Finite Group Databasebeta

← #24906129

Subgroups — #24906129

Order 10624.

344 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 75 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#127×D83C211yesmaximal, characteristic
2#19127753C211yesmaximal, characteristic, complemented
3#19127723C211yesmaximal, characteristic, complemented
4#19127445C211yesmaximal, characteristic, complemented
5#19127444C211yesmaximal, characteristic, complemented
6#19127444C211yesmaximal, characteristic, complemented
7#331×C83C211yesmaximal, characteristic, complemented
8C2×D4×D83—21no—
9C2×D4×D83—21no—
10C2×D4×D83C412yes—
11C2×#13050573—21no—
12#14581789C2211yescharacteristic
13#14581788—21no—
14#14581788—21no—
15#14581788C412yes—
16C42×D83C2211yescharacteristic
17C4⋊(C83⋊C8)C2211yescharacteristic, complemented
18C4⋊(C83⋊C8)C2211yescharacteristic, complemented
19Dic83⋊C8—21no—
20(C2×D83)⋊C8—21no—
21C8×Dic83—21no—
22#127×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic, complemented
24C4⋊C8×C83C2211yescharacteristic, complemented
25C23×D83—22no—
26C2×D332—21no—
27C2×D332—21no—
28C2×D332C2×C412yes—
29C2×C22⋊D83—22no—
30C2×C22⋊D83—22no—
31C2×C22⋊D83—22no—
32D4×D83—42no—
33D4×D83—42no—
34D4×D83—44no—
35D4×D83—42no—
36D4×D83—42no—
37C2×C4×D83C2×C411yescharacteristic
38C2×C4×D83D411yescharacteristic
39C2×C4×D83D411yescharacteristic
40#13050573—42no—
41C2×C83⋊C8—21no—
42C2×C83⋊C8—21no—
43C4×Dic83C2×C411yescharacteristic
44C4×Dic83D411yescharacteristic
45C4×Dic83D411yescharacteristic
46C2×D4×C83—21no—
47C2×D4×C83—21no—
48C2×D4×C83C2×C412yes—
49C2×C8×C83—21no—
50C2×C8×C83—21no—
51C42×C83C2311yescharacteristic
52C23×C83—22no—
53C22×D83—22no—
54C22×D83—44no—
55C22×D83—44no—
56C22×D83—44no—
57C22×D83C22⋊C411yescharacteristic
58D4×C83—42no—
59D4×C83—41no—
60D4×C83—41no—
61D4×C83—41no—
62D4×C83—41no—
63C2×C4×C83C22×C411yescharacteristic
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83C2×D411yescharacteristic
66C22⋊D83—44no—
67C22⋊D83—44no—
68C22⋊D83—44no—
69C2×Dic83C22⋊C411yescharacteristic
70C2×Dic83C22⋊C411yescharacteristic
71C2×Dic83C22⋊C411yescharacteristic
72D332—42no—
73D332—41no—
74D332—41no—
75D332—41no—
76D332—41no—
77C4×D83—21no—
78C4×D83—21no—
79C4×D83—22no—
80C4×D83D812yes—
81C4×D83SD1612yes—
82C8×C83—41no—
83C8×C83—41no—
84C83⋊C8—41no—
85C83⋊C8—41no—
86C22×C83—42no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83C2×C22⋊C411yescharacteristic
90C2×D83—21no—
91C2×D83—21no—
92C2×D83—42no—
93C2×D83—42no—
94C2×D83—42no—
95C2×D83—88no—
96C2×D83#10212yes—
97C4×C83—21no—
98C4×C83C2×D811yescharacteristic
99C4×C83C2×SD1611yescharacteristic
100C4×C83AGL(1,ℤ/8)11yescharacteristic
101C4×C83AGL(1,ℤ/8)11yescharacteristic
102Dic83—21no—
103Dic83#10012yes—
104Dic83#10212yes—
105C2×C83—82no—
106C2×C83C2×#10011yescharacteristic
107C2×C83C2×#10211yescharacteristic
108C2×C83#41811yescharacteristic
109D83—22no—
110D83—82no—
111#1545—831nomaximal, Sylow
112C83#154511yescharacteristic, complemented, Sylow
113C2×#127—831no—
114#431—831no—
115#423—831no—
116#331—831no—
117#331—831no—
118#331—831no—
119#331D8311yescharacteristic, complemented
120C22×D4—832no—
121C22×D4—1661no—
122C22×D4—1661no—
123C2×M16—1661no—
124#127—832no—
125#127—831no—
126#127—1661no—
127#127—1661no—
128#127C2×D8311yescharacteristic
129C2×C42—831no—
130C4⋊C8—831no—
131C4⋊C8—831no—
132C4⋊C8—1661no—
133C4⋊C8C2×D8311yescharacteristic, complemented
134C4⋊C8C2×D8311yescharacteristic, complemented
135C22⋊C8—1661no—
136C4×C8—1661no—
137C24—1662no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3322no—
141C2×D4—3322no—
142C2×D4—3322no—
143C2×D4—3322no—
144C2×D4—3324no—
145C2×D4—832no—
146C2×D4—1662no—
147C2×D4—1662no—
148C2×D4—1662no—
149C2×D4—1661no—
150C2×D4—1661no—
151C2×D4C4×D8312yes—
152C22×C4—831no—
153C22×C4—831no—
154C22×C4—831no—
155M16—3322no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—1661no—
159C2×C8—1661no—
160C42—831no—
161C42—831no—
162C42—831no—
163C42C22×D8311yescharacteristic
164C23—22no—
165C23—3324no—
166C23—3324no—
167C23—3324no—
168C23—831no—
169C23—1662no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—3322no—
176D4—3324no—
177D4—3324no—
178D4—3324no—
179D4—3321no—
180D4—3321no—
181D4—3321no—
182D4—3321no—
183C2×C4—831no—
184C2×C4—831no—
185C2×C4—832no—
186C2×C4—832no—
187C2×C4—831no—
188C2×C4—1662no—
189C2×C4—1661no—
190C2×C4—1661no—
191C2×C4C2×C4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C8—41no—
195C8—41no—
196C8—3321no—
197C8—3321no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—3322no—
202C22—3322no—
203C22—3322no—
204C22—832no—
205C22—6648no—
206C22—1661no—
207C22—1661no—
208C22C22⋊C4×D8311yescharacteristic
209C4—21no—
210C4—832no—
211C4—832no—
212C4—1661no—
213C4D8×D8311yescharacteristic
214C4SD16×D8311yescharacteristic
215C4#1458181611yescharacteristic
216C4#1458181911yescharacteristic
217C2—82no—
218C2—6642no—
219C2—1662no—
220C2#100×D8311yescharacteristic
221C2#102×D8311yescharacteristic
222C2#1912770411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D83
commutatorC2×C4×C83C22×C4
FrattiniC42C22×D83
Fitting#331×C83C2
socleC22×C83C2×C22⋊C4
prime radical#127×D83C2
cosocleC42×C83C23
hypercenter#331D83
hypocenterC83#1545
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #19127753C2semidirect
3. #19127723C2semidirect
4. #19127445C2semidirect
5. #19127444C2semidirect
6. #19127444C2semidirect
7. #331×C83C2semidirect
17. C4⋊(C83⋊C8)C22semidirect
18. C4⋊(C83⋊C8)C22semidirect
23. C4⋊C8×C83C22semidirect
24. C4⋊C8×C83C22semidirect
112. C83#1545semidirect
119. #331D83semidirect
133. C4⋊C8C2×D83semidirect
134. C4⋊C8C2×D83semidirect