#343—831no—
120#339—831no—
121C2×#41—831no—
122C2×M16—831no—
123C2×M16—831no—
124C4×Q8—1661no—
125C4×D4—1661no—
126C4×D4—1662no—
127#117—831no—
128#117—1661no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—1661no—
131C2×C4⋊C4C2×D8311yescharacteristic
132C2×C4⋊C4C2×D8311yescharacteristic
133C2×C42C2×D8311yescharacteristic
134#107—1661no—
135#106—1661no—
136#104—834no—
137#103—1661no—
138#102—1661no—
139#41—832no—
140#41—832no—
141#41—1661no—
142#41—1661no—
143C2×Q8—831no—
144C2×D4—831no—
145C2×D4—1661no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1662no—
149C22×C4—1661no—
150C22×C4C22⋊D8312yes—
151C22×C4C22×D8311yescharacteristic
152M16—832no—
153M16—1661no—
154M16—1661no—
155M16—1661no—
156C2×C8—1661no—
157C2×C8—1661no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—831no—
164C4⋊C4—831no—
165C4⋊C4—1662no—
166C22⋊C4—1662no—
167C22⋊C4—1661no—
168C42—21no—
169C42—1662no—
170C42—1661no—
171C42C4×D8312yescomplemented
172C23—1661no—
173C23C2×C22⋊D8311yescharacteristic
174Q8—832no—
175Q8—1661no—
176D4—832no—
177D4—1662no—
178D4—1662no—
179D4—1661no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—6642no—
188C2×C4—1661no—
189C2×C4—1661no—
190C2×C4—1661no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4C2×C4×D8311yescharacteristic
194C2×C4D4×D8311yescharacteristic
195C2×C4D4×D8311yescharacteristic
196C2×C4Q8×D8311yescharacteristic
197C2×C4#1305059011yescharacteristic
198C2×C4#1305059111yescharacteristic
199C8—3321no—
200C8—1662no—
201C22—21no—
202C22—21no—
203C22—3321no—
204C22—3321no—
205C22—1662no—
206C22C4×C22⋊D8311yescharacteristic
207C22#1458176011yescharacteristic
208C22#1458176811yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—3321no—
213C4—3321no—
214C4—1662no—
215C4C4⋊C4×D8311yescharacteristic
216C4#1458175411yescharacteristic
217C4#1458177011yescharacteristic
218C4#1458177711yescharacteristic
219C2—21no—
220C2—21no—
221C2—3322no—
222C2#1912757211yescharacteristic
223C2#1912786811yescharacteristic
224C2#1912788311yescharacteristic
subgroupG/H
centerC22#14581760
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#384×C83C2
socleC22×C83#120
prime radicalC2×#13050590C22
cosocleC22×C4×C83C23
hypercenter#384D83
hypocenterC83#1859
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #19127741C2semidirect
5. #19127470C2semidirect
6. #19127461C2semidirect
7. #384×C83C2semidirect
19. #14581720C4semidirect
38. #13050574C2×C4semidirect
58. C42×C83C2×C4semidirect
80. D332C4⋊C4semidirect
85. Dic166C4⋊C4semidirect
113. C83#1859semidirect
118. #384D83semidirect
171. C42C4×D83semidirect

Maximal subgroups

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

subgroupindex
#191281822
#191277412
C2×#145817202
#191277002
#191274702
#191274612
#384×C832
#185983