Finite Group Databasebeta

← #24920158

Subgroups — #24920158

Order 10624.

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

normal subgroups only

This group has 348 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#19128436C211yesmaximal, characteristic
2#19128426C211yesmaximal, characteristic, complemented
3#19128057C212yesmaximal, complemented
4#19128049C211yesmaximal, characteristic
5#19128000C212yesmaximal, complemented
6#19127926C212yesmaximal, complemented
7#19127902C211yesmaximal, characteristic
8#19127825C211yesmaximal, characteristic, complemented
9C4×#13050580C211yesmaximal, characteristic, complemented
10#19127788C212yesmaximal, complemented
11#475×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2211yescharacteristic
13C2×#13050579C2212yes—
14#14581790—21no—
15#14581789C2211yescharacteristic
16#14581783C2212yes—
17#14581782—22no—
18#14581781C2212yes—
19#14581780—21no—
20#14581777—22no—
21Q8×Dic83—21no—
22C4×Dic166C2211yescharacteristic
23C4×Dic166C2211yescharacteristic
24#14581757C2212yes—
25C4×D332—21no—
26#14581754C2212yes—
27#14581725C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581716C2211yescharacteristic
30#14581715C2211yescharacteristic
31#14581714C2212yes—
32#14581712C2212yes—
33#14581712C2212yes—
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4×Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166D411yescharacteristic
42C2×Dic166C2311yescharacteristic
43C2×D332—21no—
44C2×D332—21no—
45C2×D332C2312yes—
46C2×C4×D83—22no—
47#13050580—22no—
48#13050579—24no—
49C2×C83⋊C8C2311yescharacteristic
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83—22no—
52C4⋊Dic83D411yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—21no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83—21no—
65C2×Q8×C83C2311yescharacteristic
66C2×C8×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2312yes—
70C4⋊C4×C83C2311yescharacteristic
71C42×C83—21no—
72C42×C83C2311yescharacteristic
73C22×D83—22no—
74Q8×C83—21no—
75Q8×C83—21no—
76Q8×C83—42no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83C2×D412yes—
86C2×Dic83C2×D412yes—
87D332—22no—
88D332—22no—
89D332—42no—
90D332—81no—
91C4×D83—82no—
92Dic166—22no—
93Dic166—21no—
94Dic166#4112yes—
95C8×C83—22no—
96C83⋊C8—21no—
97C83⋊C8—21no—
98C22×C83C22×D411yescharacteristic
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—42no—
102C4×C83—21no—
103C4×C83—41no—
104C4×C83—42no—
105C4×C83—41no—
106C4×C83#13512yes—
107C4×C83C2×#4111yescharacteristic
108C4×C832−1+411yescharacteristic
109Dic83—22no—
110Dic83—22no—
111Dic83—22no—
112Dic83—41no—
113C2×C83#54811yescharacteristic
114C2×C83C2×#13511yescharacteristic
115C2×C83#57711yescharacteristic
116D83—82no—
117#3365—831nomaximal, Sylow
118C83#336511yescharacteristic, complemented, Sylow
119#559—831no—
120#557—831no—
121#488—832no—
122#486—831no—
123#478—832no—
124#475D8311yescharacteristic, complemented
125#464—832no—
126#460—831no—
127C8×Q8—831no—
128C4×Q16—831no—
129C4×SD16—832no—
130C2×Q16—831no—
131C2×SD16—832no—
132C4⋊Q8—1661no—
133C4⋊Q8C2×D8311yescharacteristic
134#127—831no—
135#125—832no—
136#125—1662no—
137#124—832no—
138#124—1661no—
139#123—1662no—
140C4×Q8—831no—
141C4×Q8—832no—
142C4×Q8—831no—
143C4×Q8—1661no—
144C4×Q8C2×D8311yescharacteristic
145C4×D4—832no—
146C4×D4—1661no—
147#107—831no—
148#106C2×D8312yes—
149C4⋊C8—832no—
150C4⋊C8C2×D8311yescharacteristic
151#103—831no—
152#103—831no—
153#103C2×D8312yes—
154#102—832no—
155#102—832no—
156#102—832no—
157C4×C8—832no—
158C4×C8—831no—
159C2×Q8—21no—
160C2×Q8—831no—
161C2×Q8—831no—
162C2×Q8C22×D8311yescharacteristic
163C2×D4—832no—
164C2×D4—1661no—
165C2×D4—1661no—
166C22×C4—1662no—
167Q16—1662no—
168SD16—1664no—
169C2×C8—831no—
170C2×C8—831no—
171C2×C8C22×D8312yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—831no—
175C4⋊C4—832no—
176C4⋊C4—831no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1661no—
181C4⋊C4C22×D8312yes—
182C4⋊C4C22×D8311yescharacteristic
183C22⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C42—21no—
187C42—832no—
188C42—1662no—
189C42—1661no—
190C42C22×D8311yescharacteristic
191C23—1662no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—832no—
196Q8—1662no—
197Q8—1661no—
198D4—3322no—
199D4—6641no—
200D4—1662no—
201D4—1662no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—832no—
206C2×C4—832no—
207C2×C4—6642no—
208C2×C4—1661no—
209C2×C4—1661no—
210C2×C4D4×D8311yescharacteristic
211C2×C4D4×D8311yescharacteristic
212C2×C4C23×D8311yescharacteristic
213C8—22no—
214C8—1661no—
215C8—1661no—
216C22—3322no—
217C22—3322no—
218C22—3322no—
219C22C2×D4×D8311yescharacteristic
220C4—21no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—3321no—
225C4—1662no—
226C4—1662no—
227C4—1662no—
228C4#1458181312yes—
229C4#41×D8311yescharacteristic
230C4#1458184011yescharacteristic
231C2—6642no—
232C2#1912836111yescharacteristic
233C2C2×#1458181311yescharacteristic
234C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#475×C83C2
socleC22×C83C22×D4
prime radical#14581789C22
hypercenter#475D83
hypocenterC83#3365
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #19128426C2semidirect
3. #19128057C2semidirect
5. #19128000C2semidirect
6. #19127926C2semidirect
8. #19127825C2semidirect
9. C4×#13050580C2semidirect
10. #19127788C2semidirect
11. #475×C83C2semidirect
118. C83#3365semidirect
124. #475D83semidirect

Maximal subgroups

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

subgroupindex
#191284362
#191284262
#191280572
#191280492
#191280002
#191279262
#191279022
#191278252
C4×#130505802
#191277882
#475×C832
#336583