Finite Group Databasebeta

← #24906940

Subgroups — #24906940

Order 10624.

198 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 53 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#19128077C211yesmaximal, characteristic, complemented
2#19127896C211yesmaximal, characteristic, complemented
3#19127779C211yesmaximal, characteristic, complemented
4#19127451C211yesmaximal, characteristic, complemented
5#19127445C211yesmaximal, characteristic, complemented
6#19127444C211yesmaximal, characteristic, complemented
7#459×C83C211yesmaximal, characteristic, complemented
8C2×#13050575—21no—
9C2×#13050575—21no—
10C2×#13050575—21no—
11C2×#13050574—21no—
12#14581789C2211yescharacteristic
13C4⋊(C83⋊C8)C2211yescharacteristic
14C4⋊(C83⋊C8)C2211yescharacteristic
15#14581712—21no—
16C22⋊(C83⋊C8)—21no—
17C4×C83⋊C8C2211yescharacteristic
18C2×D8×C83—21no—
19#127×C83C2211yescharacteristic
20C4×D4×C83C2211yescharacteristic
21C4⋊C8×C83C2211yescharacteristic
22#102×C83—21no—
23C2×D332—21no—
24C2×D332—21no—
25C2×D332—21no—
26#13050575—42no—
27#13050575—42no—
28#13050575—42no—
29#13050574—42no—
30C2×C83⋊C8—21no—
31C2×C83⋊C8—21no—
32C2×C83⋊C8D411yescharacteristic
33C2×C83⋊C8D411yescharacteristic
34C2×D4×C83—21no—
35C2×D4×C83—21no—
36C2×D4×C83—21no—
37C2×D4×C83D411yescharacteristic
38C22×C4×C83—21no—
39D8×C83—42no—
40C2×C8×C83—21no—
41C4⋊C4×C83D411yescharacteristic
42C22⋊C4×C83—21no—
43C42×C83C2311yescharacteristic
44C23×C83—21no—
45C23×C83—41no—
46C22×D83—41no—
47D4×C83—21no—
48D4×C83—21no—
49D4×C83—41no—
50D4×C83—41no—
51D4×C83—41no—
52D4×C83—41no—
53D4×C83—81no—
54C2×C4×C83—21no—
55C2×C4×C83—41no—
56C2×C4×C83—41no—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83#4111yescharacteristic
60D332—41no—
61D332—41no—
62D332—41no—
63D332—41no—
64D332—81no—
65C8×C83—41no—
66C83⋊C8—21no—
67C83⋊C8—41no—
68C83⋊C8—41no—
69C83⋊C8D812yescomplemented
70C22×C83—41no—
71C22×C83—41no—
72C22×C83—41no—
73C22×C83—81no—
74C22×C83—81no—
75C22×C83—81no—
76C22×C83#12111yescharacteristic
77C2×D83—81no—
78C2×D83—81no—
79C2×D83—81no—
80C4×C83—21no—
81C4×C83—41no—
82C4×C83C2×D811yescharacteristic
83C4×C83AGL(1,ℤ/8)11yescharacteristic
84C4×C83AGL(1,ℤ/8)11yescharacteristic
85C4×C83AGL(1,ℤ/8)11yescharacteristic
86C2×C83—81no—
87C2×C83—161no—
88C2×C83#45911yescharacteristic
89C2×C83#46911yescharacteristic
90C2×C83#47211yescharacteristic
91D83—161no—
92#1687—831nomaximal, Sylow
93C83#168711yescharacteristic, complemented, Sylow
94#493—831no—
95#459—831no—
96#459D8311yescharacteristic, complemented
97#436—831no—
98#334—831no—
99#331—831no—
100#331—831no—
101C2×D8—21no—
102C2×D8—1661no—
103C2×D8—1661no—
104C2×D8—1661no—
105C2×M16—1661no—
106#127—831no—
107#127C2×D8311yescharacteristic
108C4×D4C2×D8311yescharacteristic
109C4⋊C8—831no—
110C4⋊C8—831no—
111C4⋊C8C2×D8311yescharacteristic
112#102—21no—
113#102—1661no—
114C22⋊C8—1661no—
115C4×C8—831no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—1661no—
120C2×D4—1661no—
121C2×D4—1661no—
122C2×D4C22⋊D8311yescharacteristic
123C22×C4—21no—
124D8—42no—
125D8—3322no—
126D8—3322no—
127D8—3322no—
128M16—3322no—
129C2×C8—21no—
130C2×C8—831no—
131C2×C8—831no—
132C2×C8—1661no—
133C2×C8—1661no—
134C4⋊C4C22⋊D8311yescharacteristic
135C22⋊C4—21no—
136C42C22×D8311yescharacteristic
137C23—21no—
138C23—41no—
139C23—3321no—
140D4—21no—
141D4—21no—
142D4—41no—
143D4—41no—
144D4—41no—
145D4—41no—
146D4—81no—
147D4—3321no—
148D4—3321no—
149D4—3321no—
150D4—3321no—
151D4—6641no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4D4×D8311yescharacteristic
156C2×C4C2×C22⋊D8311yescharacteristic
157C2×C4#1305058911yescharacteristic
158C8—41no—
159C8—3321no—
160C8—3321no—
161C8—832no—
162C8—1661no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—6641no—
170C22—6641no—
171C22—6641no—
172C22#1458176411yescharacteristic
173C4—21no—
174C4—41no—
175C4D8×D8311yescharacteristic
176C4#1458181611yescharacteristic
177C4#1458181711yescharacteristic
178C4#1458181811yescharacteristic
179C2—81no—
180C2—161no—
181C2—13281no—
182C2#1912789411yescharacteristic
183C2#1912795211yescharacteristic
184C2#1912796911yescharacteristic
subgroupG/H
centerC22#14581764
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#459×C83C2
socleC22×C83#121
hypercenter#459D83
hypocenterC83#1687
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #19128077C2semidirect
2. #19127896C2semidirect
3. #19127779C2semidirect
4. #19127451C2semidirect
5. #19127445C2semidirect
6. #19127444C2semidirect
7. #459×C83C2semidirect
69. C83⋊C8D8semidirect
93. C83#1687semidirect
96. #459D83semidirect

Maximal subgroups

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

subgroupindex
#191280772
#191278962
#191277792
#191274512
#191274452
#191274442
#459×C832
#168783