Finite Group Databasebeta

← #24909433

Subgroups — #24909433

Order 10624.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#19128502C211yesmaximal, characteristic
2#19128142C211yesmaximal, characteristic, complemented
3#19127973C211yesmaximal, characteristic, complemented
4#19127514C211yesmaximal, characteristic, complemented
5#19127509C211yesmaximal, characteristic, complemented
6#19127484C211yesmaximal, characteristic, complemented
7C2×SD32×C83C211yesmaximal, characteristic, complemented
8#14581837—21no—
9#14581823—21no—
10#14581818—21no—
11#14581816—22no—
12#14581814C2211yescharacteristic
13#14581811—21no—
14D8×D83—21no—
15#14581798C2211yescharacteristic
16#14581738—21no—
17#14581735—21no—
18#14581732C2211yescharacteristic, complemented
19#14581728C2211yescharacteristic, complemented
20#14581711—21no—
21C2×Q16×C83C2211yescharacteristic, complemented
22C2×D8×C83C2211yescharacteristic
23SD32×C83—22no—
24C2×C16×C83C2211yescharacteristic, complemented
25#13050590—21no—
26#13050590D411yescharacteristic
27#13050589—22no—
28#13050589—41no—
29C2×C22⋊D83—42no—
30D4×D83—22no—
31D4×D83—41no—
32Dic332D411yescharacteristic
33#13050580—41no—
34#13050579—21no—
35#13050578—22no—
36#13050577—41no—
37D664D411yescharacteristic
38#13050575—22no—
39#13050574—21no—
40#13050574D411yescharacteristic
41#13050573—21no—
42C8×D83—21no—
43C83⋊C16—21no—
44C2×Q8×C83—21no—
45C2×D4×C83—21no—
46Q16×C83—21no—
47Q16×C83—21no—
48D8×C83—21no—
49D8×C83D412yes—
50C2×C8×C83C2311yescharacteristic
51C16×C83—21no—
52C23×C83—41no—
53C22×D83—42no—
54Q8×C83—41no—
55Q8×C83—41no—
56D4×C83—22no—
57D4×C83—41no—
58C2×C4×C83—41no—
59C2×C4×C83C2×D411yescharacteristic
60C22⋊D83—21no—
61C22⋊D83—42no—
62C22⋊D83—42no—
63C22⋊D83—41no—
64C2×Dic83—42no—
65D332—21no—
66D332D811yescharacteristic
67C4×D83—21no—
68C4×D83—41no—
69Dic166—21no—
70Dic166D811yescharacteristic
71C8×C83C2×D411yescharacteristic
72C8×C83C2×D411yescharacteristic
73C83⋊C8—21no—
74C83⋊C8—41no—
75C22×C83—42no—
76C22×C83—161no—
77C22×C83AGL(1,ℤ/8)11yescharacteristic
78C2×D83—21no—
79C2×D83—41no—
80C2×D83—162no—
81C4×C83—81no—
82C4×C83#12011yescharacteristic
83C4×C83C2×D811yescharacteristic
84Dic83—21no—
85Dic83—41no—
86C2×C83—21no—
87C2×C83—82no—
88C2×C83#44711yescharacteristic
89D83—41no—
90D83—81no—
91#2194—831nomaximal, Sylow
92C83#219411yescharacteristic, complemented, Sylow
93#576—831no—
94#510—831no—
95C2×SD32D8311yescharacteristic, complemented
96#473—831no—
97#361—831no—
98#359—831no—
99#350—831no—
100H(5,𝔽2)—1661no—
101#137—1661no—
102AGL(1,ℤ/8)—1661no—
103AGL(1,ℤ/8)—1662no—
104#135—831no—
105#135—1661no—
106C2×Q16C2×D8311yescharacteristic, complemented
107C2×D8—1661no—
108C2×D8C2×D8311yescharacteristic
109#131—831no—
110Q32—1661no—
111SD32—22no—
112SD32—1661no—
113M32—831no—
114C2×C16C2×D8311yescharacteristic, complemented
115#108—831no—
116#101—1661no—
117#41—3321no—
118#41—831no—
119#41—1662no—
120#41—1661no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—3321no—
124C2×D4—3322no—
125C2×D4—1662no—
126Q16—21no—
127Q16—21no—
128Q16—3321no—
129Q16—831no—
130SD16—3321no—
131SD16—1662no—
132SD16—1661no—
133D8—21no—
134D8—831no—
135D8—1662no—
136D8C22⋊D8312yes—
137M16—831no—
138M16—1661no—
139M16—1661no—
140C2×C8—1661no—
141C2×C8C22×D8311yescharacteristic
142C16—21no—
143C16—1661no—
144C23—41no—
145C23—3322no—
146Q8—41no—
147Q8—41no—
148Q8—831no—
149Q8—1661no—
150D4—22no—
151D4—41no—
152D4—3322no—
153D4—3321no—
154D4—3322no—
155D4—831no—
156D4—1661no—
157D4—1661no—
158C2×C4—41no—
159C2×C4—3322no—
160C2×C4—3321no—
161C2×C4—1661no—
162C2×C4D4×D8311yescharacteristic
163C8—3321no—
164C8—1661no—
165C8D4×D8311yescharacteristic
166C8C2×C22⋊D8311yescharacteristic
167C22—42no—
168C22—161no—
169C22—13282no—
170C22—3321no—
171C22—1661no—
172C22#1458181611yescharacteristic
173C4—81no—
174C4—3321no—
175C4—1661no—
176C4#1458176011yescharacteristic
177C4D8×D8311yescharacteristic
178C2—21no—
179C2—82no—
180C2—3321no—
181C2—6641no—
182C2#1912783311yescharacteristic
subgroupG/H
centerC2#19127833
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
FittingC2×SD32×C83C2
socleC2×C83#447
prime radical#19128502C2
hypercenterC2×SD32D83
hypocenterC83#2194
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #19128142C2semidirect
3. #19127973C2semidirect
4. #19127514C2semidirect
5. #19127509C2semidirect
6. #19127484C2semidirect
7. C2×SD32×C83C2semidirect
18. #14581732C22semidirect
19. #14581728C22semidirect
21. C2×Q16×C83C22semidirect
24. C2×C16×C83C22semidirect
92. C83#2194semidirect
95. C2×SD32D83semidirect
106. C2×Q16C2×D83semidirect
114. C2×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285022
#191281422
#191279732
#191275142
#191275092
#191274842
C2×SD32×C832
#219483