Finite Group Databasebeta

← #24912456

Subgroups — #24912456

Order 10624.

604 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#14581790C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic
3C2×D4×Dic83C211yesmaximal, characteristic
4#19127629C214yesmaximal, complemented
5#19127608C211yesmaximal, characteristic, complemented
6#19127607C211yesmaximal, characteristic, complemented
7#19127559C214yesmaximal, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C23×Dic83C2212yes—
12#14581790—24no—
13#14581779—24no—
14D4×Dic83—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2214yes—
20C2×C22⋊Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C2×C4×Dic83C2211yescharacteristic
23C83⋊#95C2212yes—
24C83⋊#95C2214yes—
25C83⋊#95C2214yes—
26C22×Q8×C83C2211yescharacteristic
27C22×D4×C83C2211yescharacteristic
28#124×C83—24no—
29C2×C22⋊C4×C83C2214yes—
30C2×C42×C83C2211yescharacteristic
31C2×Dic166—22no—
32C2×Dic166—22no—
33C2×Dic166—22no—
34C22×Dic83—24no—
35C22×Dic83—28no—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2311yescharacteristic
41C4⋊Dic83—22no—
42Dic83⋊C4—24no—
43Dic83⋊C4—24no—
44Dic83⋊C4—24no—
45C22⋊Dic83—28no—
46C22⋊Dic83—24no—
47C4×Dic83—22no—
48C4×Dic83D414yes—
49C24×C83C2312yes—
50C2×Q8×C83—22no—
51C2×Q8×C83—22no—
52C2×Q8×C83—22no—
53C2×D4×C83—22no—
54C2×D4×C83—22no—
55C2×D4×C83D414yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22⋊C4×C83—28no—
60C42×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—24no—
64C23×C83—24no—
65C23×C83C2×D414yes—
66C23×C83C2411yescharacteristic
67Q8×C83—44no—
68D4×C83—28no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83C2×D412yes—
76C2×C4×C83C2×D412yes—
77C2×C4×C83#4112yes—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—24no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—48no—
88C2×Dic83C2×D412yes—
89C2×Dic83C2×D412yes—
90C2×Dic83#4114yes—
91Dic166—44no—
92C22×C83—24no—
93C22×C83—28no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—44no—
97C22×C83C22×D411yescharacteristic
98C22×C83C22×D411yescharacteristic
99C22×C83C2×#4112yes—
100C22×C83C2×#4111yescharacteristic
101C22×C832−1+412yes—
102C4×C83—42no—
103C4×C83—42no—
104C4×C83#12114yes—
105Dic83—22no—
106Dic83—42no—
107Dic83—42no—
108Dic83—42no—
109Dic83#12414yescomplemented
110C2×C83—44no—
111C2×C83C2×#12111yescharacteristic
112C2×C83C2×#12411yescharacteristic
113C2×C83#53611yescharacteristic
114C2×C83#54712yes—
115C2×C83#55612yes—
116#2547—831nomaximal, Sylow
117C83#254711yescharacteristic, complemented, Sylow
118C2×C4⋊Q8—831no—
119C2×#124—831no—
120C2×#124D8311yescharacteristic
121C2×C4×D4—831no—
122#397—834no—
123#391—831no—
124#391—831no—
125#380—834no—
126C4×C4⋊C4—831no—
127C22×Q8—831no—
128C22×Q8C2×D8311yescharacteristic
129C22×D4C2×D8311yescharacteristic
130C23×C4—832no—
131C4⋊Q8—1664no—
132#124—24no—
133#124—1664no—
134C4×D4—1664no—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4—832no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—834no—
141C2×C22⋊C4C2×D8314yes—
142C2×C42—831no—
143C2×C42—831no—
144C2×C42C2×D8311yescharacteristic
145#95—832no—
146#95—834no—
147#95—834no—
148C24C22×D8312yes—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1662no—
153C2×Q8—1662no—
154C2×Q8—1662no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4C22⋊D8314yes—
158C22×C4—831no—
159C22×C4—831no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4—832no—
163C22×C4—1664no—
164C22×C4—1668no—
165C22×C4C22×D8312yes—
166C22×C4C22×D8312yes—
167C22×C4C22×D8311yescharacteristic
168C4⋊C4—1664no—
169C4⋊C4—1664no—
170C4⋊C4—1664no—
171C4⋊C4—1662no—
172C22⋊C4—28no—
173C22⋊C4—1664no—
174C22⋊C4—1668no—
175C42—22no—
176C42—834no—
177C42—1662no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23C2×C22⋊D8314yes—
183C23C23×D8311yescharacteristic
184Q8—44no—
185Q8—3324no—
186D4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—3328no—
194C2×C4—832no—
195C2×C4—834no—
196C2×C4—832no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1664no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4D4×D8312yes—
207C2×C4C2×C22⋊D8312yes—
208C2×C4#1305058912yes—
209C22—28no—
210C22—24no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22C2×D4×D8311yescharacteristic
215C22C22×C22⋊D8311yescharacteristic
216C22#41×D8312yes—
217C22C2×#1305058911yescharacteristic
218C22#1458183912yes—
219C4—42no—
220C4—42no—
221C4—3322no—
222C4—3322no—
223C4—3322no—
224C4—834no—
225C4—1662no—
226C4#1458176414yes—
227C2—44no—
228C2C2×#1458176411yescharacteristic
229C2#124×D8311yescharacteristic
230C2#1912827111yescharacteristic
231C2#1912835012yes—
232C2#1912842012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×#124D83
hypocenterC83#2547
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581790C2semidirect
4. #19127629C2semidirect
5. #19127608C2semidirect
6. #19127607C2semidirect
7. #19127559C2semidirect
8. C4×Dic83⋊C4C2semidirect
109. Dic83#124semidirect
117. C83#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#145817902
C2×#145817792
C2×D4×Dic832
#191276292
#191276082
#191276072
#191275592
C4×Dic83⋊C42
C2×#124×C832
#254783