Finite Group Databasebeta

← #24909102

Subgroups — #24909102

Order 10624.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#19128208C211yesmaximal, characteristic
2#19127685C211yesmaximal, characteristic, complemented
3#19127490C212yesmaximal, complemented
4#19127487C212yesmaximal, complemented
5#410×C83C211yesmaximal, characteristic, complemented
6C22×C4×D83C411yescharacteristic, complemented
7C2×#13050574—21no—
8#14581773C411yescharacteristic, complemented
9#14581773C2211yescharacteristic
10#14581768—21no—
11#14581763—21no—
12#14581760C411yescharacteristic, complemented
13#14581760C2211yescharacteristic
14C4×C22⋊D83—21no—
15C4×C22⋊D83—21no—
16#14581751C411yescharacteristic, complemented
17#14581711—21no—
18#14581709C2212yes—
19C2×#41×C83C2211yescharacteristic
20#117×C83—21no—
21#99×C83—21no—
22#99×C83C2212yes—
23C23×D83C2×C411yescharacteristic
24C2×C22⋊D83—21no—
25C2×C22⋊D83—21no—
26C22×Dic83C2×C411yescharacteristic
27C2×C4×D83—21no—
28C2×C4×D83—21no—
29C2×C4×D83—41no—
30C2×C4×D83—41no—
31#13050574—22no—
32#13050574—41no—
33C2×C83⋊C8—41no—
34C4⋊Dic83—21no—
35Dic83⋊C4—21no—
36Dic83⋊C4—21no—
37C22⋊Dic83—21no—
38C22⋊Dic83—21no—
39C22⋊Dic83C2×C411yescharacteristic
40C22⋊Dic83C2×C411yescharacteristic
41(C2×D83)⋊C4—21no—
42(C2×D83)⋊C4—21no—
43C4×Dic83—21no—
44#41×C83—41no—
45#41×C83—41no—
46C2×Q8×C83D411yescharacteristic
47C2×D4×C83C2×C411yescharacteristic
48C2×D4×C83D411yescharacteristic
49C2×D4×C83C2311yescharacteristic
50C22×C4×C83C2×C411yescharacteristic
51C22×C4×C83D411yescharacteristic
52C22×C4×C83D411yescharacteristic
53C4⋊C4×C83—41no—
54C22⋊C4×C83—22no—
55C42×C83—41no—
56C23×C83C22⋊C411yescharacteristic
57C23×C83C22×C411yescharacteristic
58C23×C83C2×D411yescharacteristic
59C22×D83—21no—
60C22×D83—42no—
61C22×D83—41no—
62Q8×C83—41no—
63D4×C83—41no—
64D4×C83—41no—
65D4×C83—41no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—21no—
72C2×C4×C83—42no—
73C2×C4×C83C22⋊C411yescharacteristic
74C2×C4×C83C22⋊C411yescharacteristic
75C2×C4×C83C22⋊C411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C22⋊D83—42no—
78C22⋊D83—42no—
79C2×Dic83—21no—
80C2×Dic83—21no—
81C2×Dic83—21no—
82C2×Dic83—41no—
83C4×D83—42no—
84C4×D83—41no—
85C4×D83—41no—
86C83⋊C8—42no—
87C22×C83—21no—
88C22×C83—21no—
89C22×C83—21no—
90C22×C83—41no—
91C22×C83—41no—
92C22×C83—41no—
93C22×C83C2×C22⋊C411yescharacteristic
94C2×D83—22no—
95C2×D83—41no—
96C2×D83—42no—
97C2×D83—41no—
98C2×D83—42no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—21no—
102C4×C83—82no—
103C4×C83#9911yescharacteristic
104C4×C83#9911yescharacteristic
105Dic83—41no—
106Dic83—41no—
107Dic83—41no—
108C2×C83—21no—
109C2×C83—41no—
110C2×C83—41no—
111C2×C83—41no—
112C2×C83C2×#9911yescharacteristic
113D83—42no—
114#2120—831nomaximal, Sylow
115C83#212011yescharacteristic, complemented, Sylow
116#525—831no—
117#413—831no—
118#410D8311yescharacteristic, complemented
119#352—832no—
120AΣL(1,𝔽16)—832no—
121C2×#41C2×D8311yescharacteristic
122C23×C4—831no—
123C2×M16—1661no—
124#123—831no—
125#123—831no—
126C22⋊Q8—1661no—
127#121—1661no—
128#120—831no—
129#120—831no—
130C4×D4—1661no—
131C4×D4—1661no—
132#117—21no—
133#117—831no—
134#101—1661no—
135#100—832no—
136#99—21no—
137#99C2×D8312yes—
138C24—831no—
139#41—41no—
140#41—41no—
141C2×Q8D33211yescharacteristic
142C2×D4—1661no—
143C2×D4—1661no—
144C2×D4C4×D8311yescharacteristic
145C2×D4C22⋊D8311yescharacteristic
146C2×D4C22×D8311yescharacteristic
147C22×C4—3321no—
148C22×C4—3321no—
149C22×C4—831no—
150C22×C4—1661no—
151C22×C4—1661no—
152C22×C4C4×D8311yescharacteristic
153C22×C4D33211yescharacteristic
154C22×C4C22⋊D8311yescharacteristic
155M16—3321no—
156M16—1662no—
157C2×C8—3321no—
158C4⋊C4—41no—
159C4⋊C4—1661no—
160C4⋊C4—1661no—
161C4⋊C4—1661no—
162C22⋊C4—22no—
163C22⋊C4—831no—
164C22⋊C4—831no—
165C22⋊C4—1661no—
166C22⋊C4—1661no—
167C22⋊C4—1661no—
168C22⋊C4—1661no—
169C42—41no—
170C42—1661no—
171C23—3321no—
172C23—3322no—
173C23—1661no—
174C23(C2×D83)⋊C411yescharacteristic
175C23C2×C4×D8311yescharacteristic
176C23C2×C22⋊D8311yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—3322no—
182D4—3322no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—3322no—
191C2×C4—3321no—
192C2×C4—3321no—
193C2×C4—3321no—
194C2×C4—1661no—
195C2×C4—1661no—
196C2×C4—1661no—
197C2×C4(C2×D83)⋊C411yescharacteristic
198C2×C4(C2×D83)⋊C411yescharacteristic
199C2×C4(C2×D83)⋊C411yescharacteristic
200C2×C4C2×D33211yescharacteristic
201C8—3322no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3321no—
209C22—3322no—
210C22—3322no—
211C22—3321no—
212C22—1662no—
213C22C2×(C2×D83)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—3321no—
219C4—3321no—
220C4—3321no—
221C4#1458170611yescharacteristic
222C4#1458170611yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—3322no—
228C2C2×#1458170611yescharacteristic
subgroupG/H
centerC4#14581706
commutatorC23×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#410×C83C2
socleC2×C83C2×#99
prime radical#19128208C2
cosocleC2×D4×C83C23
hypercenter#410D83
hypocenterC83#2120
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #19127685C2semidirect
3. #19127490C2semidirect
4. #19127487C2semidirect
5. #410×C83C2semidirect
6. C22×C4×D83C4semidirect
8. #14581773C4semidirect
12. #14581760C4semidirect
16. #14581751C4semidirect
115. C83#2120semidirect
118. #410D83semidirect

Maximal subgroups

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

subgroupindex
#191282082
#191276852
#191274902
#191274872
#410×C832
#212083