Finite Group Databasebeta

← #24906643

Subgroups — #24906643

Order 10624.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#19128336C211yesmaximal, characteristic
2#19128047C211yesmaximal, characteristic, complemented
3#19127982C211yesmaximal, characteristic, complemented
4#19127907C211yesmaximal, characteristic, complemented
5#19127428C211yesmaximal, characteristic, complemented
6#19127428C211yesmaximal, characteristic, complemented
7#331×C83C211yesmaximal, characteristic, complemented
8C2×D4×D83—21no—
9C2×D4×D83—21no—
10C2×#13050579—21no—
11#14581792C2211yescharacteristic, complemented
12#14581763—21no—
13#14581763—21no—
14#14581760—22no—
15C4×D332C2211yescharacteristic
16C4×D332C2211yescharacteristic
17#14581726—21no—
18#14581717—21no—
19#14581714—21no—
20#14581713—22no—
21C4×C83⋊C8C2211yescharacteristic, complemented
22#127×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic, complemented
24C4⋊C8×C83C2211yescharacteristic, complemented
25C23×D83—22no—
26C2×Dic166—21no—
27C2×D332D411yescharacteristic
28C2×D332D411yescharacteristic
29C2×C22⋊D83—22no—
30C2×C22⋊D83—22no—
31D4×D83—42no—
32D4×D83—42no—
33D4×D83—42no—
34D4×D83—42no—
35C2×C4×D83—21no—
36C2×C4×D83—21no—
37#13050579—42no—
38C2×C83⋊C8—21no—
39C4⋊Dic83—21no—
40C4⋊Dic83—21no—
41C4⋊Dic83D411yescharacteristic
42C4⋊Dic83D411yescharacteristic
43C22⋊Dic83—22no—
44(C2×D83)⋊C4—21no—
45(C2×D83)⋊C4—21no—
46C2×D4×C83—21no—
47C2×D4×C83—21no—
48C2×D4×C83D412yes—
49C2×C8×C83—21no—
50C2×C8×C83—21no—
51C42×C83C2311yescharacteristic
52C23×C83—22no—
53C22×D83—21no—
54C22×D83—21no—
55C22×D83—44no—
56C22×D83—44no—
57C22×D83—82no—
58D4×C83—42no—
59D4×C83—41no—
60D4×C83—41no—
61D4×C83—41no—
62D4×C83—41no—
63C2×C4×C83C2×D411yescharacteristic
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83C2×D411yescharacteristic
66C22⋊D83—44no—
67C22⋊D83—44no—
68C2×Dic83—21no—
69C2×Dic83—21no—
70C2×Dic83—41no—
71D332—21no—
72D332—21no—
73D332D812yes—
74D332SD1612yescomplemented
75C4×D83—41no—
76C4×D83—41no—
77C4×D83—41no—
78C4×D83—41no—
79Dic166—41no—
80Dic166—41no—
81C8×C83—41no—
82C8×C83—41no—
83C83⋊C8—41no—
84C22×C83—42no—
85C22×C83—42no—
86C22×C83—42no—
87C22×C83#12011yescharacteristic
88C2×D83—22no—
89C2×D83—22no—
90C2×D83—41no—
91C2×D83—41no—
92C2×D83—41no—
93C2×D83—41no—
94C2×D83—88no—
95C4×C83—21no—
96C4×C83C2×D811yescharacteristic
97C4×C83C2×SD1611yescharacteristic
98C4×C83AGL(1,ℤ/8)11yescharacteristic
99C4×C83AGL(1,ℤ/8)11yescharacteristic
100Dic83—41no—
101Dic83—41no—
102Dic83—81no—
103C2×C83—82no—
104C2×C83#44711yescharacteristic
105C2×C83#45011yescharacteristic
106C2×C83#45511yescharacteristic
107D83—42no—
108D83—42no—
109#1639—831nomaximal, Sylow
110C83#163911yescharacteristic, complemented, Sylow
111D42—831no—
112#486—831no—
113#474—831no—
114#461—831no—
115#331D8311yescharacteristic, complemented
116D4⋊C8—831no—
117D4⋊C8—831no—
118C22×D4—1661no—
119C22×D4—1661no—
120C2×SD16—1661no—
121C4⋊Q8—831no—
122#127C2×D8311yescharacteristic
123#121—1661no—
124#121—1661no—
125#120—1662no—
126C4×D4—831no—
127C4×D4—831no—
128#107—1661no—
129C4⋊C8C2×D8311yescharacteristic, complemented
130C4⋊C8C2×D8311yescharacteristic, complemented
131#103—1661no—
132#102—1661no—
133#102—1662no—
134C4×C8—831no—
135C24—1662no—
136C2×Q8—1661no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—3322no—
140C2×D4—3322no—
141C2×D4—3322no—
142C2×D4—3322no—
143C2×D4—831no—
144C2×D4—831no—
145C2×D4—1662no—
146C2×D4—1662no—
147C2×D4D33212yes—
148C22×C4—1661no—
149C22×C4—1661no—
150SD16—3322no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1661no—
154C4⋊C4—831no—
155C4⋊C4—831no—
156C4⋊C4—1661no—
157C4⋊C4—1661no—
158C22⋊C4—1662no—
159C22⋊C4—1661no—
160C22⋊C4—1661no—
161C42C22×D8311yescharacteristic
162C23—22no—
163C23—3324no—
164C23—3324no—
165C23—6642no—
166C23—1661no—
167C23—1661no—
168Q8—3321no—
169Q8—3321no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—3324no—
176D4—3324no—
177D4—832no—
178D4—832no—
179D4—1661no—
180D4—1661no—
181C2×C4—3321no—
182C2×C4—3321no—
183C2×C4—3321no—
184C2×C4—3321no—
185C2×C4—3321no—
186C2×C4—1661no—
187C2×C4—1661no—
188C2×C4D4×D8311yescharacteristic
189C2×C4D4×D8311yescharacteristic
190C2×C4C2×D33211yescharacteristic
191C8—41no—
192C8—41no—
193C8—3321no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—3321no—
198C22—3321no—
199C22—3321no—
200C22—3321no—
201C22—6648no—
202C22—1662no—
203C22—1662no—
204C22#1458176211yescharacteristic
205C4—21no—
206C4—3321no—
207C4—3321no—
208C4—6641no—
209C4D8×D8311yescharacteristic
210C4SD16×D8311yescharacteristic
211C4#1458181611yescharacteristic
212C4#1458181911yescharacteristic
213C2—82no—
214C2—3322no—
215C2—3322no—
216C2#1912783611yescharacteristic
217C2#1912785311yescharacteristic
218C2#1912788111yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#331×C83C2
socleC22×C83#120
prime radical#19128336C2
hypercenter#331D83
hypocenterC83#1639
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #19128047C2semidirect
3. #19127982C2semidirect
4. #19127907C2semidirect
5. #19127428C2semidirect
6. #19127428C2semidirect
7. #331×C83C2semidirect
11. #14581792C22semidirect
21. C4×C83⋊C8C22semidirect
23. C4⋊C8×C83C22semidirect
24. C4⋊C8×C83C22semidirect
74. D332SD16semidirect
110. C83#1639semidirect
115. #331D83semidirect
129. C4⋊C8C2×D83semidirect
130. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283362
#191280472
#191279822
#191279072
#191274282
#331×C832
#163983